Handbook of Dynamic Measurements

Dynamic Force, pressureand Acceleration Measurement

The HANDBOOK of

TheH

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ynamic

Force,pressureand

Acceleration

Measurem

ent

Endevco Corporation

30700 Rancho Viejo Road

San Juan Capistrano, CA 92675

92675-1748 USA

www.endevco.com

P/N 34666 ISBN 0-9713370-0-4 USD $45.00

Copyright 2001 by Endevco Corporation

All rights reserved.No part of this book may be reproduced, storedin a retrieval system, or transmitted, in any form or by any meanselectronic, mechanical, photocopying, recording or otherwise forcommercial purposes without the prior written permission ofEndevco Corporation.

II The Handbook of Dynamic Force, Pressure and Acceleration Measurement

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII

Part 1The Applicable Conceptual andDynamic Transducer Models . . . . . . . . . . . . . . . . . . . . . . . . . .3

Part 2The Application of Silicon andPiezoelectric Transduction Technologies . . . . . . . . . . . . . . . . .21

Part 3Signal Types: Deterministicand Nondeterministic Measurands . . . . . . . . . . . . . . . . . . . . .43

Part 4Interfacing the Transducer with its Environment . . . . . . . . . .59

Part 5Measurement System Requirements . . . . . . . . . . . . . . . . . . .81

Part 6Filtering in the Measurement System . . . . . . . . . . . . . . . . . .103

Part 7“Rules of Thumb” for Data Assessment . . . . . . . . . . . . . . . .117

Part 8Transducer Laboratory andMeasurement System Field Calibration . . . . . . . . . . . . . . . .131

Part 9Measurement System Signal Validation . . . . . . . . . . . . . . . . .151

Part 10Data Utilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .167

Part 11Measurements to Understandand Enhance Structural Dynamics . . . . . . . . . . . . . . . . . . . .189

IIIThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

Part 12Putting It All Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205

Answers to Part Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215

IV The Handbook of Dynamic Force, Pressure and Acceleration Measurement

LIST OF FIGURES

Figure 1.1

Conceptual Transducer Model . . . . . . . . . . . . . . . . . . . . . . . . .6

Figure 1.2

The Classical Dynamic Transducer Model . . . . . . . . . . . . . . .10

Figure 1.3

Dynamic Transducer ModelAmplitude Frequency Response . . . . . . . . . . . . . . . . . . . . . .11

Figure 1.4

Dynamic Transducer ModelPhase Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . .11

Figure 1.5

Real Accelerometer Frequency Response . . . . . . . . . . . . . . . .12

Figure 2.1

Accelerometer Design Based on BulkSilicon Strain Gages Mounted on a Beam . . . . . . . . . . . . . . .26

Figure 2.2

MEMS Piezoresistive Accelerometer . . . . . . . . . . . . . . . . . . .28

Figure 2.3

Serpentine Piezoresistive Gage Across Hinge . . . . . . . . . . . . .29

Figure 2.4

Microlithography Process Before Chemical Machining . . . . . .29

Figure 2.5

Chemically Machined Flexures . . . . . . . . . . . . . . . . . . . . . . .29

Figure 2.6

Various Shaped and Electroded Ceramic Elements . . . . . . . . .31

VThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 2.7

Accelerometer with Ceramic Plate Sensing Elements . . . . . . .31

Figure 2.8

Dynamic Transducer ModelAmplitude Frequency Response . . . . . . . . . . . . . . . . . . . . . .34

Figure 2.9

Dynamic Transducer ModelPhase Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . .34

Figure 2.10

MEMSVariable Capacitance Beam and Hinge . . . . . . . . . . . .36

Figure 3.1

The Same Nondeterministic, orRandom, Signal Measured By TwoTransducers to Verify Their Performance . . . . . . . . . . . . . . . .45

Figure 3.2

Package-drop Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Figure 3.3

Duality of Time and Frequency Domains . . . . . . . . . . . . . . . .46

Figure 3.4

Example of f(t) and Amplitude F(jω) . . . . . . . . . . . . . . . . . . .48

Figure 3.5

Ensemble of Random Signals . . . . . . . . . . . . . . . . . . . . . . . .49

Figure 3.6

Gaussian Probability Density Function . . . . . . . . . . . . . . . . . .51

Figure 4.1

Transducer Input and Coupling Process . . . . . . . . . . . . . . . . .60

Figure 4.2

Impedance Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

VI The Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 4.3

Measured MechanicalImpedance of an Aluminum Beamwith an Impedance Head at its Center . . . . . . . . . . . . . . . . . .62

Figure 4.4

MEMS Silicon Wafer WithNumerous Pressure TransducerDiaphragm Dies and Finished Transducer . . . . . . . . . . . . . . . .66

Figure 4.5

Pressure Transducer Coupling Model . . . . . . . . . . . . . . . . . . .67

Figure 4.6

Fundamental Natural Frequency of aColumn of Dry Air With One End Closed . . . . . . . . . . . . . . .67

Figure 4.7

Effect of Pressure TransducerCoupling in a Rocket Chamber . . . . . . . . . . . . . . . . . . . . . .69

Figure 4.8

Mounting Techniques InvolvingAdhesives, Helicoils and Interface Blocks . . . . . . . . . . . . . . . .73

Figure 4.9

Accelerometer Frequency ResponseWith Different Mounting Techniques . . . . . . . . . . . . . . . . . .74

Figure 5.1

Charge Amplifier and FET Circuit . . . . . . . . . . . . . . . . . . . . .82

Figure 5.2

DC Differential Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . .82

Figure 5.3

AC Circuit Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83

VIIThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 5.4

Simple and Complex Measurement Systems . . . . . . . . . . . . . .84

Figure 5.5

Non-linear LowPass Filter Example . . . . . . . . . . . . . . . . . . . .87

Figure 5.6

Frequency Creation in Non-linear Systems . . . . . . . . . . . .89-90

Figure 5.7

Flat Amplitude and Linear Phase Response Requirements . . . . . . . . . . . . . . . . . . . . .91-92

Figure 5.8

Classic Dynamic TransducerModel Responses to Half Sine Pulses . . . . . . . . . . . . . . . . . . .95

Figure 6.1

Amplitude Response of Ideal LowPass,High Pass, Band Pass and Band Reject Filters . . . . . . . . . . . .104

Figure 6.2

Amplitude Frequency and Phase FrequencyPlots for 6-Pole Chebyshev and Bessel Filters . . . . . . . . . . . .107

Figure 7.1

Bode Plots for Ideal Transducer . . . . . . . . . . . . . . . . . . . . . .118

Figure 7.2

Accelerometer Pyroshock Response . . . . . . . . . . . . . . . . . . .119

Figure 7.3

Limiting Cases of Resonant Frequency . . . . . . . . . . . . .123-124

Figure 8.1

Explosively Driven Shock Tube . . . . . . . . . . . . . . . . . . . . . .132

VIII The Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 8.2

Cold Gas Shock Tube With Coupled Section Open . . . . . . .135

Figure 8.3

High-Frequency AirbearingAccelerometer Calibration Shaker . . . . . . . . . . . . . . . . . . . .136

Figure 8.4

Reference “Piggyback”Accelerometer for Comparison Calibration . . . . . . . . . . . . .137

Figure 8.5

Hopkinson Bar WithPneumatically Driven Impact Projectile . . . . . . . . . . . . . . . .138

Figure 8.6

Single-Shunt Resistive Bridge Calibration . . . . . . . . . . . . . .140

Figure 8.7

Shunt Resistor Adapter Calibrator . . . . . . . . . . . . . . . . . . . .141

Figure 8.8

Typical Result Set FromShunt Resistor Adapter Calibration . . . . . . . . . . . . . . . . . . .142

Figure 8.9

Series Voltage InsertionCircuit for Charge Amplifier . . . . . . . . . . . . . . . . . . . . . . . .144

Figure 8.10

T-Junction Box for Voltage Insertion . . . . . . . . . . . . . . . . . .144

Figure 9.1

Piezoresistive orCapacitive Transducer Noise Model . . . . . . . . . . . . . . . . . . .156

Figure 9.2

Grounding Configuration for Charge Amplifiers . . . . . . . . . .158

IXThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 9.3

Impacted Plate Accelerometer Response . . . . . . . . . . . . . . .160

Figure 9.4

Non-Impacted Plate Accelerometer Response . . . . . . . . . . .161

Figure 10.1

Measurement Feedback toTrigger an Alarm or Control a Level . . . . . . . . . . . . . . . . . .168

Figure 10.2

Measurement to Support Design and Analysis . . . . . . . . . . . .169

Figure 10.3

Frequency Characteristics ofSingle Integrator and Differentiator . . . . . . . . . . . . . . . . . . .171

Figure 10.4

Seismic Event Recorded at 16 Bits Resolution . . . . . . . . . . .173

Figure 10.5

Seismic Event Recorded at 12 Bits Resolution . . . . . . . . . . .174

Figure 10.6

Response of Accelerometer Number One . . . . . . . . . . . . . .177

Figure 10.7

Energy Spectral Density Function for Figure 10.6 . . . . . . . . .178

Figure 10.8

Filtered Data of Figure 10.6 . . . . . . . . . . . . . . . . . . . . . . . . .178

Figure 10.9

Response of Accelerometer Number Two . . . . . . . . . . . . . . .179

Figure 10.10

Energy Spectral Density Function for Figure 10.9 . . . . . . . . .179

Figure 10.11

Pictorial Shock Spectra Calculation . . . . . . . . . . . . . . . . . . .181

X The Handbook of Dynamic Force, Pressure and Acceleration Measurement

Figure 10.12

Complex Field Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182

Figure 10.13

Simple Laboratory Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . .183

Figure 10.14

Envelope of Field Shock . . . . . . . . . . . . . . . . . . . . . . . . . . .183

Figure 11.1

Fan Blade With ItsDiscrete Elements for Analysis . . . . . . . . . . . . . . . . . . . . . . .193

Figure 11.2

Non-Fixed ReferenceInstrumented Hammers With Force Transducer . . . . . . . . . .194

Figure 11.3

Force-Hammer Excitation of Structural Systems . . . . . . . . . .195

Figure 11.4

Assortment of Modal Accelerometers . . . . . . . . . . . . . . . . . .195

Figure 11.5

First Vibratory Mode of a 25-Foot Tower . . . . . . . . . . . . . . .196

Figure 11.6

Assortment of Smart Modal Accelerometers . . . . . . . . . . . . .199

XIThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

LIST OF TABLES

Table 2.1

Mechanical Properties of Silicon vs Steel . . . . . . . . . . . . . . . .25

Table 2.2

Electrical and MechanicalProperties of Three Piezoelectric Materials . . . . . . . . . . . . . . .33

Table 6.1

Filter Selection Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

Table 10.1

Resolution for a 4,000 g Full Scale Channel . . . . . . . . . . . . .172

Table 11.1

Design Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189

XIIThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

XIIIThe Handbook of Dynamic Force, Pressure and Acceleration Measurement

INTRODUCTION

The measurement of time-varying physical phenomena, specifi-cally force, pressure, and acceleration, presents a significantchallenge to the engineer.This 12-part text will explain the “why”associated with the importance of various measurement-systemconsiderations. It is oriented to the senior technician or practicingengineer who has a fundamental understanding of transducers andassociated signal conditioning and who wants to design and applymeasurement systems that guarantee the acquisition of meaning-ful data from transducers. Part 1 provides both a conceptual and adynamic model for transducers.While initially these models mayappear esoteric, their understanding will help optimize measure-ment-system design for the applications of interest. Part 2 discussesthe properties of both silicon and piezoelectric materials that makethem optimum choices for incorporation into transducers intendedto measure time-varying phenomena. Part 3 describes the charac-teristics of the time-varying signals that will be measured. Part 4deals with interfacing the transducer to the environment to bemeasured.The performance of the transducer can be degraded, orthe environment to be measured modified, depending on howthis interface is accomplished. Part 5 explains the system require-ments for linearity, flat amplitude response, and linear phaseresponse when making time-varying measurements. Data filteringis then discussed in Part 6. Part 7 presents useful “rules of thumb”when designing measurement systems and interpreting data. Part8 deals both with the calibration of the types of transducers ofinterest and the calibration of measurement systems in the field.Part 9 discusses how to determine if data recordings are free ofnoise. Part 10 then looks at some applications of the measured data:integration, differentiation, power spectral densities, Fourier spec-tra, shock spectra, etc. Part 11 describes how frequency response

XIV The Handbook of Dynamic Force, Pressure and Acceleration Measurement

functions derived from acceleration data enable dynamic charac-terization of structural systems. Finally, Part 12 identifies otherinformation sources available to anyone interested in measuring thetime-varying physical phenomena that are the topic of this text. Italso provides a measurement-planning check list.

The Applicable Conceptual andDynamic Transducer Models

Part

2.52

1.5

1.00.90.80.70.60.5

0.4

0.3

0.2

0.15

0.1.04 .06 .08 .1 .15 .2 .3 .4 .5 .6 .8 1.0 1.5 2 3 4

(out

put)

/(in

put)

(forcing frequency)/(natural frequency)

The accurate measurement of force, pressure, and acceleration isquite important. If the dynamic loads applied to any linear elasticstructural system are measured, along with the system response tothese loads, the system itself becomes uniquely defined.

The force or pressure signals from transducers can define the load-ing to the structural system.These signals may be integrated withtime and averaged over the system’s surface to quantify the totallinear impulse (LI) delivered to the system. The signals fromaccelerometers mounted on the structure define its response to thisloading. If this structural response lasts for a long time relative tothe applied force or pressure loading, the loading can be mathe-matically modeled as an instantaneous impulse delivered to thesystem.This modeling uses the unit impulse time function δ(t) andis represented as LI[δ(t)].

The verification of complex structural modeling is dependent onthe quality of force, pressure, and acceleration measurements.Accel-eration measurements can also provide the basis for shock spectracalculations, enabling the generation of test specifications for vari-ous critical system components. All of these concepts will beexpanded on in succeeding parts of this text.

Many other reasons exist for measuring these specific physical phe-nomena. Machinery diagnostics enable predictive maintenance andrepair of translating and rotating equipment.The analysis of trans-ducer signals providing these diagnostics has been enhanced since

The Applicable Conceptual andDynamic Transducer Models

3The Handbook of Dynamic Force, Pressure and Acceleration Measurement

the middle 1970s by the advent of the modern digital spectrumanalyzer. Successful measurement of many other phenomenadepends on the accurate acquisition and interpretation of time-varying signals from transducers. Representative examples includeautomobile-frame and crash-dummy response in crash tests, engineout-of-balance condition, automobile ride quality, transmissionand antilock brake system performance, model performance inwind tunnel tests, effects of nuclear blast simulations, seismic vibra-tions, heart pacemaker performance, ordnance performance,parachute loads, and pressure of raw crude in oil well bore holepumps.

WHAT IS A TRANSDUCER?

There are a wide variety of transducers and transducer-related hard-ware available for the measurement practitioner to select from whenmeasuring dynamic force, pressure, and acceleration.The word“transducer” likely conjectures to this individual a mental image ofan accelerometer, pressure transducer, or force transducer.Yet thedefinition of the word transducer is not universally agreed upon.In addition, the words “sensor”and “transducer” are sometimes usedinterchangeably. Other words such as detector, instrument, trans-mitter, pickup,probe, and gage may also appear in this same context.

A survey of how current textbooks define these terms will furtherillustrate this confusion.The identification of these texts will alsoprovide additional reference sources for the reader. Fraden1.1, whenreferring to transducers, exclusively uses the word sensor.Carstens1.2

discusses electrical sensors but further defines an electrical trans-ducer as a device comprised of a sensor whose output is modified orfurther conditioned for a specific application. Doebelin1.3 uses theword transducer and categorizes transducers into passive (e.g.,piezoelectric) and active (e.g., variable impedance) types.Activetransducers are defined as those requiring auxiliary energy or power.

4 The Handbook of Dynamic Force, Pressure and Acceleration Measurement

Doebelin also defines a primary sensor as the element of a trans-ducer that first receives the energy. Sheingold1.4 follows the samepassive/active philosophy as Doebelin. Dally et al1.5 somewhat par-allel Doebelin by defining a sensor as an element (e.g., variableimpedance device) that is incorporated into a transducer. Beck-with and Marangoni1.6 describe a primary transducer (as opposed toa sensor) within a secondary transducer. Peter Stein1.7 provides uswith an uncomplicated definition of a transducer:

“A measurement system component capable of processing information and energy is a transducer.”

This is the definition that we will adopt.Wright’s1.8 recent book isconsistent with Stein’s work and is provided as a final reference.

TRANSDUCER MODELS

Having observed the confusion in attempting to define what atransducer is, one might wonder: How will it be possible to gener-ate a model for one? Peter Stein has developed a ConceptualTransducer Model that we will first present. Subsequently, the Classical Dynamic Model for the transducer types of interest will bepresented and made synergistic with Stein’s Conceptual TransducerModel. Limitations of this Classical Dynamic Model will then bediscussed.The intent is to create a logical flow so that the reader willbecome a more discriminating transducer user, particularly whenapplying them to dynamic measurements.

CONCEPTUAL TRANSDUCER MODEL

Figure 1.1 illustrates the Conceptual Transducer Model. It has threeports (x’s, y’s, and z’s) and six terminals (x1, x2, y1, y2, z1, z2).Thismodel accommodates two types of responses: self-generating (e.g.,piezoelectric) and nonself-generating (i.e., variable resistance,capacitance, or inductance).

Part 1 The Applicable Conceptual and Dynamic Transducer Models


The “measurand” is the physical quantity measured by a transducer.Two inputs, called “directly available measurands,” are required ateach port of a transducer.They are energy components, or time orspace functions of energy components.When multiplied together,their product produces dimensions of energy, energy rate, energyflux, etc.

Some elaboration will help clarify this discussion.The specific mea-surands of interest in this text are force, pressure, and acceleration.The flexure in an accelerometer experiences an inertial force pro-portional to the acceleration into the base of the accelerometer,resulting in a displacement (∆x) of its distributed mass:

F(inertial)

=m(effective flexure mass)

a(base acceleration) (1.1)


x1

z2z1

y2

y1

x2

LatentInformation

Figure 1.1ConceptualTransducer Model.Two input ports (x1,x2, z1, z2) arerequired to achievethe desiredinformation fromthe output port (y1,y2).

Figure 1.1

A pressure transducer, when loaded, experiences a force propor-tional to the applied pressure acting over the effective area of itsdiaphragm. This force results in a displacement (∆x) of itsdiaphragm:

F(load)

=p(pressure)

A(effective area) (1.2)

A force transducer, when extended or compressed, experiences aforce proportional to the displacement (∆x) of its flexure:

F=K∆x(flexure) (1.3)

Thus one directly available measurand to an accelerometer, pres-sure transducer, or force transducer is always some type of aforce(F).The force is the information processed by the transducer.It causes a deflection, ∆x, that becomes the second directly avail-able measurand at the same input port. Note the product of forcetimes displacement,F∆x,has dimensions of energy as stated in a pre-vious paragraph. In Figure 1.1 we can then set the terminals (x1,x2) at one input port equal to force and displacement (F, x). For anonself-generating transducer, the “latent information”could be thetransducer’s inductance, capacitance, or resistance.

The input force and displacement would cause this latent informa-tion to deviate from its initial value. This deviation could bedetermined by applying a supply voltage with resultant current flow(v, i) to the terminals (z1, z2) of the second input port.The dashedlines around this port would be made solid, indicating that supplyvoltage was held constant. Note, as expected, the product of volt-age and current produces units of rate of change of energy, orpower.At the output port terminals (y1, y2), transducer voltage andcurrent outputs (v, i) are present and transmit information andenergy for processing elsewhere in the system.



We can now introduce the concept of impedance at each of theseinput ports by taking the ratio of small changes in the measurands atthe corresponding terminals:

∆ F/∆ x=dF/dx=Zx

and, (1.4)

∆v/∆ I=dv/di=Zz

Thus, a nonself-generating transducer has both mechanical andelectrical input impedances and electrical output impedance. In amore general sense, the input force does not depend on a singlevariable such as x but is also frequency, ω, dependent so that dF/dxcould better be written as ∂/∂ω(∂F/∂x). Input impedances can thenbe more correctly written as Zx(jω) and Zz(jω) to represent theirfrequency-dependence.Stein refers to these impedances as the inputand output coefficients of the transducer.

Just as we want large electrical input impedances,we also want largemechanical input impedances. Our model states (Equation 1.4)that this requires the ideal transducer to have a flexure which is asstiff as possible, ∆x small, but still compatible with producing a use-able electrical signal amplitude.This large mechanical impedancewould permit all the information transfer concerning the forcebeing measured, while minimizing the energy transfer, F∆x, fromthe process being observed. Similar thought patterns can extend tothe relationships between one transducer port and another in thismodel. For example, from output terminals (v, i) to input terminals(F, x) we have ∆v/∆F=dv/dF, which is the transfer impedance,Tyx.Again acknowledging the frequency dependence of this term wecould better write ∂/∂ω(∂v/∂F), which is commonly identified asthe transfer function T(jω) of the transducer.

While the preceding discussion holds for a nonself-generating trans-ducer, a similar discussion would hold for a self-generating (e.g.,


piezoelectric) transducer.The latent information would be thepiezoelectric coefficients of the particular material. Since (x1, x2)deviate the latent information, they must be held constant, whichwould be indicated by making the circle in Figure 1.1 solid.Theinputs (z1, z2) would be (F, x).The energy-related components at theoutput port (y1, y2) would be charge and voltage (q, v),whose prod-uct is energy.The output impedance of the transducer would be∆q/∆v=dq/dv, which is C, its capacitance, and the transfer func-tion would be (∂/∂ω(∂q/∂F))=T(jω) with dimensions ofcharge/force.

We will leave this discussion for a while having established a modelfor self-generating and nonself-generating transducers.We usedthis model to develop the concept of mechanical as well as electri-cal input impedances and electromechanical transfer impedancesassociated with transducers.These impedances were noted to be fre-quency-dependent. We also noted that the product of themeasurands at the transducer terminals was energy-related and thatwe wanted high input impedances to minimize energy transfer fromthe process being measured. For flexure-based transducers, thisrequired a stiff mechanical assembly.We will come back to thismodel in future chapters, but at this point in time it is our intentto move on to the Classical Dynamic Model of a transducer.

CLASSICAL DYNAMIC MODEL

Figure 1.2 illustrates the model that transducer manufacturers andtextbook authors typically present in literature for force and pressuretransducers and accelerometers.This Classical Dynamic Model pro-vides the basis for the determination of the transfer function T(jω) justdiscussed as associated with the Conceptual Model.We should nowbe aware, when stating that any given transfer function defines theoutput/input relationship for a transducer,of the inherent assumption



that its input impedance Z(jω) is infinite.Part 4 of this text will revisitthe subject of transducer input impedances in detail.

The model in Figure 1.2 is that of a single-degree-of-freedom sys-tem which is characterized by a linear second-order differentialequation with constant coefficients of mass m, damping c, and stiff-ness k.The excitation in Figure 1.2 is harmonic. For a force orpressure transducer, x represents the absolute motion of the mass;the excitation is applied to the mass as shown.For an accelerometer,x represents the relative motion between mass and base; the excita-tion would be applied to the base of the housing.The governingequation for either situation has the form:

m(d2x/dt2)+c(dx/dt)+kx=Kejwt (1.5)

The general applicability of the model in Figure 1.2 to force trans-ducers is limited. If a force transducer is designed into a structure,it is typically modeled as a separate element of the structure itself.If the force transducer is impacted by a mass, and the duration of theimpact is such that the elasticity of the mass must be considered,the model again fails. Only when the impacting mass is “rigid” canits mass be added to the effective mass of the force transducer orload cell and the model hold. For accelerometers and flush-


x

Kejωtc

m

k

Figure 1.2The ClassicalDynamicTransducer Modelis a single-degree-of-freedom systemwith constantcoefficients ofmass (m), damping(c) and stiffness(k). The excitationis harmonic.

Figure 1.2

mounted pressure transducers, however, Figure 1.2 generally repre-sents the elementary model that they are designed to approximate.

Recall from our previous discussion on the Conceptual Model thatthe transfer function T(jω) describes the output/input relationshipfor the transducer. Figure 1.3 plots for the model of Figure 1.2 itsamplitude |T(jω)| versus frequency response and Figure 1.4 itsphase angle ∠ T(jω) versus frequency response.These plots are fora damping value ζ=0 of critical, typical of transducers designed tobe essentially undamped.



2.52

1.5

1.00.90.80.70.60.5

0.4

0.3

0.2

0.15

0.1.04 .06 .08 .1 .15 .2 .3 .4 .5 .6 .8 1.0 1.5 2 3 4

(out

put)

/(in

put)


Figure 1.3Dynamictransducer modelamplitudefrequencyresponse (zerodamping).

Figure 1.3

0 1 2 3 4 5

180°

150°

120°

90°

60°

30°

0°

phas

e la

g


Figure 1.4Dynamictransducer modelphase frequencyresponse (zerodamping).

Figure 1.4

An inconsistency exists between the simple model of Figure 1.2 andreal transducers.A flat diaphragm is the simplest form of receiver ina pressure transducer, and deflects in accord with theories applicableto circular plates. Beams can be used as flexures in either pressuretransducers or accelerometers. Piezoelectric accelerometers aredesigned with their crystal elements operating in shear, compres-sion, and other modes. In addition to flexures, transducer assembliesare comprised of housings, connectors, mounting attachments, etc.

It is apparent that real transducers are continuous structural systemsand must contain multiple resonant frequencies, not just the singleone shown in Figure 1.3.Figure 1.5 illustrates the amplitude-versus-frequency response for an accelerometer whose major resonanceoccurs low enough in frequency that the test setup is capable ofdemonstrating the presence of two additional resonances at higherfrequencies.


Figure 1.5Real accelerometerfrequencyresponse.

Mag

nific

atio

n fa

ctor

in d

B.

Log frequency in Hz.

