ME 322: InstrumentationLecture 31

April 8, 2015Professor Miles Greiner

Announcements/Reminders• This week: Lab 9.1 Open-ended Extra-Credit • New Due Date: HW 11 due Monday• Did you know?

– HW solutions are posted on WebCampus

Lab 10 Vibration of Weighted Steel and Aluminum Cantilever Beams

• 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)

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

Figure 1 VI Front Panel

Disturb Beam and Measure a(t)

• Use a sufficiently high sampling rate to capture the peaks – fS = ~400-500 Hz (>> 2fM )

• Data looks like – For under-damped vibration expect; , – How to predict ? Need , but

• 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

Fig. 5 Peak Acceleration versus Time

• The exponential decay changed at t = 2.46 sec• During the first and second periods the decay rates

are– b1 = -0.292 1/s– b2 = -0.196 1/s

Effect of Sampling Rate

• If the sampling rate is too slow, then it is likely that the peak accelerations will be missed for most of the oscillations

• Can cause a type of aliasing problem

Predict damped natural frequency and its uncertainty from mass, dimension, elastic

modulus and decay constant measurements

– Since – (predicted damped frequency)– (un-damped radial freq.)

• How to find equivalent (or effective) mass MEQ, damping coefficient lEQ, and spring constant kEQ for a weighted cantilever beam?

Equivalent Endpoint Mass

• Beam is not massless, so its mass affects its motion and natural frequency. It can be shown that for a uniform cross-section beam:

– (end mass)– (beam mass)

• mass of weight, accelerometer, pin, nut– Weight them together on analytical balance (uncertainty = 0.1 g)

MEBeam Mass MB

LT MT

LB LEClampMW

Intermediate Mass,

– How to find uncertainty in MEQ? – Power Product or Linear Sum?

– Power product or linear sum?

– Power product or linear sum?

– Power product or linear sum?

Uncertainty

Beam Equivalent Spring Constant, KEQ

• From Solid Mechanics:

– E = Elastic modulus measured in Lab 5

– Power product or linear sum?

F

d

LB

Predicted Frequencies• Undamped

– – Power Product?

• Damped

– Power product?– If , then , and

Table 1 Measured and Calculated 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

Table 2 Calculated Values and Uncertainties

• The equivalent mass is not strongly affected by the intermediate mass

• The predicted undamped and damped frequencies, fOP and fP, are essentially the same (frequency is unaffected by damping).

• The confidence interval for the predicted damped frequency fP = 9.0 ± 0.2 Hz does not include the measure value fM = 8.70 ± 0.05 Hz.

Units Value 3s Uncertainty

Equivalent Mass, MEQ [kg] 0.7631 0.0005Equivalent Beam Spring

Constant, kEQ[N/m] 2445 124

Predicted Undamped Frequency, foP

[Hz] 9.0 0.2

Measured Damped Frequency, fM [Hz] 8.70 0.05

Decay Constant, b1 [1/sec] -0.292 -Damping Coefficient, lM [Ns/m] 0.45 0.00Damped Frequency, fp [Hz] 9.0 0.2Percent Difference

(fP/fM-1)*100%   3.5% -

Decay Constant, b2 [1/sec] -0.196 -Damping Coefficient, lM [Ns/m] 0.30 0.00Damped Frequency, fp [Hz] 9.0 0.2Percent Difference

(fP/fM-1)*100%   3.5% -

Midterm 2

• Average 67

Measurements and Uncertainties

• Lengths– W, T, wW, wT: Lab 4– LT, LE, LB: Ruler w = 1/16 inch

• Masses– MT Total beam mass– MW End components measured together– Uncertainty 0.1 g