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VIBRATIONS EXPERIMENT
OBJECTIVES:1. Solve a second order non-homogenous differential equation describing the displacement of a specimen experiencing sinusoidal
excitation
2. Determine the frequency response of the specimen
3. Find viscous dampening coefficient
VIBRATIONS EXPERIMENT
OBJECTIVES:1. Solve a second order non-homogenous differential equation describing the displacement of a specimen experiencing sinusoidal
excitation
2. Determine the frequency response of the specimen
3. Find viscous dampening coefficient
VIBRATIONS EXPERIMENT
Testing equipment used:
AccelerometerForce Transducer
Signal ConditionerOscilloscopeComputer
Function Generator/ShakerSignal Amplifier
Data Acquisition:
Data Sampling:
Other Testing Equipment:
Data Acquisition:
Data Sampling:
Other Testing Equipment:
Strobe Light
Setup:
VIBRATIONS EXPERIMENT
Setup:
Testing equipment used:
Data Acquisition:
AccelerometerForce Transducer
Data Sampling:
Signal ConditionerOscilloscopeComputer
Other Testing Equipment:Function GeneratorSignal Amplifier
AccelerometerForce Transducer
Signal ConditionerOscilloscopeComputer
Signal Amplifier
Data Acquisition:
Data Sampling:
Other Testing Equipment:
TOPBASE
Accelerometer
Shaker
Sample
Signal c
onditi
oner
Data
loggin
g u
nit
Represents a physical connectionRepresents an electricalconnection
Strobe LightStrobe Light
Function Generator/Shaker
X
VIBRATIONS EXPERIMENT
Setup:
Preparing to test: Specimen Force Transducer Accelerometer Shaker Wiring:
Signal conditioner
Function generator
Signal amplifier
oscilloscope
Computer
VIBRATIONS EXPERIMENT
Objective 1:
Solving the Differential Equation:
222
21
1
nn
oF
Xk
( )m x c x kx f t
( )m x c x kx f t
Differential EquationSolver
VIBRATIONS EXPERIMENT( )m x c x kx f t
X
Data:
Having solved the differential equation, we then plotted displacement per unit forceAs a function of frequency.
222
21
1
nn
oF
Xk
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
Frequency Response:
VIBRATIONS EXPERIMENT( )m x c x kx f t
Data:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce
pe
r F
orc
e (
mft
/lb
f)
222
21
1
nn
oF
Xk
Displacement, Force, and Frequency
VIBRATIONS EXPERIMENT( )m x c x kx f t
X
Data:
Having solved the differential equation, we then plotted displacement per unit forceAs a function of frequency.
Frequency Response:
Phase angle and frequency
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Frequency (Hz)
Ph
as
e A
ng
le (
°)
( ) sinf t t
( )m x c x kx f t
VIBRATIONS EXPERIMENT( )m x c x kx f t
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Frequency (Hz)
Ph
ase A
ng
le (
°)
Phase Angle and Frequency
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
2030
40
50
60
70
80
90
100
110
-100
10
-10
-5
0
5
10
15
20
X frequency
Frequency Response
Y real X/F
Z I
mag
inar
y (X
/F)
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
2030
4050
6070
8090
100110
-100
10
-10
-5
0
5
10
15
20
X frequency
Frequency Response
Y real X/F
Z I
mag
inar
y (X
/F)
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
2030
4050
6070
8090
100110
-100
10
-10
-5
0
5
10
15
20
X frequency
Frequency Response
Y real X/F
Z I
mag
inar
y (X
/F)
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
2030
4050
6070
8090
100110
-100
10
-10
-5
0
5
10
15
20
X frequency
Frequency Response
Y real X/F
Z I
mag
inar
y (X
/F)
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
2030405060708090100110
-100
10
-10
-5
0
5
10
15
20
Frequency Response
X frequency
Y real X/F
Z I
mag
inar
y (X
/F)
VIBRATIONS EXPERIMENT( )m x c x kx f t
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
Mode 1: 64.7 Hz2 Nodes
Mode 2: 204 Hz3 NodesMode 3: 380 Hz4 Nodes
Mode shapes
Data: Frequency Response:
Mode 1: 64.7 Hz2 Nodes
Mode 2: 204 Hz2 NodesMode 3: 380 Hz4 Nodes
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Mode shapes
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Determining dampening coefficient Determining dampening coefficient:
Frequency Response Function Peak Region
0
2
4
6
8
10
12
14
16
18
20
55 60 65 70 75 80
Frequency (Hz)
Am
plit
ude
Rat
io (m
ft/lb
f)
f2f1
70.7% of Max Magnitude
Half power method
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
012365.07.64*2
1.647.65
212
Hz
HzHz
f
ff
n
(Note f2 and f1 are the Frequencies at 70.7% Of maximum magnitude)
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method Half power method
