# Vibrations

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Lab 10 – Vibrations
Objective
The objective of this lab is to measure the damped period and exponential decay for the purpose of determining
the damping ratio and natural frequency n of a thin, clamped-free beam with a mass at the free end. A
MATLAB program will be constructed to perform all data analysis.
Equipment
Beam/accelerometer assembly Calipers DAQ Provided MATLAB files
Electronic balance 30 cm rule LabVIEW
Theory (Only include numbered equations in the theory section of lab report, not all of this theory!)
Free vibrations occur when an oscillator is given an initial displacement or impulse, and then allowed to vibrate
in the absence of any external forces. Such an oscillator will then vibrate at (one of) its natural frequencies.
In any system with resistance due to internal molecular friction or viscous forces, there will be damping of the
vibrations due to energy loss. This has a very small but measurable effect on the period in most cases.
In this lab, we consider a clamped-free beam with a mass at the end.
mA = mass of mass/accelerometer assembly (kg)
mB = mass of beam (kg)
L = length of beam (m)
d = thickness of beam (m)
b = width of beam (m)
Known data:
ρ = density of beam = 2780 kg/m3
E = elastic modulus of beam = 72.5 GPa
To be calculate using MATLAB file provided:
I = area moment of inertia of beam (m4
)
=

3
12 (1)
= (2)
Fixed End
Beam
Mass and
accelerometer
assembly
MAE 3005 Measurements Lab Page 2 Lab 10
Vibrations
Equation of motion for damped oscillator:

′′() +

() + () = 0 (3)
where y = displacement of free end (m)
m = effected mass of beam (kg)
b = damping constant (Ns/m)
k = spring constant (N/m)
Using initial conditions
(0) = ,

(0) = 0
the solution to the equation of motion is
() =

cos( + ) (4)
where ωn = natural frequency (s-1
)
ζ = the damping ratio (dimensionless)
= tan−1 (

√1 −
2
)
Accelerometer
For a beam and mass assembly, the mass of the
beam is distributed uniformly. We can assume the
beam behaves like a point mass at the end of the
beam if we model the system using [1]
= +
33
140 (5)
The spring constant is then given by
=
3

3
and the natural frequency is given by
= √

The accelerometer will read a bias voltage V0 even when the beam is not vibrating as it will sense the
acceleration due to gravity. To calibrate the accelerometer, we must measure and then subtract this voltage
from the accelerometer signal. This will leave just the signal due to the vibrations.
The sensitivity of the accelerometer gives 0.222 V for every 9.8 m/s2 of acceleration, meaning the equation for
converting voltage to acceleration is
=
9.8
0.222
× ( − 0) (6)
MAE 3005 Measurements Lab Page 3 Lab 10
Analysing Data
Data analysis is to be completed using a MATLAB program which will be constructed during the lab.
The period T can be found from zero crossings
T is found using the average of the periods
measured.
The damped frequency can then be calculated using
= √1 −
2 =
2

(7)
The exponential decay can be found using a fit to the maximum values for each oscillation.
Fit the maximum points to the function
() =
− (8)
to find
= (9)
Using the T and a found in the analysis, solving simultaneously, we get
= √
1
1 + (
2

)
2
and =

(10)
0.2 0.4 0.6 0.8 1.0
5
5
10
0.2 0.4 0.6 0.8 1.0
5
5
10
MAE 3005 Measurements Lab Page 4 Lab 10
Procedure
Part 0 – Provided MATLAB code
Create a folder on the desktop, this is where ALL files for this lab need to be stored.
Some files may download as “tcross(1).m” etc…make sure you erase the (1), or the program will not run.
Part 1 – Dimensions and masses
1. Remove the mass/accelerometer assembly from the beam. You may need to use a screwdriver to gently
pry both halves open. Do not remove the accelerometer chip from the assembly.
2. Measure the mass of the mass/accelerometer assembly mA.
3. Measure the length of the beam (from the edge of the clamp…do not remove beam from the clamp).
4. Measure the thickness d and width b of the beam.
5. Put the equipment back together.
6. Enter the data with uncertainties into the appropriate section of the MATLAB file.
Part 2 – Setting up the equipment
7. Connect the 2-axis accelerometer to the DAQ using the instructions below and the wiring diagram
provided on the following page.
Vin can be supplied using the 5 V output on the Digital side of the DAQ.
Connect the ground pin to both the ground and ai0- port on the DAQ (black connector with two pins).
Connect the Y output to the ai0+ port on the DAQ.
Be very careful not to bend the pins on the chip when inserting the leads.
Mass and accelerometer
assembly
Beam
MAE 3005 Measurements Lab Page 5 Lab 10
Part 3 – Create the VI
8. Create a VI that will
a. receive an analog voltage signal.
b. take measurements for 5 s at 1000 Hz (this will be modified later).
c. convert voltage into an acceleration (insert formula box, but leave as just y = x to begin with).
d. plot data in a chart.
e. export data (time and voltage columns) to a file in the same folder as your VI and MATLAB file.
Make sure that, in the Write to Measurement File box, you have selected:
no segment headers, and one time column only.
Part 4 – Calibration
9. With the beam as still as possible (but at its rest position), take a 5 s measurement of the voltage from
the accelerometer.
10. Import the calibration data file to the MATLAB program and find the average voltage V0 from the
accelerometer by running just the relevant section (use “Run Section” rather than “Run”).
11. Insert the equation given on p. 2 into the formula box to complete the calibration of the accelerometer.
Part 5 – Vibration measurements
12. Take 5 sets of 5 s measurements at 2000 Hz (you will need to modify the DAQ in LabVIEW).
Use initial deflections of 0.5-1 cm. Larger deflections will create higher order vibrations and/or saturate
the accelerometer.
Part 6 – Data analysis program
13. Import the vibration data file to the MATLAB program to begin data analysis.
Y axis X axis
Vin
X output
Ground
Y output
MAE 3005 Measurements Lab Page 6 Lab 10
Measurements, Calculations and Data Analysis
Using the MATLAB program provided, give the following values and associated uncertainties:
1. Exponential decay constant, C, extracted from the fitted decay curve.
2. Period and damped frequency for the beam.
3. Damping ratio and the natural frequency for the beam.
4. Theoretical natural frequency of the beam.
5. The percent error in your experimental result for ωn (using the theoretical value as the true value).
6. The percent difference between damped and natural frequencies.
Deliverables
A lab summary, including, nomenclature, required theory, summary of measurements of the beam, values
extracted from the data, MATLAB plot showing data, zero-crossing and fitted decay curve, calculated results
with uncertainties (I’m expecting the uncertainty equations for and in the theory!).
The measurements and calculations sections will be quite short.
Focus on the discussion and the conclusion.
Bibliography
1. Engineering Vibration, DJ Inman, 4th Ed (Pearson, 2014) – Textbook for MAE4150 –Vibrations

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