40

30

20

10

0

–10

–200 100 1000 10,000

Figure 1.5

From this discussion and other observations relative to the ClassicalDynamic Model in Figure 1.2, the following limitations can be noted:

1. For “undamped” transducers, the model in Figure 1.2 servesonly as an approximation through the lowest resonant fre-quency of the transducer.Transducers are complex structures



in themselves, and their continuous nature indicates that theyhave multiple resonant frequencies — not just a single reso-nant frequency.

2. Flat amplitude response and linear phase response with fre-quency are requirements for all measurement systems where thetime history of the recorded signal is important.These require-ments will be established and justified in Part 5.For the present,the reader must accept them on faith. Note the transfer func-tion T(jω) in Figures 1.3 and 1.4 does not satisfy these criteria.The plot in Figure 1.3 is not flat and the plot in Figure 1.4 isnot linear with frequency.Therefore, the potential for distortionof recorded signals exists. Rules of thumb will be presented inPart 7 to minimize this distortion.

3. The transfer function in Figures 1.3 and 1.4 is intended toestablish the relationship between the output voltage or chargefrom a transducer and its input force, pressure, or accelerationmeasurand.This relationship holds only when the input imped-ance Z(jω) of the transducer is infinite.The reader must assurethis criterion is satisfied.

As we move into Part 2 we will explore why the properties of sili-con and piezoelectric materials make them particularly well suitedfor incorporation into transducers intended to measure time-vary-ing phenomena.The focus will be on their ability to provide widebandwidth and adequate dynamic range to measure a wide assort-ment of steady-state and random vibratory inputs as well asshort-duration transients.Throughout Part 2 and the remainder ofthis text, we will continue to employ and expand on the topics wehave introduced here.

REFERENCES

1.1 Fraden, Jacob, AIP Handbook of Modern Sensors,American Institute of Physics, New York, NY, 1993.

1.2 Carstens, James R., Electrical Sensors and Transducers,Regents/Prentice Hall, Englewood Cliffs, NJ, 1993.

1.3 Doebelin, Ernest O., Measurement Series:Application and Design, McGraw Hill, New York, NY, 4th ed., 1990.

1.4 Sheingold, Daniel H., Transducer Interfacing Handbook,Analog Devices, Norwood, MA, 1980.

1.5 Dally, James, Riley,William, and McConnell, Kenneth,Instrumentation for Engineering Measurements, John Wiley & Sons, NewYork, NY, 2nd ed, 1993.

1.6 Beckwith,Thomas G. and Marangoni, Roy D., Mechanical Measurements,Addison-Wesley Publishing Co., Reading,MA, 4th ed, 1990.

1.7 Stein, Peter K., The Unified Approach to the Engineering of Measurement Systems, Stein Engineering Services, Phoenix,AZ, ISBN #1-881472-00-0, 1992.

1.8 Wright, Charles P., Applied Measurement Engineering,Prentice Hall, Englewood Cliffs, NJ, 1995.


QUESTIONS

1. When using acceleration measurements to generate testspecifications for various critical system components, wecalculate ___________________ to provide a basis for thesespecifications.

a. Shock spectra.b. Fourier spectra.c. Power spectral densities.d. Peak amplitude.e. All of these.

2. Other words that are used interchangeably (and erroneously)for “transducer” include:

a. Detector.b. Pickup.c. Probe.d. Gage.e. All of these.

3. A transducer is a measurement system component capable ofprocessing _____________ and __________ (answer mostencompassing).

a. Voltage and current.b. Charge and voltage.c. Information and energy.d. Amplitude and duration.e. Frequency content and magnitude.

4. Which is not correct? Nonself-generating transducerresponses include __________________.

a. Variable capacitance.b. Variable resistance.c. Piezoelectric.d. Variable inductance.



5. The Conceptual Transducer model has _____ input and _____output ports with two directly-available measurands at each.

a. 3,0b. 2,1c. 0,3d. 4,2e. 2,4

6. The input impedances to a transducer can be both___________ and ____________. We want these inputimpedances ____________ to minimize energy transfer fromthe process being observed. These impedances are___________ dependent.

a. Mechanical, electrical, large, frequency.b. Mechanical, electrical, small, frequency.c. Mechanical, electrical, large, time.d. Electrical, mechanical, small, time.

7. The Classical Dynamic model of a transducer describes the_________________ relationship for the transducer andassumes the input impedance of the transducer is_________________.

a. Output/input, zero.b. Input/output, infinite.c. Output/input, infinite.d. Input/output, zero.

8. Force, pressure, and acceleration transducers have_________ resonant frequencies.

a. One.b. None.c. Ten.d. Five.e. Multiple.

16 The Handbook of Dynamic Force, Pressure and Acceleration Measurement 16

9. Where the time-history of the signal is important,measurement systems must have _________ amplitude and________ phase response.

a. Linear, linear.b. Flat, linear.c. Flat, flat.d. Linear, flat.

10. Part 2 of this text will focus on why __________ and___________ materials are particularly well-suited to be usedin force, pressure, and acceleration transducers.

a. Silicon, constantan.b. Piezoelectric, constantan.c. Silicon, nickel.d. Nickel, constantan.e. Piezoelectric, silicon.



The Application of Silicon andPiezoelectric Transduction

Technologies

Part


In Part 1, we concluded that an ideal transducer would have a verystiff flexure (small ∆x) but be compatible with producing a use-able electrical signal output. Satisfying these two criteria wouldresult in a transducer with high input impedance to optimize infor-mation transfer about the process being measured and minimizeenergy transfer from it.

This high input impedance, Z(jω), allows the transfer functionof the transducer,T(jω), to define its output/input relationshipcompletely.The effect of a stiffer spring or flexure is to increasethe natural frequency of the transducer. This higher natural frequency extends the range of useable flat frequency responseand linear phase response of the transducer.

A challenge exists in attempting to achieve both a stiff transducerflexure and a useable electrical signal output.This challenge canbe illustrated by considering the mechanical properties of two representative types of flexural elements used in transducer design.The definitions are provided on the following page.

The Application of Silicon and PiezoelectricTransduction Technologies

Silicon and piezoelectric materials have become the optimumchoice for inclusion in transducers intended for the measurementof the time-varying phenomena encompassed in this text. In Part2, the author discusses the technology behind their popularity.


E = Young’s modulus.L = Beam length.I = Beam area moment of inertia.a = Plate radius.t = Plate thickness.υ = Poisson’s ratio.µ = Mass/unit length.ρ = Mass/unit area.

The natural frequency (in radians/second) for the first circularmode of vibration of a fixed edge circular plate is:2.1

ωn=[(1.015)2π2/a2][Et3/(12ρ(1–υ2))]1/2 (2.1)

The natural frequency (in radians/second) for the fundamentalmode of vibration of a cantilever beam is:

ωn=3.516 [EI/(µL4)]1/2 (2.2)

The sensitivity of a transducer is proportional to the inverse stiff-ness (compliance) of its flexural element.The compliance is definedas the flexure deflection (∆x) per applied load.The compliance of acantilever beam subjected to an end load, F, is:2.2

∆x/F=L3/(3EI)=4.121/(µLωn2) (2.3)

The compliance of the center of a fixed-edge circular plate to auniform load, P, is:

∆x/W=3(1–υ2)a2/(16πEt3)=0.514/(a2 ρωn2) (2.4)

where W=Pπa2.The interaction between the resonant frequencyand the sensitivity of a transducer can now be related.

In examining Equations 2.3 and 2.4, we find that as flexure com-pliance and corresponding transducer sensitivity increases,transducer natural frequency and useable frequency response

decreases. A stiff flexure, that is, one with low compliance, isrequired to result in both a large useable transducer frequency-response range and a large linear amplitude range. It is thennecessary to select transduction techniques capable of providing alarge electrical signal output while integrated into stiff transducerflexures.These requirements focus us on semiconductor and piezo-electric transduction technologies.

DEFINING THE GAGE FACTOR

A justification can be presented for the applicability of semicon-ductor technology to transducers intended to measuretime-varying force, pressure, and acceleration measurands.Wheat-stone-resistive bridge elements are often affixed to the surface of atransducer flexure to measure its displacement, ∆x.This displace-ment, and thus the electrical bridge output, is proportional to themeasurand.This type of transducer represents the nonself-generat-ing response discussed in Part 1. The electrical resistance of aconductive element is:

R=ρL/A (2.5)

where ρ is the resistivity of the conductor, L its length, and A itscross sectional area. If we further define υ as Poisson’s ratio and ∈as longitudinal strain of the element, we can write:2.3

(dR/R)/(dL/L)=1+2υ+dρ/(ρ/∈ ) (2.6)

This is the definition of the gage factor of a strain gage.A metalstrain gage is usually a wire or foil ribbon mounted on some insu-lated backing material, although vacuum deposition is sometimesused. Metal strain-gage grids are fabricated from materials such asnichrome, advance, and iso-elastic. Strain gages comprise the resis-


Part 2 The Application of Silicon and Piezoelectric Transduction Technologies


tive elements in the Wheatstone bridge.Their gage factors are typ-ically 2-5, and the bridge electrical signal output is directlyproportional to the gage factor.

Strain gages can also be fabricated using semiconductor elements.For semiconductor gages, the dρ/(ρ/∈ ) term in Equation 2.6 ismore important than for metal gages. Gage factors for semicon-ductor gages can vary between 50-200.This high gage factor iswhat makes semiconductor strain elements attractive.They cansolve the challenge of producing a useable electrical signal outputfrom a transducer with a stiff mechanical flexure.The evolutionalhistory of the integration of semiconductor technology into trans-ducers follows.

SEMICONDUCTOR TECHNOLOGY

After Shockley’s invention of the transistor in 1947, basic semi-conductor technology continued to evolve through the 1950s and1960s. During the 1970s, focus was on the development of batchfabrication for integrated semiconductor circuits. In the 1980s weentered the miniature electro-mechanical systems (MEMS) phase.MEMS are integrated electrical and mechanical devices, the sizes ofwhich are on the order of microns; they incorporate sensing, sig-nal processing, and actuating functionalities.

The evolution of these technologies has strongly influenced thetransducer marketplace. Some indication of this influence can befound in reference 2.4 which contains approximately 70 articlesdevoted to transducers (defined as microsensors in the articles).One justification for this influence can be made when observingthe mechanical properties of single-crystal silicon, the most com-monly used semiconductor material, versus maximum-strengthsteel, as shown in Table 2.1.

Young’s Modulus for silicon is close to that of stainless and maxi-mum-strength steel, its yield strength in both tension andcompression is greater than steel, and its modulus-to-density ratiois more than three times that of steel. Single-crystal silicon alsoremains strong under repeated cycles of tension and compression,whereas polycrystalline metals tend to weaken and break becausestress accumulates at the intercrystal boundaries.These character-istics, coupled with its linear elastic properties until fracture, makeproperly grown silicon an ideal material for mechanical integrationinto transducers.

As will be discussed later, the ability of silicon to oxidize also lendsitself to transducer fabrication.To clarify semantics, a solid-state silicon resistor that changes resistance proportional to applied stressis defined as a piezoresistive element.Transducers employing siliconresistors in bridge circuits will be referred to as piezoresistive transducers.

SILICON TRANSDUCER TECHNOLOGY

Single-crystal silicon is often used in transducer manufacture. It isan anisotropic material whose atoms are organized in a lattice hav-ing several axes of symmetry.The orientation of any plane in thesilicon is provided by its Miller indices. Piezoresistive transducersmanufactured in the 1960s first used silicon strain gages fabricated



Yield Strength Young’s Modulus Density

Silicon 7.0x109 N/m2 190x109 N/m2 2.3g/cm3

Steel 4.2x109 N/m2 210x109 N/m2 7.9g/cm3

Mechanical properties of silicon vs steel.

Table 2.1

from lightly doped ingots.These ingots were sliced to form smallbars or patterns.

The Miller indices allowed positioning of the orientation of the baror pattern with respect to the crystal axes of the silicon.The bars orpatterns were often bonded directly across a notch or slot in thetransducer flexure. Figure 2.1 shows short, narrow, active elementsmounted on a beam.The large pads are provided for thermal powerdissipation and ease of electrical and mechanical connections.Therelatively short web avoids column-type instabilities in compressionwhen the beam bends in either direction.


TensionGages

SeismicMass

Compression Gages

+A

–A

Figure 2.1

Figure 2.1Accelerometerdesign based onbulk silicon straingages mounted ona beam. The largepads provide easeof electrical andmechanicalconnections. Theshort web avoidsinstabilities incompression.

Since the late 1970s we have encountered a continual evolution ofmicrosensors into the marketplace.A wide variety of technologiesis involved in their fabrication. The sequence of events whichoccurs in this fabrication process are: the single crystal silicon isgrown; the ingot is trimmed, sliced, polished, and cleaned; diffu-

sion of a dopant into a surface region of the wafer is controlled bya deposited film; a photolithography process includes etching of thefilm at places defined in the developing process, followed byremoval of the photoresist; and isotropic and anisotropic wet chem-icals are used for shaping the mechanical microstructure. Both theresultant stress distribution in the microstructure and the dopantcontrol the piezoresistive coefficients of the silicon.

Electrical interconnection of various controlled surfaces formedin the crystal as well as bonding pads are provided by thin film met-alization.The wafer is then separated into individual dies.The diesare bonded by various techniques into the transducer housing,and wire bonding connects the metallized pads to metal terminalsin the transducer housing. It is important to realize that piezore-sistive transducers manufactured in this manner use silicon both asthe flexural element and as the transduction element, since thestrain gages are diffused directly into the flexure.

The advantages of a transducer constructed in this manner includea high stiffness, Z(jω), resulting in a high resonant frequency opti-mizing its transfer function T(jω).The optimization of Z(jω) andT(jω) is attributable to the modulus-to-density ratio of silicon.Thisoptimization is desired for transducers that make time-varyingmeasurements. Other desirable byproducts are miniaturization,large signal amplitudes (semiconductor material has large gagefactor), good linearity, and improved stability.

Figure 2.2 shows a MEMS piezoresistive accelerometer. Anexpanded view of the piezoresistive gage in Figure 2.2 is shown inFigure 2.3. Figure 2.4 illustrates the microlithography process inMEMS transducer fabrication. Figure 2.5 shows an assemblage ofchemically machined accelerometer flexures.




Lid

Through Hole

PiezoresistiveGage

Support Rim

Sensitive Axis

PiezoresistiveGage

Terminals

Core

Hinge

Base

Recess

Figure 2.2Pictorial view of aMEMSpiezoresistiveaccelerometer.

Inertial Mass

Figure 2.2



Figure 2.3Serpentinepiezoresistive gageacross hinge.

Figure 2.4Microlithographyprocess beforechemicalmachining.

Figure 2.5Chemicallymachined flexures.


PIEZOELECTRIC TRANSDUCER TECHNOLOGY

Piezoelectricity is the other transduction technology particularlyapplicable to transducers intended to measure time-varying force,pressure, and acceleration measurands. Just as the research focussince the 1960s has been on silicon technology, a similar effortexisted in the 1920s through the 1950s on piezoelectric technol-ogy. Polycrystalline ceramics, discussed below, were the principalpiezoelectric materials of focus during the latter part of this timespan. The uses for piezoelectric materials continue to grow,although piezoelectric technology itself is now fairly mature.

Piezoelectricity is attributable to strain inducing a change in theshape of a crystal that possesses no center of charge symmetry.Anelectric charge results from this change in shape.Twenty-one of the32 crystal classes lack this symmetry element, and crystals in allbut one of these classes can exhibit piezoelectricity.

Quartz is a common piezoelectric material that exists in nature.Today, however, quartz is principally grown artificially.Tourmalineis piezoelectric and is acquired through mining operations. Someferroelectric polycrystalline ceramic materials can be artificiallymanufactured to exhibit piezoelectricity.This latter group of mate-rials is of particular interest since the manufacturing process cancontrol their mechanical and electrical properties.

This manufacturing process consists of the weighing and propor-tioning of the ceramic powders, calcining at high temperatures toproduce a chemical combination of the ingredients, mixing in aball mill to repowder the raw compound, adding a binder, granu-lating and then compressing the powder into pellet form, and firingthe pellets in a controlled atmosphere in a kiln.This firing trans-forms the pellets into ceramic elements.The elements are lappedand plated for subsequent polarization. A high voltage field is

applied across each pellet under controlled environmental condi-tions. The minute crystal domains within the ceramic are forced toalign themselves with the applied field, and this alignment isretained after the field is removed.

Figure 2.6 shows an array of various shaped and electrodedceramic elements. Figure 2.7 shows an accelerometer with ceramicplate sensing elements.



Figure 2.7

Figure 2.6

Figure 2.6Various shapedand electrodedceramic elements.

Figure 2.7Accelerometer withceramic platesensing elements.

PIEZOELECTRIC PROPERTIES

There are many important properties of piezoelectric materials.Thepiezoelectric constant for a material expresses the amount of chargegenerated per unit applied force, or the deflection per unit appliedvoltage.This constant is typically provided in tensor notation, suchas d33,with the first subscript identifying electrical direction and thesecond subscript identifying mechanical direction.Typical units arecoulombs/Newton (C/N) or meters/volt (m/V).

The coupling coefficient provides the energy conversion efficiencyof the piezoelectric material; a high value is desired.The dielectricconstant determines the capacitance of the material.The Curietemperature is that temperature above which the crystal latticemodifies its structure.Young’s modulus is a measure of material stiff-ness. The material resistivity must be high to keep the chargegenerated in the material from leaking off.The open-circuit voltageis the voltage generated at the output of the piezoelectric elementper unit of applied force.A piezoelectric transduction element iseffectively a capacitor that produces a charge across its plates pro-portional to the force applied to it. Unlike piezoresistive transducers,piezoelectric transducers do not have response to dc or 0 Hz.

Table 2.2 provides some electrical and mechanical properties for afew piezoelectric materials.These properties indicate why thesematerials are desirable for application as transduction elements.

The moduli of elasticity of these materials is quite high, between 50and 150% of the values presented previously for steel and silicon.Similarly, their modulus-to-density ratios are between 60 and 130%that of steel.Therefore, analogous to silicon, they can be incorpo-rated into flexures resulting in transducers with large inputimpedances Z(jω) and resonant frequencies.




Young’s Modulus (C11) Density d33 (10–12)

Quartz2.5 86.7x109 N/m2 2.65 g/cm3 –2.3 C/N (d11)

Tourmaline2.6 270x109 N/m2 3.1 g/cm3 2.1 C/N

PZT-5A2.7 121x109 N/m2 .75 g/cm3 374 C/N

Electrical and mechanical properties of threepiezoelectric materials.

Table 2.2

If the properties of PZT 5A (lead zirconate titanate) are used as anexample, the d33 value is 374 picocoulombs/Newton. It is appar-ent that a small force input to this ceramic material would resultin a large electrical output signal.Therefore, these materials enableminiaturization of transducer design while still permitting opera-tion over large amplitude ranges.

The models of Part 1, along with the preceding discussion, allow usto understand why piezoelectric and silicon transduction tech-nologies are particularly well suited for incorporation intotransducers intended for time-varying measurements.These tech-nologies, when integrated into transducers, can operate overamplitude ranges from fractional pounds, psi, or g’s to over 100,000pounds, psi, or g’s.Their useable frequency response can extend tothousands or tens of thousands of Hz, depending on their ampli-tude range.

DAMPING

We now want to return briefly to the Classical Dynamic Model wediscussed in Part 1 and its transfer function,T(jω), to address onefinal issue.T(jω) was presented in those figures for the undampedcase. Figures 2.8 and 2.9 plot the same transfer function for a vis-cous damping value of 0.707 critical.A damping value of 0.707

critical results in the maximum flat amplitude frequency responseand near-optimum linear phase frequency response for T(jω).

For example, in Part 1 we showed how the amplitude responsewas flat within five percent to only one-fifth (20%) of theundamped natural frequency, while in Figure 2.8 it is flat withinfive percent to three-fifths (60%) of the undamped natural fre-


2.52

1.5

1.00.90.80.70.60.5

0.4

0.3

0.2

0.15

0.1.04 .06 .08 .1 .15 .2 .3 .4 .5 .6 .8 1 1.5 2 3 4


(out

put)

/(in

putf)

Figure 2.8

180°

150°

120°

90°

60°

30°

0°


phas

e la

g

0 1 2 3 4 5

Figure 2.9

Figure 2.8Dynamictransducer modelamplitudefrequencyresponse (.707damping).

Figure 2.9Dynamictransducer modelphase frequencyresponse (.707damping).

quency.These amplitude and phase requirements will be justified inPart 5. If damping enhances transducer performance, the questionthat must be answered is why all transducers are not damped.

The achievement of damping in any mechanical system requiresenergy dissipation; energy dissipation requires mass motion.There-fore, it is easier to damp transducer flexures if they are highlycompliant. However, caution has to be exercised when using anytransducer with a highly compliant flexure.The flexure can beeasily overstressed and driven non-linear if not mechanicallystopped. Equations 2.3 and 2.4 illustrate that every parameter thatincreases the compliance of the flexure also reduces its natural fre-quency.This results in less useable transducer frequency response.

Commercial force transducer and pressure transducer flexures(with the exception of microphones) are not compliant enoughto be damped.Accelerometers can only be damped for relativelylow full scale ranges. Damping in accelerometers has historicallybeen achieved by filling them with silicone fluid.The viscosity ofsilicone fluid is highly temperature-dependent.Thus, 0.707 damp-ing is achieved only to an approximation within ±20º or 30°F ofroom temperature. Damped accelerometers often become objec-tionably large.

MEMS technology, in a variable capacitance electrical bridge con-figuration, solves a number of these problems.Figure 2.10 illustratesa MEMS variable-capacitance beam element for an accelerome-ter.The detection of acceleration requires a pair of silicon sensors.The sensing elements experience a change in capacitance attrib-utable to minute deflections resulting from the inertial accelerationforce.The single-crystal nature of the silicon, the elimination ofmechanical joints, and the ability to chemically machine mechan-ical stops, result in a transducer with a high over-range capability.



The damping characteristics for this type of transducer can beenhanced over a broad temperature range if a gas is employed forthe damping medium as opposed to silicone oil. A series ofgrooves, coupled with a series of holes in the central mass, squeezegas through the structure as the mass displaces.The thermal vis-cosity change of a gas is small relative to that of silicone oil.Capacitive MEMS accelerometers currently operate to hundreds ofg’s and frequencies to 1 kHz.The MEMS technology results insignificant transducer size reduction.

Having discussed transducers and transduction technologies inParts 1 and 2, it is now important to understand the characteristicsof the signals being measured.This issue is the focus of Part 3.


Figure 2.10MEMS variablecapacitance beamand hinge (partialphoto).

Figure 2.10

REFERENCES

2.1 Meirovitch, Leonard, Analytical Methods in Vibrations,The Macmillan Co., New York, NY, pp. 163 and 187, 1967.

2.2 Roark, R. J., Formulas for Stress and Strain, McGraw-Hill Book Co., New York, NY, ch. 8, p. 104 and ch. 10, p. 217,1965.

2.3 Khazan,Alexander D., Transducers and Their Elements,Prentice Hall, Englewood Cliffs, NJ, 1994.

2.4 Muller, Richard S. et. al., Microsensors, IEEE Press, New York, NY, 1991.

2.5 Valpey-Fisher Corp., User’s Guide to Ultrasound & Optical Products, Hopkinton, MA, 1996.

2.6 Mason,Warren P., Piezoelectric Crystals and Their Applicationto Ultrasonics,Van Nostrand, New York, NY, 1950.

2.7 Hellwege (Editor), Landolt-Bornstein New Series,Ferroelectrics and Related Substances, Springer-Verlag,Vol. 16, 1981.



QUESTIONS

1. Increasing the stiffness of the flexure of a force, pressure, oracceleration transducer also:

a. Increases its resonant frequency.b. Extends its useable flat frequency response.c. Decreases its useable electrical signal.d. Extends its range of linear phase response.e. Does all of these.

2. Which is not correct? To increase the natural frequency of a circularplate flexural element, I can:

a. Increase its modulus of elasticity.b. Increase its density.c. Increase its thickness.d. Use a material with a larger Poisson’s ratio.

3. For a semiconductor material, the gage factor or sensitivityincreases with:

a. Resistance.b. Gage length.c. Unit change in resistivity.

4. Silicon based materials are used in transducer manufacture because(select all that apply):

a. Their electrical properties are constant with temperature.b. The modulus of elasticity of silicon is relatively high.c. They do not fatigue under repeated stress cycling.d. Silicon has very linear elastic properties.e. When configured in a strain gage, their high gage factor

enables large electrical output signals to be acquired from Wheatstone bridge circuits.

5. MEMS, a common technology used in transducers today,stands for:

a. Miniature energy memorization systems.b. Maximum energy monitoring systems.c. Miniature electro-mechanical systems.d. Miniature electro-magnetic systems.e. Maximum excited mechanical systems.


6. Piezoelectricity is attributable to:

a. Surface charge generation.b. Crossed electric and magnetic fields.c. Current in a resistor.d. Strain-inducing change in shape in a crystal with no center

of charge symmetry.

7. Unlike quartz and tourmaline that are naturally piezoelectric,some ceramic materials can be artificially manufactured todisplay piezoelectric properties. Steps of this manufacturingprocess include all but:

a. Weighing, proportioning, and mixing ceramic powders.b. Adding a binder, granulating, and compressing the powder

into pellets.c. Firing the pellets in a kiln.d. Repressing the pellets.e. Lapping and plating.f. Aligning the crystal domains under controlled conditions.

8. Piezoelectric and silicon transduction technologies areparticularly well suited for incorporation into transducers fortime-varying measurements. Which statement is not true?

a. They have a large dynamic range (operate over wide amplitude range).

b. They enable transducer miniaturization.c. These technologies both operate from –450° to +1300°F.d. They produce large electrical signal outputs when

mechanically loaded.e. They have a large useable frequency response.

9. Which statement concerning the addition of damping totransducers is not true?

a. It extends the flat frequency response range.b. It is temperature dependent.c. It requires energy dissipation through mass motion.d. It is dependent on a stiff transducer flexure.



