36601

© 2012 The MITRE Corporation. All rights reserved. Approved for Public Release: 12-00003. Distribution Unlimited

MODAL, HARMONIC, AND RANDOM VIBRATION ANALYSIS

TECHNIQUES FOR CONDUCTING AND VERIFYING FINITE

ELEMENT ANALYSIS

Dr. Anthony A. DiCarlo R. Paul Normandy

[email protected] [email protected]

mailto:[email protected]

mailto:[email protected]


About MITRE

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– Department of Defense (National Security Engineering Center)

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■ MITRE has 7,600 scientists, engineers and support specialists


■ Improper analysis can lead to the amplification of loads perhaps resulting in the failure of the electronic equipment’s ability to function properly or at all.

■ Most designs are too complicated to analyze with hand calculations and simplifying them may lead to a biased solution that conceals other ill-fated phenomena.

■ The finite element (FE) method, which can be implemented through computerized software, provides an efficient solution while retaining most of the design details.

■ This presentation works through several finite element techniques starting with a single degree of freedom model and progressing to a pseudo-continuous model. Throughout these examples, theoretical checks of the FE simulations are presented.

Introduction


Two Simple Oscillators – Fixed Base Modal Analysis

m 1

k 1

m 2

k 2

x

y

z 0 1

0 2

0

1

0

1 2

3 4

Node

Numbers

0

Y 1 Y 2

Variable Name Value

m1

m2

k1, k2

Mass 1

Mass 2

Spring Constant

2.085 lbf-s2/in

0.1303 lbf-s2/in

18500 lbf/in

nnn mkf /

2

1

nn mY /1

Resonant

Frequency (Hz)

Max. Modal Disp.

Mode Location Calculated FEA Calculated FEA

1 Y1 (Node 3) 15 14.99 0.693 0.693

2 Y2 (Node 4) 60 59.97 2.77 2.77

2slbfin

Modal Analysis

ANSYS commercial software used for FEA model


Two Simple Oscillators – Fixed Base Harmonic Analysis

22 n

nf

gQY

2/1max Q

Y1 (Node 3) Y2 (Node 4)

Calculated FEA Calculated FEA

Resonant Frequency(Hz) 15 15.085 60 60.04

Maximum Displacement (in) 2.17 2.17 0.136 0.136

Maximum Acceleration (in/s2) 19304 19312 19304 19314

Finite Element Amplitude at Node 3

15 Hz.

2.17 in

15 Hz.,

19312.37 in/sec2

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

FE Amplitude of the Acceleration and Displacement

at Node 3

Finite Element Amplitude at Node 4

60 Hz.

0.136 in.

60 Hz.

19314.45 in/sec2

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

FE Amplitude of the Acceleration and Displacement

at Node 4

1% damping


Two Simple Oscillators –Fixed Base Harmonic Analysis

Finite Element: Real and Imaginary at Node 3

-2.25

-1.75

-1.25

-0.75

-0.25

0.25

0.75

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz.)

Am

pli

tud

e (

in.)

Imaginary

Real

Finite Element: Real and Imaginary at Node 4

-0.15

-0.1

-0.05

0

0.05

0.1

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz.)A

mp

litu

de (

in.)

Imaginary

Real

FE Results of the Real and Imaginary Components of the Displacement

Node 3 Node4


Two Simple Oscillators –Fixed Base PSD Analysis

QSfY onrmsn2

)(

2

)(

)(

n

rmsn

rmsn

YY

Y1 (Node 3) Y2 (Node 4)


Resonant Frequency (Hz) 15 14.99 60 59.97

Max. 1σ Displacement (in) 0.0149 0.0150 0.00187 0.00187

Max. 1σ Acceleration

(in/s2) 132.6 132.9 265.1 265.7

Displacement Response PSD

1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00

1 10 100

Frequency (Hz.)

Am

pli

tud

e (

in2/H

z.)

Node 3

Node 4

0.474e-3 in2/Hz.

0.185e-5 in2/Hz.

PSD Displacement Responses for Nodes 3 and 4

Illustration of the Maximum Displacements at

Nodes 3 and 4 via Square Root of the Integrated

PSD

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

0 20 40 60 80 100 120

Am

plitu

de (

in)

Frequency (Hz.)

Displacement Based on Response PSD

Node 3

Node 4

1.87e-3 in.

0.0150 in.

