u“

'JP {1‘ has!

'“ ’- A»
. ,- ,3»

"1‘:

I .591?

 

 

 

 

 

 

'4 1.
*u. ..,. V

‘1' 531'?" :1!" 4' "

f;

1

J. 4
r: .
:guLn

fin. w. »

. . ‘x a ’ .
5-,. ,' » . ‘ , <

, .. _ 2;.“ '4' ‘

k
'5‘ {:7
'4‘
hwfifig‘

.u

. ‘ qlfixw L

.-. . - -

-. ‘ 1 2; ark 2‘?

FCdazfimfi”

agfgficg '23:;
"k ‘ 29E?“ .

‘93:.

..
”91:2":
1. t:
u:
N z z?
a:

.9 u

,-
u.‘ . r...,< .
‘:‘-,:.:s‘. Lu ~.
§A=su~$u W.
5’ ‘ — ~ r

t i . ..
zaiefi‘t‘g;

vex.
aw?“ '
K .
at!“
t.”

‘1‘

. v .
_. m .
~.,,,

“' 1.!-
3.3L!
'3,“ .- «Mr ‘
. » .
% "'-;~
~ 19"

Ki?
,3

v “115.3232: ‘
u a... twp
: “usfia

fly 31:,
M far-7“}: 31*}:
fit “*7-

 

a.
1“»..-
-~

‘A.
w-
~

3.
{Kc-1‘3“
- ’1‘...” ‘
- ”:5 5,;
-,. km.

‘9' w‘

1;. a?

«5
h a; , w:
iugvgfigfiv?

MICHIGAN STATE U NERSITY LIBRARIES

ltll(lilll'lillH/ll/lHill!llllll/lllfl/l/lUH

3 1293 01020 4968

 

 

 

 

ll

This is to certify that the

dissertation entitled

NOISE AND VIBRATION DESIGN MODELING
WITH STATISTICAL ENERGY ANALYSIS

presented by

XIANLI HUANG

has been accepted towards fulfillment
of the requirements for

 

 

Datew

MS U is an Affirmative Action/Equal Opportunity Institution

0-12771

 

 

 

LIBRARY
Mlchlgan State
University

 

 

PLACE ll RETURN DOXtommmbmmm yourrocord.
TO AVOID FINES Mum onorbdoroddoduo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU I. An Namath. Action/Equal 0990mm Induction
Wan-n1

_ . ,

NOISE AND VIBRATION DESIGN MODELING
WITH STATISTICAL ENERGY ANALYSIS

By
Xianli Huang

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1 994

ABSTRACT

NOISE AND VIBRATION DESIGN MODELING
WITH STATISTICAL ENERGY ANALYSIS

By

Xianli Huang

Statistical Energy Analysis (SEA) was initially developed for predicting sound
pressure and vibration levels in interior spaces and structures at audio frequencies. The
work presented here includes important results on four topics: internal loss factor
Identification, sound pressure sensitivity analysis, SEA response variance analysis and
SEA response distributions. Internal loss factors for each element are identified from
measured vibro-acoustic responses and the identified internal loss factors yield models
which accurately predict vibro-acoustic responses. An SEA response sensitivity algorithm
is formulated to relate variability of design parameters to SEA response variation. The
method can be used to sort the design parameters which have greatest effects on sound
pressure in acoustic spaces. The SEA response variance analysis relates response
variances directly to variances of design parameters through linear approximation to SEA
response variances. The SEA response distributions due to randomness of parameters are
shown to be approximately Gaussian. The Gaussian distributions with the SEA response
variance analysis provide a means to estimate the response confidence levels for existing
SEA models or develop specifications on design parameters given variance specifications
for sound and vibration performance. With these features SEA becomes a powerful tool

for engineering acoustic design.

Copyright

Copyright by

Xianli Huang

1994

Dedication

To my parents: H. Huang and W. Wang, to my wife: Xiaowei Li and to my

daughters: SiRui and Rachel.

iv

Acknowledgments

I would like to express my sincere appreciation to Professor Clark J. Radcliffe not
only for his inspiration provoking advice all the way through the dissertation, for his
training on me in professional faith, skills and insights of engineering, for his generous
support, but also for his personal advice as a mentor. I would like to thank Dr. Joseph
Wolf for his advice and help. I would like to thank Professor C. MacCluer for his advice
and insight on derivations of SEA response distributions.

I would like to thank my committee members, Professors R. Rosenberg, A.
Haddow and RM. FitzSimons, for their advice, education, comments and help during my
studies here.

I would like to thank my parents and sisters for their encouragement for my
overseas studies. I would like to thank my father and mother-in-law, Mr. Zhou Li and
Ms. L.M. Zheng, for taking care of my daughter in China when my wife and I were in The
United States. I would like to thank my wife and my daughter for their support.

I would like to thank my officemate, Sachin Gogate for his friendship and his help
over the years. I would like to thank my friends during my studies here. They include:
Lu-ping Chao, Chris Eusebi, Blane Hansknecht, Steve Connolly, Jon Iwamasa, Chang-b0
Chao, Jerome Palazzolo and Matt Brach.

Finally I gratefully acknowledge the technical assistance and support of Dr. Steve
Rohde, Dr. P. Oh and Dr. Joseph Wolf of the General Motors Systems Engineering
Center, especially for their discussions of statistical variances. I also gratefully

acknowledge the assistance and support of Dr. W. Resh of Chrysler Technology Center.

LIST OF TABLES ........................................................................................................... viii

