Download Random Vibration Analysis Using Miles Equation and Workbench

Random Vibration Analysis Using
Miles Equation and ANSYS
Workbench
August 2010
Owen Stump
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1
Purpose
• The purpose of the following testing was to determine if
there was a significant difference in the results of a random
vibration problem using Miles Equation followed by a static
analysis in ANSYS or using the ANSYS random vibration
analysis.
• Additionally, it was desirable to see if either analysis
accounted for the octave rule.
– The Octave Rule: In a coupled system undergoing random
vibration, there is an amplification to the output of the system.
– The frequency range where the amplification exists is
dependent on the weight ratio, frequency ratio, and damping
ratio between the input and the output of the coupled system.
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2
Methods for Solving
The following methods were compared against each other to determine which is the
more accurate method to solve random vibration problems:
•
Miles Equation
– Perform modal analysis to find natural frequency of system.
– Use Miles Equation in order to find 3 Sigma GRMS for the system.
– Multiply 1.0 G by the 3 Sigma GRMS value and apply that product as an acceleration in desired
direction.
– Perform a static structural analysis to find deformations and stresses.
Miles Equation:
3 GRMS  3 *
Q
•

2
* Q * f * PSD
1
2 *
Random Vibration analysis in Workbench
– Perform modal analysis to find natural frequency of the system.
– Perform a random vibration analysis using the modal analysis as the initial condition
environment, with the PSD Base Excitation applied in the desired direction.
– Evaluate desired stresses and deformations at 3 sigma values.
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3
Two Bar System
• Initially, a two bar system was tested.
• Three geometries were used for the initial testing.
• Further testing was performed on eight geometries with a varying
output bar width.
• One end of the bar has a fixed boundary condition placed on one
face (Zero Displacement in X, Y, and Z).
• The same material (Structural Steel; E = 2.9E7 psi; Density =
0.28383 lbm/in^3) was used for all bars.
• Q was held constant at 10 by changing the constant damping ratio
for the random vibration analysis to 0.05.
• The acceleration used in the static analysis was applied in the –Z
direction.
• The PSD base excitation was applied in the +Z direction for the
random vibration analysis.
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4
PSD Levels
The following PSD Levels were used for all trials:
Frequency PSD Level
20
0.0200
50
6.1900
80
20.0000
120
20.0000
630
1.2000
1000
1.2000
2000
0.0200
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5
X
Z
Y
First Geometry
Fixed Face on
End of Bar
Input Bar
1”x1”x10”
Output Bar
0.5”x0.5”x10”
Bonded
Connection
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Results for the First Geometry
2 Bar Model
Solutions using
Miles Equation
Solutions using
Random Vibration
Analysis in ANSYS
Workbench
Frequency (Hz)
114
114
Abort PSD Level (G^2/Hz)
20
20
Q
10.0
10.0
3 Sigma GRMS
568
N/A
3 Sigma Maximum Equivalent Stress (psi)
212750
286430
3 Sigma Maximum Deformation in bending
direction (in)
0.80536
0.85146
CP Time for Modal Analysis (sec)
1603
1603
CP Time for Random Vibration Analysis (sec)
92
665
Total Run CP time (sec)
1694
2268
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7
X
Z
Y
Second Geometry
Fixed Face on
End of Bar
Input Bar
1”x1”x10”
Output Bar
1”x1”x10”
Bonded
Connection
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Results of the Second Geometry
2 Identical Bar Model
Solutions using Miles
Equation
Solutions using Random
Vibration Analysis in
ANSYS Workbench
Frequency (Hz)
80
80
Abort PSD Level (G^2/Hz)
20
20
Q
10.0
10.0
3 Sigma GRMS
476
N/A
3 Sigma Maximum Equivalent Stress (psi)
247540
203840
3 Sigma Maximum Deformation in bending
direction (in)
1.11560
1.05190
CP Time for Modal Analysis (sec)
2240
2240
CP Time for Random Vibration Analysis (sec)
151
1270
Total Run CP time (sec)
2392
3510
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9
Y
X
Z
Third Geometry
Input Bar
1”x1”x10”
Output Bar
0.5”x0.5”x10”
Bonded
Connection
Fixed Face on
End of Bar
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Results of the Third Geometry
2 Bar Model - Reverse Boundary Conditions
Solutions using
Miles Equation
Solutions using Random
Vibration Analysis in
ANSYS Workbench
Frequency (Hz)
21
21
Abort PSD Level (G^2/Hz)
0
0
Q
10.0
10.0
3 Sigma GRMS
9
N/A
3 Sigma Maximum Equivalent Stress (psi)
24151
95992
3 Sigma Maximum Deformation in bending
direction (in)
0.28172
0.38378
CP Time for Modal Analysis (sec)
1527
1527
CP Time for Random Vibration Analysis (sec)
90
675
Total Run CP time (sec)
1616
2202
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Further Testing with a Variable
Two Bar System
Y
X
Z
Fixed Face on
End of Bar
Input Bar
10” long bar
f2
h1 = 1”
Output Bar
10” long bar
Bonded
Connection
h2
f1
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Results of Variable Two Bar System
Q
3 Sigma
GRMS
Miles Eq. Stress
(psi)
Miles Eq.