2100

0

dYY RPSD

[1] Crandall

So = 0.0001 g2/Hz


Two Simple Oscillators Attached to a Large Mass Modal Analysis

2slbfin

Variable Name Value

m1

m2

m3

k1, k2

Mass 1

Mass 2

Mass 3

Spring Constant

2.085 lbf-s2/in

0.1303 lbf-s2/in

2215 lbf-s2/in

18500 lbf/in

x

y

z 0

10 2

1

m 3

1

rigid

m 1

m

2

k 2

1

2

3 4

Node Y

1 Y

2

Y 3

10 20

k 1

3

2

1

3

2

1

1221

22

11

3

2

1

3

2

1

)(

)(

)(

)(

0

0

00

00

00

tF

tF

tF

Y

Y

Y

kkkk

kk

kk

Y

Y

Y

m

m

m

KM

02

YI

M

K

97.59

00.15

0

2

1

3

2

1

3

2

1

f

f

fslbf

in

0

77.2

0

,

0

0

692.0

,

021.0

021.0

021.0

Resonant Frequency (Hz) Max. Modal Disp.


1 All 0 0.335E-05 0.021 0.021

2 Node 3 15 14.99 0.692 0.692

3 Node 4 60 59.97 2.77 2.77


Two Simple Oscillators Attached to a Large Mass Modal Shapes

x

y

z 0

10 2

1

m 3

1

rigid

m 1

m

2

k 2

1

2

3 4

Node Y

1 Y

2

Y 3

10 20

k 1

Zero Value of Natural

Frequency (Rigid Body

Mode)

15 Hz Natural Frequency

slbf

in

0

77.2

0

,

0

0

692.0

,

021.0

021.0

021.0

60 Hz Natural Frequency


Two Simple Oscillators Attached to a Large Mass Harmonic Analysis

x

y

z 0

1

2

1

m 3

15

rigid arms

m 1

m

2

k 2

1

2

3

4

Node Y

1 Y

2

Y 3

1

2

k 1

F= m 3 g Finite Element: Amplitude at Node 3

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

19294 in/sec2

15 Hz.

2.17 in

FE Displacement and Acceleration of Node 3 in

Response to Excitation Frequency

Finite Element: Amplitude at Node 4

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

60 Hz.

19293 in/sec2

60 Hz.

0.136 in

FE Displacement and Acceleration of Node 4 in

Response to Excitation Frequency


Two Simple Oscillators Attached to a Large Mass PSD Analysis

QSfY onrmsn2

)(

2

)(

)(

n

rmsn

rmsn

YY

Node 3 Node 4


Resonant Frequency (Hz) 15 14.99 60 59.97

Max. 1σ Displacement (in) 0.0149 0.0150 0.00187 0.00187

Max. 1σ Acceleration (in/s2) 132.6 138.01 265.1 266.0

FE Displacement PSD Response of Node 3 and 4 FE Acceleration PSD Response of Nodes 3 and 4


1.0E-11

1.0E-09

1.0E-07

1.0E-05

1.0E-03

1.0E-01

1.0E+01

1 10 100

Frequency (Hz.)

Am

pli

tud

e (

in2/H

z.)

Node 3

Node 4

0.472e-3 in2/Hz.

0.185e-5 in2/Hz.

Acceleration Response PSD

1.0E-06

1.0E-04

1.0E-02

1.0E+00

1.0E+02

1.0E+04

1.0E+06

1 10 100

Frequency (Hz.)

Am

pli

tud

e (

in/s

ec

2)2

/Hz.)

Node 3

Node 4

37297.4

(in/sec2)2/Hz.37221.59 (in/sec2)2/Hz.


Two Simple Oscillators Attached to Two Large Masses - Modal Analysis

02

YI

M

K

Variable Name Value

m1 Mass 1 2.085 lbf-s2/in


m3 Mass 3 2215 lbf-s2/in


k1, k2 Spring Constant 18500 lbf/in

x

y

z 0

1 0

m 3

m 1 m 2

k 2

1 2

3 4

Node

Y 1 Y 2

Y 3 m 4

Y 4

k 1

4

2

3

1

4

3

2

1

22

11

22

11

4

3

2

1

4

3

2

1

)(

)(

)(

)(

00

00

00

00

000

000

000

000

tF

tF

tF

tF

Y

Y

Y

Y

kk

kk

kk

kk

Y

Y

Y

Y

m

m

m

m

KM

Hz

f

f

f

f

60

0

15

0

2

1

4

3

2

1

4

3

2

1

,

0

/1

0

0

,

/1

0

/1

0

,

0

0

0

/1

,

0

/1

0

/1

2

42

42

1

31

31

m

mm

mm

m

mm

mm

Resonant Frequency

(Hz) Max. Modal Disp.