LIST OF FIGURES ......................................................................................................... x
Chapter 1 Introduction ................................................................................................... 1
1.1 Statistical Energy Analysis (SEA) Overview ................................................ l
1.2 New Developments for SEA ......................................................................... 3
Chapter 2 Identification of Internal Loss Factors of SEA ............................................ 4
2.1 Introduction .................................................................................................... 4
2.2 SEA Parameters and Identification ................................................................ 4
2.3 Input Power and Optimal Internal Loss Factor Identification ....................... 6
2.4 Application Examples ................................................................................... 12
2.5 Summary ....................................................................................................... 24
Chapter 3 Noise and Vibration Design Sensitivity from SEA ...................................... 25
3.1 Introduction .................................................................................................... 25
3.2 Sound Pressure (SP) Sensitivity .................................................................... 26
3.3 Modal Energy Derivatives to Parameter Variation ....................................... 28
3.4 Application of SP Sensitivity to Damping ..................................................... 30
3.5 Additional Design Examples ......................................................................... 32
3.6 Conclusions .................................................................................................... 36
Chapter 4 Putting Statistics into SEA ............................................................................ 37
4.1 Introduction .................................................................................................... 37
4.2 SEA Response Variance Analysis ................................................................. 38
4.3 Examples and Monte Carlo Validation .......................................................... 39
4.4 Conclusions .................................................................................................... 45

vi

vii

Chapter 5 Probability Distribution of SEA Responses .................................................. 47
5.1 Introduction .................................................................................................... 47
5.2 Statistical Distribution of SEA Responses .................................................... 48
5.3 Distribution of Total Pressure and Velocity .................................................. 53
5.4 Monte Carlo Test ........................................................................................... 54
5.5 Hypothesis Test ............................................................................................. 57
5.6 Conclusions .................................................................................................... 63
Chapter 6 Summary and Conclusions ............................................................................ 64
6.1 Dissertation Summary ................................................................................... 64
6.2 Conclusions .................................................................................................... 65
APPENDIX A SEA EQUATIONS ................................................................................ 66
APPENDIX B GEO METRO MODEL DATA ............................................................. 68
B.1. GEO METRO Design Parameters ............................................................... 68
B.2. GEO SEA Model Connectors ...................................................................... 69
B.3. GEO METRO Test Results .......................................................................... 70
B.4. Model Predictions and Comparison ............................................................. 71

BIBLIOGRAPHY ............................................................................................................ 76

LIST OF TABLES

Table 2.1 Number of Independent SEA Tests versus Model Size shows that for

very small models it is impossible to identify all SEA parameters

because there are more parameters than tests .................................................. 5
Table 2.2 Identified and Optimized Damping Ratio and Reverberation Time ............... 21
Table 2.3 Comparisons of Predicted and Measured Total Energies (in dB) .................. 21
Table 3.1 Interior SP Sensitivities ................................................................................... 35
Table 5.1 lz-Test for Space 2 Total ms Sound Pressure ............................................... 59
Table 5.2 Hypothesis Test Result Summary for Spaces ................................................. 60
Table 5.3 12 -Test for Total ms Structural Velocity of the Plate ................................... 61
Table 5.4 X2 Test for Structural Velocity of The Plate .................................................. 62
Table A1 Geometric and Material Properties Used in Example 1 ................................. 67

Table B1 List of Design Parameters for The GEO SEA Model Body Structures .......... 68

Table B2 List of Design Parameters for The GEO SEA Model Acoustic Spaces ......... 69

Table B3 List of Connectors between panels and spaces for The GEO SEA Model ..... 69

Table B4 List of Connectors between panels for The GEO SEA Model ....................... 70

viii

ix
Table B5 List of Space to Space Connector for The GEO SEA Model ......................... 70

Table B6 List of Measured Acceleration and SPL of State 1 for GEO SEA Model ...... 70

Table B7 List of Measured Acceleration and SPL of Operating State 2 for The
GEO SEA Model ............................................................................................. 71

LIST OF FIGURES

Figure 1.1 Conceptual SEA Model ................................................................................. 1
Figure 1.2 3-Element SEA Model Schematic ................................................................. 1
Figure 2.1 Internal Loss Factor Optimal Identification Algorithm ................................ 9
Figure 2.2 SEA Automotive Vehicle Model .................................................................. 15
Figure 2.3 SEA Automotive Vehicle Model Element Coupling Chart .......................... 15

Figure 2.4 Identified and Computed Internal Loss Factors Using Computed

Reverberation Time for Under Car Volume ................................................. 16
Figure 2.5 Identified and Average Internal Loss Factors for Front of Dash .................. 17
Figure 2.6 Geo Metro Body Photo ................................................................................. 18
Figure 2.7 GEO METRO SEA Model Chart .................................................................. 19
Figure 2.8 Vibrator Mount Assembly Photo .................................................................. 19
Figure 2.9 Geo Model Non-Optimal/Optimal/Measured SPL in dB of State 1 ............. 22

Figure 2.10 Geo Model N on-optimal/Optimal/Measured ms Velocity in dB(re 1
m/s) ................................................................................................................ 22

xi

Figure 2.11 SEA Model Verification Procedure ............................................................. 23
Figure 2.12 GEO SEA Model Verification Plot .............................................................. 24
Figure 3.1 Modal Energy Sensitivities in Space 2 to Damping Parameters ................... 33
Figure 3.2 Sensitivity Rankings for Interior Modal Energies ......................................... 34

Figure 4.1 Plate Thickness Histogram and Corresponding Gaussian Probability
Density ........................................................................................................... 41

Figure 4.2 Gaussian Analytical Probability Density Function (PDF) Prediction for
Acoustic Energy in Space #2 Compared with Histogram and PDF for
2,000 Model Monte Carlo Test Results ......................................................... 42

Figure 4.3 Floor Panel Thickness Histogram and Probability Density Function ........... 43

Figure 4.4 Histogram and Gaussian Probability Densities from Analytical and
Monte Carlo Tests for Standard Deviation 5% of Floor Panel Thickness
at 5,000 Hz. .................................................................................................... 44

Figure 4.5 Comparison of Standard Deviations of Response Energy in the Vehicle

Interior at 5,000 Hz. from Analytical Prediction and Monte Carlo Test

due to Deviation in Floor Panel Thickness .................................................... 45
Figure 5.1 Lyapunov Condition Terms for SEA Responses .......................................... 55
Figure 5.2 Space 1 ms Sound Pressure @ 1,000 Hz. Histogram (X104P32) ................ 56
Figure 5.3 Space 1 Total ms Sound Pressure Histogram (X10'3 13212) ............................ 56

Figure 5.4 Plate ms Velocity @ 1,000 Hz. Histogram (X104mm2/Secz) ....................... 56

xii

Figure 5.5 Plate Total ms Velocity Histogram( “04 mmz / secz) .................................... 56
Figure 5.6 Space 2 ms Sound Pressure @ 1,000 Hz. Histogram (X 104 P32) ................. 57
Figure 5.7 Space 2 Total ms Sound Pressure Histogram ( X 1041’?) ............................. 57
Figure 5.8 Normal Cumulative Graph of Total ms Velocity of Plate ............................ 62
Figure B1 GEO Front of Dash Velocity Prediction and Measurement .......................... 71
Figure B2 GEO Engine Compartment SPL Prediction and Measurement ..................... 72
Figure B3 GEO Hood Velocity Prediction and Measurement ....................................... 72
Figure B4 GEO Under Car Volume SPL Prediction and Measurement ........................ 73
Figure B5 GEO Floor Panel Velocity Prediction and Measurement .............................. 73
Figure B6 GEO Over Car Volume SPL Prediction and Measurement .......................... 74
Figure B7 GEO Body Panel Velocity Prediction and Measurement .............................. 74

