Vibrating Beam Analysis
Team #77: Tristan DeRycke 50%, Eusha Rahman 50%
11/23/2025
Theory & Background
Results
Curve Fitting
In excel, curve fitting is done through minimizing a certain value, in our case the sum of the residuals, by only changing a certain
number of other values, in our case it was A, b, t0, ω, yss. The result of this process are shown below in Figure 3.
Damped Frequencies
In curve-fitting for the tip displacement graph uses Equation 1
Equation 1
𝑦 𝑡 = 𝐴𝑒
(
)
cos 𝜔 𝑡 − 𝑡
+𝑦
𝐴 = 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒
𝑏 = Damping coefficient
𝑡 = 𝑇𝑖𝑚𝑒
𝜔 = 𝑃𝑒𝑟𝑖𝑜𝑑
𝑦 = 𝑆𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒
The amplitude is the starting height form the first peak measured from the equilibrium position, 𝑦 . The damping coefficient
controls how quickly the oscillations die out. A larger damping coefficient means that the system loses energy faster and comes to rest
more quickly. The damping coefficient is estimated from an exponential decay curve to the peak values of the displacement vs. time
graph. The period is the angular frequency of the damped oscillation. The equation of period is shown in Equation 2, where Td is the
time between two consecutive peaks.
Table 1
Aliasing
We discovered that aliasing occurred when the
sampling rate was not high enough, undersampling.
Aliasing made our frequency look wrong and our data
seem incorrect, see Figure 4. To counteract this, we
increased our sampling rate, in conjunction with the
low-pass filter in the signal conditioning circuit to
produce data and a sinusoidal frequency, Figure 3.
Figure 3
Signal aliasing is a phenomenon where a high-frequency signal appears as a lower, false frequency when sampled at an insufficient rate.
The most direct method to combat aliasing is to increase the sampling rate. When using a data acquisition system that samples very
rapidly, you raise the frequency, making it less likely that any signal component will exceed it. For example, sampling with 500 samples
per second provides a more accurate graph compared to a graph with 10 samples per second.
Procedures
LabVIEW VI
The VI uses a DAQmx read function to read the waveform data that was output. The waveform data is then calibrated to our specific
sensor using a “waveform scale and offset” function. The waveform data was then split using “get waveform components”. The Y,
voltage, and dt, time, data arrays are combined into a single 2D array and written to a spread sheet using the “write delimited
spreadsheet” function. The waveform data is also routed to a power spectrum function, where it is saved to a waveform graph to show
variance in frequencies.
Calibration
To calibrate our data set properly, we took 3 different data sets with masses 50g, 150g, and 250g. For each of these trials, the tip height,
of the beam, was measured and the tip displacement was calculated based on the measured voltage values we received from the VI
data. Figure 2 shows the trendline which was used to get the offset and scale for our waveform data.
Acquiring Data
Figure 2
To acquire our data, we would load and release the beam to create vibrations,
with either mass or no mass attached to the end of the beam. The setup is
depicted in Figure 1. The data is measured by the strain gauge mounted to the
beam and read by the myDAQ device. The signal conditioning circuit is a
Wheatstone bridge configuration, which includes; excitation, amplification, and
low-pass filtering.
Figure 4
Conclusions
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•
•
Figure 1
[mycourses]
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The values are consistent with the physical changes. A beam with a higher mass, when deflected, should have a higher value of
angular period compared to a beam without mass, as it should take more time for the oscillating deflection to complete a wave
cycle, making a higher frequency value, thus a lower period value.
The different values of frequency from both LabVIEW and the curve-fit results in Excel agree as the they are extremely precise to
each other
Strain gauges require a complex Wheatstone bridge circuit with a stable DC voltage to detect tiny resistance changes, whereas
regular accelerometers use a simpler constant current-powered integrated circuit that outputs a voltage signal directly proportional
to acceleration
While accelerometers are easier to install than strain gauges, they cannot measure static strain