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 200 400 600 800 1000 1200
Time (ms)
Dis
tan
ce (
ft)
Log decrement method Log decrement method
0
2
1 2ln
1n
x
n x
Solving for zeta where x0 and xn are amplitudes
10 cycles (n) apart:
012305.0
012306.01
012306.02
2
Log decrement method
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method Half power method Log decrement method
Best guess
0
2
4
6
8
10
12
14
16
18
20
0 20 40 60 80 100 120
Log decrement method
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method Log decrement method
Best guess
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Angular Velocity (rad/s)
Ph
ase
An
gle
(°)
Data Best Guess Half Power Log Decrement
0
5
10
15
20
25
0 20 40 60 80 100 120
Frequency (Hz)
Am
plit
ud
e R
atio
(m
ft/lb
f)
Data Best Guess Half Power Log Decrement
Log decrement method
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method Log decrement method
Best guess
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Angular Velocity (rad/s)
Ph
ase
An
gle
(°)
Data Best Guess Half Power Log Decrement
0
5
10
15
20
25
0 20 40 60 80 100 120
Frequency (Hz)
Am
plit
ud
e R
atio
(m
ft/lb
f)
Data Best Guess Half Power Log Decrement
VIBRATIONS EXPERIMENT
0
5
10
15
20
25
0 20 40 60 80 100 120
Frequency (Hz)
Am
plitu
de R
ati
o (
mft
/lb
f)
Data Best Guess Half Power Log Decrement
VIBRATIONS EXPERIMENT
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Angular Velocity (rad/s)
Ph
ase A
ng
le (
°)
Data Best Guess Half Power Log Decrement
Log decrement method
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method Log decrement method
Best guess
0
20
40
60
80
100
120
140
160
180
200
0 20 40 60 80 100 120
Angular Velocity (rad/s)
Ph
ase
An
gle
(°)
Data Best Guess Half Power Log Decrement
0
5
10
15
20
25
0 20 40 60 80 100 120
Frequency (Hz)
Am
plit
ud
e R
atio
(m
ft/lb
f)
Data Best Guess Half Power Log Decrement
Log decrement method
Determining dampening coefficient Determining dampening coefficient:
VIBRATIONS EXPERIMENT
Mode shapes
Data: Frequency Response:
Half power method
Best guess Data Validity
Most data was soundThe specimen began to bounce around at frequencies below 55 Hz. We therefore ruled that data as invalid.
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 20 40 60 80 100 120
Frequency (Hz)
Dis
tan
ce p
er
Fo
rce (
mft
/lb
f)
Belo
w 5
5 H
z
VIBRATIONS EXPERIMENT
Frequency ResponseDomain Change
Error Analysis O-Scope Traces
Domain change
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
VIBRATIONS EXPERIMENT( )m x c x kx f t
Complete Frequency ResponseTrying to express the frequency response on one graph:
Change domains:
Vector with length X/F
At angle to Y Axis
At any given frequency , we have…..
Error Analysis
Finding the error in the dampening coefficient:
Error AnalysisDAMPENING CALCULATION ERROR: HALF POWER METHODWe calculated the quantitative errors for the half power method dampening coefficient below:
2 1
2 1
2 22 1
2 1
2
2
2
To simplify the equation, we first found the error in the numerator, a difference of frequencies.
65.7 64.1 1.6
( ) ( )
2 0.1 2 0.1414
We then found the err
n
f f
f
Let
R f f
R f f
Since
f f
R f
or in the deonminator.
2 2 64.7 129.4
2 2(1) 2n
n
Let
S f
S f
Error AnalysisDAMPENING CALCULATION ERROR: HALF POWER METHODWe calculated the quantitative errors for the half power method dampening coefficient below:
Since the dampening equation is a quotient of two smaller pieces, we found the relative error in the dampening.2 2 2 2
0.1414 20.08973
1.6 129.4
To find the absolute error in the dampening we multiplied by the value of the dampening.
(0.08973)0.012365 .00111
Therefor
R S
R S
e, the dampening coefficient by the half power method was found to be
=0.012365 0.00111
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0 1 2 3 4 5 6 7 8 9 10
Time (ms)
Fo
rce
(lbf)
-10
-8
-6
-4
-2
0
2
4
6
8
10
Dis
tan
ce (
pft
)
Force Acceleration
64.7Hz
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0 1 2 3 4 5 6 7 8 9 10
Time (ms)
Fo
rce
(lbf)
-10.000000
-8.000000
-6.000000
-4.000000
-2.000000
0.000000
2.000000
4.000000
6.000000
8.000000
10.000000
Dis
tan
ce (
nft
)
Force Acceleration
130 Hz
-0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
0 1 2 3 4 5 6 7 8 9 10
Time (ms)
Fo
rce
(lb
f)
-10.000000
-8.000000
-6.000000
-4.000000
-2.000000
0.000000
2.000000
4.000000
6.000000
8.000000
10.000000
Dis
tan
ce (
pft
)
Force Acceleration
630 Hz
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