Figure B8 GEO Interior SPL Prediction and Measurement ........................................... 75

Chapter 1 Introduction

1.1 Statistical Energy Analysis (SEA) Overview

Separate Plate

 

    

In... 40$ $\\\

Figure 1,1 Conceptual SEA Model Figure 1. 2 3-Element SEA Model
Schematic

   

3
Pdlss1 Pdissz Pdlss3

SEA can be illustrated with a simple SEA model: two spaces are separated by an
aluminum plate (Figures 1.1 and 1.2). Stored energies are represented by El, Ezand 53 in
the element blocks. Power flows between the elements are shown by the arrows labeled
P12, P13, and P23 where the subscripts refer to the elements connected. The model
includes one external power input, P1. Power dissipated by damping and acoustic
treatment is represented by Pdissl, Pdiss2 and Pdim.

SEA model analysis is a set of simultaneous, frequency dependent, algebraic

equations for total element energies.

N(fk’x)e(fk’x)=_ P(fk) (1-1)

21tfk

where x = [x 1 xM], the vector of SEA design parameters

2
f k is the kth analysis band center frequency (Hz.)

e( f k,x) is a modal energy vector (Joule Hz/mode),

P(fk) =[P,(fk) PK(fk)]T,inputpowervector(Watt).

N( f bx) is a symmetric and diagonally dominant matrix.
17,- = 7M f bx), internal loss factors (non-dimensional)
17,]. = 770‘ ( f k,x) , coupling loss factor (non-dimensional)

"i

= n,( f bx) , the element modal density (mode/Hz) and

N( f k,x) is a K xK symmetric, diagonally dominant and definite positive matrix.

' K
"101+ ,2 "1’7” “1’71: “"I’IIK
J??-
t-l K
N(fk,X)= _nln]i nini+21njnji+ . z+lninij -niniK (1.2)
1= 1:1
16-1
‘"1"1K “"i’lix "MK" _21”j’7jK
J= .

 

 

The fundamental assumption of SEA is that the energies in all independent modes

equalize at steady state within each, narrow, frequency band. Total energy in "i‘h"

element over "kth" frequency band, E,.k(f,‘,x), is derived from the product of the

element’s modal energy, 31,1: ( f bx), and modal density, ni(fk,x).

Ei.k(fk’x) = ni(fk’x)ei,k(fk’x) (1:3)
The energy in the "ith" element in the "k’h" frequency band is related to mean-

squared pressure or velocity.

fi Pi2,k ( f k,x) for acoustic space

I l
Ei,k(fk’x) =4 (1.4)
miVi2,k (fk,x) for structure

 

L
where V,- is acoustic volume (In3 ). P,- is mass density (kg/m3)

c,- is speed of sound in medium (m/s), m,- is mass (kg)

p2,, is expected value of ms sound pressure (Paz).

3

vgk is expected value of mean-squared velocity (mz/sz).

Both mean-squared (total) acoustic pressure and mean-squared (total) velocity are

the sums of independent frequency components in each frequency band.

 

r c2 N]
p-2(x)= 2p, k (fk,x)= _p_,V__c ‘ —2E, k (x)fk, )for acoustic spaces
< V’ k: 1 (1.5)
viz(x)=NZv,-2’k(fk,x )=—N2E,k( fk,x) for structures
L k=l ml 1:: I
where: N f is the number of response frequency bands.

1.2 New Developments for SEA

The new developments for SEA presented in this dissertation include optimal
internal loss factor identification, total sound pressure sensitivity analysis, SEA response
variance analysis, and confidence level prediction for SEA responses. These
developments on SEA extend the SEA method into a powerful tool for engineering vibro-
acoustic design and modeling at audio frequencies. The SEA developments allow users to
obtain accurate SEA responses and sound pressure sensitivity to design parameter
variations. Users can compute SEA response variances via design parameter variances
and use the variances with the Gaussian distributions to compute SEA response

confidence levels and/or develop design parameter specifications for SEA models.

Chapter 2 Identification of Internal Loss Factors of SEA‘

2.1 Introduction

Statistical Energy Analysis (SEA) is becoming a powerful method for quality
acoustic design of automobile vehicle acoustic spaces. SEA predicts vehicle sound
pressure and vibration response levels and is most accurate at high frequencies where
differential equation based methods often fail. The accuracy of SEA predictions are
dependent on the accuracy of SEA model parameters, such as modal density, internal loss
factor and coupling loss factor. SEA has a long usage history in aerospace and naval
engineering and analytical expressions for SEA parameters are available for regular
geometric shapes such as beams, plates, and volumes. Many of the structural details used
by naval and aerospace engineers are not applicable to automobiles (Vail, CF, 1972).
Automobile structures are often made of spot welded sheet metals and the parameters for
real assemblies are not accurately approximated by available idealized analytical model
subsystems. Currently, measured data are not generally available for automotive vehicle
structures. SEA models for automobile applications need to be validated with measured
automotive vehicle responses and accurate automotive SEA model parameters determined.
2.2 SEA Parameters and Identification

Total energies are the mean-rms energies in each element. The input power is the
external power into the element. The internal loss factors quantify the rates of energy

dissipation from each element. The coupling loss factors govern the power flow from

 

I This Chapter is based on the paper titled "Identification of Internal Loss Factors During Statistical Energy
Analysis of Automotive Vehicles" in Proceedings of the 1993 Noise and Vibration Conference, SAE P-264

4

5
one element to another. The SEA analysis problem is to use (1.1) and (1.3) to solve for

the energy responses of each element given the element SEA parameters and input
powers. The SEA parameter identification problem is an inverse problem which uses the
power balance equations and measured values of element response energies and input
powers to compute SEA parameter values. The SEA equations are linear in the SEA
coefficients Na: For each test with a single measurable power input, a maximum of K
values in the coefficient matrix N can be computed. With access to all element input
powers , K independent tests are possible, and K 2 values in the coefficient matrix can be
computed. For a general SEA model with all possible element connections, there are K
internal loss factors, K modal densities and [K (K — 1)/2] independent coupling loss
factors. For the simplest practical case of two elements (K = 2) , there are five SEA
parameters to identify at each band frequency: two internal loss factors, two modal
densities and one coupling loss factor. With a two element model there are only four
possible tests with which to find these parameters (Table 2.1). The noteworthy fact here is
that independent identification of all SEA parameters requires models with three or more
elements for a model when all element couplings are present.