Deflection
(in)
WB Stress
(psi)
WB Deflection
(in)
78
10
455
234040
1.299
237680
1.295
161
115
10
569
158220
0.805
212820
0.850
320
241
100
10
532
142470
0.879
140780
0.877
1
320
320
80
10
476
209150
1.115
172650
1.052
1.25
320
399
66
10
340
212150
1.135
187140
1.102
1.5
320
476
55
10
247
209840
1.136
191460
1.123
2
320
628
42
10
111
158040
0.871
155470
0.919
2.5
320
774
34
10
52
112420
0.627
117310
0.685
h2/h1
f1
(Hz)
f2
(Hz)
f_system
(Hz)
0.25
320
80
0.5
320
0.75
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Conclusions from the Two Bar System
• This initial testing was inconclusive, because too many
variables were uncontrolled.
• The results show that the Workbench method is clearly
different than the Miles Equation method, so there
should be one that is preferable to the other.
• The Miles Equation method is the quicker of the two
methods.
• Additional testing needed to be performed to gain a
better understanding of the differences between the
results of the two method of solving random vibration
problems in ANSYS.
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Mass and Spring System
• The advantage of using the mass spring
system in testing is that each variable (weight,
spring stiffness, and damping) can be easily
controlled.
• This allows the results of the testing to be
easily compared to previously created graphs
showing the effects of the octave rule at
various weight ratios and frequency ratios.
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Y
Z
Geometry
X
Output Block
W = 1 lb
For all trials
Grounded
Block
For W2/W1 = 0.05, W = 0.05 lbs
For W2/W1 = 0.25, W = 0.25 lbs
For W2/W1 = 0.50, W = 0.50 lbs
Input Block
Three 1” x 1” x 1” cubes located 10” apart along Z axis.
Density of the output block was altered in each trial to change the
weight ratio. Density of input block and grounded block set to 1 lb/in3.
Young’s Modulus of each block set to 1E7 psi.
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Y
Z
Boundary Conditions
X
Fixed Boundary
Condition (Zero
Displacement on X,
Y, and Z) on one face
on grounded block.
4 Springs connected to each corner of two sides (+Y side and +X side)
of input block and output block (8 springs total for each block). Each
Spring connected to ground 100” away from block and has a stiffness
of 100,000 lbs/in. Springs were added to prevent rotation and
translation in the X and Y directions for the two free blocks. They do
no prevent motion in the Z direction.
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Y
Z
Connections
X
k2
k1
The stiffness of the spring
connecting the input block to
the output block was changed
for each trial in order to
change the frequency ratio.
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Spring connecting the input
block to the grounded block
kept at a constant stiffness of
10000 lbs/in.
18
Y
Z
Applied Loads
X
Acceleration in –Z
direction applied in static
analysis for the Miles
Equation analysis.
PSD base excitation applied in +Z direction to
fixed boundary condition for the Random
Vibration analysis.