10. When damping a transducer to 0.707 of critical, the statementthat is not true is:

a. The transducer’s amplitude response is flat within 5% to three-fifths of its natural frequency.

b. The transducer’s frequency response peaks or is a maximum at its natural frequency.

c. The transducer’s phase is very linear to three-fifths of its natural frequency.

d. The transducer attenuates frequencies above three times its natural frequency by more than a factor of ten.


Signal Types: Deterministic andNondeterministic Measurands

Part


All signal types can generally be classified as either deterministicor nondeterministic (random). A random time function is onecomprised of a continuous distribution of sine waves over somerange of frequencies.The amplitudes and time delays of these sinewaves vary in an unpredictable manner as a function of time. Ran-dom vibration can be thought of as a vibratory process in whichthe particle being vibrated encounters irregular motion that neverexactly repeats. Since this motion does not repeat, an infinitely longtime record would be needed to completely describe the motion.From the instantaneous value of a random time function, it isimpossible to predict what its value will be at some later time.

A deterministic signal is the opposite of a nondeterministic signal.On the other hand, the instantaneous value of a deterministic sig-nal enables its value to be predicted at any other time.

Additional clarification can be provided through examples of ran-dom and deterministic processes. Rainfall impacting a structure

It is important to have an understanding of the signal types thatcharacterize the time-varying measurands of interest. Thisunderstanding is necessary to enable the measurement system tobe designed to pass these measurands without distortion. In Part3 we will discuss these signal types. The design of themeasurement system will be the focus of Part 5, and thesubsequent processing and use of the resultant data will be thefocus of Part 10.

Signal Types: Deterministic andNondeterministic Measurands


imparts a random force-time function to the structure.Earthquake-induced motion is typically characterized as random with time.Transportation environments impart random loads to structures,resulting in random vibration of the structure.A truck travelingdown a highway at varying speeds illustrates a transportation envi-ronment.The truck encounters random excitations due to highwayroughness. Similar analogies can be presented for a train travelingdown a track or an airplane flying through turbulence.At launch ofa rocket system, random acoustic pressures are generated and sub-sequent random fluctuations in thrust occur during flight; these areboth random inputs to the structure of the rocket system. It shouldby now be apparent that random time functions must be discussedin terms of their statistical properties.

The world is full of deterministic as well as random processes, andmore examples can easily be provided. When its free end isimpacted, the structural vibration and decay of a cantilever beamresults in a deterministic signal from an accelerometer mountedon its surface.The pressure-time history encountered by a pres-sure transducer in the breech of a gun at firing results in adeterministic signal.The load-time history encountered by a forcetransducer during a single metal-punching operation is determin-istic. Steady-state vibrations induced in a structure due to anout-of-balance condition in a motor/generator set are also deter-ministic. Figure 3.1 shows the same nondeterministic signalmeasured by two different transducers to verify their perform-ance. Note, as expected, there is no repetition in the waveform ofthe individual signals. Figure 3.2 shows a controlled drop-testwhere a deterministic pulse would result. The package wouldencounter the same dynamic loading with each drop.


Part 3 Signal Types: Deterministic and Nondeterministic Measurands

Figure 3.1The samenondeterministic,or random, signalmeasured by twotransducers toverify theirperformance.

Time

Am

plitu

de

Figure 3.2Package-droptesting results in adeterministicsignal.

Figure 3.1

Figure 3.2

DETERMINISTIC TIME FUNCTIONS

We will initially provide a detailed discussion of deterministic timefunctions. Mathematical formulas will be presented for complete-ness, but they are not necessary for an understanding of thisdiscussion.As opposed to providing specific references, references3.1 and 3.2 are provided as general sources of information to sup-port the mathmatics presented with this chapter. A number ofsimilar references exist.

Figure 3.3 provides a fundamental concept that must be under-stood by designers of measurement systems intended to respondto time-varying measurands.A duality exists between the time andfrequency domains.That is, signals can be thought of as existing inboth time and frequency space. Not every signal can be mappedfrom one space to the other. For the purpose of this discussion,however, those signals of interest to us will all possess this duality.


Time domain Frequencydomain

F(jω)

f (t)

Figure 3.3Duality of time andfrequencydomains. F(jω)has real andimaginary parts asa function offrequency or canbe represented asamplitude andphase versusfrequency.

Figure 3.3

Any signal that has an initial value of zero and a final value of zerocan exist in both spaces. For example, any mechanical shock pulsethat starts at zero amplitude and ends at zero amplitude satisfies thiscriterion. If part way through the shock pulse the cable of thetransducer should break, the pulse amplitude would not be zeroand the recorded portion of the pulse could not be represented infrequency space.

Other unique functions, such as step functions, can also be repre-sented in frequency space but will not be discussed here. Periodicfunctions also have this dual representation but possess a discrete(or line) spectrum in frequency space as opposed to a continuousfrequency spectrum.The Fourier transform pair is the mathemat-ical tool that allows us to convert a function f(t) to the frequencyspace F(jω) and back again:

F(jω)= ∫+∞

f(t)e-jωtdt, and (3.1)

f(t)=1/2π ∫+∞

F(jω)e+jωtdω

–∞

–∞

In the workplace of today, the above calculations are performedeither by stand-alone computers or computers embedded in spec-trum analyzers. Looking at the first equation of the pair inEquation 3.1, we observe that the Fourier transform of a timefunction,when transformed into the frequency domain, results in acomplex function of frequency. Its real and imaginary parts repre-sent this complex function.

These real and imaginary parts can be combined so that the trans-form of a time function can be plotted in terms of its amplitudefrequency content and its phase frequency content.Thus, to repro-duce any time function accurately, we must preserve bothamplitude and phase information. Figure 3.4 illustrates the ampli-tude frequency content of a complex shock pulse.This is derivedfrom the two parts of the frequency-space representation containedin Equation 3.1.

Mathematics also provides us with a basis to explain other facts thatcorrelate with physical observations.The time or frequency scal-ing of Fourier transforms can be expressed as:

f(bt)↔(1/ b )F(ω/b) (3.2)

Equation 3.2 states that the transform of f(bt) is proportional to theoriginal transform of f(t) with all the frequencies divided by b.Physically this means that as pulse durations get shorter (i.e., bbecomes less than 1) their frequency content gets larger.The con-verse of this statement is also true. For example, if the duration ofa pulse shifts from 1.0 second to 0.01 second and its waveformremains unchanged, the frequency component at 1 Hz is scaled andshifted to 100 Hz.The fact that a deterministic pulse can be rep-resented in terms of its Fourier transform F(jω) will be referred tonumerous times in this text.




Figure 3.4Example of f(t) andamplitude F(jω).

300.0

200.0

100.0

0.0

–100.0

–200.00.0 100.0 200.0 300.0

Acc

eler

atio

n (g

)

f (t)

Time (ms)

0.0 100.0 200.0 300.0 400.0

4.0

3.0

2.0

1.0

0.0

Fou

rier

mod

ulus

(g/

Hz)

Frequency (Hz)

Amplitude F(jωω)

Figure 3.4

RANDOM TIME FUNCTIONS

Here we will highlight in a few paragraphs what is covered inentire textbooks on random processes and random data analysis.Figure 3.5 illustrates a collection or ensemble of records containingrandom signals that could have been recorded by an array ofaccelerometers measuring seismic vibration. Each record can beconsidered to have originated from a single input/output measure-ment system, as in the upper portion of the figure.The collectionof all time histories that could have been measured is called therandom process. Referring to Figure 3.5, at any point in time twe could calculate an average and an average squared value forthe ensemble.We could also calculate an average value of the prod-uct of the data at time t and t+T.This last function would be calledan autocorrelation and, if the number of records was N, could bedefined as:

Rxx(t,T)=limN→∞(1/N)∑i=1,N(xi(t)xi(t+T)) (3.3)



Figure 3.5Ensemble ofrandom signals.

Figure 3.5

Am

plitu

de

system

Time

t+Tt


If the average, average squared, and autocorrelation values are allindependent of the time t when they are calculated, the randomprocess is defined as stationary. If any of these values vary with t, theprocess is defined as nonstationary.The discussion of nonstation-ary random processes is beyond the intent of this text.

The previous discussion implies that an infinite number of recordsis required in the random process to acquire a complete statisticalaverage.For most stationary data,we can calculate the average, aver-age squared, and autocorrelation for a single record in the ensembleover time and get the same statistic as calculating across the entireensemble.A system is considered ergodic if it enables us to definethe statistics of a random process by dealing with a single recordover time. The key point here is that most random processesare treated as stationary and ergodic, enabling us to compute thestatistics of the process from a single record.

PROBABILITY DENSITY FUNCTION

We will deal with one more statistical function associated with ran-dom signals — the probability density function. For our case, theprobability density plotted in Figure 3.6 gives the probability thatthe random signal will have a value of x. In Figure 3.6, the mostprobable amplitude of a random signal is zero,while the least prob-able amplitudes are those large positive and negative values.Thearea under the entire curve must equal one, which of course is thetotal probability that the signal has any value.The area under thecurve from zero to infinity is the same as the area under the curvefrom zero to negative infinity, which means that the average ormean value is zero.This, as stated previously, is also the most prob-able value.


POWER SPECTRAL DENSITY

The last function discussed in Part 3 is the power spectral density(PSD). Since for stationary data Rxx(t,T) is independent of t, wecan write it as Rxx(T).The Fourier transform of Rxx(T) in Equa-tion 3.1 is the PSD function Φ(ω) of the process; that is:

Rxx(T)↔Φ(ω) (3.4)

In the ergodic case, this calculation originates from a single record.With modern computational techniques, the PSD is calculateddirectly from digital Fourier transforms applied to the original datarecords; Φ(ω) can then be plotted as a function of frequency.

The nomenclature “power spectral density” is a historical one andis somewhat misleading since the units associated with the PSD aretypically not those of power. For a voltage signal, the amplitude ofthe PSD is V2/Hz.When normalized to a 1Ω load and integratedover frequency, the PSD has units of normalized power. However,when a voltage signal is scaled in units of acceleration in g’s, pres-


Figure 3.6Gaussianprobability densityfunction.

Figure 3.6

–4σ –3σ –2σ –1σ 0 +1σ +2σ +3σ +4σ

0.4

0.3

0.2

0.1

0

p xx

( ) exp= −

12

12

2

2σ π σ

Amplitudedensity function

sure in psi, or force in pounds, a PSD with dimensions of g2/Hz,psi2/Hz, or pounds2/Hz has no dimensional relationship to power.

Nevertheless, the PSD does have physical significance.When inte-grated over frequency, the PSD yields the mean squared signallevel.The square root of this resultant integral is the rms signalmagnitude. For example, the square root of the area under ag2/Hz versus frequency curve is the grms signal level.There is anadditional observation we will make about the PSD. If the trans-ducer, or any element of the measurement system has a transferfunction T(jω), its input/output relationship is:

Φ(ω)0=T(jω) 2Φ(ω)i (3.5)

Equation 3.5 states that the output PSD Φ(ω)0 is related to theinput PSD Φ(ω)i of the linear measurement system or componentby the amplitude squared of the transfer function T(jω) . Recallthat for deterministic signals we had to preserve both amplitudeand phase information. For random signals, where a PSD calcula-tion is the desired result, the measurement system’s phasecharacteristics are not important.

Previously, we developed models for the transducers of interest.We also established a basis to select the most appropriate trans-duction techniques to integrate into these transducers. Now weunderstand the properties of the signal types we are attempting tomeasure. In the next part we will investigate how to interface thesetransducers to the measurands of interest.


REFERENCES

3.1 Phillips, Charles L. and Parr, John M., Signals, Systems,and Transforms, Prentice Hall, Englewood, NJ, 1995.

3.2 Hirsching, Paul H., Paez,Thomas L. and Ortiz, Keith,Random Vibration-Theory and Practice, John Wiley and Sons, New York, NY, 1995.




QUESTIONS

1. Which statement is not true for a nondeterministic or randomsignal?

a. The signal is comprised of a continuousdistribution of sine waves over some rangeof frequencies.

b. The amplitude and time delays of the sinewaves in the signal vary in an unpredictable manner.

c. An infinitely long record is needed tocompletely describe the signal.

d. Knowing the value of the signal at any onetime enables its value to be stated at afuture time.

2. Which of the following is not an example of a randomprocess?

a. The forces associated with a man with a sledge hammerbreaking up concrete.

b. The sound pressure associated with an organplaying Yankee Doodle.

c. The vibration associated with a train travelingdown a track at varying speeds.

d. The response of a building due to buffetingwind.

3. Which of the following is not an example of a deterministicprocess?

a. The cylinder pressure in an automobile engine at constantrpm.

b. The force applied by a stationary rivet gun attaching aircraft skin.

c. The deceleration-time pulse experienced by a box being dropped on its flat surface on a concrete floor.

d. The acceleration-time response of a speaker enclosure when dropped down a flight of stairs.

4. For deterministic signals, the tool that enables us to easilyinteract with either the time or frequency domain is the:

a. Fourier transform.b. Convolution integral.c. Laplace transform.d. Maclaurin series.e. Taylor series.

5. If a half-sine deceleration pulse of 100 milliseconds durationhas a spectral content at 1 Hz of 5 g/Hz, what spectralcontent does a 1 millisecond pulse have at 100 Hz?

a. 2 g/Hz.b. .05 g/Hz.c. 50 g/Hz.d. 20 g/Hz.e. 1 g/Hz.

6. Ideally, to acquire the complete statistic for a random processwe need an infinite number of records. If we average over asingle record and claim the resultant statistics define theprocess, it must be:

a. Deterministic.b. Ergodic.c. Low frequency.d. Positive.e. Proportional.

7. The fact that unpredictability is associated withnondeterministic signals requires their amplitude to bedescribed statistically in terms of a:

a. Transform.b. Linear straight line.c. Probability density function.d. Curve fit.e. Binomial distribution.

8. True or False: The power spectral density (PSD) functionderived from a force transducer measuring a random force-time input has dimensional units of power.

9. True or False: If the sole objective of a measurement is toenable computation of a PSD, the phase response of themeasurement system is not important.

10. True or False: Thus far in this text we have developed modelsfor the transducers of interest, described the transductiontechniques best suited for transducers that acquire the



measurands of interest, and provided understanding of thesignal types that exist in nature as transducer inputs. Next, wewill discuss how to interface the transducer to the processbeing measured.


Interfacing the TransducerWith Its Environment

Part


In Part 1, we established the fact that two directly-available mea-surands exist at each input port to a transducer. One of thesemeasurands was shown to carry the information to be processed bythe transducer, and the second coexisting measurand caused energyto be transferred from the process being measured.We used theexample of force (F) and displacement (∆x) inputs to the flexureof a transducer.The product of these measurands (F∆x) providesthe value of the energy being transferred from the process underobservation.This is an important point, as it implies that the processbeing observed or measured will always be changed by the act ofmeasurement. Just as we attempt to minimize energy transferbetween electrical components in a measurement system by select-ing high input and low output impedances, we also demand highmechanical input impedances Zx(jω) for our transducers.A highvalue of Zx(jω) causes the input signal (Sx(jω)) delivered to thetransducer to be equal to the value of the measurand (Ss(jω)) withimpedance Zs(jω). Equation 4.1 establishes this relationship.

Sx(jω)=Ss(jω)[Zx(jω)/(Zx(jω)+Zs(jω))] (4.1)

or

Sx(jω)=Ss(jω), for high Zx(jω)

Part 4 of this text deals with the need to consider the inputimpedance of and coupling to a transducer when interfacing it toa process.

Interfacing the Transducer withIts Environment


When this last criterion is satisfied, the transducer transfer func-tion T(jω) from the Classical Dynamic Model (discussed in Part 1)can be used to define the relationship between the transducer out-put (So(jω)) and the signal (Ss(jω)) at the measurement source:

So(jω)=T(jω)Ss(jω) (4.2)

If this transfer function T(jω) has flat amplitude response and lin-ear phase response over the entire range of frequencies contained inthe measurement source signal, the transducer output exactly repli-cates the process being measured.The goal of every measurementis to establish the relationship:

So(jω)=Ss(jω) (4.3)

Even if the criterion for a high input impedance is met, anotherproblem can arise, depending upon the manner in which we cou-ple the transducer to the process being observed or measured.Thiscoupling process, as shown in Figure 4.1, can modify the overalltransfer function between the signal source and the transducer out-put so that Equation 4.2 has to be expressed as:

So(jω)=T1(jω)T2(jω)Ss(jω) (4.4)

Process Being

Observed

SS(jω)ZS(jω)

Transducer

SO(jω)ZO(jω)

SX(jω)ZX(jω)

T1(jω)T2(jω)

Figure 4.1Transducer inputand couplingprocess.

Figure 4.1


T2(jω) is the transfer function of the transducer as characterized inthe calibration laboratory, and is defined as T(jω) in Equation 4.2.T1(jω) is the transfer function associated with the coupling of thetransducer to the process being observed. Physical examples ofvarious couplings are: a mounting block under an accelerometer, alength of tubing connecting a pressure transducer to a pressuresource, and a stud coupling a force transducer to a structural inter-face. Ideally, one would like T1(jω) equal to one so that it has noeffect on the transducer output So(jω). Restated in nonmathemat-ical terms, the two difficulties we will deal with in this chapterare:

1. The degree to which the presence of the transducer has animpact upon the process being measured, i.e., (Zx(jω)≠∞) inEquation 4.1, and

2. The distortion of the dynamic response of the transducerdue to the manner in which we couple it to the process, i.e.,(T1(jω)≠1) in Equation 4.4.

In the following discussion, the nomenclature we will associatewith item #1 is Transducer Input Impedance Considerations. Item#2 will be referred to as Transducer Coupling Considerations.Wewill provide some practical examples of these considerations forforce, pressure, and acceleration measurands.

FORCE

Transducer Input Impedance Considerations. The mechan-ical input impedance of a force transducer was shown in Part 1 tobe:

Z=dF/dx (4.5)

Part 4 Interfacing the Transducer with Its Environment


Figure 4.2Impedance head.

Frequency

Mec

hani

cal I

mpe

denc

e,

lb.

in./s

ec.

20 100 1000 5000

104

103

102

10

1

Figure 4.3Measuredmechanicalimpedance of analuminum beamwith an impedancehead at its center.

Figure 4.2

Figure 4.3

Recall our previous conclusion that all input impedances are frequency-dependent, which now allows us to write Equation 4.5as Z=Z(jω). Equation 4.5 reinforces the fact that all force trans-ducers have elasticity associated with them (i.e., they deflect underload).

For example, a mechanical impedance head is a commerciallyavailable transducer. It combines a force transducer and anaccelerometer, coupled as close together as possible, into a singleinstrument.The mechanical impedance measured by this device isdefined as the ratio of sinusoidal force to velocity,

ForceTransducer

Accelerometer

Through Mounting Hole

Connectors

Z(mech)=(F/V)ejθ (4.6)

and provides a ratio of the resistance of a structural system tomotion.At any frequency (ω), the point impedance of a structurecan be determined by taking the ratio of the peak output (F) fromthe force transducer to the peak output (Xω2) from the accelerom-eter and then multiplying by the frequency.This multiplicationconverts the acceleration measurement to velocity and results inthe magnitude (F/Xω) of the mechanical impedance. In the earlyto mid 1960s, the aerospace industry performed a significantamount of analysis and testing using mechanical impedance heads.Figure 4.2 conceptually illustrates an impedance head, and Figure4.34.1 illustrates the measured mechanical impedance of a beam thatis free at both ends and excited at its center. The peak mechanicalimpedance values in Figure 4.3 occur when the peak motion (X)of the beam and associated velocity (Xω) are small. If under load(F) the elasticity of the impedance head results in an associateddeflection (∆x), the measured velocity would be falsely determinedto be (X+∆x)ω.The peaks in Figure 4.3 would erroneously beportrayed with lessened values, and the beam would be assumedto be less resistant to motion than it really is! An infinitely stiffimpedance head would eliminate this erroneous portrayal. (Note:At the minimum impedance values in Figure 4.3 the mass of theimpedance head below the force transducer induces inertial loadsand is also an error contributor.)

Transducer Coupling Considerations. The coupling of amechanical impedance head or a force transducer to a test structurealmost universally occurs by means of a bolt or a stud.This bolt orstud essentially “sandwiches” the transducer housing (a spring), tothe test structure (another spring).Thus, the spring constants orstiffness (K1 of the transducer,K2 of the test structure, and K3 of thebolt or stud) govern the elastic properties of the assembly.As long




as preloads are not exceeded, K3 can be viewed to be in parallelwith K1 and K2.The overall assembly stiffness can be envisionedas:

[K1K2/(K1+K2)]+K3 (4.7)

In calibrating transducers such as impedance heads, it is always sug-gested to use the same stud and preload torques that will be used inservice.This ensures that the dynamic measurements of the cali-bration laboratory will apply to field service.

PRESSURE

Transducer Input Impedance Considerations. The approxi-mate expression for the input impedance of a pressure transducerwas derived4.2 as early as 1958. For nomenclature purposes, p ispressure, F is the resultant force on the pressure receiver,V and Kare the volume and the elastic constant of the pressure receiver, xis the pressure receiver displacement, and Aeff is the effectivereceiver area.The mechanical impedance of a pressure transducercan be defined as:

Z=dp/dV (4.8)

In this case, p and V can represent the directly available measur-ands at the input to our Conceptual Transducer Model in Part 1.Aswe now expect, the product of (p∆V) has dimensions of energy.The elastic constant of the pressure receiver is:

K=dF/dx (4.9)

An effective receiver area can be defined such that:

dp=dF/Aeff and dV=Aeffdx (4.10)

Equations 4.8 through 4.10 can be combined to define mechanicalimpedence of the pressure transducer as:

Z=K/A2eff (4.11)

Again, the frequency dependence Z=Z(jω) is implied. Equation4.11 states that to approximate an ideal input impedance, Z=∞,we require a pressure transducer receiver that is as stiff as possibleand has as small an effective receiver area as possible.Our discussionon silicon and piezoelectric transduction technologies in Part 2identified them both as capable of satisfying these criteria.

When considering time-varying gas pressure measurements, exceptin a very small chamber, the energy absorbed by a pressure trans-ducer is minimal. This justifies the assumption of an infinitemechanical impedance for the pressure transducer. However thisassumption is not valid for liquid media in chambers that are smalland rigid. In the latter situation, even when the response of thetransducer is adequate to reproduce the time history, the resultwill not represent the conditions that exist when no transducer isattached to the chamber. For a particular fluid measurement appli-cation, any associated error would depend on both the fluid bulkmodulus and the chamber dimensions.

Very few pressure transducer specifications include a value formechanical input impedance. Wright4.3 provides a value ofZ=10,000 psi/in3 for a specific 50 psia metal strain gage transducer.The material properties and reduced geometries of pressure transducers employing the silicon or piezoelectric technologies discussed in Chapter 2 would result in more than an order of magnitude increase in this value. Figure 4.4 shows a silicon wafercontaining approximately 75 miniature silicon dies to be used asforce collectors in pressure transducers along with a finished transducer.




Figure 4.4MEMS siliconwafer withnumerous pressuretransducerdiaphragm diesand finishedtransducer.

Figure 4.4

Transducer Coupling Considerations. The ideal configura-tion for a pressure transducer measuring time-varying events is tohave its diaphragm mounted flush with the structural surface towhich it is affixed and normal to the incident pressure wave.TheClassical Dynamic Model of Figure 1.3 then becomes applicable. Itmay be necessary, however, to measure rapidly changing pressuresfrom a pressure tap where a pressure transducer cannot be flush-mounted. It may also be necessary to mount the pressuretransducer remotely from the process being observed or meas-ured. In particular, remote mounting might be required to isolatea transducer from a high heat source or other harsh environment.In such cases, the transducer is coupled to the process by a lengthof tubing or other intermediate fittings.We will look at a few selectsituations for both gas and fluid measurements.These considera-



d

L

Volume=V

Transducer Diaphragm

Figure 4.5Pressuretransducercoupling model.

in.

2.25

1.5

.75

0

1,000 10,000 100,000

Frequency (Hz)

Col

umn

Leng

th

Figure 4.6Fundamentalnatural frequencyof a column of dryair with one endclosed(Top=500°C,Mid=25°C,Lower=–50°C).

Figure 4.5

Figure 4.6

tions are very important because they can greatly decrease thedynamic response capability of a pressure transducer.

In Figure 4.5, the diaphragm represents the force collector in thepressure transducer.The volume shown is that associated with thecavity in front of the transducer diaphragm.The tubing of diame-ter (d) provides an interconnect. Based on the assumption that alldimensions are much less than the wavelength of sound at the fre-quency at which this system is designed to operate, an analysis ofthis cavity as a second-order-single-degree-of-freedom system canbe performed. For a short tube, the Helmholtz4.4 resonator modelyields a natural frequency in Hz of:

fn=c/(4π)[πd2/(V(L+.85d))]1/2 (4.12)

In this equation, c is the velocity of sound of the gas being meas-ured (≅ 1100 feet/second for room temperature air). If the tube inFigure 4.5 is lengthened to cause the volume V to become negli-gible, and if the tube is sufficiently narrow that the displacementof the gas at any instant is the same at all points on its cross sec-tion, it can be modeled by the wave equation.The result4.4 of thisis:

f=[(2n–1)c]/4L (4.13)

where n=1 corresponds to the first natural frequency. Figure 4.6plots this first frequency for dry air. It is interesting to look brieflyat a representative value from Figure 4.6.Assume that a pressuretransducer with a resonant frequency of 100 kHz is mounted at theend of a 1.5 inch long standpipe measuring room temperature(25°C) air. Figure 4.6 indicates that the overall system resonancewould effectively be lowered to 2,200 Hz! Equations 4.12 and 4.13then enable calculation of some limiting values as to how thedynamic response of a pressure transducer can be reduced when



coupled through a gas-filled cavity. Reference 4.5 suggests that forgas pressure measurements the Helmholtz resonator model shouldtransition to the wave equation model when the volume of thetube is about one-half the volume of the chamber.When perform-ing explosive gas measurements these equations may yieldimprecise results because gas composition and temperature, thussound velocity as well, are often unknown.