Table 2.1 Number of Independent SEA Tests versus Model Size shows that for very small

models it is impossible to identify all SEA parameters because there are
more parameters than tests

 

 

 

 

 

 

Model Possible Loss Modal Coupling Total SEA Equations-
Size Tests Factors Densities Factors Parameters Parameters
K K2 K K K(K-l)/2 K(K+3)/2 K(K-3)/2

1 1 l 1 0 2 -1

2 4 2 2 l 5 -1

3 9 3 3 3 9 0

4 l6 4 4 6 l4 2

5 25 5 5 10 20 5

 

 

 

 

 

 

 

 

 

Previous investigations of SEA parameter identification have assumed input power
is measurable, SEA models limited to two or three elements and elements with regular

geometric topology. Bies and Hamid (1980) conducted an situ experiment to identify

6
internal loss factors and the coupling loss factor for two coupled plates. Clarkson and

Pope (1981) developed indirect experimental method to determine the modal density and
internal loss factors for flat plates and cylinders. Ghering and Raj (1987) conducted an
experimental investigation on a cylinder—plate-beam structure. Norton and Greenhalph
(1986) identified the internal loss factors of lightly damped SEA model of a pipeline
system by both steady-state power flow and burst random-noise techniques.

The work discussed here will discuss parameter verification for more general, and
larger, SEA models. For this initial investigation only internal loss factors will be
identified and the simplifying assumption will be made that all coupling loss coefficients
are known . This assumption is justified for many structures because SEA coupling
coefficients are related to structure geometry, mass and stiffness properties. These values
are typically well determined by the structure’s design. In contrast, structure energy
dissipation models are typically not well determined by structure geometry and material
properties. Structure energy dissipation models are more typically determined empirically
and it is these dissipation models which are needed to determine the SEA internal loss
factors. The discussion below will present a method for identifying the K independent
SEA internal loss factors of a K element model from a test with one known power input
through a single SEA element. Additionally, the measurement of input power is often
difficult and the identification of input power from measured element energies will be
presented.

2.3 Input Power and Optimal Internal Loss Factor Identification

SEA response energies, E,, are measurable quantities in terms of rms vibration

velocity and sound pressure level. In contrast input powers, R, are not easily measurable

without using an impedance head and external excitation source. We will assume here
that responses from all elements in an SEA model are measurable at each band center
frequency of interest, and that all coupling loss factors and modal densities are known at

these frequencies. For convenience, we will assume with generality that power is input to

7
element 1. We will now use the SEA model equations solve for input power to element

one and internal loss factors for all other elements.
Rewrite the SEA equation (1.1) to explicitly remove the input element’s energy

response from the SEA coupling matn’x.

 

 

 

 

 

 

erl‘ ”N12 N13 NlK-VeZ‘ i.
271f
N12 N22 N23 Nzx 6’3 o
t . >e1+ . . . . 4M“ * (2'1)
(NlK, _NK2 NK3 NKK, fix, L 0 J

 

 

Removing the power input element’s equation from the SEA model equations
removes the input power from the model equations. The power input element’s equation
in (2.1) is rearranged to give the elements input power in terms of known element
energies, coupling losses, and the input element’s internal loss factor.

K
P1 = 2nfi§lNliei (2.2)
The internal loss factor for the powered element, 11,, appears in the right hand side

of this equation in NH and is needed to solve for input power, P,. The internal loss

factor for the powered element with unknown input power cannot be identified from
measured element energies and we must use other methods to estimate it.

The remaining SEA model equations in rows 2 to K of (2.1) are rearranged in a
symmetric form which parallels the original SEA model equation but has the SEA model

size reduced by one.

 

PN22 N23 Nzx- r32‘ erZ‘
N N N e N
:23 :33 .. :BK < :3 }=_< :13 ’31 (2.3)
_N2K N3K NKK_ (ex, LNIK)

 

 

 

 

 

The modified coupling matrix, N is produced by removing the row and column

associated with the power input element. The reduced response energy vector, 5, is

8
produced by removing the power input element’s energy. An equivalent power input

vector, P, is then written in terms of the input element’s response energy, known coupling
factors for that element and the input element’s internal loss factor. The above
formulation allows an SEA identification problem with unknown input power to be recast

into the form (2.3) with known inputs.

 

 

 

 

NE=P (2.4)
erz‘ r—Nizei‘
.. N —N e
where: Pz—J :13 >el =< :‘3 ' >
kNlKJ t_ lKel)

In the revised SEA model, the vector P is the new SEA model input vector.

Internal loss factors in SEA govern the power dissipation in SEA model element.
The internal loss factors are defined by damping ratio for structures and reverberation time
for acoustic volumes. No analytical expressions exist for damping ratios and
reverberation times and they must be identified experimentally.

The optimal internal loss factor identification algorithm developed here identifies
the internal loss factors in such a way that the internal loss factors yield SEA models
which most accurately predict the total SEA energy responses. Optimal internal loss
factor identification is achieved by three steps. Step 1 is to identify an initial internal loss
factors in each frequency band from measured energies in each element, measured power
inputs, analytical modal densities and coupling loss factors. Step 2 is to determine
element-wise constant damping ratios for structures and reverberation times for acoustic
spaces from the identified internal loss factors in step 1 by using the least square method.
Step 3 uses the SEA model and optimizes constant damping ratios and reverberation times
with step 2 initial values to minimize the mean squared (ms) normalized error between
predicted and measured total energies. The internal loss factors obtained in the
optimization process yield an SEA model which predict the SEA responses with the least

difference from measured total energy responses. The algorithm is illustrated in Figure

2.1 and explained below in detail.

Step 1 Step 2 Step 3

lEm(fk)’P(fk)’ 71.),"i->-—)i Ci’TRi ‘9 y

@each I, Vall )‘k

 

 

 

 

 

+ -

w ,. 1
ENE NE (Winn.

