ME 322: InstrumentationLecture 30

April 6, 2015Professor Miles Greiner

Announcements/Reminders• This week in lab

– Open ended Lab 9.1 – 1%-of-grade extra-credit for active participation

• HW 10 due Friday– I will revise the Lab 10 Instructions, so don’t start it

until Wednesday.

Piezoelectric accelerometer

• Seismic mass increases/decreases compression of crystal, – Compression causes electric charge to accumulate on its sides– Changing charge can be measured using a charge amplifier

• High damping, stiffness and natural frequency– Good for measuring high frequency varying accelerations

• But not useful for steady acceleration

Accelerometer Model

• Un-deformed sensor dimension y0 affected by gravity and sensor size• Charge Q is affected by deformation y, which is affected by acceleration a • If acceleration is constant or slowly changing, then F = ma = –ky, so

– Only the spring is important: yS = (-m/k)a; – Static transfer function

• What is the dynamic response of y(t) to a(t)? (Damper become important)

y0

Charge Q=fn(y)= fn(a)

a(t) = Measurand

k [N/m] l [N/(m/s)]

y = Reading

a

y

-m/k

Moving Damped Mass/Spring System

• Want to measure acceleration of object at sensor’s bottom surface

• Forces on mass, – z(t) = s(t) + yo + y(t) (location of mass’s bottom surface)– Fspring = -ky, Fdamper = -lv = -l(dy/dt)

Inertial Frame

z

s(t)

Response to Impulse (Step change in v)

• Huge a at t = 0, but a(t) = 0 afterward – Ideally: y(t) = -(m/k)a(t)= 0 for t > 0

• my’’+ ly’ + ky = 0• Solution:

– depend on initial conditions

• Depends on damping ratio:

v

t

a

t

Response

• Undamped – t +Dcost , – oscillatory

• Underdamped – , – damped sinusoid (observe this in Lab 10)

• Critically-damped , and Over-damped – not oscillatory

Response to Continuous “Shaking”

– A = shaking amplitude– = forcing frequency

• Find response y(t) for all – For quasi-steady (slow) shaking,

• Expect – For higher , expect lower amplitude and delayed response

• my’’+ ly’ + ky = -ma(t) = -m• y(t) = yh(t) + yP(t)

– Homogeneous solutions yh(t) same as response to impulse– yh(t) 0 after t ∞

• How to find particular solution to whole equation?

Particular Solution• myP’’+ lyP’ + kyP = -m• Assume yP(t) = Bsin+Ccos (from experience)

– Find B and C – yP

’ = cosCs– yP

’’ = Bscos

• m(sCcos)+ l(BcosCs) + k(Bsin+Ccos) = -m• s() = 0

– – – Two equations and two unknowns, B and C

Solution• yP(t) = Bsin+Ccos

–

– ; – For no damping (l = 0), and : AP

• For : –

Compare to Quasi-Steady Solution

– Insert Undamped Natural Frequency ; Damping ratio: ; • (Want this to be close to 1)• with ,

𝐴𝑃

𝐴𝑆

Problem 11.35 (page 421)

• Consider an accelerometer with a natural frequency of 800 Hz and a damping ratio of 0.6. Determine the vibration frequency above which the amplitude distortion is greater than 0.5%.

Problem 11.35 (page 421)

• Solution:

• ?• Find f =?

Lab 10 Vibration of Weighted Steel and Aluminum Cantilever Beam

• This lab can be on the course Final• Accelerometer Calibration Data

– http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm

– C = 616.7 mV/g– Use calibration constant for the issued

accelerometer– Inverted Transfer function: a = V*1000/C

• Measure: E, W, T, LB, LE, LT, MT, MW – Estimate uncertainties of each

W

LT MT

T

LB LE

Accelerometer

Clamp MW

E (Lab 5)

Table 1 Measured and Calculated Aluminum Beam Properties

• The value and uncertainty in E were determined in Lab 5• W and T were measured using micrometers whose uncertainty were

determined in Lab 4• LT, LE, and LB were measured using a tape measure (readability = 1/16 in)• MT and MW were measured using an analytical balance (readability = 0.1

g)

Units Value3s

UncertaintyElastic Modulus, E [Pa] [GPa] 63 3

Beam Width, W [inch] 0.99 0.01Beam Thickness, T [inch] 0.1832 0.0008

Beam Total Length, LT [inch] 24.00 0.06End Length, LE [inch] 0.38 0.06

Beam Length, LB [inch] 10.00 0.06Beam Mass, MT [g] 196.8 0.1

Intermediate Mass, MI [g] 21.9 1.5Combined Mass, Mw [g] 741.2 0.1

Figure 2 VI Block Diagram

Formula Formula: v*1000/c

Statistics Statistics This Express VI produces the following measurements: Time of Maximum

Spectral Measurements Selected Measurements: Magnitude (Peak) View Phase: Wrapped and in Radians Windowing: Hanning Averaging: None

• Very similar to Lab 5• Add Formula Block• Suggestion: To get

practice and prepare for final, re-write the entire VI

Figure 1 VI Front Panel

Disturb Beam and Measure a(t)

• Use a sufficiently high sampling rate to capture the peaks – fS = ~400 Hz (>> 2fM )

• Looks like – Expect ,

• Measure f from spectral analysis ( fM )

• The sampling period and frequency were T1 = 10 sec and fS = 200 Hz. – As a result the system is capable of detecting frequencies between 0.1 and 100 Hz, with a resolution of 0.1 Hz.

• The frequency with the peak oscillatory amplitude is fM = 8.70 ± 0.05 Hz. – Easily detected from this plot.

• Find b from exponential fit to acceleration peaks

Time and Frequency Dependent Data• http

://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm

• Plot a versus t – Time increment Dt = 1/fS

• Plot aRMS versus f– Frequency increment Df = 1/T1

• Measured Damped (natural) Frequency, fM – Frequency with peak aRMS – Uncertainty

• Exponential Decay Constant b (Is it constant?)– Show how to find acceleration peaks versus time

• Use AND statements to find accelerations that are larger than the ones before and after it • Use If statements to select those accelerations and times• Sort the results by time• Plot and create new data sets before and after 2.46 sec

– Fit data to y = Aebx to find b

Uncertainty Calculation

Dynamic (high speed) Accelerometer Response

y(t)

y0 +y(t)

s(t)

z(t) = s(t) + y(t) + y0

Lab 10 Vibration of a weighted cantilever Beam

Measure a(t)Find damping coefficient and damped natural frequency, and compare to predictionsHow to predict?

t (s)

Fit to data: find b and f