The analysis of liquid-filled tubing becomes even more complexthan gas-filled tubing.Two references are provided4.5,4.6 includingexperimental data. In the simplest analysis, the ringing whichoccurs can be attributable to the mass of the liquid and the springrate of the transducer diaphragm. Figure 4.74.5 shows qualitativemeasurements acquired from two symmetric-mounted pressuretransducers in a rocket engine chamber. One is from a flush-mounted transducer (T1(jω)=1 in Equation 4.4) and provides thecorrect pressure-time history, and one is from a 48 inch water-filledline (T1(jω)≠1 in Equation 4.4) in front of the same type of trans-ducer.The latter produces an erroneous result!


Water-Filled Tubing (W)

Flush-Mount (F)

Figure 4.7 Effect of pressuretransducercoupling in a rocketchamber.

Figure 4.7


The final item to be mentioned concerning pressure transducercoupling is the averaging effect that a sine wave sees as it passes atright angles over the circular diaphragm of a transducer.The cir-cular diaphragm spatially integrates the sine wave. Thisphenomenon occurs when measuring time-varying static pressure(the pressure exerted by the flow field normal to the containmentsurface).This is not a problem when measuring stagnation pressure,the sum of the static pressure and the dynamic pressure, where thetransducer faces directly into the incoming pressure wave.

The analysis of the spatial averaging of the transducer diaphragmhas been performed.4.7 For example,when measuring time-varyingstatic pressure in dry air, a 0.3 inch diameter diaphragm would pro-duce a five-percent amplitude error at 9,200 Hz and a 0.1 inchdiameter diaphragm the same error at 24,000 Hz.

ACCELERATION

Transducer Input-Impedance Considerations.In considering how the impedance of an accelerometer can mod-ify the response of a structure to which it is affixed, it is convenientto return to the mechanical impedance concept contained inEquation 4.6. For this discussion, we will consider the directlyavailable measurands at our accelerometer input to be force (F) andvelocity (V).This is consistent with Part 1 since velocity (V) anddisplacement (x) can be related through frequency as V=jωx wherex=Xejωt.Also consistent with Part 1, the product (F∆V) has unitsof energy rate or power.

If a linear structural system is excited by a force (F) at point 1, itsvelocity response (Vs) at point 2 is:

Vs=F/Zs (4.14)


where Zs is the complex transfer impedance between points 1and 2. If we place an accelerometer with impedance Za to meas-ure this velocity (obtained by integrating the response of theaccelerometer) at point 2, and if we keep F constant,Vs becomesthe new value Vs

*. This modification is due to the mechanicalimpedance of the accelerometer.The ratio of the measured veloc-ity to the actual velocity (Vs

*/Vs) of the structure is:

Zs/(Zs+Za) (4.15)

The question now arises as to the nature of Za so that its influ-ence can be evaluated.A properly designed accelerometer will havea housing that is stiffer than its contained flexural element.Thus, anaccelerometer can be modeled as a pure mass to frequenciesapproaching the fundamental resonant frequency of its flexural ele-ment.When a displacement excitation Xejωt is applied to a puremass (m), the resultant peak inertial force is –mXω2 and the peakvelocity is jωX. From Equation 4.6, the mechanical impedance ofa properly designed accelerometer is:

Za=jωm (4.16)

Equations 4.15 and 4.16 are important because they show not onlythat the presence of an accelerometer modifies the measuredmotion of a structure, but also that the amount of this modificationincreases with frequency.The Society of Automotive Engineers(SAE) recognized these facts during construction of a test appara-tus designed to measure the vibration transferred to the occupantsof automobiles at the body-to-seat interface.Accelerometers had tobe located between the body and the seat and not alter theirdynamic properties. An SAE pad, SIT-BAR (1974), was designedto make this impedance match.



The following illustrative example can assess the importance ofEquations 4.15 and 4.16. We will investigate the ratio of thevelocity response Vs

*/Vs to determine the influence an eightgram accelerometer has on the response of the free end of auniform aluminum bar at 50 percent of the fundamental resonantfrequency. For a 30 inch bar this frequency is 1,677 Hz, and for aneight-inch bar it is 6,289 Hz.The bar will be analyzed as drivenby a sine wave at its free end. Results4.7 of the analysis are:

Rod Length (inches) Vs*/Vs

30 1.00615 1.0468 1.406

It is apparent from these results that large accelerometers at highfrequencies can greatly modify the response of structural systems. Itis also apparent that a careful assessment of measurement results isnecessary when an attempt is made to acquire acceleration data atmany 10’s of kHz (the upper limit today).

In Part 11, we will briefly discuss “Modal Testing.” Experimen-tal modal analysis involves vibration testing in which the structureis vibrated with a known excitation.This is accompanied by dataacquisition and subsequent analysis.This testing normally employsan array of accelerometers relocated on the structure at time inter-vals over the duration of the testing.The accelerometers perturbthe dynamics of the system and introduce errors in the measuredvibrations.The global curve fitting algorithms for modal processingcan become inconsistent due to accelerometer relocation. Refer-ence 4.8 develops techniques to correct these algorithms foraccelerometer loading effects in modal testing and analysis. Anexample is shown in which two accelerometers representing 0.4%of the total weight of an automobile-engine support, produce sig-nificant frequency errors when measuring the fourth and fifthvibration modes of the support between 250 and 300 Hz.

If any of the above testing can be repeated, the addition of adummy mass alongside the measuring accelerometer will enable adetermination as to whether the impedance of the accelerometer isperturbing the response of the structure.As a first attempt, select amass equal to that of the accelerometer. If test results are identicalwhen the mass is present and removed, the input impedance ofthe accelerometer is not a concern.

Transducer Coupling Considerations.The information gener-ally supplied by a manufacturer of an accelerometer includes themaximum frequency to which its amplitude response is flat withina few percent when the optimum mounting technique is used.Typically, this mounting is by means of a threaded stud onto a flatsurface of fine finish.The initial mounting torque must preloadthe accelerometer to a degree that is sufficient to prevent separationfrom a test structure in application. In actual practice, it may notbe possible to tap a hole to attach an accelerometer, the mountingsurface may not be flat, or the structure may not possess adequatethread strength when tapped.To overcome these limitations, cou-pling to the structure can be by means of helicoils, adhesives andinterface blocks (Figure 4.8). Insulated studs are used in some sit-



Figure 4.8Mountingtechniquesinvolvingadhesives, helicoilsand interfaceblocks.

Figure 4.8


25

20

15

10

5

0

–5

Mag

nific

atio

n F

acto

r (d

B)

0.1 1.0 10.0 100.0

Frequency (kHz)

InsulatedStud

Solid Stud

Aluminum Block+Epon 9340.625" diam, 0.80" long Cylinder

Aluminum Block0.625" Cube

Figure 4.9Accelerometerfrequencyresponse withdifferent mountingtechniques.

Figure 4.9

uations for electrical noise considerations when using piezoelectricaccelerometers.All of the variations in mounting technique tend todegrade the response of the accelerometer from that specified bythe manufacturer by introducing lower frequency resonanceswhich distort the frequency response of the accelerometer.Whencoupling to a structure, as documented in reference 4.9,mounting-surface finish is another variable in accelerometer performance.Figure 4.9 shows how the frequency response of an accelerome-ter is degraded from that achievable by using a solid stud whenthe accelerometer is mounted using either interface blocks of vary-ing geometry or an insulated stud.

This part of the text has sensitized us to the need to consider boththe input impedance of and coupling to a transducer when inter-facing it to a process. Having gained an understanding ofappropriate transduction technologies in Part 2 and signal types ofinterest that pass through our measurement systems in Part 3, weare ready to consider measurement system components that fol-low the transducer in Part 5. Our focus remains the same: toacquire successful time-varying force, pressure, and accelerationmeasurements.


REFERENCES

4.1 Bouche, R. R., Endevco Corp., Instruments and Methods for Measuring Mechanical Impedance, Shock and Vibration Bulletin, Part II, pp.18-28, Jan. 1962.

4.2 Barton, J. R., A Note on the Evaluation of Designs of Transducers for the Measurement of Dynamic Pressures in LiquidSystems, Statham Instrument Notes, No. 27, Oxnard, CA,Oct. 1958.

4.3 Wright, C. P., Applied Measurement Engineering, Prentice Hall, Englewood Cliffs, NJ, p. 108, 1995.

4.4 Morse, P. M., Vibration and Sound, McGraw Hill, New York, NY, ch. 22, pp. 221-235, 1948.

4.5 Thomson,T. B., The Effect of Tubing on Dynamic Pressure Recording, Rocketdyne, Canoga Park, CA,Technical Report 61-3, Feb. 1961.

4.6 Fowler, R. L., An Experimental Study of the Effects of LiquidInertia and Viscosity on the Dynamic Response of Pressure Transducer-Tubing Systems, M. Sc.Thesis,The Ohio State University, Mechanical Engineering Department, 1963.

4.7 Walter, P. L., Limitations and Corrections in Measuring Dynamic Characteristics of Structural Systems, Ph. D.Thesis,Arizona State University, pp. 140-141 and 205-208,Dec. 1978.



4.8 Decker, J. and Witfeld, H., Correction of Transducer Loading Effects in Experimental Modal Analysis, Proceedings 13th International Modal Analysis Conference, Nashville,TN,Sponsored by Union College and SEM,Vol. 2, pp. 1604- 1608, Feb.1995.

4.9 Mangolds, B., Effect of Mounting-Variables on Accelerometer Performance, Shock and Vibration Bulletin,Vol. 33, No. 3,pp. 1-12, Mar. 1964.



QUESTIONS

1. If S0(jω) is the transducer output in the frequency domain,Ss(jω) its input, and T(jω) its transfer function, the goal ofevery measurement is to be able to express:

a. S0(jω) = Ss(jω)T(jω).b. S0(jω) = Ss(jω).c. S0(jω) = T(jω).d. Ss(jω) = T(jω).

2. In Equation 4.4 in Part 4, examples of where the transducer’stransfer function T2(jω) = T(jω) is modified by the coupling ofthe transducer to the process being measured include:

a. mounting blocks under an accelerometer.b. a length of tubing interfacing a pressure transducer.c. a compliant stud coupling a force transducer to a

structure.d. all of the above.

3. Excess elasticity in the case or housing of an impedance headresults in impedance measurements which are:

a. too high.b. too low.c. unaffected.

4. The performance of a pressure transducer is optimized by:

a. a stiff transducer receiver.b. a small receiver area.c. high output signals.d. all of the above.

5. If a pressure transducer is interfaced to its environment as inFigure 4.5, which of the following won’t enhance its frequencyresponse:

a. high gas velocity c.b. small cavity volume V.c. long interconnect tube length L.

6. True or False: When measuring time-varying staticoverpressure, a larger diameter diaphragm produces less error.

7. True or False: A properly designed accelerometer will have ahousing stiffer than its contained flexural element.

8. The mechanical impedance of an accelerometer should:

a. be as small as possible.b. decrease with increasing frequency.c. be independent of mass.d. none of the above.

9. True or False: If test results are the same when a second testmass equal to the accelerometer is placed adjacent to it, theaccelerometer’s impedance is not adversely influencing themeasurement.

10. Variations in mounting technique that can degrade theresponse of an accelerometer include:

a. helicoils.b. insulated studs.c. mounting blocks.d. all of the above.


Measurement SystemRequirements

Part


The output from piezoelectric transducers typically interfaces intoeither a charge amplifier or a field-effect transistor (FET) based cir-cuit (Figure 5.1). In the latter case, the FET may be packaged in aseparate module or contained within the housing of the transducer.Resistance-based silicon transducers typically require a dc differen-tial amplifier (Figure 5.2) while capacitive-based silicon transducersrequire an ac carrier demodulator amplifier or equivalent circuit(Figure 5.3).These amplifiers can also be external or internal to thehousing of the transducer. A knowledgeable amplifier designermust be familiar with terms such as slew rate, source current, com-mon mode rejection ratio, operating temperature range, zerostability, overload recovery, gain, etc.

The transducer signal output is delivered to the next component inthe measurement system through its cable. Coaxial or twisted paircables, dependent on amplifier type/location, are used with piezo-electric transducers. Multiconductor shielded cable assemblies areused with silicon transducers.A knowledgeable cable designer must

Measurement System Requirements

This part focuses on system considerations for time-varyingmeasurements. The chapter first looks at some generalizedmeasurement system requirements, then specifically justifies therequirements for linearity, flat amplitude response, and linearphase response in measurement systems.


Figure 5.1Charge Amplifierand FET Circuit.

Piezoelectric-Based Transducer

FET Circuit

V

τ = +( )R C Cs s a

C Rs siconst

Z 1/jw Co a= ω

Figure 5.2DC DifferentialAmplifier.

V1

V2Resistance-

BasedTransducer

Z Ro = R2

R1

R2

R3

R3

R4

R4

Figure 5.2

Vo

Piezoelectric-Based Transducer Charge Amplifier

V Z 1/jw Co a=

Rf

Ccable

Cf

ω

τ = R Cf f

Figure 5.1

V


be familiar with triboelectric effects in coaxial cables of chargesensing circuits, distributed capacitance effects in multiconductorcables, operating temperature ranges of various dielectrics, materi-als flexibility and strength properties, connector types, etc.

Numerous design disciplines focus on measurement system com-ponents such as those just discussed, but few focus on theend-to-end design of an entire measurement system. Figure 5.4contains two measurement system examples.One is simple and oneis complex.The top example is a transducer whose signal is con-nected through a cable directly to a recorder. To describe thissystem Equation 4.4 would have to be expanded as:

So(jω) = T1(jω)T2(jω)T3(jω)T4(jω)Ss(jω) (5.1)

where T3(jω) accounts for the transfer function of the cable andT4(jω) accounts for the transfer function of the recorder.The bot-

Part 5 Measurement System Requirements

Capacitance-Based

Transducer

VARIABLE CAPACITANCEMICROSENSOR

ZERO BALANCE

OUTPUT SIGNAL

REFERENCE VOLTAGE

EXCITATION VOLTAGE

APPLICATION SPECIFIC INTEGRATED CIRCUIT

TRIANGLEWAVE

GENERATOR

FREQUENCY SETTING

CURRENTDETECTOR ANDSUBTRACTOR

OPERATIONALAMPLIFIER

VOLTAGEREFERENCE AND

REGULATOR

GAIN SETTINGAND

OUTPUT FILTER

Figure 5.3

Figure 5.3AC CircuitAmplifier.


tom example is a complex measurement system similar to one thatmight be encountered in space telemetry.Time-varying force, pres-sure, and acceleration are often recorded during space flights. In thisexample, Equation 4.4 would be modified as:

So(jω) = T1(jω).........T16(jω)Ss(jω) (5.2)

Clearly,making So(jω) = Ss(jω) in Equation 5.2 can be a significantchallenge.There are 16 transfer functions to account for!

Transducer

Cable

Recorder

Simple Measurements System

Transducer

Multicoupler

Filter Encoder VCOAmplifier

Pre-Amp Receiver DiversityCombiner

Discriminator #1

Recorder Decomutator Discriminator #2

Flight Antenna

TransmitterGroundAntenna

Complex Measurement System

Figure 5.4Simple andComplexMeasurementSystems.

Figure 5.4


We have ignored the input impedances of the various measurementsystem components in our discussions thus far.The development ofthe operational amplifier in the mid-1950s has enabled electricalcircuits to be designed with very high input impedances. As aresult, circuit loading rarely occurs.

The measurement systems we are interested in are time-invariant;their characteristics do not change with time. Furthermore, allmeasuring systems that are intended to record dynamic data mustbe linear5.1,5.2,5.3. If an input i1(t) to a single-input single-output sys-tem produces an output o1(t), and an input i2(t) produces an outputo2(t), then concurrent inputs ai1(t) and bi2(t) must result in an out-put ao1(t) + bo2(t) for the system to be linear. Linear systems satisfythis principle of superposition.All linear systems have associatedwith them a unique overall transfer function Toverall(jω).As can beseen from Equations 5.1 and 5.2,Toverall(jω) is the product of theT(jω)s of numerous measurement system components.TypicallyToverall(jω) is characterized by its log amplitude and phase versus logfrequency responses as Bode plots.

Next, an improperly-designed filter provides an example of a non-linear system. Filters will be discussed in detail in the next chapter.Curve A in Figure 5.5A (measured with a rms meter) shows theamplitude frequency response of a lowpass filter with a –3dB fre-quency of approximately 1200 Hz.Figure 5.5B shows the phase lagversus frequency for this filter.Both of these curves were generatedwith a sinusoidal input voltage applied to the filter.This inputresulted in a low frequency output equal to 60 percent of the lin-ear amplitude range of the filter.Curve B of Figure 5.5A shows theamplitude frequency response of this same filter generated with asinusoidal input voltage that resulted in a low frequency output of20 percent of the linear amplitude range of the filter. Note thatwhen normalized the two curves are different (superposition is notsatisfied).The difference between these two curves in this particu-



lar design is due to slew induced distortion. The result is anonunique Toverall(jω).A nonunique transfer function is a charac-teristic of all non-linear system.

Figure 5.5C provides further characterization of the non-linearitiesassociated with this particular filter.At 100 Hz the two curves ofFigure 5.5A superpose indicating the filter to be linear at this fre-qucncy. The test data presented in Figure 5.5C confirm thisobservation.At higher frequencies, however, the two curves of Fig-ure 5.5A differ. The test data taken at 1600 Hz (Figure 5.5C)provide confirmation of inadequate filter design.Thus, checkinglinearity at only one frequency does not confirm a system to belinear!

The principal problem with non-linear systems is that they createfrequencies in their output not present in their input.The conceptof how they do this is simple. If a non-linear system’s input-out-put relationship is smooth and well behaved, it can beapproximated over a finite interval by a polynomial relationshipas:

o = c0 + c1i + c2i2 + c3i3 +... (5.3)

For a linear system only coefficients c0 and c1 are nonzero.For non-linear systems, coefficients above c1 also become involved.

Transient signals contain a continuous frequency spectrum (Equa-tion 3.1); however, the troublesome effect of system non-linearitiescan be demonstrated by considering an input signal comprised ofonly two frequencies:

i(t) = sin(ω1t) + sin(ω2t) (5.4)

If only coefficients c0 and c1 exist (linear system), the system



Figure 5.5AAmplitude vsFrequency

AMPLITUDE FREQUENCYRESPONSE1.2

1.0

0.8

0.6

0.4

0.2

0.00 1000 2000 3000

FREQUENCY (Hz)

A

B

Figure 5.5BPhase vs

Frequency

PHASE FREQUENCYRESPONSE

0 1000 2000 3000FREQUENCY (Hz)

Figure 5.5CLinearity Check

LINEARITY

1.8

1.5

1.2

0.9

0.6

0.3

0.00.0 0.02 0.04 0.06

INPUT (VRMS)

100Hz

1600Hz

Figure 5.5Non-linearLowPass FilterExample.

Figure 5.5

response to this input signal occurs at the same frequencies as theinput.The c0 coefficient generates an additional dc component. Ifthe coefficient c2 exists, trigonometric identities show frequenciesto be created at 2ω1, 2ω2, ω1–ω2, and ω1+ω2. Similarly, if the coef-ficient c3 exists, frequencies are created at ω1, 3ω1,ω2, 3ω2, 2ω2–ω1,2ω2+ω1, 2ω1–ω2, and 2ω1+ω2. Note that non-linear systems cre-ate frequencies at, above, and below the system-input frequencies.

This frequency creativity is further illustrated in Figure 5.6. Figure5.6A contains a decaying sinusoidal signal analytically describedby the following expression:

3e-60πtsin(1999πt) (5.5)

This type signal is a typical transient excitation response in elec-trical oscillator circuits and structural mechanics systems. Figure5.6B describes three output (o) versus input (i) relationships; thefirst is linear and the second two are non-linear.The relationshipsare:

o = i, (5.6)

o = i for -1 ≤ i ≤1 and o = sgn(i) for |i| > 1 and

o = tanh(i)

The signal of Figure 5.6A is individually passed through each of thesystems described in Figure 5.6B. In Figure 5.6C, the system out-put looks just like the input when passed through the linear system.The Fourier amplitude spectrum of the signal in Figure 5.6C isshown in Figure 5.6D. Note the spectral content has a peak. Fig-ure 5.6E describes the output signal that results from passing theinput signal through the non-linear system described by sgn(i) for|i|> 1. Its amplitude spectrum is described in Figure 5.6F. Note



that additional spectral content is generated with peaks occurringat more than one frequency. Figure 5.6G shows the result of pass-ing the same input signal through the non-linear system describedby tanh(i).Again, as shown in Figure 5.6H, additional spectral con-tent and peaks are generated. Figures 5.6F and 5.6H could beerroneously interpreted to indicate the presence of multiple reso-nant frequencies in an electrical or structural system where onlyone exists.We have made a case for the requirement for linearityin measurement systems for time-varying signals.We now justify


DECAYING SINUSOIDALSIGNAL

SYSTEM DESCRIPTIONS

AM

PL

ITU

DE

OU

TP

UT

3

2

1

0

-1

-2-30.000 0.008 0.016

TIME (s)-3

INPUT-2 -1 0 1 2 3

3

2

1

0

-1

-2-3

0=SGN(I) 0=TAHN(I)O=SGN(1)

O=I

O=TANH(I)

Figure 5.6A: Decaying Sinusoid

Figure 5.6B: Linear and Non-linear Systems

OUTPUTLINEAR SYSTEM

AMPLITUDE SPECTRUMLINEAR SYSTEM OUTPUT

AM

PL

ITU

DE

UN

ITS

/Hz

3

2

1

0

-1

-2-30.000 0.008 0.016

TIMES (s)

0.010

0.008

0.006

0.004

0.002

0.000102 103 104

FREQUENCY (Hz)Figure 5.6C: Linear

OutputFigure 5.6D: Spectrum

Linear System

Figure 5.6ADecaying Sinusoid.

Figure 5.6BLinear and Non-linear Systems.

Figure 5.6CLinear Output.

Figure 5.6DSpectrum LinearSystem.

Figure 5.6FrequencyCreation in Non-linear Systems.


the requirement for flat amplitude response and linear phaseresponse in measurement systems.5.4

The fact that a requirement exists for flat amplitude versus fre-quency response in measurement systems is somewhat intuitive tounderstand. In Part 3 we talked about the concept of visualizing asignal in both the time and frequency domains.Therefore, it isapparent that if we modify the amplitude of the frequency con-tent of a signal, we will also modify the time representation of thesignal. Passing a signal through a measurement system that does nothave a flat (constant) |Toverall(jω)| modifies this amplitude-fre-quency content. However, intuition does not provide us with a

Figure 5.6GNon-linear TanhOutput.

Figure 5.6HSpectrum Non-linear System.

OUTPUT SGN(I)SYSTEM

AMPLITUDE SPECTRUMSGN(I) SYSTEM OUTPUT

AM

PL

ITU

DE

3

2

1

0

-1

-2

0.000 0.008 0.016TIME (s)

-3

Figure 5.6E: Non-linearSGN Output

Figure 5.6F: SpectrumNon-linear System

UN

ITS

/Hz

0.010

0.008

0.006

0.004

0.002

0.000102 103 104

FREQUENCY (Hz)

OUTPUTTANH(I) SYSTEM

AMPLITUDE SPECTRUMTANH(I) SYSTEM OUTPUT

AM

PL

ITU

DE

UN

ITS

/Hz

3

2

1

0

-1

-2-30.000 0.008 0.016

TIMES (s)

0.010

0.008

0.006

0.004

0.002

0.000102 103 104

FREQUENCY (Hz)

Figure 5.6G: Non-linearTanh Output

Figure 5.6H: SpectrumNon-linear System

Figure 5.6ENon-linear SGNOutput.

Figure 5.6FSpectrum Non-linear System.



Figure 5.7Flat Amplitude andLinear PhaseResponseRequirements.

Figure 5.7APulse Train.

Figure 5.7B: Boxcar Filter

AM

PL

ITU

DE

RA

TIO

0.0 5.0 1.00

FREQUENCY (Hz)

1.0

0.015.0 20.0 25.0

Figure 5.7A: Pulse Train

AM

PL

ITU

DE 1.5

0.0 0.5 1.0

TIME (sec)

1.0

0.5

0.01.5 2.0

Figure 5.7BBoxcar Filter.

Figure 5.7C: Non-linearPhases

PH

AS

E L

AG

(ra

diu

s)

0.0 5.0 10.0

FREQUENCY (Hz)

35.0

15.0 20.0 25.0

30.025.020.015.010.0

5.00.0

φ πω= 2φ ω= 0 2.

φ ωπ

=2

250

φ ωπ

=3

212 500,

Figure 5.7CNon-linear Phases.

Figure 5.7

n

n

n

n


Figure 5.7EThrough LinearPhase.

Figure 5.7FThrough Non-linearPhases.

Figure 5.7E: Through Linear Phase

AM

PL

ITU

DE

0.0 0.2 0.4TIME (sec)

0.6 0.8 1.0

1.5

1.0

0.5

0.0

0.5

Figure 5.7F: Through Non-linear Phases

AM

PL

ITU

DE

0.0 0.2 0.4TIME (sec)

0.60.8

1.0

1.5

1.0

0.5

0.0

0.5

φαω3φαω2

φα ω

Figure 5.7DThrough BoxcarFilter.

Figure 5.7D: Through Boxcar Filter

AM

PL

ITU

DE

0.0 0.2 0.4TIME (sec)

0.6 0.8 1.0

1.5

1.0

0.5

0.0

0.5

Figure 5.7 continued


similar understanding as to how non-linear phase response modi-fies the time representation of a signal.The pulse train representedin Figure 5.7A will provide us with this insight.