K{N’ E'm(fk)_E'P(fk Y) 2‘
F = l I ’
algal h
I

minimize F ( y)

 

 

 

 

 

 

 

 

 

 

 

subject to: 0 < C,- < upper bound
0 < Tm < uppper bound

 

 

 

Figure 2.1 Internal Loss Factor Optimal Identification Algorithm

Step 1: Internal Loss Factor Identification. Internal loss factors in each
frequency in each element can be solved as an inverse SEA problem from measured
energy responses. Since the SEA equations are linear the internal loss factors can be
determined uniquely from the measured energies, power inputs, analytical modal density

and coupling loss factors.

To identify values for the internal loss factors, 11,-, from measurements of the

element energies, Ei, use (1.2) and solve for 11,. in N“ .

10

 

' i—l K
2 "knki(ek-ei)+ 2’finik(ek"'ei) i=2,---,K—I
Th: ="—< (2.5)
2 "knki(ek_ei) i=K
where: e,- = Ei/ni

Step 2: Least Square Damping Ratios and Reverberation Time Computation.
The damping ratios for structural element and reverberation time for acoustic spaces are
assumed to be constant over all frequencies in an SEA model (Lyon, 1975). The element
based damping, Q, and reverberation time,TR,-, can be determined from the identified
frequency-dependent internal loss factors step 1 by minimizing the ms error.

Internal loss factors for acoustic spaces at frequency, fk, are defined to be

inversely proportional reverberation time, T Ri, for the ith element and at the kth
frequency, fk.

2. 2
TRif k

where N f is the number of frequencies

 

"i (fk) = (2-6)

77‘0") is the identified internal loss factor for the ith element at the kth
frequency

TR,- is the reverberation time to be identified for be im element and

S is the mean squared difference

The reverberation time, Tm, can be found from the identified internal loss factors over N f
frequency bands by the least square method.

NI
2.221/ f3
T — i=1
R. ._

l

 

(2.7)

2
\

ni(fk)/fk

R‘
II

The internal loss factor for structure is defined to be twice of the damping ratio.

11
n.(fk)= 2C.- (2.8)

The ms difference between identified and computed internal loss factors can be written as:

N’ 2

The damping ratio can be found from the identified internal loss factors over N f

frequency bands by the least square method.

Ci: 1],- ()fk (2.10)
m, —NX,

Step 3: Optimization of Damping Ratio and Reverberation Time. This step
optimizes the damping ratios and reverberation times to minimize the ms difference
between predicted and measured total energy in the system. The constant damping ratios
and reverberation time in step 2 are used as an initial estimation for the optimization. Let
y be a vector whose components are damping ratios for structural elements and
reverberation times for acoustic spaces in the SEA model. The predictions of SEA
responses from the SEA model are functions of the vector y. The objective function for
the optimization is the normalized ms difference between predicted and measured total

energy in the system.

 

(31%): N, “(EN fE)m(ff)(fk’y))2 (2.11)

i=1

1M?

where E{" ( f k) is the measured energy in the ith element at the kth frequency f k
P
Ei (fk ,y) is the predicted energy in the ith element at the kth frequency fk

The normalization is important for the objective function. The reason is that the
energies in the elements near power input sources are larger than the energies in the
elements which is far from the power input sources. The normalization will weigh the
elements with smaller energy response equally in the optimization, otherwise, the

difference will be dominated by the energies in a few elements near sources. The most

12
important elements often have the lowest element energies because they are the elements

at the end of the power flow chain, e.g. a vehicle's interior.
2.4 Application Examples

The first example is a simple conceptual model (Figures 1.1 and 1.2) to illustrate
the details of internal loss factor identification. The SEA model consists of two identical
cubes, separated by an aluminum panel with a flanking path between the cubes to model
energy transfer by paths other than through the panel. Acoustic powers are input to the
space 1 and the working medium is air (Table A1 in Appendix A). Space #1 is designated
as element 1, the panel as element 2 and space #2 as element 3. Geometric and Material
properties are listed in Table A1 in Appendix. The pressures in Space #1 and #2 are 39.13
and 19.41 (N / m2) respectively and the velocity in the plate is 8.85 x104(m/s) at 5,000
Hz.

Computation of the SEA model parameters, such as modal densities and coupling
loss factors for every element and coupling, is the first step in internal loss factor

identification. The modal density (A1, Appendix A) for structural element 2 is

 

n _«/3A_ {330.52
2 he, 0.007-5182 (2.12)

= 11.9 X 10-3 (modes / Hz)

The modal densities at 5,000 Hz.(A2, Appendix A) for acoustic elements 1 and 3 are

47rf2V+7rfAs+££

 

 

n =7! =
1 3 c3 2c2 86
_ 47z-50002 .053 +
- 3443 (2.13)

nOSOOOOI.5 + 6
3442 80344
= 1.10 (modes/ Hz)

 

The critical frequency is 1786 Hz. and the band frequency is 5,000 Hz.. The
coupling loss factors (A3, Appendix A) from the structural element 2 to the acoustic

element 3 are

13

 

 

c A
7721 = 7723 : pAIR AIR 1
f /2
Mpw(l——£]
f
: 1.244-34400.25 1786 (2.14)
2700-0.0070015027115000. 1—————
5000

= 9.0 x 10"
The coupling loss factor from acoustic element 1 to the structural element 2 can be

computed by the reciprocity relationship (Lyon, RH, 1975).

 

—3
17,2 :_71_2_n21 :11.94x10 9.0“04
n1 1.1 (2.15)
=1.0><10‘S

The coupling loss factor for the flanking path between element 1 and element 3 is assumed
constant for this example.
11,3 = 0.0001 (2.16)

The internal loss factor identification is carried out by using (2.5) to compute the

internal loss factors. For this example assume the measured energy values at 5,000 Hz.

 

V 0.125
E, =—,-( 2)= 2(39.132)
pc 1.244 . 344 (2.17a)
= 1.3 X 10’3 (Joules / Hz.)
E2 = m<v2)
= 2700 . 0.0075 - 0.52((8.85 x 104)?) (2.17b)

= 3.7 x10‘6 (Joules / Hz.)

 

__V_ 2 = 0.125 ,
_ch< > 1.244-3442(19'41> (2.17c)

= 3.2 ><10‘4 (Joules / Hz.)