1 Node 2 & 4 0 0 0.0876 0.0876

2 Node 1 & 3 0 0.476 E -8 0.212 0.0219

3 Node 3 15 15 0.69 0.69

4 Node 4 60 60 2.77 2.77

2slbfin


Two Simple Oscillators Attached to Two Large Masses - Mode Shapes

Rigid Body Motion of Masses 2 and 4 Rigid Body Motion of Masses 1 and 3

15 Hz Natural Frequency 60 Hz Natural Frequency


Two Simple Oscillators Attached to Two Large Masses - Harmonic Analysis

x

y

z 0

1 0

m 3

m 1

m 2

k 2

1

2

3 4

Node Numbers

Y

1 Y 2

Y 3 m

4

Y 4

k 1

F3=m3g F4=m4g


1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

19303 in/sec2

15 Hz.

2.17 in

FE Displacement and Acceleration of Node 3


1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

60 Hz.

19301.55 in/sec2

60 Hz.

0.136 in

FE Displacement and Acceleration of Node 4


Two Simple Oscillators Attached to Two Large Masses - PSD Analysis


1.0E-11

1.0E-09

1.0E-07

1.0E-05

1.0E-03

1.0E-01

1.0E+01

1 10 100Frequency (Hz.)

Am

pli

tud

e (

in2/H

z.)

U3rpsd

U4rpsd

0.471E-3 in2/Hz.

0.184E-5 in2/Hz.

PSD Displacement Response for Node 3

(U3rpsd) and Node 4 (U4rpsd)

Acceleration Response PSD

1.0E-06

1.0E-04

1.0E-02

1.0E+00

1.0E+02

1.0E+04

1.0E+06

1 10 100

Frequency (Hz.)

Am

pli

tud

e (

in/s

ec

2)2

/Hz.)

A3rpsd

A4rpsd

37196 (in/sec2)2/Hz.37196 (in/sec2)2/Hz.

FE Acceleration PSD Response for Node 3

(A3rpsd) and Node 4 (A4rspd)


Simply Supported Beam Modal Analysis

Variable Name Value

W Weight per unit length 26 lbf/in

E Young’s Modulus 107 lbf/in2

I Area moment of Inertia 3267 in4

L Length 270 in.

x

y

z 0

Node Numbers

1 12 21

270 in

xmW

dx

xWdEI

dx

d 2

2

2

2

2

A

EInn

2 1, 2, 3 …= , 2 , 3 …

Hzf

f

60

15

2

1

2

1

2

1

Resonant Frequency

(Hz)

Max. Modal Disp.

Mode Calculated FEA Calculated FEA

1 15 15.01 0.332 0.332

2 60 60.05 0.332 0.332

2slbfin

Simply Supported Beam Mode Shapes

0.3320.332

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 30 60 90 120 150 180 210 240 270

Beam Length (in.)

Am

pli

tud

e (

lb.

sec

2/i

n2)

Mode Shape 1

Mode Shape 2

LxrmLxW r /sin/2 r = 1, 2 First and Second Calculated Mode Shapes for a

Simply Supported Beam


Simply Supported Beam Harmonic Analysis

x

y

z 0

Node Numbers

1 12 21

270 in

tmg

t

txym

x

txyEI sin

,,2

2

4

4

txXtxy sin,

2

sincossinhcosh

g

xaDxaCxaBxaAxX

4 2 / EIma

La

gLagD

gC

La

gLagB

gA

sin2

cos,

2,

sinh2

cosh,

2 2222

Simply Supported Beam: Harmonic Response

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Natural Frequency (Hz.)

Am

pli

tud

e (

in)

Calculated Harmonic Response

Finite Element: Amplitude at Node 12 (Mid Span)

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

24549 in/sec2

15 Hz.

2.76 in


Simply Supported Beam Harmonic and PSD Analyses


1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

24549 in/sec2

15 Hz.

2.76 in

323 2

24

1xLxL

EI

xQgmxy

2xyxy

Model Displacement (in)

Acceleration

(in.sec2)

Node Location Calculated FEA Calculated FEA

7 L/4 1.962 1.95 24460 24549

12 L/2 2.754 2.76 17430 17359

323 224

1xLxL

EI

xymxy

rmsrms

Mode

l

Displacement

(in)

Acceleration

(in/sec2)


12 L/2 0.019 0.019 167.9 169


LMM Applied to a Beam Modal Analysis

x

y

z 0

Node Numbers

1 12 21

2 270 in

x

y

z 0

1

12

21

270 in

22

Rigid Beam Large Mass

(a) Undeformed

(b) Deformed

Variable Name Value

W

E

I

L

M1

Weight per unit length

Young’s Modulus

Area moment of Inertia

Length

Large Mass

26 lbf/in

107 lbf/in2

32.67 in4

270 in.

