Dynamic Response Of A Vibrating Structure To Sinusoidal Excitation
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Transcript of Dynamic Response Of A Vibrating Structure To Sinusoidal Excitation
Experiment #4Dynamic Response of a Vibrating Structure
to Sinusoidal Excitation
Objectives
Perform a standard vibration test to measure the frequency response of a structural system
Solve a differential equation describing the motion of a structure with one degree of freedom under sinusoidal excitation
Calculate the equivalent viscous damping coefficient (ζ) of a single degree of freedom structure
1.
2.
3.
Test Specimen and Test Setup
Load Cell
Specimen
Accelerometer
Exciter
Part I – Frequency Response
Forced Response Free Response
Mechanical Excitation Manual Excitation
Forced Response
Node First Mode – 10.1 Hz
First Resonance – displays one node
Forced Response
Second Mode – 68.8 Hz
Nodes
Second Resonance – displays two nodes
Third Mode – 115 Hz
Nodes
Forced Response
Third Resonance – displays three nodes
Forced Response
Force
Acceleration
Before 68.6 Hz Resonance is small
In phase
Forced Response
Force
Acceleration
Before 68.6 Hz Resonance is small
In phase
Forced Response
Force
Acceleration
At 68.6 Hz Resonance is large
90°phase shift
Forced Response
After 68.6 Hz Resonance
Force
Acceleration
is small
Back in phase
Magnitude Ratio vs. Frequency
30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.00.000E+00
5.000E-04
1.000E-03
1.500E-03
2.000E-03
2.500E-03
3.000E-03
Frequency (Hz)
Mag
nitu
de R
atio
(ft/l
b)
Experimental data indicates that there is a resonance ~ 68 Hz
Phase Angle vs. Frequency
30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0
-200.00
-150.00
-100.00
-50.00
0.00
50.00
Frequency (Hz)
Phas
e A
ngle
(°) Experimental data
indicates that there is a phase shift of 90° at
~68 Hz
Free Response
Acceleration
Decreasing acceleration and
decreasing displacement
Part II – Lumped Parameter Model
The diagram below describes the motion of our beam:
x
F = applied force by the exciter
X = beam displacement
Single DOF Spring-Mass-Damper System
The mathematical model we used to describe the motion of the beam was a Single DOF Spring-Mass-Damper System.
X(t) = displacement
F(t) = applied load
m = mass
k = spring constant
c = damping coefficient
The lumped parameter model can be modeled by a non-homogeneous differential equation:
𝑚�̈�+𝑐 �̇�+𝑘𝑥=𝐹 (𝑡)We developed two particular solutions to this DE:
𝜙= tan−1
−2𝜁 ( 𝜔𝜔𝑛 )1−( 𝜔𝜔𝑛 )
2
- Phase angle between forcing function and the displacement of the beam
𝑋𝐹0
=1
𝑘√[1−( 𝜔𝜔𝑛 )2]
2
+[2𝜁 ( 𝜔𝜔𝑛 )]2
- Magnitude ratio of displacement and applied force
Single DOF Spring-Mass-Damper System
Part III – Equivalent Viscous Damping Coefficient (ζ)
Three Methods for Finding ζ
Half-Power Method
Log Decrement
Method
Best Guess Method
Half-Power Method
𝜁 𝐻𝑃=𝑓 2− 𝑓 1
2 𝑓 𝑛
𝜁 𝐻𝑃=69.1142−68.5295
2∗68.9
𝜻𝑯𝑷=𝟎 .𝟎𝟎𝟒𝟐𝟒𝟗
The half-power method utilizes frequencies on either side of the natural frequency along with the natural frequency to approximate the viscous damping coefficient (ζ).
Half-Power Method
68.3 68.5 68.7 68.9 69.1 69.3 69.50
0.0005
0.001
0.0015
0.002
0.0025
0.003
68.5381, 0.001934307 69.1003, 0.001934307
Frequency (Hz)
Magnitude Ra-tio (ft/lb)
70.7%
Half-Power Points
Resonance
𝑋𝐹0
Log Decrement Method
𝛿= 1𝑛 ln ( 𝑥0
𝑥𝑛 )
The log decrement method utilizes frequencies at different points along the Free Response result in Part I.
- This is the log decrement
The log decrement is then used to find the viscous damping coefficient (ζ):
𝜁 𝐿𝐷=1
√1+( 2𝜋𝛿 )
2𝜁 𝐿𝐷=
1
√1+( 2𝜋0.07597 )
2
𝜻𝑳𝑫=𝟎 .𝟎𝟏𝟐𝟎𝟗
Best Guess Method
The Best Guess Method involved simply picking a value for ζ and then plotting the theoretical curves alongside the experimental data. The correct value of ζ is found when the theoretical curves match the experimental data.
𝜻𝑩𝑮=𝟎 .𝟎𝟏
Comparison of HP and LD ζ Values
Differential error analysis shows that:
Therefore, we conclude that the Log Decrement Method is a much more accurate method of calculating the viscous damping coefficient (ζ).
Frequency Response Function Curves - Magnitude
34.0 44.0 54.0 64.0 74.0 84.0 94.0 104.00
0.0005
0.001
0.0015
0.002
0.0025
0.003
Magni-tude RatioBest GuessHalf PowerLog Decrement
Frequency (Hz)
Mag
nitu
de R
atio
All curves agree as to the location of the resonant frequency
The value of ζ affects both the height of the curve and the
slope leading up to the resonance
68.6
Magnitude Ratio vs. Frequency, ω
35 45 55 65 75 85 95 105-10
10
30
50
70
90
110
130
150
170
190
Phase ShiftBest GuessHalf PowerLog Decrement
Frequency (Hz)
Phas
e Sh
ift (°
)Frequency Response Function Curves –
Phase Angle (φ)
FRF curves don’t correlate well with the
experimental phase shift in this region
All curves indicate that there is a
phase shift of ~90° at 68.6 Hz
Phase Shift, φ vs. Frequency, ω
68.6
Conclusions
We successfully performed experiments to find both the forced and free response of our test specimen.
The magnitude ratio curves produced by our model correlate very well with our experimental data.
Our theoretical phase angle curves fit the experimental data well up until the resonance, and show the expected phase shift.
Recommendations
I recommend that there be a more permanent test setup for the frequency response experiments in such a way that it can’t be altered between lab sessions.