E3

The element modal energies, ei, are computed by dividing measured total energy,

E, by the modal density, 71,-.

l4

 

e -51- 1.3x10‘3
1 n1 1.1 (2°183)
= 1.2 x 10-3 (Joules / mode)
6 _ g; _ 4.0><10‘6
2 n2 11.94x10'3 (2.18b)
=3.1x10‘4 (Joules/mode)
e _ g _ 3.2x10'4
3 n3 1.1 (2.180)

= 3.0 x 10—4 (Joules / mode)

Equations (2.12) through (2.18) are substituted into (2.11) to yield the values of internal

loss factors, 172 and n3.
172 = 8.1x10‘5. (2.19a)

173 = 3.1x10". (2.1%)
The internal loss factor in element 1 cannot be independently computed because we have
used measured energy in that element as our model input.

The second example is an automotive vehicle to further demonstrate the internal
loss factor identification algorithm (Figures 2.2 and 2.3). The automotive vehicle SEA
model consists of 11 elements. "ENGINE" represents the engine surface vibration,
"ENGCOMP" represents the engine compartment acoustic pressure, "LRAIL" and
"RRAIL" represent the frame vibration, "UNDCARVOL" represents the under car volume
acoustic pressure, "HOOD" represents the hood vibration, "DASH" (dash panel)
represents the front of dash vibration, "FLOORPAN" represents the floor panels vibration,
"OVERCARVOL" represents the over car volume acoustic pressure, "BODYPAN"
represents the body panels vibration and "INTERIOR" represents the car interior acoustic
pressure. The engine is assumed to be the only noise source and every element has at least
one path to another element in the model. The physical parameters are measured from the

car and then used to compute modal densities and coupling loss factors. The vibro-

15
acoustic energies from every elements are measured and then used to identify the internal

loss factors. The computed modal densities, coupling loss factors and identified internal
loss factors are used to form the car SEA model. The power emitted by the engine surface
is very difficult to measure, however the surface vibration can be easily measured through

accelerometers on a dynamometer test.

00006
00909-
~ &? -

I“.

 

 

———L_lmm‘
Ssfiflfififldbq5”m”””m“§§9 fife

AA

 

 

 

 

I l
IENGCOMPI I LRAIL I IRRAIL UNDCARVOL
l L————1‘————|
I 11000 Wimp} FLOORPAN
I
EVRCARVOLH BODYM—I INITERIOR I

 

Figure 2.3 SEA Automotive Vehicle Model Element Coupling Chart

The laboratory test condition used was vehicle operation under load at 3000 rpm
on a dynamometer. The RMS sound pressure levels in acoustic volumes and RMS
acceleration levels in structural elements are measured in 4-12 spatial positions
corresponding to each element in the model and the measured RMS velocity and pressure

responses are used to identify the internal loss factors. The identified internal loss factors

16
for under car volume and front of dash are presented in Figures 2.4 and 2.5 respectively.

Constant internal loss factors for structural elements are typical modeling
assumptions for most SEA models. Structural internal loss factor corresponds to twice the
linear viscous damping ratio for these structures (Lyon, 1975). Identified internal loss
factor values for our automotive structural element example show that the internal loss
factors are not a constant at all frequencies. Specifically, the average internal loss factor
for the front of dash (FOD) is 0.013 with a large standard deviation of 0.008 (Figure 2.5).
The average internal loss factor for the FOD results corresponds to a damping ratio of
0.6% which is very small. There is no apparent trend in the identified internal loss factors
for the FOD element and the other elements in our model displayed no regular trend as
well This observation indicates that the actual internal loss factor values for automotive
structural elements can not be accurately modeled either by Coulomb or viscous friction
models. The very small damping measured may partially explain the inability to detect

data trends. These results indicate that substantially more experience with identification

I

3.00E~02 .

2.50E-02 -1- ' ‘\‘/ Identified lntemal Loss Factor

2.005-02-I I \

\
1.50E-02 . I Computed lntemal Loss Factor
Using Reverberation Time Identified

   

1.00E-02 - '

Internal Loss Factors

I

5.0013-03 --I
I

1 ' ‘ A '

— '
l l 1 l 1 1 1 "f I
I I I

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
Third Octave Center Frequency (Hz)

 

 

0.00E+00

IF
all
u
d
u
d
d

Figure 2.4 Identified and Computed lntemal Loss Factors Using Computed
Reverberation Time for Under Car Volume

17

3505-02 .
3.005-02 ' I \ Identified Internal Loss Factor

2.50E-02 - I \

2.00E-02 « I \ , - “. Averaged lntemal Loss Factor

1.505-02

lntemal Loss Factor
\ -
’ -
\Q
\
\
/
I
/
’

1.00E-02 ‘I \ o \ l -

5.00E-03 -

 

0.00E+OO

db
..
d
.1
d
..
d
.-
.1
..

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000

Third Octave Center Frequency (Hz)

Figure 2.5 Identified and Average lntemal Loss Factors for Front of Dash

of automotive structural dissipation is required before these sources of internal energy loss
can be accurately modeled.

The internal loss factor for an acoustic element is often modeled using
reverberation time (2.6). The identified internal loss factor values for the under car
volume (UNDCARVOL) are presented in Figure 2.4. A least-squared-error fit to a
reverberation time from the identified internal loss factors can be computed (2.7).

TR = 0.27 (second) (2.20)
The internal loss factors computed (solid line) from the identified reverberation time are
also presented in Figure 2.4 .

The acoustic internal loss factors identified for the under car volume
(UNDCARVOL) for under car volume are very different from the reverberation time
model at lower frequencies: 500 Hz. and 630 Hz. At higher frequencies: 800 to 5,000 Hz,

identified internal loss factors closely match the trend corresponding to the values for a

18
reverberation time of 0.27 seconds. This result at high frequencies was true in general for

other model acoustic volumes with the measured automotive data. These results indicate
that a reverberation time based model of automotive vehicle acoustic spaces is appropriate
at higher frequencies where modal density is larger. Our automotive data appeared to
follow fixed reverberation times when modal densities were above approximately 0.5
modes per Hertz. Again, additional automotive data are required before modeling trends

can be accurately determined.

