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Unit 15

SDOF Response to Base Input in the Frequency Domain

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Introduction

Steady-state response of an SDOF System

Base Input:PSD – stationary with normal distribution

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Miles Equation

Miles Equation is the simple method for calculating the response of an SDOF to a PSD

Assume white noise, flat PSD from zero to infinity Hz

As a rule-of-thumb, it can be used if PSD if flat within + 1 octave of the natural frequency

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Miles Equation

The Miles equation is a simplified method of calculating the response of a single-degree-of-freedom system to a random vibration base input, where the input is in the form of a power spectral density. The Miles equation is

P is the power spectral density level at the natural frequency

f n is the natural frequency

GRMSx

GRMSx P f Qn2

where

is the overall response

Q is the amplification factor

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SDOF System, Base Excitation

1 kfn

2 m

The natural frequency fn is

The amplification factor Q is

The damping coefficient C is

mk2C

)2/(1Q

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SDOF Free Body Diagram

The equation of motion was previously derived in Webinar 2.

yz2nωznω2ξz

yzx

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Sine Transmissibility Function

Either Laplace or Fourier transforms may be used to derive the steady state transmissibility function for the absolute response.

After many steps, the resulting magnitude function is

nf/fwhere

where f is the base excitation frequency and fn is the natural frequency.

222

2

21

)2(1

y

x

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0.1

1

10

20

0.1 1 10

Q = 10Q = 2Q = 1

Frequency Ratio ( f / fn )

Tra

nsm

issi

bilit

y (G

out

/ G

in )

TRANSMISSIBILITY MAGNITUDESDOF SYSTEM SUBJECTED TO BASE EXCITATION

The base excitation frequency is f. The natural frequency is fn.

Frequency Ratio (f / fn)

VibrationdataTransmissibility Curve Characteristics

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The transmissibility curves have several important features:

1. The response amplitude is independent of Q for f << fn.

2. The response is approximately equal to the input for f << fn.

3. Resonance occurs when f fn.

4. The peak transmissibility is approximately equal to Q for f = fn and Q > 2.

5. The transmissibility ratio is 1.0 for f = 2 fn regardless of Q.

6. Isolation is achieved for f >> fn.

VibrationdataExercises

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vibrationdata > Miscellaneous Functions >

SDOF Response: Steady-State Sine Force or Acceleration Input

Practice some sample calculations for the sine acceleration base input using your own parameters.

Try resonant excitation and then +/- one octave separation between the excitation and natural frequencies.

How does the response vary with Q for fn=100 Hz & f =141.4 Hz ?

Vibrationdata “Better than Miles Equation”

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Determine the response of a single-degree-of-freedom system subjected to base excitation, where the excitation is in the form of a power spectral density

The “Better than Miles Equation” is a.k.a. the “General Method”

VibrationdataMiles Equation & General Method

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The Miles equation was given in a previous unit

Again, the Miles equation assumes that the base input is white noise, with a frequency content from 0 to infinity Hertz

Measured power spectral density functions, however, often contain distinct spectral peaks superimposed on broadband random noise

The Miles equation can produce erroneous results for these functions

This obstacle is overcome by the "general method"

The general method allows the base input power spectral density to vary with frequency

It then calculates the response at each frequency

The overall response is then calculated from the responses at the individual frequencies

VibrationdataGeneral Method

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nii

N

1iiiPSDA

21

)2(1nGRMS

f/f

,f)f(Y,fx2

i22

i

2i

The general method thus gives a more accurate response value than the Miles equation.

The base excitation frequency is f i and the natural frequency is f n

The base input PSD is

)f(Y iPSDA

VibrationdataNavmat P-9492 Base Input

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0.001

0.01

0.1

100 100020 2000

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

PSD OVERALL LEVEL = 6.06 GRMSPSD Overall Level = 6.06 GRMS

Frequency (Hz)

Accel (G^2/Hz) Frequency

(Hz) Accel

(G^2/Hz)

20 0.01

80 0.04

350 0.04

2000 0.007

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Apply Navmat P-9492 as Base Input

fn = 200 Hz, Q=10,

duration = 60 sec

Use:

vibrationdata > power spectral density > SDOF Response to Base Input

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4.5-sigma is maximum expected peak from Rayleigh distribution

Miles equation also gives 11.2 GRMS for the response

Relative displacement is the key metric for circuit board fatigue per D. Steinberg (future webinar)

SDOF Acceleration Response

= 11.2 GRMS = 33.5 G 3-sigma = 49.9 G 4.5-sigma

SDOF Pseudo Velocity Response

= 3.42 inch/sec RMS = 10.2 inch/sec 3-sigma = 15.3 inch/sec 4.5-sigma

SDOF Relative Displacement Response

= 0.00272 inch RMS = 0.00816 inch 3-sigma = 0.0121 inch 4.5-sigma

VibrationdataPseudo Velocity

The "pseudo velocity" is an approximation of the relative velocity

The peak pseudo velocity PV is equal to the peak relative displacement Z multiplied by the angular natural frequency

Pseudo velocity is more important in shock analysis than for random vibration

Pseudo velocity is proportional to stress per H. Gaberson (future webinar topic)

MIL-STD-810E states that military-quality equipment does not tend to exhibit shock failures below a shock response spectrum velocity of 100 inches/sec (254 cm/sec)

Previous example had peak velocity of 15.3 inch/sec (4.47-sigma) for random vibration

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ZPV n

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Peak is ~ 100 x Input at 200 Hz Q^2 =100 Only works for SDOF system responseHalf-power bandwidth method is more reliable for determine Q.

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Peak Design Levels for Equivalent Static Load

Author Design or TestEquation Qualifying Statements

Himelblau, et al 3sHowever, the response may be non-linear and non-Gaussian

Fackler 3s3s is the usual assumption for the equivalent peak sinusoidal level

Luhrs 3sTheoretically, any large acceleration may occur

NASA3s for

STS Payloads2s for

ELV Payloads

Minimum Probability Level Requirements

McDonnellDouglas 4s

Equivalent Static Load

Scharton & Pankow 5sSee Appendix C

DiMaggio, Sako, Rubin nsSee Appendices B and D for the equation to calculate n via the Rayleigh distribution

Ahlin Cn See Appendix E for equation to calculate Cn

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Rayleigh Peak Response Formula

Tfnln2nc

nc

5772.0ncnC

nnC Maximum Peak

fn is the natural frequency

T is the duration

ln is the natural logarithm functionis the standard deviation of the oscillator responsen

Consider a single-degree-of-freedom system with the index n. The maximum response can be estimated by the following equations.

a.k.a. crest factor

VibrationdataConclusions

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The General Method is better than the Miles equation because it allows the base input to vary with frequency

For SDOF System (fn=200 Hz, Q=10) subjected to NAVMAT base input…

We obtained the same response results in the time domain in Webinar 14 using synthesized time history!

Response peaks may be higher than 3-sigma

High response peaks need to be accounted for in fatigue analyses (future webinar topic)

VibrationdataHomework

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Repeat the exercises in the previous slides

Read

T. Irvine, Equivalent Static Loads for Random Vibration, Rev N, Vibrationdata 2012

T. Irvine, The Steady-state Response of Single-degree-of-freedom System to a Harmonic Base Excitation, Vibrationdata, 2004

T. Irvine, The Steady-state Relative Displacement Response to Base Excitation, Vibrationdata, 2004