Over any one period, this pulse train-time function f(t) can be rep-resented as:

f(t) = 1, .3 ≤ t ≤ .7 (5.7)

f(t) = 0 elsewhere in 0 ≤ t ≤ 1

The Fourier series representation of this periodic function is:

f(t)= 0.4 + (2/π)∑n=1,∞ [(-1)n/n] sin(0.4πn)cos(2πnt + φn) (5.8)

where φn is initially equated to zero. Summing its first 25 har-monics will approximate this function.This is equivalent to passingthe function through the ideal “boxcar” filter described in Figure5.7B with no accompanying phase shift.The resultant filtered sig-nal can then have the various linear and non-linear phase responsesof Figure 5.7C associated with it.These are respectively:

φn = (2πωn)1/2, (5.9)

φn = .2ωn,

φn = ωn2/(250π), and

φn = ωn3/(12,500π2)

where ωn = 2πn.These phase responses are all normalized to pro-duce a shift of 10πradians at n=25 Hz, the highest harmonic in thefiltered signal. Figure 5.7D shows the result over one period of thetime function when it is passed through the filter in Figure 5.7B.



The ripple on top of the pulse is solely due to truncation of thehigher frequencies (due to deviation from flat amplitude response)and is called Gibb’s phenomenon. Figure 5.7E shows the effect ofshifting the phase of the signal in Figure 5.7D in the linear fash-ion described by the second of Equations 5.9. As predicted bytheory, only a constant time delay occurs and further signal distor-tion is avoided.

Figure 5.7F superposes results of shifting the phase of the signal inFigure 5.7D in a non-linear fashion according to the 1st, 3rd, and4th of Equations 5.9. (In Figure 5.7F, α denotes “proportional”).The distorted waveforms, erroneous signal amplitudes, and modi-fied pulse durations are solely attributable to the various non-linearfilter phase responses!

This part has effectively summarized the justification for linearity,flat amplitude response, and linear phase response in measurementsystems where accurate reproduction of a time-varying signal isimportant. Recall, for random signals, where a power spectral den-sity (PSD) calculation is the desired result, the measurementsystem’s phase characteristics are not important. In this latter situa-tion, we require only measurement system linearity and flatamplitude response.

In concluding Part 1, we noted that the transfer function T(jω) inFigures 1.4 and 1.5 for the Classical Dynamic Model of a trans-ducer does not have flat frequency response and linear phaseresponse at all frequencies. Specifically, these criteria are satisfiedonly at the lower frequencies.Thus, there is the potential for sig-nal distortion to be produced even by an optimally mounted andapplied force or pressure transducer or accelerometer.Equation 3.2stated that wide pulses have low frequency content and short pulseshave high frequency content.Thus one would expect force and


pressure transducers and accelerometers to preserve the fidelity oftime-varying signals associated with wide pulses and distort thetime-varying signals associated with narrow pulses.This is exactlythe case.

Figure 5.8 illustrates this phenomenon for a half-sine pulse.Thepulse input of unit amplitude and duration is illustrated.The recip-rocal of the transducer natural frequency (fn) is T. For a fixed pulseduration (tc), as the transducer natural frequency gets higher, and


Non-dimensional

Figure 5.8

Figure 5.8Classical DynamicTransducer ModelResponses to HalfSine Pulses.

Nondimensional Time (t/tc)


thus T and the ratio of T/ tc smaller, pulse reproduction improves.Note that the value of T/ tc of 0.2222 provides a response withless than ten percent error on pulse peak. Smaller values yet (highernatural frequencies) would continue to improve the reproductionfidelity.

In some instances it may be useful in measurement system designto tailor the phase and/or amplitude response versus frequencyover a select frequency range.The inclusion of analog filtering inmeasurement systems accomplishes this tailoring.Part 6 will discussfiltering applicable to the signals of interest in this text. Part 7 willthen provide “rules of thumb” that the engineer can easily imple-ment to guide overall measurement system design and/or dataanalysis.

REFERENCES

5.1 Stein, P. K., Dynamic Measurements on and With Non-LinearSystems: Problems and Approaches, Proceedings First International Modal Analysis Conference, Orlando, FL,Sponsored by Union College and SEM, Schenectady,NY, Nov. 1982, 358-389.

5.2 Wright, C. P., Dynamic Data Invalidity Due to Measurement System Non-Linearity,Western Regional Strain Gage Committee Proceedings, SEM, Phoenix,AZ, Feb. 1985.

5.3 Walter, Patrick L., Problems With Frequency Creation inNon-linear Measurement Systems, Proceedings of the 11th Triennial World Congress of the International Measurement Confederation (IMEKO), Houston,TX, Oct. 1988,425-428.

5.4 Walter, P. L., Effect of Measurement System Phase Response on Shock Spectrum Computation,The Shock and VibrationBulletin, Part I, May 1983, 133-142.




QUESTIONS

1. The signal output from a piezoelectric transducer typicallyinterfaces into a (circle as many as applicable):

a. charge amplifier.b. FET based circuit.c. dc differential amplifier.d. ac carrier amplifier.

2. In writing Equations 5.1 and 5.2, we account for the transferfunctions of the various measurement system componentsand assume each component’s input impedance is____________ relative to the output impedance of thepreceding component.

a. very low.b. very high.c. equal to.

3. Whether a measurement system is simple or complex, anylinear measurement system has a unique overall transferfunction that can be represented by:

a. its resistance.b. its power dissipation characteristics.c. its amplitude and phase versus frequency responses.

4. A characteristic of all non-linear measurement systems is:

a. a nonunique transfer function.b. identical frequency response at low frequencies.c. identical frequency response at high frequencies.

5. If the frequency response of a linear measuring system isverified by sinusoidal sweeps at two different amplitude levels,when normalized the resultant response curves should be:

a. different.b. identical.



6. When recording dynamic signals, non-linear measurementsystems should not be used because they:

a. attenuate signals.b. amplify signals.c. generate frequencies in their output not present in

their input.

7. In Equation 5.3 the effect of the i3 term for an input signalcomprised of 2 discrete frequencies is to generate:

a. 5 output frequencies.b. 6 output frequencies.c. 7 output frequencies.d. 8 output frequencies.

8. Over the range of frequencies that we wish to maintain dataintegrity, we want the amplitude-frequency response of ameasurement system to be:

a. linear.b. non-linearc. flat or constant.d. uniformly decreasing.

9. Over the range of frequencies that we wish to maintain dataintegrity, we want the phase-frequency response of ameasurement system to be:

a. linear.b. non-linear.c. flat or constant.

10. An accelerometer can be expected to properly respond toacceleration pulses only in a frequency band:

a. above its resonant frequency.b. below its resonant frequency.c. centered around its resonant frequency.

Filtering in the MeasurementSystem

Part


Filters will be defined as devices that greatly attenuate theunwanted portion of an input signal in the frequency domain.Filters are implemented as lowpass, high pass, band pass, or bandreject types. Figure 6.1 depicts the ideal responses of these commontypes. Filters, however, are not ideal; they start to attenuate slowlyand rolloff at an increasing rate that approaches an ultimate value.This ultimate value is not even approximated until far past thecutoff value of the filter. The cutoff value of an actual filter isdefined, through common acceptance, as the point at which themagnitude of the amplitude response of the filter is attenuated 3 dB(|T(jω)|= 1/√2).The passband of an actual filter is defined asbelow the cutoff frequency for a lowpass filter, above the cutoff fre-quency for a high pass filter, between cutoff frequencies for a bandpass filter, and below the low and above the high cutoff frequencyfor a band reject filter. The number of filter poles (6 dB/octave/pole) ultimately specifies the value of the filterrolloff. For example, a six-pole (three sections built around opera-tional amplifiers) filter would achieve an ultimate rolloff of 36dB/octave. Filters can cause both amplitude and phase distortion,

Filtering in the MeasurementSystem

Filters are used to attenuate unwanted parts of signals in thefrequency domain. They are often used with signals from forceand pressure transducers and accelerometers in an attempt toremove high frequency distortion. However, if not properlyselected, the filters can introduce additional distortion. This partfocuses on proper filter selection and application in themeasurement system.

along with time delay, in their output signal.Two references6.1,6.2 areprovided for the reader seeking additional information on filtertheory beyond that given in this chapter.


Tjω (

)T

jω ()

Tjω (

)T

jω ()

(a)

(b)

(c)

(d)

1

1

1

1

Frequency

Frequency

Frequency

Frequency

fc

fc

fc1 fc2

fc1 fc2

Figure 6.1AmplitudeResponse of Ideal a. Lowpass; b. Highpass; c. Band Pass; andd. Band RejectFilters.

Figure 6.1


The principles of electric wave filters were documented about1915.These early filters were all passive types.That is, they exclu-sively used resistors, inductors, and capacitors. In the 1950s theactive filter, built around resistors, capacitors, and operationalamplifiers, came into play.Although numerous types of active filtershave been proposed and built due to the ease of designing with anoperational amplifier, a few popular configurations continue to bedominant.

Our focus will be on the lowpass filter.The reason for this is sim-ple. Looking at the transfer function T(jω) for the ClassicalDynamic Model of the transducer in Figure 1.2 (Figures 1.3 and1.4), we see it satisfies the measurement system requirements forflat amplitude response and linear phase response at low frequen-cies.The lowpass filter at the top of Figure 6.1, at least in idealform,would preserve the desirable low frequency response of T(jω)for the transducer and eliminate the undesirable high frequencyportion which causes distortion. Recall that this distortion wasdocumented in Figure 5.8.

The Butterworth lowpass filter configuration was suggested in1930. Its amplitude response is characterized as:

[1/(1 + ω2n)]1/2 (6.1)

where n is the number of filter poles. Its emphasis is on maximallyflat amplitude response.Another commonly used lowpass config-uration is the Chebyshev.This configuration makes use of theChebyshev polynomial and produces an equal-ripple amplitudevariation in the passband. Outside the passband the gain alsodecreases monotonically, but at a faster rate than the Butterworthcharacteristic.The class of filter that has equal ripple in both thepassband and the stop band is the elliptic-function or Cauer filter.Its gain does not decrease monotonically outside the passband, but

Part 6 Filtering in the Measurement System

it offers further improvement in attenuation than the Chebyshevlowpass filter.The Bessel-Thompson filter configuration empha-sizes filter phase linearity as opposed to flat amplitude response.

Group delay is defined as the derivative of the phase frequencyresponse [-(dφ/dω)] of a filter. It is of importance when constanttime delay is needed in a filter. For a Bessel-Thompson filter, groupdelay is a constant.The Bessel-Thompson amplitude frequencyresponse, however, is far inferior to that of the other filter config-urations discussed.We will focus our attention on those lowpassfilters with monotonic decreasing response outside their passband:the Butterworth, Chebyshev, and the Bessel-Thompson (some-times just referred to as Bessel). Figure 6.2 plots theamplitude-frequency and phase-frequency responses of two ofthese lowpass filters.The six-pole Chebyshev is the best approxi-mation of an ideal “boxcar” filter based on its amplitude response(Figure 6.2A), but its phase response is more non-linear than thatof the six-pole Bessel as shown in Figure 6.2B. Recall that Part 3established the fact that phase characteristics are not importantwhen the goal of the measurement system is to acquire and processnondeterministic (random) data resulting in a power spectral den-sity function Φ(ω)o. Figure 6.2A would then indicate theChebyshev filter shown to be the best for random data. It providesthe closest approximation to the ideal lowpass filter. For deter-ministic data, the filter choice is not as apparent.


Before establishing criteria for filter selection, which will con-clude this part, it is next desired to decide the location in themeasurement system where it is optimum to insert the lowpassfiltering.This filtering should be provided between the measuringtransducer output and the first-stage gain of the signal condition-ing.6.3 This location is optimum because it provides three benefits:



NORMALIZED FREQUENCY (f/f-3dB)

AM

PL

ITU

DE

RA

TIO

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 1.0 2.0 3.0

CHEBYSHEV

BESSEL

NORMALIZED FREQUENCY (f/f-3dB)

PH

AS

EL

AG

(d

egre

es)

600.0

500.0

400.0

300.0

200.0

100.0

0.00.0 1.0 2.0 3.0

CHEBYSHEV

BESSEL

Figure 6.2B

Figure 6.2A

Figure 6.2AmplitudeFrequency andPhase FrequencyPlots for 6-PoleChebyshev (0.1dBripple) and BesselFilters.

1. It prevents subsequent signal conditioning components frombeing driven non-linear and causing distortion of the time-varyingsignal. Realistic measurement-system calibration levels can beestablished once signal distortion is eliminated. Otherwise, analyt-ically unpredictable signal magnitudes can be initiated throughthe measurement system due to discontinuities in loading and/orsudden impulses that excite the resonant frequencies of the trans-ducer.

2. It enhances the measurement system signal-to-noise ratio. Ifinitially filtered, all signal amplification can occur in the first avail-able gain stage, and a high first-stage gain maximizes thesignal-to-noise ratio of the entire data channel.

3. It narrows the frequency spectrum occupied by data trans-mission. If numerous channels are multiplexed, as in space radiofrequency transmission, data frequency content can be made com-patible with individual channel bandwidth. Otherwise, bandwidthlimitations of the measurement system may arbitrarily limit trans-mission of the entire signal,which could include transducer-causeddistortion.The overall effect of filtering in this situation is to makemore effective use of the information capacity of the measure-ment system.

If additional channels are available, it is also prudent to record themeasurement system output in its unfiltered state in order to assurethe transducer has operated within its linear range.The aforemen-tioned unpredictable signal magnitudes from the transducer can,however, overrange subsequent measurement system components.Caution should be exercised if conservatively calibrating the datachannel to preclude this overranging.The time-varying signal ofinterest can end up competing with the noise floor of the meas-urement system.



The criterion recommended for filter selection for nondetermin-istic signals is easy to state since phase is not a consideration inpower spectral density processing.The filter selected should bearthe closest approximation to the ideal “boxcar” filter while passingthe low-frequency signal content of interest. Of the three filtersdiscussed, this is typically the Chebyshev.


Table 6.1: Filter Selection Type

Number of Poles Butterworth Bessel 0.1 dB Ripple Chebyshev

2 .573 .399 .522*

4 .575* .399 .489*

6 .541* .392 .418*

8 .506* .389 .372*

* upper limit due to phase nonlinearity

The above Table 6.1,5.4 based on calculations by the author, pro-vides filter selection guidelines for deterministic signals. Filtersconsidered are the Butterworth,Chebyshev, and Bessel-Thompsonwith two, four, six, and eight poles of attenuation. Numeric valuespresented are the ratio of the upper frequency limit at which thefilter should be used to that of its –3 dB frequency.This upper fre-quency limit is based on the lesser of the two values at which thefilter deviates either five percent from a flat amplitude response orfive degrees from phase linearity based on its initial phase slope.These criteria should assure waveform reproduction without distortion.

An example will show how to use this table.A six-pole Butter-worth and a six-pole 0.1 dB Chebyshev filter with a 1,000 Hz,–3dB frequency would be limited in application to 541 and 418Hz respectively because of phase non-linearities.A six-pole Besselwould become limited at 392 Hz due to its deviation from a flatamplitude response. In general, for deterministic data from force


and pressure transducers and accelerometers, the lowpass Butter-worth will always be the preferred filter of the three configurationsconsidered.

Throughout the first six parts of this text we have provided mod-els of, and application considerations for, numerous measurementsystem components up to and including the entire measurementsystem. In practice, however,we often have to make quick decisionsduring measurement system design or while assessing the outputdata from measurement systems. Part 7 presents rules of thumb toenable both a designer of systems and a data user to make thesedecisions concerning the time-varying signals of interest.



REFERENCES

6.1 Waters,Allan, Active Filter Design, McGraw Hill, Inc.,New York, NY, 1991.

6.2 Su, Kendall L., Analog Filters, Chapman & Hall, London,1996.

6.3 Walter, P. L., and Nelson, H. D., Limitations and Correctionsin Measuring Structural Dynamics, ExperimentalMechanics,Vol. 12, No. 9, Sept. 1979, 309-316.

QUESTIONS

1. Which is not a filter type?

a. lowpass.b. high pass.c. band pass.d. band reject.e. differential pass.

2. An 8-pole lowpass filter would have an ultimate rolloff of howmany dB/octave?

a. 48. b. 40. c. 32. d. 24.

3. As a minimum, accomplishment of two filter poles requireshow many operational amplifiers?

a. 4.b. 3.c. 2.d. 1.

4. Which filter has the most constant group delay?

a. Butterworth.b. Bessel-Thompson (often referred to as Bessel).c. Cauer.d. Chebyshev.

5. Which is not correct? Filtering signals from resonant transducers can:

a. enhance the measurement system signal-to-noise ratio.b. eliminate noise within the data bandwidth.c. narrow the frequency spectrum of the data transmission.d. prevent subsequent signal conditioning components from

being driven non-linear.


6. If one were to select a 6-pole Butterworth lowpass filter with a-3dB frequency of 1000 Hz, what is the highest frequency towhich he/she could maintain both flat amplitude and linearphase response?

a. 575 Hz.b. 541 Hz.c. 392 Hz.d. 418 Hz.

7. If interest is solely in calculating the power spectral density ofa nondeterministic signal, which filter is the most effective?

a. Butterworth.b. Bessel-Thompson (often referred to as Bessel).c. Chebyshev.

8. If one were to select a 4-pole Bessel lowpass filter with a -3dBfrequency of 2000 Hz, what is the highest frequency to whichhe/she could maintain both flat amplitude and linear phaseresponse?

a. 856 Hz.b. 644 Hz.c. 798 Hz.d. 399 Hz.

9. What are active filters built around?

a. resistors and inductors only.b. resistors and capacitors only.c. resistors and operational amplifiers.d. resistors, capacitors, and operational amplifiers.

10. Where should gain be maximized in a measurement system tooptimize its signal-to-noise ratio?

a. last stage.b. first stage.c. intermediate stage.



“Rules of Thumb” forData Assessment

Part


Part 2 explained why transducers that measure the vast majorityof dynamic force, pressure, and acceleration data use either sili-con-based (i.e., piezoresistive or capacitive) or piezoelectrictransduction elements.These elements have wide dynamic signalranges, enabling them to provide useable signals when integratedinto stiff transducer flexures. Stiff flexures result in high resonantfrequencies with an associated increase in useable frequencyresponse. Part 4 discussed concerns that must be addressed whencoupling the transducer to the process it is measuring.The Classi-cal Dynamic Model of Part 1 is capable of characterizing thetransducer only if it is properly coupled. Flush-mounted pressuretransducers incident to the pressure wave, force transducersimpacted by rigid masses, and accelerometers optimally attached tothe test structure can satisfy this model.

The ideal transducer of the type being considered would be a per-fectly linear system,have an infinite input impedance, respond onlyto the measurand for which it is designed (i.e., not respond toundesired environments such as temperature, electromagneticfields, etc.), have flat amplitude response over all frequencies, andhave linear phase response over all frequencies. If the transducer isproperly coupled to the process being observed or measured, its

“Rules of Thumb” for DataAssessment

The intent of this part is to enable both the test engineer and thedata analyst to perform a quick initial assessment of the validityof recorded force, pressure, and acceleration signals. The signals areassumed to be free of any spurious noise. This assumption willbe discussed further in Part 9.

output will be an exact time-delayed replica of its desired force,pressure, or acceleration measurand. Figure 7.1 shows the Bodeplots for this ideal transducer.


Figure 7.1

frequency frequency

(out

put)

/(in

put)

phas

e la

g

Figure 7.1Bode plots for idealtransducer.

Realizable transducers, however, are not ideal.They become lim-ited by their first resonant frequency,which, if properly designed, isthat of their flexural sensing element.The Bode plots for the non-ideal transducer were presented in Figures 1.3 and 1.4. Theyapproximate the response of a properly designed “real” transducerat frequencies up to and including this first resonant frequency. (Fortransducers with zero damping, the natural and resonant frequen-cies are identical.) Note in these figures the deviations from flatfrequency response and linear phase that occur as we approachthis resonant frequency. Part 5 justified the requirements for flatamplitude and linear phase response with frequency when record-ing time-varying measurands. The deviations from theserequirements in Figures 1.3 and 1.4 are responsible for signal dis-tortion at high frequencies. Short-duration input pulses to atransducer can excite its resonant frequency(s), causing signalamplification and phase reversal (demonstrated in Figure 5.8).

Figure 7.2A shows the distorted response of an accelerometermeasuring a pyroshock environment.Pyroshock will be briefly dis-

cussed in Part 10.The “hash” in the figure is the accelerometer’sresonance superposed on the response of the structural system.Fig-ure 7.2B contains the same data in filtered form, with the effect ofthe accelerometer’s resonance eliminated.The guidelines that areprovided in Part 6 should be applied in filter selection.The ampli-tude of the filtered data becomes 81% peak-to-peak of theunfiltered data.


Part 7 “Rules of Thumb” for Data Assessment

Time (ms)

Acc

eler

atio

n (g

x 1

03)

00

-5

-10

55

10

00 1 22 3 4 5

Figure 7.2BFiltered response.

Figure 7.2Accelerometerpyroshockresponse.

Figure 7.2ADistorted response.

Time (ms)

Acc

eler

atio

n (g

x 1

03)

00

-5

-10

55

10

00 1 22 3 4 5

Figure 7.2


When dealing with resonant systems, two limiting cases can occur:(1) the transducer’s resonant frequency is below the upper fre-quency limit of its recording electronics (as in dashed Figure 7.3A)or (2) the recording electronics obscure the presence of the reso-nant frequency of the transducer (Figure 7.3B).

A unique characteristic of piezoelectric transducers is that, in addi-tion to high-frequency constraints, they also possess alow-frequency constraint! They do not have response to 0 Hzbecause, at low frequencies, they become limited by the time con-stant τ of their recording electronics. This time constant isdetermined either by the charge amplifier or FET circuit that con-ditions the output of the piezoelectric device. (Figure 5.1).Piezoresistive or capacitive transducers, however, do not have thislimitation.

It is apparent that we must be in a position to evaluate recorded sig-nals as to whether or not a measurement system possessed adequatefrequency response to reproduce them with fidelity.Without pro-viding detailed justification, we show how to answer the followingquestions:

1. Did my force, pressure, or acceleration measurementsystem have adequate low-frequency response?

For a piezoresistive- or capacitive-based transducer, dc or 0 Hzresponse is assured. For a piezoelectric transducer, ascertain thecircuit time constant τ.This can be determined from manufac-turer’s specifications, or measured by putting an electrical stepfunction through the system, or calculated as 1/ωc where ωc cor-responds to the low-frequency limit in radians/second at which-3dB attenuation occurs.7.1 Next, determine the recorded pulseduration. If τ is greater than 10 times this pulse duration, the errorsin the pulse peak and associated undershoot are probably less than


a few percent.7.2 Alternately, look at the Fourier spectrum of therecorded signal. The low-frequency limit in Hertz at which apiezoelectric measuring system will attenuate this spectrum by 10percent is 0.3/ τ.7.1

2.The resonant frequency of my recording transducer isbelow the high-frequency limit of my measuring system.Did my measuring system have adequate high-frequencyresponse?

Ratio the pulse duration tc to the natural period T of the trans-ducer.The natural period T is the reciprocal of the fundamentalresonant frequency fn (T = 1/fn) of the transducer. If (tc/T) isgreater than 5, there will be less than 10 percent overshoot of thepulse peak. As this ratio increases, the amount of overshootdecreases.7.3 The potential for this overshoot to occur was illus-trated in Figure 7.2A. If the pulse risetime tr is shorter thanone-half the pulse duration, assess the data based on the value ofthe risetime.The value of (tr/T) should be greater than 2.5, againlimiting overshoot to less than 10 percent of the pulse peak. In allof the preceding, the pulse duration is measured at its 10% ampli-tude value, and its risetime is the time required to go from the 10-to the 90-percent level.7.3

Alternatively, look at the Fourier spectrum of the recorded signal.The upper frequency limit in Hertz at which the response of theaccelerometer possesses 4 percent gain is fn/5.The spectrum ofthe recorded signal should be below this limit.

3.The resonant frequency of my recording transducer isabove the high-frequency limit of my measuring system.Did my measuring system have adequate high-frequencyresponse?



The resonant frequency of the transducer will not appear in thedata. For any measuring system that monotonically rolls off at highfrequencies, the relationship between its risetime and upper -3dBlimit is7.4:

trf-3dB = 0 .35 to 0 .45. (7.1)

The lower limit holds for a first-order RC low-pass system, and thehigher limit would hold for an ideal “box car” or rectangular fil-ter. For example, the filters in Figure 6.2 have the same -3dB point,but very different attenuation at high frequencies, yet both satisfythis criterion.Again, the risetime is that required for the signal levelto increase from the 10- to the 90-percent value.

For clarity in the above equality, use a value of 0.4.To apply thisequality, first ascertain the pulse risetime. If the risetime multipliedby the -3dB frequency in Hertz is close to 0.4, the pulse peak haslikely been attenuated by the measuring system. If this product issignificantly greater than 0.4 (e.g., 0.6), no attenuation exists. Onthe other hand, the upper frequency limit of the measurement sys-tem may not be known, but the risetime of its individualcomponents (e.g., tr1, tr2, tr3, ...) may be known.To assure no atten-uation of the pulse peak, the measured risetime should be greaterthan 5 times the square root of the sum of the squares of these indi-vidual risetimes7.4.

One brief example might be illustrative as to how to apply some ofthese rules.A measured acceleration pulse has a duration of 200microseconds and a risetime of 40 microseconds. It will be assumedthat noise validation of the channel has been performed and thechannel has an adequate signal/noise ratio.The fundamental reso-nant frequency of the recording piezoelectric accelerometer is 100kHz.The upper -3dB frequency response of the measurement sys-tem is 10 kHz.The lower -3dB frequency is 0.1 Hz or ωc = 0.628



rad/sec which is equivalent to τ = 1.6 sec. Perform a quick assess-ment of the recorded data. First, assess the adequacy of the lowfrequency response of the measurement system. Since τ is 8,000times the pulse width (the requirement is 10 times), low frequencyresponse is not a concern. Next, the accelerometer’s resonant fre-quency can be proven not to be a system constraint.The naturalperiod of this resonant frequency is 10 microseconds.As a check,since the risetime is less than one-half the pulse duration, we willbase our calculations around this 40 microsecond value. 40/10must be greater than 2.5, which it is. However, the risetime mul-tiplied by the -3dB frequency is 0.4, indicating that the upper -3dBfrequency of the measurement system has constrained this risetimeand pulse peak.