 

Figure 2.6 Geo Metro Body Photo

An SEA model of a 1992 Geo Metro XFI body (Figure 2.6 and Figure 2.7) is used
as the third example to illustrate the optimal internal loss factor identification algorithm.
The Geo's engine, suspension and power train systems are removed and the body is
supported by a frame support. The vehicle SEA model has 16 connectors and 8 elements.
The structural elements, FOD, HOOD, FPAN and BPAN, represent dash panel, hood,
floor panel and body panel surface vibration respectively. The acoustic spaces,

ENGCMP, UND, OVER and INT, represent engine compartment, under car volume, over

19
car volume and interior acoustic pressure respectively. The physical design parameters

and connectors for Geo SEA model are listed in Appendix B.

 

 

I HOOD H ENGCMPI—I UND I
I L

% I
BPAN I—-I OVERJ I FPAN I_
| l

 

 

 

 

Figure 2.7 GEO METRO SEA Model Chart

 

Figure 2.8 Vibrator Mount Assembly Photo

Tests of the Geo body were conducted in Controls Research Laboratory of the
Department of Mechanical Engineering at Michigan State University. A Model 400

vibrator from Ling Dynamic Systems is mounted in the engine compartment (Figure 2.8).

20
The vibrator table is connected to the front of dash panel (FOD). The tests were

conducted under two operating states: the two operating states were corresponding to two
input signals. The operating state 1 is that the random signal was generated with an
amplitude of 0.9 V over frequency range from 500 Hz. to 5000 Hz. from 35660A HP
dynamic signal analyzer and sent to the vibrator through a power amplifier, Model
PAlOOE from Ling Dynamic Systems. The operating state 2 is that the recorded Mazda
626 engine noise was played back to excite the GEO front of dash panel. The
accelerations and SPL for all panels and acoustic spaces were measured for the operating
state 1. The front of dash acceleration and interior SPL were measured for the operating
state 2. The vibration signals were picked up by accelerometers between 6 and 16
locations for each structural elements. The sound pressures were picked up by
microphones over 6-8 locations in each of the acoustic spaces. The measured
accelerations and sound pressures were averaged over the measurement locations to obtain
the mean values of the GEO vibro-acoustic responses corresponding to each model
element. The standard deviations about the mean values were also computed from the
measured responses over the different locations. The measured vibro-acoustic responses
from the operating state 1 were used to generate the SEA model for the GEO through the
optimal algorithm. The Mazda acceleration for front of dash panel was used as an input
and was fed to the generated GEO SEA model to predict the interior SPL. The predicted
interior was compared with the measured interior SPL. The measured SEA responses for
all elements are shown in Appendix B. The optimal identification algorithm developed

here was implemented in MATLAB.

21

Table 2.2 Identified and Optimized Damping Ratio and Reverberation Time

 

FOD ENGCMP HOOD UND FPAN OVER BPAN INT
(%) (second) (%) (second) (%) (second) (%) (second

 

identified N/A 0.2907 33.65 0.3487 5.948 6.532 34.90 0.4111
optimized N/A 0.4917 40.33 0.5914 3.997 6.3875 63.05 0.2111 I

 

 

 

 

 

 

 

 

 

 

 

The test data from the operating state 1 were used to identify the damping ratios
and reverberation times in the optimization. The identified (from step 2) and optimized
(in step 3) damping ratios and reverberation times are shown in Table 2.2. The damping
ratios are not available for Front of Dash because the power input is unknown. the energy
and damping ratio are model inputs.

Two GEO SEA models were formed by two sets of damping ratios and
reverberation times obtained without (step 2) and with the optimization (step 3)
respectively. The two GEO SEA models were used to predict the GEO vibro-acoustic
responses. The predicted and measured ms responses were shown in Table 2.3. The
results in Table 2.3 were plotted in Figure 2.9 for the SPL comparison and in Figure 2.10

for ms structural velocity comparison.

Table 2.3 Comparisons of Predicted and Measured Total Energies (in dB)

 

 

 

 

 

 

 

 

 

 

 

Front 0‘ Engine Under floor Over Body J
Dash Compartment Hood Car Panel Car Panel Interi
Volume Volume r
measured -66 88 -94 82 -90 80 -102 78
redicted (no optimal) -66 84 -90 78 -91 79 - 100 80
predicted (optimal) -66 86 -91 81 -90 81 - 102 78

 

 

 

 

22

 

 

/////////

 

 

n-Optimal/Optimal/Measured SPL in dB of State 1

Model No

Figure 2.9 Geo

 

Figure 2.10 Geo Model Non-optimal/Optimal/Measured ms Velocity in dB(re 1

d

 

‘ I I predicted w/o optimal E] predicted w/ optimal measure

ppppppppp
........

ssssssss
ssmqsamsmo

 

m/s)

time improved the accuracy of the

The optimized damping ratio and reverberation

23
prediction. The improvement for the Hood and Floor Panel is 1 dB. There are 2 dB

improvement in Engine Compartment, Body Panel and Interior and 3 dB improvement in
Under Car Volume. The improvement on SEA prediction shows that the algorithm

developed here is powerful.

operating State 2
(independent of State 1)

operating I I
State 1 ,
—->I GEO Body I—. Optimization P>I GEO SEA Model

Measurements

 

 

 

 

 

 

prediction
> Match ? <—~

 

 

Figure 2.11 SEA Model Verification Procedure

The optimally generated SEA model for the GEO METRO body needs to be
verified. The GEO METRO Front of Dash was excited by a Mazda 626 engine noise
recording which is independent of the excitations and the responses used for the SEA
internal loss factor optimization. The Front of Dash acceleration responses were measured
and used as the SEA model power input. The Mazda power input was fed into the
generated GEO SEA model to yield the interior SPL prediction. The interior SPL due to
the Mazda excitation was measured and used as the SEA model output. The verification
is to compare the GEO interior response prediction by the Mazda power input through the
optimized SEA model to the measured GEO interior responses due to the same power
input (see Figure 2.11). The predicted and measured interior SPL spectra were shown in
Figure 2.12. The comparison shows that the predicted interior SPL is 74 dB and measured
interior SPL is 75 dB. The predicted spectrum has good trend compared to the measured

spectrum.

24

 

I —o— predicted mean 0 measured mean

 

8

on
Q

\I
?
I
l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I

    
 

SPL in dB (re 2X10"(-5)Pa)

 

60.-.. ._._.-._._._---_- ....................... _
50.)— .............................................. .—
40. ..............................................................
3G ‘ i J I ‘ i i i ‘ 4

 

1600 2000 2500 3150 4000 5000
Third Octave Band Center Frequency (Hz.)