18168 lbf s2/in

Mode

Resonant Frequency

(Hz)

Max. Modal Disp.


1 0 0 0.007415 0.007415

2 0 0.252E-4 0.235 0.234612

3 15 15 0.333 0.331338

4 60 59.77 0.332 0.330239

2slbfin

Third FE Mode Shape Fourth FE Mode


LMM Applied to a Beam Harmonic and PSD Analysis


1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

24492 in/sec2

15 Hz.

2.76 in

Model Displacement (in) Acceleration

(in/sec2)


7 L/4 1.962 1.95 24460 24492

12 L/2 2.754 2.76 17430 17315


1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00


Am

pli

tud

e (

in2/H

z.) 7.61E-4 in2/Hz.

Acceleration PSD Response

1.0E-04

1.0E-02

1.0E+00

1.0E+02

1.0E+04

1.0E+06


Am

pli

tud

e (

in2/H

z.)

60018 (in/sec2)2/Hz.

FE PSD Displacement Response at Mid-Span

of the Beam’s Length

FE PSD Acceleration Response at Mid-Span


Beam with End Masses Mode Shape

0.332

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 30 60 90 120 150 180 210 240 270

Beam Length (in.)

Am

pli

tud

e (

lb.

sec

2/i

n2)

Mode Shape 1

Flexible Beam with Masses Pinned at the Ends Modal Analysis

x

y

z 0

Node Numbers 1 12 21 2

270 in

Variable Name Value

W Weight per unit length 26 lbf/in

E Young’s Modulus 107 lbf/in2

I Area moment of Inertia 3267 in4

L Length 270 in.

mend End Mass 9084 lbf s2/in

mb Beam Mass w L/g

bendb

endbn

mmmL

mmEIf

188.02 3

endsb

bb

b mm

mm

L

x

mxY

2

/22sin

2

Resonant Frequency (Hz) Max. Modal Disp.

Mode Calculated FEA Calculated FEA

1 0 0 0.01049 0.01047

2 0 5.634E-7 0.01049 0.01047

3 0 6.446E-6 0.24194 0.23612

4 15 15.004 0.33145 0.33135

5 Not Calc. 59.799 Not Calc. 0.33032

[3] Steinberg

2slbfin


Flexible Beam with Masses Pinned at the Ends Harmonic Analysis


1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz.)

Am

pli

tud

e

Displacement

Acceleration

15 Hz.

24485 in/sec2

15 Hz.

2.76 in

FE Displacement and Acceleration at Mid-Span

Finite Element: Real and Imaginary at Node 12 (Mid Span)

-1.2

-0.7

-0.2

0.3

0.8

1.3

1.8

2.3

2.8

3.3

0 10 20 30 40 50 60 70 80 90 100

Frequency (Hz.)

Am

pli

tud

e (

in.)

IMAGINARY

REAL

FE Real and Imaginary Displacement

Components at Mid-Span


Flexible Beam with Masses Pinned at the Ends PSD Analysis


1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00


Am

pli

tud

e (

in2/H

z.) 7.61 E-4 in2/Hz.

The FE PSD Displacement Responses at Mid and Quarter-

Span of the Beam’s Length

Displacement Based on Response PSD

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

1.8E-02

2.0E-02

0 20 40 60 80 100 120

Frequency (Hz.)

Am

pli

tud

e (

in)

0.0190 in.

2100

0

dYY RPSD

Integration of the PSD Displacement Response Produces

the Maximum Displacement for the Beam

Results of the Hand Calculations and ANSYS Simulations of the Beam with End Masses

Model Displacement (in) Acceleration (in/sec2)


12 L/2 0.019 0.019 168.7 175

QSfy onrms2

)(

323 224

1xLxL

EI

xymxy

rms

rms 2rmsrms xyxy


■ A complete work-through starting with simple oscillators and ending with pseudo-continuous modeling has been presented.

■ The models were designed such that they produce responses in agreement for equivalent excitations.

■ FE programs with an outline for simulating and verifying modal, harmonic, and random vibration models

■ Closed form solutions are available and should be implemented to check FEA

Concluding Remarks


1. S. S. Rao, Mechanical Vibrations, 3rd ed., Addison-Wesley, Reading, Massachusetts, 1995.

2. S. H. Crandall, Random Vibration, MIT Press and John Wiley & Sons, New York, 1958.

3. Dave S. Steinberg, Vibration Analysis for Electronic Equipment, John Wiley & Sons, 1973.

References

36601

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