Actual transducer records are often complex, with many crossingsof their zero reference line (Figure 3.4). In these situations, the def-initions of pulse width and risetime can become ambiguous.Thesolution is to look for limiting cases.That is, assess both the widestand the narrowest single-sided portion of the record as well as theshortest risetime contained in the record.

Figure 7.3AResonance belowelectronic’s frequencylimit.

log frequency

ampl

itude

fn

Figure 7.3Limiting cases ofresonantfrequency.

Figure 7.3


log frequency

ampl

itude

fn

Figure 7.3BResonance aboveelectronic’sfrequency limit.

Figure 7.3 continued

While the guidelines presented are not all mathematically rigorous,they are easy to employ. In summary:

1. If measurement system noise is minimal, perform a quick lookat the recorded transducer record. Undershoot can indicate inade-quate low-frequency response for piezoelectric transducers.Evidence of ringing can indicate inadequate transducer high-fre-quency response.Also, look for pulse clipping.

2.Assess the pulse characteristics for risetime and duration.Applythe applicable rules “on the back of an envelope”. Ensure that τ >10 tc; that the frequency spectrum is above 0.3/τ; that (tc /T) > 5;that (tr/T) > 2.5; that the frequency spectrum is below fn/5; thattrf-3dB > 0.4; etc.

3. If these tests are passed, it has been quickly determined that thedata are of reasonable quality for comparison with analysis.

Thus far we have described a number of simple assessments that canbe performed to establish data quality.The energy spectral densityfunction will be discussed later as another analysis tool.

REFERENCES: PART 7

7.1 Hambley,Allan R., Electrical Engineering, Prentice Hall,NJ, p. 303, 1997.

7.2 Pennington, Dale, Piezoelectric Accelerometer Manual,Endevco Corp., San Juan Capistrano, CA, pp. 73-75,1965.

7.3 Bickle, Larry W. and Keltner, Ned, Estimate of Transient Measurement Errors, Sandia National Laboratories SAND78-0497,Albuquerque, NM,Aug. 1978.

7.4 Stein, Peter K., The Unified Approach to the Engineering of Measurement Systems, Stein Engineering Services, Inc.Phoenix,AZ, ISBN# 1-881472-00-0,Apr. 1992.




QUESTIONS

1. The ideal transducer would not respond to measurands otherthan that for which it was designed and would have___________ frequency response and __________ phaseresponse over all frequencies.

a. flat, flat.b. linear, linear.c. flat, linear.d. linear, flat.

2. Force and pressure transducers and accelerometers can haveresonant frequencies associated with their mounting, case,connector, internal lead wires, flexural sensing element, etc. Ifproperly designed and mounted, the limiting resonantfrequency should be the:

a. case.b. internal lead wires.c. connector.d. flexural sensing element.

3. DC response is guaranteed in all of the following types oftransduction techniques in force and pressure transducers andaccelerometers but:

a. capacitive.b. piezoresistive.c. inductive.d. piezoelectric.

4. If the low frequency -3dB point of a piezoelectric transducercircuit is 2 radians/second, its time constant t is:

a. 0.5 seconds.b. 2 seconds.c. 0.05 seconds.d. 1 second.



5. If it is desired to record a 27 millisecond long pulse, the low-frequency measurement system time constant τ should begreater than ______ milliseconds.

a. 27.b. 135.c. 270.d. 2.7.

6. In order to record a blast pressure measurement with a 10-90% risetime of 25 microseconds, the resonant frequency of thetransducer should be greater than __________ Hz.

a. 1,000.b. 100,000.c. 1,000,000.d. 25,000.

7. If the frequency spectrum of the dynamic force, pressure, oracceleration signal that is to be recorded extends to but islargely below 10,000 Hz, the resonant frequency of thetransducer should be above _______ Hz.

a. 10,000.b. 20,000.c. 30,000.d. 50,000.

8. The shortest 10-90 % duration zero crossing of a complexacceleration signal is 40 microseconds. The measuringaccelerometer should have a resonant frequency greater than_______ Hz.

a. 12,500.b. 25,000.c. 100,000.d. 125,000.

9. A measuring system with an upper -3 dB frequency of 100,000Hz is intended to record a signal with a risetime of 10microseconds. Will this measurement be successful?

a. yes.b. no.

10. A measuring system with an upper -3 dB frequency of 100,000Hz is intended to record a signal with a risetime of 1microsecond. Will this measurement be successful?

a. yes.b. no.


Transducer Laboratory andMeasurement System Field Calibration

Part


By now, we have models for the transducers of interest, we under-stand their dynamic behavior, and we understand their transductiontechnologies (silicon and piezoelectricity).We understand the criti-cality of the proper coupling of these transducers to the environmentthey are measuring.We also understand system requirements fortime-varying measurements as well as data filtering. Last, we are ableto perform a quick-look assessment of the validity of the recordedsignals from these transducers.

Next, we turn our focus to the laboratory calibration of force, pres-sure, and acceleration transducers for dynamic measurements.Subsequent to that, we will discuss the field calibration of the entiremeasurement system. Many reference sources are available that

Transducer Laboratory andMeasurement System Field

Calibration

In order for the recorded output signal from a measurementsystem to be equated to its physical input, the measuringtransducer must be calibrated. For the transducer to becomecalibrated, it must be exposed to a “pure source” in the calibrationlaboratory. Once the transducer is calibrated, the entiremeasurement system must subsequently be “end-to-end”calibrated in the configuration that it will be used in application.Ideally, a known value of the measurand would be applied to thetransducer at the front of the measurement system immediatelybefore a test. Practically, an electrical simulation of the transduceris usually provided to the measurement system. This part alsodistinguishes between the words calibration and evaluation.

describe laboratory calibration. Only a few references are availablethat deal with calibration of measurement systems under field con-ditions.

It is first important to differentiate between laboratory calibrationand evaluation of transducers.A laboratory calibration occurs whena pure source (force, pressure, or acceleration) is applied to thetransducer and a relationship is established between the source andthe electrical output of the transducer (e.g., mV or V or pC perpound, psi, or g). In the case of force or acceleration sources, anadded constraint is that it must be unidirectional and parallel withits sensing axis. Force transducers and accelerometers all possesssome limited transverse response, which becomes a variable athigher frequencies.

The only time a force, pressure, or acceleration transducer encoun-ters a pure source is in the calibration laboratory8.1. For example,in explosive pressure environments (e.g., Figure 8.1) the pressuretransducer also encounters thermal transients, ionized gases,mechanical shock and vibration, strain induced in its housing, andother adverse environmental factors in addition to pressure.All ofthese other environmental factors attempt to couple into the


Figure 8.1Explosively drivenshocktube (SandiaNational Labs).

Figure 8.1


signal from the transducer as additive or multiplicative noise. In Part 9 we will focus on how to validate the extent to which theacquired field measurements are contaminated by this noise.

The laboratory evaluation of a transducer involves subjecting it to lev-els and frequencies of its intended measurand above and belowthose expected in its field application. In addition, the response ofthe transducer to the adverse environments expected in the fieldshould also be quantified or observed. For example, a pressuretransducer could be mounted as it ultimately will be in its fieldapplication, allowing the test engineer to quantify the influence(ideally zero) on its signal output due to strain induced by torquingor shocking its mechanical mount. Similarly, force transducerscould be evaluated as to the effect of off-axis force components.Before a transducer is applied in the field, it should be both cali-brated and evaluated. We now summarize how the laboratorycalibrations of interest are performed.

All calibrations must be traceable; i.e., they must have some hier-archical relationship to a central standards laboratory (NationalInstitute of Standards and Technology (NIST) in the USA).Thetypes of calibration can be either absolute or comparison; defini-tions follow.

Absolute: The calibration factor of the transducer isobtained directly from a measurement of the quantitiesthat form the base units of a system of units. For example,force could be traceable directly to mass (M), accelerationto length (L) and time (T) since its units are L/T2, etc.This calibration type typically encompasses high costs.

Part 8 Transducer Laboratory and Measurement System Field Calibration

Comparison: The calibration factor of the transducer isobtained by comparing simultaneous measurements of theoutputs of the transducer under test and a reference trans-ducer of known and stable characteristics.

A summary of the dynamic calibration capabilities of the measur-ands of interest is disappointing. No traceable dynamic forcecalibration system has gained universal acceptance8.2,8.3. Reference8.2 describes a method of placing a known mass and a dynami-cally traceable accelerometer on top of a force transducer andemploying rigid-body mechanics to derive a calibration value atlow frequencies.The effective end-mass of the force transducermust be considered in the calibration.The end-mass is the portionof the force transducer that acts on itself in an inertial manner toproduce an output when the transducer is subjected to accelera-tion.References 8.2 and 8.3 both identify other work that has beenperformed. Fortunately, as mentioned in Part 1, force transducersare typically integrated into structures where they must be dynam-ically analyzed as part of the structure, and their independentdynamic characterization is not a requisite for most experimentalwork. Experimental modal analysis, which will be discussed inPart 11, is one of the few exceptions.

For pressure transducers, except at acoustic pressure levels, again notraceable dynamic calibration system exists that has gained univer-sal acceptance. Considering the frequent requirement for dynamicpressure measurements in pumps, compressors, combustionprocesses, etc., this lack of traceability poses a significant challengeto the credibility of dynamic pressure measurements. Recognizingthis challenge, a group was formed under the Subcommittee onPressure of Standards Committee B88,American National Stan-dards Institute.This group was originally going to propose standardmethodologies for dynamic pressure transducer calibration, but


after the members reviewed the vast assortment of methods thatwere being attempted in government laboratories and industry,they decided to generate ANSI B88.1-1972, A Guide for theDynamic Calibration of Pressure Transducers.This Guide did notsolve the problems associated with the lack of traceability but didspecify preferred calibration techniques.The Guide was reaffirmedin 1987 and is currently being updated by the ISA (Instrumenta-tion, Systems, and Automation Society) standards subcommitteeSP37.16. Summarized within this Guide are periodic, pulsed, andshock tube calibration techniques over the amplitude range of 0-100,000 psi and the frequency range of 0-10,000 Hz.

The cold gas shock tube (Figure 8.2) is probably the most versa-tile tool for both pressure-amplitude and frequency-responsedeterminations. Conceptually it consists of two straight sections ofpipe rigidly coupled together, with their far ends capped or sealed.A frangible diaphragm material spans the cross section of the pipesat this coupled interface. In operation, one pipe section is pressur-ized, typically with helium, causing the diaphragm material to



Figure 8.2Cold gas shocktube with coupledsection open(Texas ChristianUniversity).

Figure 8.2

rupture.The resultant shock wave travels into the second section,which usually contains air or nitrogen.The shock wave superposesa submicrosecond risetime pressure step on the face of a pressuretransducer flush-mounted in the far, capped end.The magnitude ofthe step can be calculated from gas-dynamics theory.The frequencyresponse of the pressure transducer can be assessed based on timeand frequency analysis of its response. Reference 8.4 describes onesuch technique. If frequency assessment is not required, the pressuretransducer to be calibrated is usually mounted in the wall section ofthe second section.

In contrast to dynamic pressure and force, laboratory dynamic cal-ibration of accelerometers is well supported by NIST8.5.Absolutecalibration is accomplished either by reciprocity, laser interferom-etry, or other8.6 techniques.Reciprocity requires a vibration exciterwith a velocity coil or reference accelerometer.The sensitivity ofthe accelerometer being calibrated is related to measurements ofvoltage ratio, transfer admittance (current/voltage), frequency, andmass.Comparison vibration calibrations, based on absolute calibra-


Figure 8.3High-frequency air-bearingaccelerometercalibration shaker(Endevco).

Figure 8.3

tions, can subsequently be performed by NIST over frequencyranges to 20,000 Hz. Mechanical shock calibrations can be per-formed to levels of 5,000 g.Vibration accuracy is typically betterthan 3 percent, and shock accuracy is better than 5 percent. Figure8.3 is illustrative of the small, high-frequency shakers used foraccelerometer calibration.An air-bearing sleeve around the shakerarmature provides uniaxial motion to the maximum extent possi-ble.The moving armature is of a stiff, lightweight material such asberyllium, ceramic, or magnesium.

Comparison calibration involves transferring the sensitivity of oneaccelerometer to another. Figure 8.4 shows a reference or “piggy-back”8.7 accelerometer, which is one method of effecting thistransfer.The test accelerometer is mounted on the top surface ofthis reference accelerometer, away from its mounting stud. Forsmall shaker systems such as in Figure 8.3, the “piggyback” is incor-porated in the shaker armature.



Figure 8.4Reference“piggyback”accelerometer forcomparisoncalibration.

Figure 8.4

For particularly severe test environments, the shock levels encoun-tered may exceed the traceable calibration capability of NIST.Commercial accelerometers exist with stated full-scale ranges to200,000 g. Significant experimental work8.8, 8.9, 8.10, based on themeasurement of compression stress waves traveling in long, slen-der (Hopkinson) bars, has enabled the establishment of a calibrationcapability in various laboratories to 100,000 g. Figure 8.5 showsone such system.

After any transducer has been calibrated, an end-to-end calibra-tion of the entire measurement system is required. Ideally, a knownvalue of the measurand is applied to the transducer, and the meas-urement system output is recorded. However, typically the outputof the transducer is electrically simulated to the measurement sys-tem.A complete listing of methods to accomplish these simulationshas been compiled in a single reference8.11. Selected methods arepresented below.

Piezoresistive transducers typically contain four individual resistivearms in a Wheatstone bridge.The resistance values of these arms


Figure 8.5Hopkinson bar withpneumatically drivenimpact projectile(Endevco).

Figure 8.5

change dramatically with temperature.The shunt-calibration ofthese transducers requires two of the four arms to be configuredwith temperature-insensitive, metal-film resistors. Inserting a resis-tor in parallel with the temperature-insensitive arm of the bridge issingle-shunt calibration. Figure 8.6 shows a system configured inthis manner.The shunt calibration resistor RC is inserted acrossthe arm opposite the conditioning system.The conditioning sys-tem may contain a balance potentiometer, a limit resistor, modulusresistors, and temperature-compensation resistors, all in the excita-tion leads (+/- input) to the bridge. Standard practice is to insertthe shunt resistor between the negative-excitation input and thenegative-signal output (Figure 8.6).This reduces errors caused byshunting some of the bridge-conditioning resistors.

The value of RC is determined by first applying a value of the mea-surand to the transducer and monitoring the voltage change atthe transducer output terminals.With the measurand removed, adecade box is substituted for RC, and the resistance is adjusted untila voltage change results with a magnitude equal to that caused bythe measurand.

Once this is accomplished, for subsequent calibrations a compara-ble value of the fixed resistor RC can be substituted for the decadebox.When the switch in series with RC is closed, a step voltage willbe produced in the measuring system of amplitude equal to thatproduced by the measurand.When shunting one arm of the bridge,the resistance change produced in that arm is:

∆R = -R2/(RC + R) (8.1)

where:

RC = shunt value in ohms and R = value of bridge resistance.



In the calibration laboratory, the small lead-length associated withthe transducer introduces no error in establishing RC.However, theapplication of the bridge transducer in the field can require signif-icant lengths of cable with associated transmission line resistance(RL). Figure 8.6 shows RC being applied remotely. In this situation,RL must either be accounted for, or RC must be applied directlyat the bridge. (Note.The capacitance associated with long cable lengthsmust also be considered; it may result in unintended filtering at high fre-quencies.)


Figure 8.6Single-shuntresistive bridgecalibration.

An accurate bridge transducer model would have the same out-put impedance (Z(j)) as the transducer, and would provide a fastand simple method of generating static and dynamic outputsequivalent to that generated by the transducer for a given physicalload.The shunt resistor adapter (Figure 8.7) serves this function.The adapter is inserted between the transducer and the rest of themeasurement system. It performs three primary functions:

4 Arm Transducer4 Arm Transducer

BridgeBalanceNetwork

InputInput

Output

Output

Input

ShuntRL

RL

RL

RL1

2

3

4

RC

R R

R R

Figure 8.6


1. It supplies the stimulus for the performance of end-to-end calibrations. Shunting the arms of a transducer bridge with theappropriate resistors produces an imbalance in the bridge equiva-lent to that produced by a given measurand.The adapter providesa convenient method of applying these shunt resistors directly tothe bridge to perform a measurement system linearity test.

2. It performs a frequency response test of the measurement system. Selecting the appropriate shunt resistor, and sweeping theadapter’s AC power supply over the desired range, is a convenientway of determining the system’s frequency response. Figure 8.8displays a set of typical results.

3. Since the adapter shunt resistors are applied directly to thebridge, lead resistance and other variables will not affect the equiv-alency of the adapter shunt resistors.


Shunt Resistor/Dynamic Adapter

Transducer

AC VariableFreq. Source

S1 = Shunt Resistor Selector SwitchS2 = Signal Polarity Selector SwitchS3 = System DC Power/AC Power Selector SwitchRC = Adapter Calibration Shunt Resistors

3 3

1 1

2 2

4 4

RT

RT

R1

R2R3

R4

S2

S1

RC

RL

RL

RL

RLS3

DC

DC

AC

AC

Figure 8.7Shunt resistoradapter(linearity/frequencyresponse)calibrator.

Figure 8.7

We next turn our attention to the end-to-end calibration of meas-urement systems for piezoelectric transducers. In the 1960s, NASAoriginated a contract for the design of a group of accelerometerscontaining dual back-to-back piezoelectric crystals. During quies-cent periods, a voltage applied to one crystal would physicallyvibrate the other. A resultant calibration signal of known value

would emanate from the accelerometer.This end-to-end calibra-tion technique was novel when contrasted to those describedpreviously for piezoresistive transducers. However, capacitive cou-pling of input and output signals could cause errors, physicalvibration inputs induced additive errors in the resultant signal, andthe mechanical complexity of the resultant transducer reduced itsreliability.Therefore, most piezoelectric measurement channels arecalibrated by electrical simulation.

For transducers containing FET circuits, system calibration viaelectrical simulation is only possible through substitution of a low-impedance voltage source for the transducer. If the bias voltage ofthe FET can be monitored in front of the output coupling capac-itor (Figure 5.1), a check of the FET can be obtained. For chargeamplifiers, however, series voltage insertion enables end-to-endmeasurement system calibration.


DC 100 Hz 500 Hz 750 Hz 1000 Hz 1500 Hz

Figure 8.8

Figure 8.8Typical result setfrom shunt resistoradapter calibration.


Figure 5.1 illustrated the piezoelectric-based transducer model tobe electrically equivalent to a voltage source (V) with a seriescapacitance (Ca).This is also equivalent to a charge generator Q =VCa in parallel with Ca ( Figure 8.9). In Figure 8.9, the total capac-itance of the cable attached to the piezoelectric transducer is equalto C1 + C2.The fact that this capacitance is divided into two partsreflects that a series resistor Rseries is inserted in the ground side ofthe signal. C1 represents that part of the capacitance in front ofthis resistor, and C2 that following it. Rseries should be selected tosatisfy:

Rseries << 1/ω(Ca + C1) (8.2)

at the highest frequency of interest.The voltage to insert acrossRseries is:

Q/(Ca + C1) (8.3)

For example, assume that a given piezoelectric force transducer hasa sensitivity of 10 picocoulombs (pC)/pound and an internalcapacitance of 100 pF. Further assume it to have 100 pF of cable(C1) in front of the series resistor.To calibrate the channel for 10pounds force would require a charge Q of 100 pC (10 pC/poundx 10 pounds). Using equation 8.3, 100 pC/(100 pF + 100 pF) =0.5 volts 0-peak or 1.0 volts peak-peak. This voltage should beinserted across the series resistor.



A question may arise as to how to insert Rseries in the cable fromthe transducer. Figure 8.10 shows a T-junction box used to accom-plish this insertion.The insertion voltage is usually ground-isolatedto assure a single-point system ground.Transformer coupling canachieve this isolation.

With this overview on transducer and measurement system cali-bration complete, it is next desirable in Part 9 to discuss how tocertify data channels as being noise-free. Once high-quality dataare assured, we can begin to look at data utilization.

Signal fromTransducer

This connector insulated fromjunction box case Calibration Signal

To Amplifier

Figure 8.10T-junction box forvoltage insertion.

Figure 8.10

QCa C1 C2 RL

Rseries

Figure 8.9Series voltageinsertion circuit forcharge amplifier.

Figure 8.9

REFERENCES: PART 8

8.1 Stein, Peter K., The Unified Approach to the Engineering of Measurement Systems, Stein Engineering Services,Phoenix,AZ, ISBN# 1-881472-00-0, pp. 29-30, 1992.

8.2 Krumme, R.,“Dynamic Investigation of ForceTransducers,” Experimental Techniques, pp. 13-16,November/December 1993.

8.3 Dixon, Michael J.,“A Traceable Dynamic ForceTransducer,” Experimental Mechanics, pp. 152-156,June 1990.

8.4 Favor, John D. and Stewart, Ralph,“Primary Calibration of Pressure Transducers to 10,000 Hz,” Proceedings of the International Instrumentation Symposium (TMD and ASD),Instrument Society of America (ISA), 197X.

8.5 Robinson, Serbyn, Payne,“A Description of NBSServices in Mechanical Vibration and Shock,” National Bureau of Standards (NBS, now NIST) Tech Note 1232,1987.

8.6 Mechanical Vibration and Shock, International Standards Organization (ISO) Handbook,Vol. 1, 2nd ed., Geneva,Switzerland, 1995.

8.7 Bouche, R.R.,“Accurate Accelerometer Calibrations by Absolute and Comparison Methods,” Endevco TP 233,San Juan Capistrano, CA, March 1966.

8.8 Brown, G.Wayne,“Accelerometer Calibration with the Hopkinson Bar,” Instrument Society of America, 18th AnnualConference, preprint 49.3.63, Chicago, IL, Sept. 9-12, 1963.



8.9 Sill, R. D.,“Shock Calibration of Accelerometers at Amplitudes to 100,000 g Using Compressional Waves,”Endevco TP 283, San Juan Capistrano, CA, Feb. 1984.

8.10 Bateman,V. I., Leisher,W. B., Brown, F.A.,“Calibration ofa Hopkinson Bar with a Transfer Standard,” Journal of Shock and Vibration,Vol. 1, No. 2, pp.145-152, Nov/Dec 1993.

8.11 “End-to-End Test Methods for Telemetry Systems,” Test Methods for Telemetry Systems and Subsystems,Vol. 1,118-79,Telemetry Group, Range Commanders’ Council,WSMR, NM, 1979.



QUESTIONS

1. Calibration involves the application of a pure source to atransducer. For a force transducer, which of these inputsrepresents a pure source?

a. force input oriented at 45 degrees to its sensing axis.b. force input parallel to and directly into its sensing axis.c. force input parallel to but offset from its sensing axis.d. same as (b) but at some known, elevated temperature.

2. Assessing the response of a pressure transducer to thermaltransients, ionized gasses, mounting strain, etc. is known aswhat?

a. evaluation. b. calibration.

3. Pressure transducers can be calibrated by dead weighttesters. This process involves inserting a known mass ofcontrolled geometry into a known cross sectional area andcorrecting for the buoyancy of air. This is an example of whattype of calibration?

a. absolute.b. comparison.

4. Of the three dynamic measurands discussed, which is the onebest supported by NIST?

a. force.b. pressure.c. acceleration.

5. At very high shock levels, accelerometers are calibratedbased on the measurements of compressive stress waves inwhat?

a. thick plates.b. shells.c. beams.d. long, slender bars.



6. Advantages of the shunt resistor adapter include which of thefollowing?

a. system linearity can be checked.b. system frequency response can be checked.c. lead resistance will not affect its equivalency.d. all of these.

7. Some indication of the state of health of a piezoelectrictransducer with a contained FET can be determined bymeasuring what?

a. its capacitance.b. its bias current.c. its input impedance.

8. A series voltage insertion for a piezoelectric transduceroperating into a charge amplifier is performed. Thecapacitance of the piezoelectric transducer is 300 pF. Itssensitivity is 3 pC/unit. The series voltage is inserted into a“tee” at the end of 6 feet of cable with capacitance of 30pF/foot. What voltage should be inserted to simulate 10 unitsout of the transducer?

a. .0625 v 0-pk.b. .100 v 0-pk.c. .167 v 0-pk.

9. After calibration of the transducer, the calibration of the entiremeasurement system is called what?

a. total.b. measurand-based.c. absolute.d. end-to-end.

10. Shunt calibration of bridge transducers requires which ofthese (circle as many as needed)?

a. temperature-insensitive bridge resistors.b. a known shunt calibration resistor.c. avoidance of bridge arms in parallel with resistors in the

conditioning system.d. a known voltage source.

Measurement SystemSignal Validation

Part


Part 7 provided rules of thumb to enable the test engineer and/orthe data analyst to perform a quick initial assessment of the valid-ity of recorded force, pressure, and acceleration signals.The signalswere assumed to be free of any spurious noise.This part describeshow to validate that assumption.

Part 1 defined a transducer as “a measurement system componentcapable of processing information and energy.”We further pre-sented both a Conceptual (Figure 1.1) and a Classical (Figures 1.2,1.3 and 1.4) Transducer Model. Most of our discussion until thistime has involved the Classical Dynamic Model.The focus willnow turn to Stein’s Conceptual Transducer Model.The intent isto use this model as a foundation to establish techniques to ascer-tain if the data resulting from measurement systems containingforce, pressure, or acceleration transducers are of good quality or arecorrupted by noise.This article tells how to perform that deter-mination. If the data are found to be noise-free, the “rules ofthumb” provided in Part 7 can be applied to assess the adequacyof the dynamic performance of the entire measurement system.

Measurement System SignalValidation

In addition to the desired signal, the output from a measurementsystem can contain noise. If this noise contamination is notidentified, erroneous data can be accepted as valid.The goal of thispart is to enable the reader to incorporate check-channels intotheir measurement system for signal validation. Once data arevalidated as noise free, the "rules of thumb" of Part 7 can beapplied to assess the adequacy of the dynamic performance of themeasurement system.