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Variables
d, D = Displacement
f = Frequency
k = Spring stiffness
p, P = Spring force
W = Weight
D2
f system,
3 Sigma GRMSf_system
W2
k2, P2
d1
f1,
3 Sigma GRMSf1
d2
W1
p1, k1
Input Block
Uncoupled
D1
W2
f2,
3 Sigma
GRMSf2
p2, k2
Output Block
Uncoupled
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W1
k1, P1
Input and Output
Block coupled
20
Equations
Frequency Ratio 
Weight Ratio 
f2
f1
W2
W1
Displacement Amplification Ratio 
D2
d2
P2
p2
3 Sigma GRM Sf system
Force Amplification Ratio 
3 Sigma GRM S Ratio 
3 Sigma GRM Sf 2
Third Amplification Ratio Displacement 
Third Amplification Ratio Force 
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Displacement Amplification Ratio
3 Sigma GRM S Ratio
Force Amplification Ratio
3 Sigma GRM S Ratio
21
Details of First Trial
•
•
•
•
Weight Ratio (W2/W1) = 0.05
q2/q1 = 1 (Assumed Value)
Q = 10 (Assumed Value)
Displacement amplification ratio is the ratio of the coupled response of the output block to
the uncoupled response of the output block.
–
–
•
Force amplification ratio is the ratio of the force at the spring of the coupled output block to
the force at the spring of the uncoupled output block.
–
–
•
This shows the increase in the response of the output block as a result of being connected to the input block
compared to the uncoupled output block response.
The coupled output block response was calculated by subtracting the input block’s displacement from the total
displacement of the output block.
For Random Vibration analysis in Workbench, force was calculated based on F=k*x for the coupled and uncoupled
analysis.
For Static analysis, force was found using the spring probe on the appropriate spring.
The third amplification ratio is the ratio of one of the previous two amplification ratios and a
3 Sigma GRMS ratio
–
–
–
The 3 Sigma GRMS ratio is a ratio between the 3 Sigma GRMS value of the coupled system and the 3 Sigma GRMS
value of the uncoupled output block.
This is an attempt to show a ratio that does not include the effects of the varying PSD Level, as well as the varying
frequency values.
This ratio shows the degree to which coupling amplifications to the two available measurements.
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Details of Second and Third Trials
•
•
•
•
•
Second Trial: Weight Ratio (W2/W1) = 0.25
Third Trial: Weight Ratio (W2/W1) = 0.50
q2/q1 = 1 (Assumed Value)
Q = 10 (Assumed Value)
The amplification ratios are calculated in the same manner as in the first
trial.
• The frequency ratios were not held constant from the first trial in order to
better capture the peak of the amplification curve for each trial.
• The values for W1, k1, and f1 were held constant for all trials.
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Results of First Trial
W2/W1 = 0.05
q2/q1 = 1
Displacement Amplification Ratio
3.000
Miles Equation
Workbench Method
2.500
2.000
D2/d2
1.500
1.000
0.500
0.000
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Results of First Trial
W2/W1 = 0.05
q2/q1 = 1.0
Force Amplification Ratios
3.00
Miles Equation
2.50
Workbench Method
2.00
P2/p2
1.50
1.00
0.50
0.00
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Results of First Trial
Amplification Ratio divided by 3 Sigma GRMS ratio
W2/W1 = 0.05
q2/q1 = 1.0
3.000
2.500
Miles Equation - Force
2.000
Amplification Ratio divided
3 Sigma GRMS Ratio
1.500
Workbench Method - Force
1.000
0.500
0.000
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Results of Second Trial
W2/W1 = 0.25
q2/q1 = 1
Displacement Amplification Ratio
1.800
1.600
1.400
1.200
1.000
D2/d2
0.800
0.600
Miles Equation
0.400
Workbench Method
0.200
0.000
0.00
0.50
1.00
1.50
2.00
2.50
3.00
f2/f1
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Results of Second Trial
W2/W1 = 0.25
q2/q1 = 1.0
Force Amplification Ratios
1.80
1.60
1.40
1.20
1.00
P2/p2
0.80
Miles Equation
0.60
Workbench Method
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
