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Figure 1 VI Front Panel
Citation preview
MECH 322 Instrumentation
Lab 10 Damped Vibration of a Weighted Cantilever Beam
Performed: 4/1/15
Group 0Miles Greiner
Lab Instructors:Marissa Tsugawa
ABSTRACT• The goal of this experiment was to measure the
dependence of oscillatory frequency of a weighted cantilever beam on the beam length, and compare it with predictions based on measured beam dimensions, elastic modulus and masses.
• The oscillations were measured using an accelerometer and computer data acquisition system.
• The frequency decreases as the beam length increased. The predicted frequencies were roughly 15% larger than the measured values, and their confidence intervals did not overlap.
• The damping rate was not constant during the experiment, and had little effect on the beam frequency.
Figure 1 VI Front Panel
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
Table 1 Measured Steel 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 5.• LT, and LE were measured using a tape measure
– readability = 1/16 in• MT and MW were measured using a analytical scale
– readability = 0.1 g
Units Value3 s
UncertaintyElastic Modulus, E GPa 206.3 7.5
Beam Width, W inch 0.9968 0.0015Beam Thickness, T inch 0.1220 0.0015
Beam Total Length, LT inch 23.813 0.063End Length, LE inch 0.469 0.063
Beam Mass, MT g 377.2 0.1Combined Mass, MW g 333.9 0.1
• Acceleration versus time measurements were acquired at a sampling rate of 600 Hz for two beam lengths, LB = 7 and 13 inches.
• The sampling rate was sufficiently high so the peaks decreased monotonically
• The longer beam clearly had fewer peaks and a lower oscillatory frequency.
Figure 3 Acceleration versus Time
Figure 4 Oscillatory Amplitude Versus Frequency
• The sampling time and frequency were T1 = 10 sec and fS = 600 Hz, so the system is capable of detecting frequencies between 0.1 and 300 Hz, with a resolution of 0.1 Hz.
• For beam lengths of LB = 13 and 7 inches, the peak frequencies are, respectively, fM = 7.50 ± 0.1 and 18.60 ± 0.1 Hz. – These frequencies are easily detected from this plot.
Fig. 5 Peak Acceleration versus Time
• The average exponential decay constants for the beam lengths of LB = 13 and 7 inches, are b = -0.176 and -0.192, respectively
• The magnitude of these constants (slope of the curves) decreased slightly with time.
Table 2 Calculated Values and Uncertainties
• The intermediate mass is small compared to the equivalent mass. • For both beam lengths, the damping is sufficiently low so that the predicted
undamped and damped frequencies, fOP and fP, are nearly the same
• The predicted damped frequencies are roughly 15% higher than the measured values, and their confidence intervals do not overlap.
Value 3s Uncertainty Value 3s
Uncertainty
Beam Length, LB [m] 0.1778 0.0016 0.3302 0.0016Intermedate Mass, MI [kg] 0.033 0.001 0.055 0.001
Equivalent Mass, MEQ [kg] 0.367 0.001 0.389 0.001Equivalent Beam Spring
Constant, kEQ[N/m] 6912 402 1079 58
Predicted Undamped Frequency, foP
[Hz] 21.8 0.6 8.4 0.2Measured Damped
Frequency, fM [Hz] 18.60 0.05 7.50 0.05
Decay Constant, b [1/sec] -0.192 - -0.176 -Damping Coefficient, lM [Ns/m] 0.1409 - 0.1368 -
Damped Frequency, fp [Hz] 21.8 0.6 8.4 0.2Percent Difference
(fP/fM-1)*100% 17% - 12% -
LB = 7 inch LB = 13 inch
Units
Extra (not part of report)
Warning: Be careful to check your data before processing
• For example, see oscillations between 2 and 4 seconds
Very repeatable frequency for same beam length (7 and 15 inch)
• 7 inch
Class Summary 2006Measure versus Predicted Frequency
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35 40
fM [Hz]
f P [H
z]
Damping Coefficient
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50
fP [Hz]
CM
[Ns/
m]