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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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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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:

• 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.

3

*2

1

***2

*33 GRMS

Q

PSDfQ

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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.


PSD Levels

Frequency PSD Level

20 0.0200

50 6.1900

80 20.0000

120 20.0000

630 1.2000

1000 1.2000

2000 0.0200

The following PSD Levels were used for all trials:


First Geometry

6

Fixed Face on End of Bar

Bonded Connection

Input Bar1”x1”x10”

Output Bar0.5”x0.5”x10”

YX

Z


Results for the First Geometry

7

2 Bar ModelSolutions 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


Second Geometry

8


Bonded Connection


Output Bar1”x1”x10”

YX

Z


Results of the Second Geometry

2 Identical Bar ModelSolutions using Miles

Equation

Solutions using Random Vibration Analysis in ANSYS Workbench



Q 10.0 10.0

3 Sigma GRMS 476 N/A



1.11560 1.05190





Third Geometry

10


Bonded Connection


Output Bar0.5”x0.5”x10”

YX

Z


Results of the Third Geometry

11

2 Bar Model - Reverse Boundary ConditionsSolutions using Miles Equation

Solutions using Random Vibration Analysis in ANSYS Workbench



Q 10.0 10.0

3 Sigma GRMS 9 N/A



0.28172 0.38378





Further Testing with a Variable Two Bar System

12


Bonded Connection

h2

Output Bar10” long bar

Input Bar10” long bar

h1 = 1”

f2

f1

YX

Z


Results of Variable Two Bar System

13

h2/h1f1

(Hz)f2

(Hz)f_system

(Hz)Q

3 Sigma GRMS

Miles Eq. Stress(psi)

Miles Eq. Deflection

(in)

WB Stress(psi)

WB Deflection(in)

0.25 320 80 78 10 455 234040 1.299 237680 1.295

0.5 320 161 115 10 569 158220 0.805 212820 0.850

0.75 320 241 100 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


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.


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.


Geometry

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.

Y

X

Z

W = 1 lbFor all trials

For W2/W1 = 0.05, W = 0.05 lbsFor W2/W1 = 0.25, W = 0.25 lbsFor W2/W1 = 0.50, W = 0.50 lbs

Grounded Block

Input Block

Output Block


Boundary Conditions

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.

17

Y

X

Z


Connections

Spring connecting the input block to the grounded block kept at a constant stiffness of 10000 lbs/in.

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.

k1

k2

18

Y

X

Z


Applied Loads

PSD base excitation applied in +Z direction to fixed boundary condition for the Random Vibration analysis.

19

Acceleration in –Z direction applied in static analysis for the Miles Equation analysis.

Y

X

Z


Variables

Input Block Uncoupled

Output Block Uncoupled

Input and Output Block coupled

p1, k1 p2, k2

k2, P2

k1, P1

W1 W1W2

W2

d1 D1d2

D2

f1, 3 Sigma GRMSf1

f2, 3 Sigma GRMSf2

f system, 3 Sigma GRMSf_system

20

d, D = Displacementf = Frequencyk = Spring stiffnessp, P = Spring forceW = Weight


Equations

Ratio GRMS Sigma 3

Ratioion Amplificat ForceRatioion Amplificat Third

Ratio GRMS Sigma 3

Ratioion Amplificatnt DisplacemeRatioion Amplificat Third

GRMS Sigma 3

GRMS Sigma 3 Ratio GRMS Sigma 3

p2

P2 Ratioion Amplificat Force

d

DRatioion Amplificatnt Displaceme

W

W RatioWeight

f

fRatio Frequency

Force

ntDisplaceme

f

f

2

2

1

2

1

2

2

system


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.– 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.

• 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.– 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.


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.


Results of First Trial

24

0.000

0.500

1.000

1.500

2.000

2.500

3.000

0.00 0.50 1.00 1.50 2.00 2.50

D2/d2

f2/f1

Displacement Amplification Ratio

Miles Equation

Workbench Method

W2/W1 = 0.05q2/q1 = 1



25

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50

P2/p2

f2/f1

Force Amplification Ratios

Miles Equation

Workbench Method

W2/W1 = 0.05q2/q1 = 1.0



26

0.000

0.500

1.000

1.500

2.000

2.500

3.000

0.00 0.50 1.00 1.50 2.00 2.50

Amplification Ratio divided3 Sigma GRMS Ratio

f2/f1

Amplification Ratio divided by 3 Sigma GRMS ratio

Miles Equation - Force

Workbench Method - Force

W2/W1 = 0.05q2/q1 = 1.0


Results of Second Trial

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

1.800

0.00 0.50 1.00 1.50 2.00 2.50 3.00

D2/d2

f2/f1


Miles Equation

Workbench Method

W2/W1 = 0.25q2/q1 = 1



28

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

0.00 0.50 1.00 1.50 2.00 2.50 3.00

P2/p2

f2/f1


Miles Equation

Workbench Method

W2/W1 = 0.25q2/q1 = 1.0



29

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

1.600

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Amplification Ratio /3 Sigma GRMS

f2/f1

Amplification Ratio divided by 3 Sigma GRMS Ratio

Miles Equation - Force

Workbench Method - Force

W2/W1 = 0.25q2/q1 = 1.0


Results of Third Trial

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50

D2/d2

f2/f1


Miles Equation

Workbench Method

W2/W1 = 0.5q2/q1 = 1



31

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50

P2/p2

f2/f1


Miles Equation

Workbench Method

W2/W1 = 0.5q2/q1 = 1.0



32

0.000

0.200

0.400

0.600

0.800

1.000

1.200

1.400

0.00 0.50 1.00 1.50 2.00 2.50

Amplification Ratio/3 Sigma GRMS Ratio

f2/f1

Amplification Ratio divided by 3 Sigma GRMS ratio

Miles - Force

Workbench - Force

W2/W1 = 0.5q2/q1 = 1.0


Combined Results

0.500

1.000

1.500

2.000

2.500

3.000

0.00 0.50 1.00 1.50 2.00 2.50 3.00

D2/d2

f2/f1

Displacement Amplification Ratio at Different Weight Ratios

Miles Equation W2/W1=0.5

Workbench Method W2/W1=0.5





q2/q1 = 1


Combined Results

34

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00

P2/p2

f2/f1

Force Amplification Ratios at Different Weight Ratios

Miles Equation W2/W1 = 0.05

Workbench Method W2/W1 = 0.05





q2/q1 = 1


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.


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 post-processing for the random vibration analysis.


Geometry

37

Bracket

Housing

Clamps

Y X

Z

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.


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.


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.


Modal Analysis – Deformation Plot


Miles Equation

42

• 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).


Miles Equation – Directional Deformation Plot – Y Direction


Miles Equation –Equivalent Stress Plot


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.


Workbench Method – Directional Deformation Plot – Y Direction


Workbench Method – Equivalent Stress Plot


Results

MeasurementsMiles Equation

ResultsWorkbench Method

Results

Ratio of Workbench Method to Miles

Equation

DirectionalDeformation in Y

0.0156 in. 0.035404 in. 2.269

Equivalent Stress 30,401 psi 50,112 psi 1.648

52

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.


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.


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.


Random Vibration Analysis Using Miles Equation and Workbench

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