Part 8 alluded to the fact that transducers only encounter puresources in the calibration laboratory. Numerous technical agenciessuch as the ISA,The Instrumentation, Systems, and AutomationSociety and Control (formerly the Instrument Society of America),have written specifications and test guides for various type trans-ducers. One such publication (ISA-RP37.2-1982 9.1) is the “Guidefor Specifications and Tests for Piezoelectric Acceleration Trans-ducers for Aero-Space Testing”. Included within this documentare specifications to minimize the response of accelerometers tosuch undesired environments as steady-state and transient temper-ature, base strain, acoustic pressure, magnetic fields, humidity, radiointerference, and nuclear radiation. Usually, but not always, theresponse of the accelerometer to these undesired environments issmall compared to its acceleration response.

Another ISA publication (ISA-S37.10 (R 1982) 9.1) is entitled“Specifications and Tests for Piezoelectric Pressure and Sound-Pressure Transducers”. It is interesting to note that whileISA-RP37.2 contains specifications to minimize the response ofpiezoelectric accelerometers to acoustic pressure, ISA-S37.10 con-tains specifications to minimize the response of piezoelectricacoustic pressure transducers to acceleration. In this case, one trans-ducer’s desired environmental stimulus is another transducer’sundesired environmental stimulus.Thus, the concept of desired andundesired environmental stimuli has been developed.The desiredstimulus is always the same as the pure source used to calibrate thetransducer.

Part 1 also alluded to the fact that two transducer types exist: self-generating and nonself-generating. Nonself-generating typesrespond to changes in the material properties or geometries withinthe transducer. For example, the inertial force due to accelerationmodifies the resistance (R)

(9.1)

of the piezoresistive strain gages on the sensing flexure of anaccelerometer by changing the gage geometry (where is resistivity,L is resistor length, and A is cross sectional area). Similarly, thediaphragm of a pressure transducer deflects when loaded, and inthat way it modifies the resistance of the gages affixed to its sur-face or diffused into it.

Voltage must be applied to the resistive Wheatstone bridge of aforce, pressure, or acceleration transducer to elicit a nonself-gen-erating response.This resistance change results in a signal of typicalmagnitude of tens or hundreds of millivolts. A parallel thoughtprocess exists for capacitive transducers. Here the movement ofthe plates of a capacitor unbalances the bridge. An ac voltageapplied to the bridge results in a signal output proportional to themeasurand (force, pressure, or acceleration).

Self-generating transducers are not dependent upon voltage beingapplied to them. Photovoltaic, thermoelectric, pyroelectric, andpiezoelectric phenomena are examples of transduction techiquesused in self-generating transducers. Force, pressure, and accelerationtransducers typically depend on piezoelectric technology.

Thus, at this juncture we have reinforced the concept of self-gen-erating and nonself-generating transducers. These transducersrespond to desired and undesired environments. Before providingspecific noise-documentation techniques for the measurement sys-


Part 9 Measurement System Signal Validation

tems of interest,we need to acknowledge one additional truism:Alltransducers contain both self-generating and nonself-generating responses!For example, consider a piezoresistive bridge transducer.Variousmaterials in a spatially distributed Wheatstone bridge networkinterconnect the gages. Even with the bridge power removed, thistransducer is capable of producing an electrical output. For exam-ple, the transducer can produce a response to electromagneticallyinduced noise. In addition, the dissimilar materials in the transducerrepresent thermocouple junctions.These junctions can produce athermoelectric response to thermal-transient noise.The semicon-ductor materials in the transducer, if exposed to light, can alsoproduce a photoelectric response.These are just a few of the unde-sired environments the transducer will respond to even when novoltage is applied to its bridge.

Now we are able to advance a definition of noise in measurementsystems. In any measurement system we have four possible responsecombinations.There are two transducer types: self-generating andnonself-generating. There are two response types: desired andundesired.The type/response combinations are:

nonself-generating - desired,

nonself-generating - undesired,

self-generating - desired, and

self-generating - undesired.

Consider the case of a piezoresistive accelerometer.This examplecan be generalized to any bridge-type transducer.The nonself-gen-erating response (resistance change) to the desired environment(acceleration) is defined as signal. It is the object of the test.Thenonself-generating response to the undesired environments, as well



as the self-generating response to both the desired and undesiredenvironments, is noise. Figure 9.1 illustrates the paths associatedwith these four combinations, path 4 being signal, and paths 1, 2,and 3 being noise.

The goal in any measurement system is to assure that only the sig-nal path is significantly present. Some question may arise as tohow to implement this verification.An acceptable method wouldbe to field three accelerometers in close proximity.The first couldbe mounted without electrical power applied to document paths1 and 3. Note that without power, paths 2 and 4 are not possible.The second accelerometer could have power applied but bemounted on a piece of foam (or suspended in air) to isolate it fromthe acceleration environment, resulting in documentation of paths1 and 2. Note that without the desired environment (acceleration)present, paths 3 and 4 are not possible.The third accelerometercould be mounted with power properly applied to measure theacceleration environment. If the first two accelerometers producedno output, paths 1, 2, and 3 were not present and the output from the thirdaccelerometer would be path 4,which is the noise-free signal. Data worthyof subsequent analysis would have been acquired!

For force and pressure transducers, the same strategy applies. Sim-ply install three force or pressure transducers in close proximity.Apply power to one, don’t apply power to the second, and applypower but isolate the third from its intended force or pressure envi-ronment. For example, a pressure transducer could be mounted ina “blind hole” to assure its diaphragm is not exposed to pressure.It would still be exposed to vibration, strain, electromagnetic fields, and other undesired environments to which it could poten-tially respond.


For piezoelectric transducers, it is not as apparent how to imple-ment this procedure.Yet, a similar method exists.Most piezoelectrictransducers are designed using ferroelectric polycrystalline ceramicmaterials that can be manufactured to exhibit piezoelectricity.Themechanical and electrical properties of these materials can be con-trolled during manufacture. This manufacturing process waspresented in Part 2. Briefly, it consisted of weighing and propor-tioning the ceramic powders, calcining at high temperatures toproduce a chemical combination of the ingredients, mixing in aball mill to repowder the raw compound, adding a binder, granu-lating and then compressing the powder into pellet form, and firingthe pellets in a controlled atmosphere in a kiln.This firing trans-forms the pellets into ceramic elements.The elements are lappedand plated for subsequent polarization. A high-voltage field isapplied across each pellet under controlled environmental condi-tions.The minute crystal domains within the ceramic are forcedto align themselves parallel to the applied field and this alignmentis retained after the field is removed.A key point to note is that ifthis high-voltage field is not applied, no piezoelectric characteris-tics are exhibited.These polycrystalline ceramics are unpoled andno electrical output results if they are mechanically stressed. These

156

DesiredEnvironment

UndesiredEnvironment

3

1

4

2

Self-GeneratingResponse

Non-Self-Generating

Response

Power

Figure 9.1

Figure 9.1Piezoresistive orcapacitivetransducer noisemodel.

unpoled ferroelectric polycrystalline ceramic materials can then be incorpo-rated in transducers that will respond only to noise. When transducersof this type are employed, and no signals are elicited from them, itcan be ensured that data worthy of subsequent analysis has beenacquired from nearby active piezoelectric transducers!

It should also be noted that quartz, which is a natural crystal,can also result in an element which produces no output whensqueezed.This requires the quartz to be cut along select crystalaxes.

The author has used this procedure over the years (often with sur-prising results).The following example is from his work at SandiaNational Laboratories,Albuquerque, NM.

It was desired to measure random vibration (acceleration)inside an ordnance system.The system was being testedon a rocket-sled track. A parachute was to be deployedwhen the ordnance system was ejected from the sled.Theanticipated vibration environment was ±10 g.The decel-eration pulse associated with the parachute deploymentwas 150 g. If the measurement channel were to accom-modate the deceleration pulse, resolution of therandom-vibration environment would be poor.

The width of the deceleration pulse was known to beabout 200 milliseconds.The low-frequency -3 dB pointof the charge amplifier was adjusted to 50 Hz to providea high-pass filter.This filtering then removed the influenceof the pulse from the data electrically.This filtering wasachieved by placing a resistance in parallel with the inputto the charge amplifiers used in the test. (See top portion




of Figure 9.2.) The shunt resistor was placed in a “tee” inthe low-noise coaxial input cable of the charge amplifier.Electrically, all amplifiers had a common input ground.

Eight charge amplifiers were used.These were divided intotwo groups of four each. (See bottom portion of Figure9.2.) Each group was multiplexed into a radio-frequencytransmitter for data transmission through space. Of keyimportance, each transmitter had assigned to it three activeaccelerometers and one depolarized accelerometer.

Before parachute deployment, a gas generator functionedin the ordnance system imparting large structural loadsinto it.These loads caused all(!) eight (8) channels to satu-

RshuntPiezoelectric

AccelerometerCrystal

Piezoelectric Accelerometer/Charge Amp with Rshunt of 6-8 KΩ

Rshunt

Rshunt

CapacitiveCoupled

ac Ground

PiezoelectricAccelerometer

Crystal

PiezoelectricAccelerometer

Crystal

Piezoelectric Accelerometer/Charge Amps Cascaded (4 total - 2 shown)

Figure 9.2

Figure 9.2Groundingconfiguration forcharge amplifiers.


rate.This saturation obscured the desired event. However,it wouldn’t have been identified as erroneous data if thedepolarized monitoring channels weren’t present and haddisplayed a signal.

Because the monitoring channels had produced an output,the data were known to be invalid.To determine the causeof the erroneous signals, after the test one set of fouraccelerometers attached to the metal skin of the ordnancesystem was dismounted and then remounted on a sepa-rate metal plate.This plate was kept electrically coupledto the vehicle skin. Impacting the plate caused all fourchannels to respond! In addition, concurrent signals weredisplayed from the second set of four accelerometers thatremained on the skin of the ordnance system even thoughit was not impacted! Figures 9.3 and 9.4 display theseresults.

The cause of the erroneous data was identified as follows:All of the “tees” in the cables of the accelerometers to bothtransmitters were common at the metal block to whichthey were attached, which was isolated from ground.However, when the gas generator was fired, the resonantfrequency of the active accelerometers at 40 kHz wasexcited.An ac ground path was then formed, which cou-pled into all accelerometers. The key point in thediscussion is that the depolarized noise-measuring chan-nels precluded bad data from being accepted as good. Inaddition, the cause of the noise could be resolved and cor-rective action taken before the next test.

Based on checks such as these, it can become readily apparentwhen the recorded output from a force, pressure or accelerationtransducer is worthy of further analysis.Having described the opti-



mization of the design of the measurement system through theprevious eight parts of this text, as well as the type data we areinterested in measuring, attention will now be focused on the uti-lization of the acquired measurement system signals.

Figure 9.3Impacted plateaccelerometerresponse.

Figure 9.3

CHAN 2 SYS 1

CHAN 1 SYS 1

CHAN 3 SYS 1

MONITOR SYS 1

Endevco Model 2221 M2ADepolarized Piezoelectric

3/29/80



Figure 9.4

CHAN 1 SYS 2

CHAN 2 SYS 2

CHAN 3 SYS 2

MONITOR SYS 2

Endevco Model 2221 M2ADepolarized Piezoelectric

3/29/80

Figure 9.4Non-impacted plateaccelerometerresponse.

REFERENCES

9.1 Standards and Recommended Practices forInstrumentation and Control, 10th Edition,Vol. 1,Instrument Society of America, Research Triangle Park,NC, 1989.



QUESTIONS

1. The "rules of thumb" from Part 7 that access the adequacy ofthe dynamic performance of the measurement system dependon:

a. assuming the signal is filtered.b. assuming the signal has an adequate time constant.c. assuming the signal is near the full-scale limits of the

channel’s range.d. assuming the signal is noise-free.

2. Accelerometers, pressure transducers, and force transducersrespond to environments such as:

a. magnetic fields.b. steady-state and transient temperature.c. humidity.d. all of these.

3. The desired environment for a transducer is always the sameas what?

a. the pure source used to calibrate the transducer.b. the ambient temperature.c. the ambient humidity.d. the same signal conditioning as will be used in laboratory

or field testing.

4. Which phenomenon is not an example of the response of aself-generating transducer?

a. photovoltaic.b. pyroelectric.c. piezoresistive.d. piezoelectric.e. thermoelectric.

5. In Figure 9.1, signal is what path?

a. 3.b. 1.c. 4.d. 2.



6. The goal in any measurement system is:

a. acquire an output signal.b. trigger correctly.c. assure only the signal path is present.d. not overrange.

7. To use a piezoelectric transducer in a noise check mode,during manufacture it should:

a. be made of quartz.b. have high temperature properties.c. not be polarized.d. not be plated.

8. The cause of the noise in Figures 9.3 and 9.4 was:

a. electromagnetic interference.b. ac ground path.c. radio frequency interference.d. electrostatic discharge.

9. The data in Figures 9.3 and 9.4 are known to be invalidbecause:

a. the resonant frequency of the accelerometers was 40 kHz.b. the monitoring channels produced an output.c. the plate was electrically coupled to the skin.d. "tees" were in the cables.

10. To elicit a nonself-generating response from a Wheatstonebridge based transducer, what must be applied:

a. supply voltage.b. thermoelectric voltage.c. piezoelectric voltage.

Data Utilization

Part


Part 1 described some of the many reasons for measuring dynamicforce, pressure, and acceleration data. Part 2 indicated that trans-ducers using silicon and piezoelectric technology enablemeasurements over ranges from fractional pounds, psi, and g’s togreater than 100,000 pounds, psi, and g’s. Part 3 clarified the dif-ference between deterministic and nondeterministic timefunctions. Parts 4 through 9 described how to interface a trans-ducer to its measurand of interest; system requirements whenacquiring dynamic measurements; data filtering; data assessment;transducer and/or measurement system calibration; and measure-ment system noise documentation.

At this juncture of the text, we focus on data utilization. This partwill discuss some of the various uses data are put to, as well as sig-nal-processing techniques to accomplish these uses. Based onguidance provided thus far, it is assumed that distortion-free, noise-free, transducer-based signals have been recorded for processing.The challenges associated with signal differentiation and integra-tion will be noted, along with uses for the power spectral densityfunction (PSD-described in Part 3), energy spectral density func-tion (ESD), and shock spectra. Part 11 will then describe additionalutilization of data in the frequency domain, which will enable an

Data Utilization

The goal of recording valid dynamic force, pressure, or accelerationdata is either to enable control of a process or system or to supportdesign and analysis. This part begins to discuss some of the waysin which we use successfully recorded data.

introduction to experimental modal analysis. Experimental modalanalysis is dependent on dynamic force and acceleration measure-ments.

When performing measurements, we are supporting one of twoactivities. Figure 10.1 illustrates measurements being acquired toenable control of a process or a system. For example, the pressurein a pump could be the process being measured. If a transducerdetermined that the pressure was too high, a feedback signal wouldbe generated to either shut down the pump or reduce the pumpstroke. Similarly, an accelerometer could be mounted on the bear-ing raceway of a motor-generator10.1 set. Excessive vibrationscould indicate an out-of-balance condition, and the signal from theaccelerometer would be used to shut down or rebalance themotor-generator set.

Alternately, measurements can be acquired to support design andanalysis10.2 as illustrated in Figure 10.2. At the component level,design is highly dependent on analysis. Figure 10.2 shows thisdependency; testing is used to validate this modeling and analysis.For large systems, the assemblage of components becomes diffi-cult to model accurately. For example, the thermal and structuralinterfaces between components may not be repeatable or may benon-linear. Thus, at the system level, we depend more heavily on


Feedback/Alarm

ProcessInIn Out

Figure 10.1Measurementfeedback to triggeran alarm or controla level.

Figure 10.1


testing and the test measurements to verify performance. In sum-mary, test measurements are important when validating componentmodels, and the measurements become more important when ver-ifying system performance. Force, pressure, and accelerationtransducers are crucial to the validation and verification processassociated with structural systems.

Once recorded, time-varying signals representing force, pressureor acceleration can be integrated or differentiated. Differentiationcan enable determination of the rate of loading or response of astructural system. However, integration of signals is a more com-mon operation. Integrating force-time signals can quantify thetotal impulse delivered to a system. Integrating pressure-time sig-nals over both time and area can also quantify the total impulsedelivered to a system. An example would be the quantification ofthe total impulse delivered to a structure, due to blast overpressurefrom an explosion. Similarly, integration of the output from anaccelerometer quantifies the velocity change of the structure towhich it was affixed. A second integration yields the local changein displacement of the structure.

Part 10 Data Utilization

Design

Testing &Measurements

Validate

Figure 10.2Measurement tosupport design andanalysis.

Modeling/Analysis

Verify

Figure 10.2


Both signal integration and differentiation can be challenging tasks.The following equations and figures help us understand theseprocesses. The symbol x(t) will represent displacement in the timedomain and v(t) and a(t) will represent velocity and accelerationrespectively. X(jω) will be the Fourier transform of x(t), and soon for the other parameters.

Displacement = x(t), transforms as X(jω) (10.1)

Velocity = v(t), transforms as V(jω)

Acceleration = a(t), transforms as A(jω)

Once transformed, the relationships between them are shownbelow.

V(jω) = (jω) X(jω) (10.2)

A(jω) = (jω)2X(jω)

V(jω) = A(jω)/(jω)

X(jω) = A(jω)/(jω)2

The first equation in 10.2 shows that as we differentiate displace-ment to get velocity, we multiply by frequency. If we perform thisoperation twice to get acceleration, we multiply by frequencysquared. Thus, if noise is present in the displacement signal at highfrequencies, its effect is amplified. For anyone who has tried toperform signal differentiation, the fact that noise amplification hasoccurred has been obvious. The last two equations in 10.2 showthat as we integrate acceleration to get velocity, we divide by fre-quency. A second integral to get displacement entails dividing byfrequency squared.Thus, integration mitigates high frequency noise


and differentiation amplifies it. Figure 10.3 illustrates this sameeffect for a single integrator and differentiator. Note that a phaseshift is also involved.

The previous discussion is general to any single/double integrationprocess. From it we may assume that integration is always well-behaved, and differentiation is the only challenging operation.Thefollowing example shows this not to be the case. Data offset canbe a problem.

In this example, a data channel for recording acceleration has beencalibrated so that ±5 volts corresponds to ±2,000 g. Dependingon the number of bits to which the channel is digitized, the dataresolutions of Table 10.1 are possible.


log

ampl

itude

log

ampl

itude

θ = −1

θ = +1

phas

eph

ase

+90 deg

-90 deg

logω

Differentiation

Integration

logω

logω

Figure 10.3Frequencycharacteristics ofsingle integratorand differentiator.

Figure 10.3

logω

Figure 10.4A shows a seismic event acquired by an accelerometerand recorded on this data channel.10.3 This record was digitized toa resolution of 16 bits. Note its peak-peak amplitude is about 600g, or less than one-sixth the channel full-scale range. Thus, morethan two bits resolution relative to the signal peak-peak value hasbeen lost. Figure 10.4B shows the result of the first signal integra-tion to determine velocity. Figure 10.4C shows the displacementsignal resulting from the second integration. Note this gross dis-placement is 200 cm.

Figures 10.5 A, B and C show the same set of data, but recordedand digitized to 12 bits resolution. Notice that Figures 10.4A and10.5A look essentially identical. However, when they are double-integrated, the results in Figures 10.4C and 10.5C indicate a 10%difference between them! Why is this?

The characteristic of this particular record is that it’s at a near-zerovalue for a considerable time period.The accuracy with whichthis level can be defined is ± one-half bit resolution.When con-sidering the effect of a half-bit bias error on double integrationover the 1.6-second duration of the record, it is understandablehow its influence can become significant.Thus, when integratingdata, a definite focus must be maintained on bit resolution.


+/- 5 V = +/-2000 g full scale

BITS

8101216

DIGITAL VALUE

2551023409565535

g’s/COUNT

7.84311.955.488.031

Table 10.1

Table 10.1Resolution for a4,000 g full scalechannel.



200

100

0

-100

-200

-300

-400

-500

-600.2 .4 .6 .8 1.0 1.2 1.4 1.6

ACCELERATION

G

Time (sec.)

800

400

0

-400

-800

-1200

-1600

-2000

-2400

VELOCITY

CM/SEC

.2 .4 .6 .8 1.0 1.2 1.4 1.6Time (sec.)

Figure 10.4AAcceleration vs.Time.

Figure 10.4BVelocity vs. Time.

.2 .4 .6 .8 1.0 1.2 1.4 1.6Time (sec.)

200

160

120

80

40

0

-40

DISPLACEMENT

CM

Figure 10.4CDisplacement vs.Time.

Figure 10.4Seismic eventrecorded at 16 bitsresolution.


200

100

0

-100

-200

-300

-400

-500

-600.2 .4 .6 .8 1.0 1.2 1.4 1.6

ACCELERATION

G

Time (sec.)

800

400

0

-400

-800

-1200

-1600

-2000

-2400

VELOCITY

CM/SEC

.2 .4 .6 .8 1.0 1.2 1.4 1.6Time (sec.)

.2 .4 .6 .8 1.0 1.2 1.4 1.6Time (sec.)

180

140120

80

40

0

-40

DISPLACEMENT

CM

-20

20

60

100

160

Figure 10.5AAcceleration vs.Time.

Figure 10.5Seismic eventrecorded at 12 bitsresolution.

Figure 10.5BVelocity vs. Time.

Figure 10.5CDisplacement vs.Time.


accelerometer was of the same type as that which recorded data inFigure 10.6, and was overranged by a factor of three.The ampli-fier was also overranged. If someone was reviewing these datawithout knowing that overranging had occurred, the data mightsubsequently be filtered.However, the energy spectral density func-tion in Figure 10.10 indicates that there is no separation betweenthe response of the structure and the structural response of theaccelerometer.The reason this separation did not occur is that non-linearities in the data (attributable to the accelerometer and theamplifier) created frequencies in the output signal not present inthe input.The energy spectral density function served as a usefultool to warn us of problems with the data.While the precedinganalysis was performed on an accelerometer record, it is equallyapplicable to force and pressure data.

We now turn our attention to shock spectra.10.4 Mechanical shocktypically occurs because of an energy release of short duration andsudden onset.A package dropped onto a floor, an automobile crash,


10

5

0

-5

-10

Acc

eler

atio

n (g

x 1

03)

Time (ms)

0.0 1.0 3.02.0 5.04.0

Figure 10.6Response ofaccelerometernumber one.

Figure 10.6


2.0

1.6

1.2

0.8

0

Ene

rgy

spec

trum

(g2

/ H

z2)

Frequency (kHz)

0.0 10 3020 40

0.4

Figure 10.7Energy spectraldensity function forFigure 10.6.

10

5

0

-10

Acc

eler

atio

n (g

x

103

)

Time (ms)

0 1 32 54

-5

Figure 10.8Filtered data ofFigure 10.6.

Figure 10.7

Figure 10.8



Figure 10.9Response ofaccelerometernumber two.10

5

0

-5

-15

Acc

eler

atio

n (g

x

103

)

Time (ms)

0 1 32 54

-10

15

2.5

2.0

1.5

1.0

0

Ene

rgy

spec

trum

n (g

2 /

Hz2

)

Frequency (kHz)

0 10 3020 5040

0.5

3.0

60

Figure 10.10Energy spectraldensity function forFigure 10.9.

Figure 10.9

Figure 10.10

an aircraft’s hard landing, provide a few examples of mechanicalshock. A shock can last for several periods of vibration in a struc-tural system, or it can be short, relative to the natural period ofoscillation of the system. Pyroshock, a subclass of shock, is a short-duration, high-amplitude, and high-frequency transient structuralinput. The forces causing the transient input are usually a combi-nation of explosive and impact events within an ordnance item, aswell as forces associated with the release of stored strain energy.

Shock-spectrum calculations are tools to enable replication in thetest laboratory, based on damage potential, the effects of any com-plex acceleration pulse.The shock-response spectrum is a plot ofthe peak response of an infinite number of lightly damped oscilla-tors to an input shock transient. It consists of a primary spectrum(peak response during the pulse), a residual spectrum (peakresponse after the pulse), and a maximax spectrum (envelope of thegreater of the primary or the residual). Under this analysis, a shocktest loading is considered satisfactory if its selected shock responsespectrum envelops the shock spectra of the field data. To performthis analysis, it is imperative that the accelerometer that character-ized the field environment was properly coupled to the structureand that its noise-free time history was recorded with fidelity.

Figure 10.11 illustrates the process behind a shock-spectra calcu-lation. The input time history for a given pulse is shown alongwith the response of a specific tuned oscillator to this input. Pointh is the oscillator’s maximum positive response during the pulse;point j is its maximum positive response after the pulse; and pointi is its maximum negative response after the pulse. These are allplotted at the tuned frequency of the oscillator. A similar calcula-tion at another oscillator frequency would provide three morepoints. Additional calculations provide additional points. Eventu-ally, these can be enveloped as shown.




Mag

nific

atio

n,R

espo

nse

yIn

put

x

b

a

d

f

g

h

j

i

e

c

kl

m

n

Time Ratio, τ/T

y

2.0

x

h

j

i

Mag

nific

atio

n,R

espo

nse

yIn

put

x

0

1.0

-1.0

-2.0

b

a

d

fg

h

j

i

e

c

0 0.25 1.0 1.5 2.0

Time Ratio, τ/T

TIME HISTORY SPECTRUM

Figure 10.11Pictorial shockspectra calculation.

Figure 10.11


Figure 10.12 illustrates a complex pulse that a component couldencounter in the field.Again, fidelity of the accelerometer datamust be reliable. Figure 10.13 illustrates a shock pulse that can eas-ily be approximated on laboratory test equipment. Figure 10.14shows that the shock spectrum of the laboratory pulse of Figure10.13 envelops the field data of Figure 10.12 and would becomethe laboratory test equivalent (with conservatism) of the field envi-ronment.