f2/f1
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Results of Second Trial
W2/W1 = 0.25
q2/q1 = 1.0
Amplification Ratio divided by 3 Sigma GRMS Ratio
1.600
1.400
1.200
1.000
Amplification Ratio /
3 Sigma GRMS
0.800
Miles Equation - Force
0.600
Workbench Method - Force
0.400
0.200
0.000
0.00
0.50
1.00
1.50
2.00
2.50
3.00
f2/f1
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Results of Third Trial
W2/W1 = 0.5
q2/q1 = 1
Displacement Amplification Ratio
1.60
1.40
1.20
1.00
D2/d2 0.80
Miles Equation
0.60
Workbench Method
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Results of Third Trial
W2/W1 = 0.5
q2/q1 = 1.0
Force Amplification Ratios
1.60
1.40
1.20
1.00
P2/p2 0.80
Miles Equation
0.60
Workbench Method
0.40
0.20
0.00
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Results of Third Trial
Amplification Ratio divided by 3 Sigma GRMS ratio
W2/W1 = 0.5
q2/q1 = 1.0
1.400
1.200
1.000
Amplification Ratio/
3 Sigma GRMS Ratio
0.800
0.600
Miles - Force
0.400
Workbench - Force
0.200
0.000
0.00
0.50
1.00
1.50
2.00
2.50
f2/f1
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Combined Results
Displacement Amplification Ratio
at Different Weight Ratios
q2/q1 = 1
3.000
Miles Equation W2/W1=0.5
Workbench Method W2/W1=0.5
2.500
Miles Equation W2/W1=0.05
Workbench Method W2/W1=0.05
2.000
Miles Equation W2/W1=0.25
D2/d2
Workbench Method W2/W1=0.25
1.500
1.000
0.500
0.00
0.50
1.00
1.50
2.00
2.50
3.00
f2/f1
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Combined Results
Force Amplification Ratios at Different Weight Ratios
3.00
q2/q1 = 1
Miles Equation W2/W1 = 0.05
Workbench Method W2/W1 = 0.05
2.50
Miles Equation W2/W1 = 0.25
Workbench Method W2/W1 = 0.25
2.00
Miles Equation W2/W1 = 0.5
Workbench Method W2/W1 = 0.5
P2/p2 1.50
1.00
0.50
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
f2/f1
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Observations from Mass Spring Testing
•
•
•
•
For each weight ratio, there is a larger amount of amplification due to the coupling of the masses
using the workbench random vibration analysis, rather than the method using Miles Equation and
static analysis.
Although not perfect, the amplification curves from the analysis method using the workbench
random vibration analysis resemble the curves on pages 152 and 153 of Vibration Analysis for
Electronic Equipment (Steinberg).
The amplification curves from the Miles Equation method do not resemble these curves.
Based on the third amplification ratio, the Miles Equation responses’ amplification is based entirely
on the variations to the 3 Sigma GRMS value, which is expected, but the workbench method retains
an amplification once the variations to the 3 Sigma GRMS value are taken out.
–
•
•
•
•
•
This would indicate that there are additional coupling effects added in the workbench method for solving random
vibration systems.
Outside of the Octave Rule range (about 0.5<f2/f1<2), the Miles Equation method and the
workbench method produce similar results.
The Octave Rule range did not shift as much as expected when the weight ratio was changed, based
on the graphs on pages 152 and 153 of Vibration Analysis for Electronic Equipment (Steinberg).
The amplification is changed when the weights are altered, even when the frequency ratio and the
weight ratio remain the same.
The difference between the amplifications between the two solution methods is greater when the
weight ratio is lower.
The Miles Equation force amplification ratio is equal to the 3 sigma GRMS ratio.
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Testing with Real Model
• A more complex model was run in ANSYS to see if the
observations made from the previous testing are
conserved.
• The results for the limiting area of the model will be
compared to determine if the Workbench method
produces results different to the Miles Equation
results.
• Equivalent Stress, Normal Stress, and Directional
Deformation will be used to compare the two methods
of analysis.
– These were chosen because they are the most meaningful
measurements that can be evaluated in the postprocessing for the random vibration analysis.
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36
Y
X
Geometry
Z
Clamps
Bracket
Housing
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6 Faces Fixed (Zero Displacement in X, Y, and Z) on
Housing. The 3 Faces not shown are the similar faces
on the opposite side of each housing part.
37
Details of testing
• A modal analysis was run to find the natural frequencies of the
model
• These natural frequencies were used to perform the random
vibration analysis in Workbench
– Random Vibration analysis was performed using the same PSD level
chart as in the previous experiments.
• The natural frequency of the bracket was used in the Miles
Equation analysis
– The same PSD levels were used for the Miles Equation as the previous
experiments.
– Miles Equation analysis included a static structural analysis with an
acceleration applied to the tool.
• Weight Ratio between bracket (W2) and the housing and clamps
(W1) is 0.11.
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Modal Analysis
• The modal analysis indicated the natural
frequency of the bracket is 493.34 Hz.
• 6 Faces on the housing parts were fixed (Zero
displacement in X, Y, and Z).
• The modal solution was limited to frequencies
between 10 Hz and 750 Hz in order to reduce
the time for solving the analysis.
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39
Modal Analysis – Deformation Plot
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40
Modal Analysis – Deformation Plot
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41
Miles Equation
• The Miles Equation was used to analyze the
model.
• Based on the 493 Hz found in the modal analysis
and the PSD Levels chart used in the previous
experiments, the 3 Sigma GRMS value was found
to be 356.
• A static structural analysis was performed on the
model with an applied acceleration of 137416
in/s^2 in the -X direction.
• 6 Faces on the housing were fixed (Zero
displacement in X, Y, and Z).
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42
Miles Equation – Directional Deformation
Plot – Y Direction
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43
Miles Equation – Directional Deformation
Plot – Y Direction
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44
Miles Equation –Equivalent Stress Plot
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45
Miles Equation –Equivalent Stress Plot
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46
Workbench Method
• Random Vibration analysis was performed in
Workbench using the PSD level chart used in the
previous experiments.
• PSD G acceleration applied in +X direction.
• Modal analysis results were used for the random
vibration analysis.
• 6 Faces on the housing were fixed (Zero
displacement in X, Y, and Z). These faces are
where the PSD base excitation is applied during
the analysis.
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47
Workbench Method – Directional
Deformation Plot – Y Direction
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48
Workbench Method – Directional
Deformation Plot – Y Direction
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49
Workbench Method – Equivalent
Stress Plot
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50
Workbench Method – Equivalent
Stress Plot
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51
Results
Measurements
Miles Equation
Results
Workbench Method
Results
Ratio of Workbench
Method to Miles
Equation
Directional
Deformation in Y
0.0156 in.
0.035404 in.
2.269
Equivalent Stress
30,401 psi
50,112 psi
1.648
Note: Equivalent Stress for Miles Equation is the Von Mises stress, and for the
Workbench Method it is a modified version of the Von Mises stress.
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52
Conclusions from Test with Real Model
• The workbench method of solution results in a more
conservative result than the Miles Equation solution.
• The difference seen in the ratio of the results for the
workbench method to the results of the Miles Equation
method is consistent with the expected difference resulting
from the octave rule.
• The workbench method was not able not solve the model
as it was originally constructed, although using the Miles
Equation method, the solution could be found.
– Multiple connections had to be changed slightly to allow the
model to solve successfully using the random vibration analysis.
– This could prove to be an issue when trying to solve more
complicated models.
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53
Overall Conclusions
• The workbench method is the more conservative method
to use when solving random vibration problems.
• Special considerations regarding which solving method is
appropriate to use, should be made for any system where
the PCB and chassis have a frequency ratio that is within
the Octave Rule region (0.5<f2/f1<2 for W2/W1 = 0.05).
• The results from the workbench method more closely
match the expected results based on the octave rule for the
mass spring system and the real model.
• Further testing could be performed to find the limitations
to the random vibration analysis in ANSYS.
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54