In this part, we have discussed the uses of and challenges associ-ated with signal differentiation and integration; the uses of thepower spectral density function (PSD), energy spectral densityfunction (ESD), and shock spectra.We will next discuss experi-mental modal analysis in Part 11.

.00

Acc

eler

atio

n (g

)

Time (sec)

-400

0

400

.01 .02 .03 .04 .05

Figure 10.12Complex fieldpulse.

Figure 10.12



Figure 10.13Simple laboratorypulse.

Figure 10.13

.0000

Acc

eler

atio

n (g

)

Time (sec)

400

800

1200

1600

2000

.0001 .0002 .0003 .0004 .0005

102

Res

pons

e (g

) 103

Frequency (Hz)

102

103 104

DAMPING = .03

2000g x 0.5msHAVERSINE

FIELD SHOCK(Figure 10.12)

Figure 10.14Envelope of fieldshock (from Baca/Sandia Labs).

Figure 10.14


REFERENCES

10.1 Barkov,Alexej and Barkov, Natalja, “Condition Assess-ment and Life Prediction of Roller Bearing Elements,”Sound and Vibration, pp. 10-17, June 1995.

10.2 Smallwood, David O., “The Correct Balance Between Test and Analysis,” Sound and Vibration,Vol. 34, No. 3,pp. 6-7, March 2000.

10.3 Stough,T.A., “Effect of Digital Word Size on Precision of Data Recovery from Field Instrumentation,” Defense NuclearAgency report DNA 001-79-C-0298, Nov.1980.

10.4 Harris, Cyril M., Shock and Vibration Handbook, McGraw-Hill, New York, NY, 3rd edition, pp. 23-10 to 23, 1988.



QUESTIONS

1. When we measure with an accelerometer the mechanicalresponse of a laptop computer to a prescribed accelerationinput when the computer is affixed to a mechanical shockmachine, we are measuring to:

a. control a process or system.b. support design and analysis.

2. When we measure time varying pressure across an orificeplate and feed the pressure transducer’s signal back to aregulator to adjust this flow, we are measuring to:

a. control a process or system.b. support design and analysis.

3. Structural testing is important throughout the design process.The level of design at which testing becomes the mostimportant is the:

a. component.b. subsystem.c. system level.

4. If when recording dynamic force, pressure, or accelerationdata, residual electrical noise is on the recording after thesignal stops, differentiating the data will cause the record tobecome:

a. noisier.b. quieter.

5. If when recording dynamic force, pressure, or accelerationdata, residual electrical noise is on the recording after thesignal stops, integrating the data will cause the record tobecome:

a. noisier.b. quieter.

6. When integrating dynamic force, pressure, or accelerationdata, the only consideration in selecting the channel bitresolution should be the transducer calibration accuracy.

a. true.b. false.

7. When integrating dynamic data, the effect of inadequate bitresolution will be accentuated most in its:

a. original time history.b. first integral.c. second integral.

8. The areas under the graphs of Figures 10.7 and 10.10 areproportional to the:

a. electrical signal energy.b. mechanical energy in the structure or its forcing function.

9. Figure 10.8 represents the:

a. dynamic response of the test structure.b. dynamic response of the structure as “colored” by its

measuring transducer.c. dynamic response of the filter applied to the data.

10. Shock spectra are a tool to replicate in the test laboratory:

a. acceleration-time histories.b. equivalent damage potential of complex acceleration

pulses.c. equivalent velocity changes to components.


Measurements to Understand andEnhance Structural Dynamics

Part


Part 4 of this series indicated that the current part would providean introduction to experimental modal analysis. This analysis is avery important tool in structural design.We will first provide a def-inition of structural dynamics, next elaborate on the computationalaspects of the Fourier transform first mentioned in Part 3, thensummarize the evolution of the modern spectrum analyzer, andsubsequently introduce the topic of experimental modal analysis.This analysis is largely dependent on accurate measurements fromforce and acceleration transducers. Last, we will describe "smart"transducers.

Measurements to Understand andEnhance Structural Dynamics

Continuing with our theme of data utilization, we first discussstructural dynamics and then look at the role of experimentalmodal analysis in design. Smart transducers are introduced.

Table 11.1Design space

Force Absent Force Present

Motion Absent Style Statics

Motion Present Kinematics Structural Dynamics

Design Space

Table 11.1

Table 11.1 defines the design space.Any structural system can betotally constrained (motion absent) or free to move in one or moredirections (motion present). Similarly, external forces can either


be absent from the structure (force absent) or acting (force pres-ent) on it. Contained in this design space are four distinct designproblems.

1. Forces and Motion Absent:The design problem is one of style.

2. Forces Present and Motion Absent:The design problem is oneof statics.

3. Forces Absent and Motion Present:The design problem is oneof kinematics.

4. Forces and Motion Present:The design problem is one of struc-tural dynamics.

Structural dynamics can then be defined as the design problem inwhich we concern ourselves with the time-varying forces actingon a structure, and the response of the structure to these forces.Thedegree of ride comfort we achieve in our automobiles on the high-way, the potential fatigue of the blades of large windmills excited bywind-induced vibrations, and the quality of the harmonious soundsof stringed instruments (e.g., guitars, violins) are all influenced bystructural dynamics.

The Fourier transforms of force and acceleration signals readilyenable the identification of critical structural frequencies inmechanical systems.Whereas the Fourier transform has existed fora number of years, it wasn’t until 196511.1 that it became practical toimplement on a computer.The Fourier transform could always bewritten in discrete form as the discrete Fourier transform (DFT),however the fast Fourier transform (FFT) eliminated redundantcomputations in the DFT, allowing its more rapid calculation.Withthe memory and speed capability of the computers of today, the

FFT does not play such an important role.However, its importancewas very significant in enabling machine calculations in the 1960sand 1970s.

In the late 1960s, 6-bit digitizers and displays were just becomingcommercially available from companies such as Biomation. In ashort period of time, the bit resolution of these recorders began toincrease. In the early 1970s, these recorders, combined with theFFT, enabled the first all-digital spectrum analyzers to enter themarketplace. Both amplitude and phase information could be presented.

Two-channel analyzers followed shortly.Two-channel machinesenabled the force input to a structure to be measured at one loca-tion, along with the acceleration response at a second location, sothat a frequency-response function (FRF) for a structure could becalculated.The FRF was derived from the ratio of the output tothe input FFTs.These two-channel machines created such a greatdemand that by the mid-1970s, FFT-based machines with four ormore channels became available.

Software followed that allowed the animation of the dynamicmotion of a structure. Once the designers saw their systems indynamic motion, they had many ideas for optimizing their design.This initially involved building a new prototype to test. Subse-quently, the idea evolved of extracting a mathematical model fromthe FRF data describing the actual structural system. Once thismodel was extracted, the designer’s modifications could be approx-imated within it.These modifications to the structural system couldbe proposed and evaluated by software within minutes! Thus,multi-channel analyzers and their processing capability haveenabled the field of experimental modal analysis to continue toevolve through today11.2.This entire process is further describedon the next page.


Part 11 Measurements to Understand and Enhance Structural Dynamics

Analytical modal analysis starts with the measurement of the geom-etry of a structure, its boundary conditions, and the characteristicsof its materials.The mass, stiffness, and damping of the structureare expressed in terms of the matrices for these three parameters(M, K, and C, respectively). Depending on the complexity of thestructure, these matrices can contain thousands, tens of thousands,or even more elements! Equations of motion describing the struc-ture are formed:

M x+ C x+ Kx = f (11.1)

where x and f are respectively the generalized response and forc-ing function vectors. Each dot represents a time derivitive.

Individual equations of motion are first formed for each discreteelement (Figure 11.1) of the structure.The coordinates for theseindividual elements are then aligned in system space through a lin-ear transformation, after which the elements are combined, resultingin matrix Equation 11.1.The generalized displacement and forcematrices can be combined into specified and unspecified segments,and Equation 11.1 can then be transformed so that its left side isexpressed only in terms of unspecified coordinates. Once this finalmatrix formulation is achieved,without providing additional detail,sufficient information is available to extract the system modalparameters (natural frequencies, damping factors, and vibratorymode shapes). Subsequently, the system frequency response matrixcan be determined in terms of these modal parameters.

The experimental modal analysis approach starts by measuringdirectly this frequency-response-function matrix.Measurements ofconcurrent dynamic input forces to and responses of the struc-ture, when they are transformed and have their ratio taken, ideallyresult in the same frequency-response functions as determined


through the analytical process. However, this does not occur inpractice: Assuming that good-quality experimental data areacquired, an update of the analytical finite element model is per-formed. Disagreement between models can occur due to one ormore of the following factors: the measured degrees of freedomdo not coincide with the number of degrees of freedom in thefinite element model; the set of experimental modal data is incom-plete due to limited bandwidth; or damping cannot be accuratelyincluded in the finite element model. Updating results in a finiteelement model that yields accurate and reliable predictions of thedynamic behavior of the structural system, greatly enhancing thedesign process.



Figure 11.1

Figure 11.1Fan blade with itsdiscrete elementsfor analysis.


To perform experimental modal analysis, a vibratory force has to beapplied to the structure.This force can be either deterministic ornondeterministic as discussed in Part 3.A vibratory exciter affixedto both the test object and either the ground or a reference framecan be used.Alternately, the exciter alone can be affixed to thestructure, and the mass of the shaker then serves as the reference(as opposed to the ground or the frame).The exciter is driveneither by electrodynamic or hydraulic force.A non-fixed excitationsource (e.g., instrumented hammer) can also be used, as is illustratedin Figure 11.2. Figure 11.3 shows such an application. Each exci-tation technique has relative advantages and disadvantages.

Figure 11.2Non-fixedreferenceinstrumentedhammers withforce transducer.

Figure 11.2


A piezoelectric force transducer is typically used to measure theinput force.The response of the structure is usually measured bypiezoelectric accelerometers. For very large, low-frequency (lessthan one Hertz) structures, variable-capacitance accelerometers areoften used. Figure 11.4 shows an assortment of accelerometersthat might be used in modal applications.


Figure 11.3

Figure 11.3Force-hammerexcitation of structuralsystems.

Figure 11.4Assortment of modalaccelerometers.

Figure 11.4


Various combinations of input-response measurements arerecorded from the force transducers and accelerometers.As noted,the signals from the transducers are processed to estimate fre-quency-response functions. These functions are stored in diskmemory for model updating as previously described.Animatedgraphical tools can simulate on a computer screen the modal defor-mation of the structural system. Figure 11.5 shows the firstcantilever-mode shape associated with a TV tower vibrating at 9.4 Hz.

Figure 11.5

Figure 11.5First vibratorymode of a 25-foottower (9.4 Hz).

More than half of all modal tests use a simple force-hammer inputand a few accelerometers to measure structural response.These testsare intended only to identify resonant frequencies in the struc-ture.The remainder of the modal testing can require anywherefrom tens to hundreds of accelerometer measurements. In dealingwith so many transducers, numerous problems can arise: cable bun-dles can change the dynamic performance of the structural system;these bundles can also increase the susceptibility of low-level ana-log transducer to cross-talk and noise; and cables can accidentallybecome interchanged between channels. Smart transducers aredealing with these problems11.3, 11.4.

A smart transducer contains an internal transducing mechanism(resistive or capacitive bridge, piezoelectric element, etc.), analogsignal conditioning, analog-digital (A/D) converter, digital mem-ory chip, and communications interface. If the signal conditioningand A/D cannot survive the operating environment of the trans-ducer, they can be packaged separately in something called atransducer bus interface module (TBIM). For example, transduc-ers monitoring turbine or internal combustion engine pressuresand vibrations at temperatures above 400 degrees Fahrenheit couldperform these functions in a "smart" mode if their associated elec-tronics were isolated from the high-temperature environment andplaced in a remote TBIM.These smart transducers are ideally net-work-independent.A processor provides an interface between anetwork and the smart transducers.A common bus can addressthe multiple, physically separated, smart transducers.

Referring to the aforementioned challenges in modal testing, smarttransducers in the form of accelerometers can offer many solutions.The location of the signal conditioning and A/D in the transducercan reduce susceptibility to electrical noise and interference. Cablebundles can be eliminated by having multiple transducers com-municate on a common bus.The number of equipment racks can




be greatly reduced.A transducer electronics data sheet (TEDS)within the smart transducer can store transducer sensitivity, type,position, output polarity, model number, serial number, calibrationdata, and more. For example, steady-state thermal corrections canbe preprogrammed and, if internal temperature measurements aremade, data can be corrected in near-real time. If transducers areexchanged during a test, they simply report their new identificationto preclude system positional and calibration errors. In addition,information such as position location on the structure can beentered into the TEDS and updated as needed.

The IEEE has collected standardization of this technology underspecifications IEEE1451. Subservient standards either do or willdefine: standard network independent interfaces; a standard digitalinterface for connecting physically separated transducers in a mul-tidrop configuration; transducer electronic data sheets [TEDS (thebrains of the transducer)]; communications protocols on the net-work bus; and much more.

Reflecting on smart transducers, measurement-system electronicshave steadily migrated into the transducer for over 50 years.Theearly electromechanical transducers were largely stand-alonedevices with contained inductive, capacitive, resistive, or piezo-electric transduction elements.The advent of the transistor in the1950s started the movement of the signal conditioning towardsthe transducer.The 1960s–1970s saw the replacement of metalstrain gages by deposited metal film, bulk silicon, and diffused sili-con strain gages in many transducer applications. Miniatureelectromechanical systems (MEMS) technology,which emerged inthe latter part of this same time period, enabled resistive and capac-itive sensors to become integrally manufactured assemblies. In themid-1960s, field effect transistor (FET) circuits migrated intopiezoelectric-based transducers to provide signal conditioning



Figure 11.6Assortment ofsmart modalaccelerometers.

Figure 11.6

within the transducer housing. Smart transducers are just the nextstep in the continuous movement of enhanced electronic capabil-ities into the transducer. Figure 11.6 shows an array of smartaccelerometers.

Discussing smart transducers, a relatively new technology, is a fittingtransition from Part 11 to Part 12.Part 12 will summarize the total-ity of our learning in this series and provide a set of guidelines tooptimize the successful acquisition of dynamic force, pressure, andacceleration measurements.

REFERENCES

11.1 Cooley, J.W. and Tukey, J.W., "An Algorithm for the MachineCalculation of Complex Fourier Series," Math. Comput.,Vol.19, p. 297,April, 1965.

11.2 Heylen,Ward, Lammens, Stefan, Sas, Paul, Modal Analysis Theory and Testing,Katholieke Universiteit Leuven,Leuven,Belgium, 1997.

11.3 Gen-Kuong, Fernando and Swanson, Bruce, "Smart Sensor Network System," International Test and EvaluationWorkshop,April 19-22, 1999.

11.4 Chu,Anthony, Gen-Kuong, Fernando, and Swanson,Bruce, "Smart Sensors and Smart Sensor Network System,"The 70th Shock and Vibration Symposium,Albuquerque,NM, Nov. 15-19, 1999.



QUESTIONS

1. The design problem in which we consider the time varyingforces acting on a structure and the response of the structureis:

a. style.b. kinematics.c. statics.d. structural dynamics.

2. True or false. The fast Fourier Transform (FFT) producesdifferent results than the discrete Fourier Transform (DFT).

a. true.b. false.

3. The combination of what two things enabled the digitalspectrum analyzer to enter the market place?

a. DFT, DC power supply.b. FFT, digitizer.c. memory scope, analog filters.d. DFT, analog filters.

4. Analytical modal analysis starts with measurements of astructure’s:

a. geometry.b. boundary conditions.c. material characteristics.d. all of these.

5. True or false. Experimental modal analysis is often used toupdate the analytical finite element model.

a. true.b. false.

6. The type of force transducer used to measure the input forcesto a structural system when performing experimental modalanalysis is:

a. piezoelectric.b. piezoresistive.c. variable capacitance.d. force-balance.


7. The type of accelerometers most often used when performingexperimental modal analysis is:

a. piezoelectric.b. piezoresistive.c. variable capacitance.d. force-balance.

8. True or false. At very low frequencies, variable capacitance accelerometers can be used.

a. true.b. false.

9. Smart transducers can solve problems associated with:

a. stiffening due to cable bundles.b. channel cross-talk.c. interchanged channels.d. all of these.

10. True or false. Smart transducers are just the next logical stepin a constant evolution of technology.

a. true.b. false.


Putting It All Together

Part


Throughout this text, reference articles and other texts have beencited to make available information to complement the material pre-sented. Professional societies serve as another valuable informationsource.Professional societies originate and maintain specifications fortransducers.They also house technical subgroups and divisions thatindividuals may query on various measurement system topics. Fromthe many societies available, the subset of societies judged most use-ful to the measurement engineer is listed below.

International Foundation for Telemetering (IFT)5959 Topanga Canyon Blvd., Suite 150 Woodland Hills, CA 91364

(Sponsors of technical forums, educational activities, and technical publications in telemetering.)

Putting It All Together

This final portion of the text is primarily intended to be a recapor summary. It identifies additional resources that can assist inmeasurement system design. It concludes by providing a practicalMeasurement Checklist for the senior technician or practicingengineer (identified in Part 1 as the primary audience for thistext) who has a fundamental understanding of transducers andassociated signal conditioning and who wants to design and applymeasurement systems that guarantee the acquisition ofmeaningful data.

The Instrumentation, Systems, and Automation Society (ISA)67 Alexander DrivePO Box 12277Research Triangle Park, NC 27709

(Maintains standards and contains committees focused on measurementdevices and transducers.)

International Organization for Standards (ISO)Geneva, SwitzerlandRepresentative U. S. body:American National Standards Institute (ANSI)11 West 42nd Street13th floorNew York, NY 10036

(Contains technical committee TC108 on mechanical vibration and shock.)

Society for Experimental Mechanics, Inc. (SEM)7 School StreetBethel, CT 06801-1405

(Contains a transducers-and-sensors division, and hosts The InternationalModal Analysis Conference (IMAC).)

Shock and Vibration Information Analysis Center (SAVIAC)Park Drive8th FloorFalls Church,VA 22042

(Maintains information resources for shock and vibration; has hosted 72 Shock and Vibration Symposia to date.)


A Measurement Checklist is presented next,based primarily on Parts1–11 of this sequence.The checklist is intended to provide guidance,in an orderly and logical fashion, to enable the design of measure-ment systems for dynamic force, pressure, and acceleration.The partof the course that complements each item in the list is referencedso that it can easily be consulted for supporting material.

CHECKLIST FOR THE SUCCESSFUL MEASUREMENTOF DYNAMIC FORCE, PRESSURE, AND

ACCELERATION

The test requester is responsible for providing the guidance requiredin 1 below.

(1) All programmatic objectives should be explained. In addition,the measurement objectives of the test or control processshould be provided in written form to the individual respon-sible for the success of the measurement(s). Informationprovided should include but not be limited to:

• Test item definition (drawings and schematics as required).

• Measurement types desired: force, pressure, and/or acceleration.

• Measurement locations.• Measurement directions (if applicable).• Measurement channel identification requirements.• Measurement accuracy required.• Environmental operating conditions.• Estimate, based on experience and analysis, of the

anticipated types, levels, and frequency requirements of the measurand(s) to be expected.


Part 12 Putting It All Together

• Resources available (financial, manpower, physical [electrical power, communications, mechanicalfixturing]).

• Reporting format required.

The measurements engineer is responsible for the following information and tasks.

(2) The force, pressure, and/or acceleration transducer types andmodels must be selected. Criteria that should be consideredinclude:

• Transducer full-scale range/sensitivity (see 1, this Part and Part 2).

• Transducer frequency-response requirements (see 1, this Part and Parts 1, 2, 5, and 7).

Piezoelectric technology will not satisfy DC response (see Parts 5 and 7).

• Transducer effect on the process being measured (see 1, this Part and Parts 1 and 4).

Size, stiffness, mounting torque, etc.(see Part 4).

• Transducer known environmental operating conditions (see 1, this Part and Part 8).

Damping/sensitivity dependency withtemperature as applicable (see Part 2).


Evaluated response to known extraneous environmental inputs (see Part 8).

• Transducer off-axis (transverse) response (see 1, this Part and Part 8).

(3) The design of the remainder of the measurement system must be optimized:

• Select signal conditioning (see 1 and 2, this Partand Part 5).

Customize measurement-system’s transferfunction (see Parts 4 and 5).

• Consider expected signal types (see 1, this Part and Part 3).

• Select data-sampling rate/bit resolution (see 1, this Part and Part 10).

• Consider applicability of smart transducers (see Part 11).

• Select type/location of signal filtering (see Part 6).

• Interface the transducer and the signal conditioning via cable (see Parts 5 and 12).

• Avoid unwanted filtering due to cable capacitance (see Part 8).



(4) The transducer(s) should be appropriately interfaced to the test item or control process (see 1, this Part and Part 4):

• Avoid modifying the transducer dynamic response because of the measurand coupling process (see Part 4).

Consider effects of Helmholtz resonator,helicoils, adhesives, etc. (see Part 4).

(5) The transducers and the entire measurement system should be appropriately calibrated. Specifically, the following should be performed (see 1, this Part and Part 8):

• Transducer laboratory calibration (see Part 8).

• Measurement-system field calibration (see Part 8).

(6) The measurement system should be verified to be noise-free (see Part 9):

• In addition to the active data channels, measurement-system check channels should be incorporated into thetest (see Part 9).

(7) The data should be validated as worthy of subsequent analy-sis after it is verified to be noise-free (see Part 7):

• Apply rules of thumb for data assessment (see Part 7).

Validate low- and high-frequency response (see Part 7).


Dependent on limitations of selectedtransducers and electronics (see Part 7).

(8) The data should be reduced and analyzed (see 1, this Part and Parts 10 and 11):

• Integration, differentiation, shock spectra, energy spec-tral density (ESD), power spectral density (PSD), modalanalysis (see 1, this Part and Parts 3, 10, and 11).

(9) The data should be reported and the measurement system con-figuration and hardware should be documented for future reference (see 1, this Part).

CONCLUSION

Having developed this Measurement Checklist to provide textclosure, this rhetorical question is posed:What would I do differ-ently if I were developing this text again? Fortunately, the answer is,"Not much".The topic of cables probably was not covered withthe intensity it deserved. In particular, noise effects (triboelectriccharge generation) in cables used with high-impedance piezo-electric transducers deserved discussion that was not received.Fortunately, however, the increasing incorporation of FET circuits(Part 5) in the piezoelectric-transducer housing results in conver-sion of the high-impedance piezoelectric signal into alow-impedance source before it exits the cable; thus, the cablenoise concern is greatly lessened.Where cable noise is a concerndue to cable-whip or stress, high-quality cables can be acquiredwith internal graphite coatings to preclude charge buildup. Cabletie-down, with a strain-relief loop, should typically be performedwithin a few inches of a transducer to avoid cable-strain couplinginto the transducer housing. If the relief loop bends down, it forms



a "drip" loop for condensation that might form on the cable andthus precludes moisture from following the cable into the con-nector.

Another topic not definitively covered was data sampling and alias-ing. However, this topic is generic to all dynamic measurementsand well covered in any communications text.The concept of fil-tering, applicable to aliasing considerations, was covered in Part 6.

Last, it is appropriate to philosophize about the larger lessonlearned in this text. Consider a simple situation where we have asystem, its input and its output.There are three problem types con-tained in such a format.

INPUT SYSTEM OUTPUT

The first problem (below) occurs when we have knowledge ofthe input and the system and the question is:What is the output?This problem type is known as analysis. It arises when we try topredict the response of a given system to a given input.

INPUT SYSTEM ?

The second problem type (below) occurs when we know the sys-tem’s input and output and the question is:What is the system?


INPUT ? OUTPUT

Three subproblems come under this category. (A) When we knowthe input and output desired and the objective is to create the sys-tem, the problem is one of design. (B) When we apply a given inputand measure the system output, the problem is one of calibration.(C) When we apply a given input to a target and measure thereflected signal from the target, the problem is one of target iden-tifiction.

The third and final problem type (below) occurs when we knowthe system output and have knowledge of the system and the ques-tion is:What is the input? This problem type is MEASUREMENT.We record the output of the force, pressure, or acceleration meas-uring system in mV,V, or picocoloumbs (pC), and the question tobe answered is:What is the input force, pressure, or acceleration thatcaused this output? The dynamic characteristics of the system(transducer and its mounting, cable, signal conditioning, filter, etc.)all impact what we finally assess the measurement system input tohave been.

? SYSTEM OUTPUT

Parts 1, 2, 4, 5, and 6 have dealt directly with this measurementproblem while Part 8 has covered the calibration problem. Part 9discussed the integrity verification of the system output (noise-ver-ification check-channels). Part 7 addressed data validation on thebasis of knowledge of the measurement system. Part 3 dealt withsignal types and Parts 10 and 11 with data utilization. Hopefullythese topics, combined with the Measurement Checklist provided,will enable the dynamic input forces, pressures, and accelerations tomeasurement systems to be determined accurately.



Answers to Part Questions


ANSWERS

PART 11. a2. e3. c4. c5. b6. a7. c8. e9. b10. e

PART 21. e2. b3. c4. b,c,d,e5. c6. d7. d8. c9. d10. b

PART 31. d2. b3. d4. a5. b6. b7. c8. false9. true10. true

PART 41. b2. d3. b4. d5. c6. false7. true8. a9. true10. d

PART 51. a and b2. b3. c4. a5. b6. c7. d8. c9. a10. b

PART 61. e2. a3. d4. b5. b6. b7. c8. c9. d10. b

PART 71. c2. d3. d4. a5. c6. b7. d8. d9. a10. b


PART 81. b2. a3. a4. c5. d6. d7. b8. a9. d10. a, b, c

PART 91. d2. d3. a4. c5. c6. c7. c8. b9. b10. a

PART 101. b2. a3. c4. a5. b6. b7. c8. a9. a10. b

PART 111. d2. b3. b4. d5. a6. a7. a8. a9. d10. a

Dynamic Force, pressureand Acceleration Measurement

The HANDBOOK of

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Force,pressureand

Acceleration

Measurem

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Handbook of Dynamic Measurements

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