500 630 800 1000 1250

Figure 2.12 GEO SEA Model Verification Plot

2.5 Summary

The SEA internal loss factor identification problem for automotive vehicle model
applications is addressed and an optimal algorithm for identifying the internal loss factors
is developed. The optimal internal loss factor identification algorithm is demonstrated by
both simple illustrative example and two more complicated automotive vehicle SEA
models. Preliminary results for the identification without optimal process show that the
common assumption of constant internal loss factors for structural elements may not
accurately represent actual system responses, however, reverberation time models for
acoustic volume elements are probably justified at higher frequencies. Additional
automotive measurement experience is required before more accurate modeling trends can
be determined. The results show and verify that the optimization on identification of
damping ratios and reverberation time yields an SEA model which accurately predict the

SEA model ms vibro-acoustic responses.

Chapter 3 Noise and Vibration Design Sensitivity from SEA“

3.1 Introduction

The SEA method predicts the expected value of vibro-acoustic response for modes
grouped over a narrow frequency band. With the SEA assumption of equipartition of
modal energy, this modal grouping significantly reduces the number of degrees of
freedom in SEA models. The grouping also improves response prediction accuracy for
modes with similar resonant frequencies through averaging of modal responses. The SEA
method is a useful design tool for preliminary noise and vibration design studies because
models can be developed at the prototype stage without detailed structural design
specifications. Models can then be easily refined to increase prediction accuracy as the
design matures. A specific example is the design of complex automotive structures.

SEA development started with prediction of the mean-rms values of vibro-acoustic
response (Lyon, 1975). SEA response sensitivity analysis was initiated for optimal
damping selection (Lu, 1987). This chapter will present a sound pressure (SP) sensitivity
analysis method which relates SP variability to the variations in automotive vehicle model
design parameters. The methodology proposed here can rank SP sensitivities and allow
users to select the parameters which have the greatest effect on the SP in the vehicle
interior.

SPL in an automotive vehicle interior is an accepted measure (ANSI S 1.4-1971)

of perceived noise loudness and is used as one of the criteria for evaluating sound quality

 

I This Chapter is partially based on the paper titled " Sensitivity of the Statistical Energy Analysis for
Automotive Vehicles, published in Proceedings of the Fourth Symposium on Advanced Technologies of the
1993 ASME Winter Annual Meeting

25

26
in many acoustic designs including automotive vehicle interiors. Acoustic design

objectives often require that SPL be efficiently reduced as low as feasible. SPL is a
logarithm of SP via a standard reference pressure, me = 2011Pa. Reducing SP means
reducing SPL. The question arises: "Which parameter in the vibro acoustic SEA model
has the maximum effect on SP?"
3.2 Sound Pressure (SP) Sensitivity

SP sensitivity relates SP variations to the variability of design parameters. SP
sensitivities quantify the effect of design parameters on acoustic performance. The design
parameters which have greatest and least effect on SP can be determined by ranking the
SP sensitivities to design parameters.

Sensitivity of a function F to its variable x is defined as (Frank, 1978):
_ aF/ax
F /x

2
SP Sensitivities, Sfmm, with respect to a design parameter are derived by taking

 

sf (3.1)

partial derivatives of (1.5) with respect to each design parameter and dividing by the ratio

of the mean-squared pressure to that parameter.

 

 

 

8122 x.f- ax,
SSP : Sp (X) 3p2(x)/axn = j=l ( J)/
x," x" p2(X)/xm NI 2
.21” (x’fj)/x..
J:
N!
2(P2(X,fJ)/x )(8p2(x,fj)/ax‘)/(p2(x,fj)/xn)
' I
__ I:
_ NI
21p2("’fj)/x.
J:
N! 2
p2(x.f,-)Sf (m) (3.2)
i=1 "'

 

27

2
The SP sensitivities, 5;; (x), has now been expressed in terms of the pressure

sensitivities in each frequency band, 5:122:1(mf, ). Sensitivity of the sound pressure Sf", p2( f’ )

is equal to sensitivity of acoustic space energy, EIx, f j) In each frequency band.

p.(x,,j)_ap2(x,f,)/a. _xaE( ,f,)/a.,, _SE M) (3.3)

S): _ 2 — x
p (X’s-Va Ex( MW ..
SP sensitivity can now be expressed in terms of the sensitivities of energy in each
frequency band by substituting (3.3) into (3.2).
’ E x f
NXEIX’fJ'ISx,( N, E( f)
55”: F' = a s "’ 1 3.4
I. N, g 1 x, ( )
250’ij
i=1

 

=—£—)-<land [Ea-=1
N, , 1
E314 f.)

where 01,-

a,- is a normalized SP spectrum and represents a ratio of element energy in the jth
frequency band to the total energy over all frequency bands. SP sensitivity is therefore a
weighted sum of sensitivities of total energies to the design parameters over all frequency
bands. Frequency band energies are predicted by SEA equations and their sensitivities in

each frequency band can be applied to the SP sensitivity analysis.

The sensitivity of the total energy is equal to the sum of the sensitivities of the
modal energy and modal density because energy (1.3) is equal to product of modal energy
and modal density (Frank, 1978).

550.4) ___ 84:14.- )/a. Haul f.).(, ) /3x
1"" E(x.f,-)/x., "(x’fIIeIX’fJ'I/xm (3.5)
= 51f " "I + 53:“ ")

SP sensitivities can be expressed using (3.4-3.5) in terms of the sensitivities of the

28
modal energy and modal density.

X

N!
sff’ = Zaszjfx’f’) + 5"("f 1)] (3.6)
j=l

SP sensitivity in an element depends on three terms in (3.6). The normalized SP

spectrum, 01,-, and modal energy response sensitivity, Six'f’ ), are dependent upon system

models and the modal density sensitivity, 51"“), is dependent upon element models

only. Suppose that modal energy and modal density sensitivities in an element are
computed for several design parameters and the sensitivity to a particular design parameter
has the greatest value in all frequency bands, then one can say that design parameter has

the greatest effect on SP without invoking (3.6) because a ,-'s are the same for those

design parameters. This case is quite useful and convenient in early design stage.

Sensitivities of acoustic response modal energy,e(x, f j), and modal density,
n(x, f j), to design parameters, x, can be used to complete the analysis of SP sensitivities
to design parameters. The sensitivity of modal density, n(x, f j), is defined by acoustic

element definition equations. The sensitivity of acoustic modal energy, e(x, f j), requires

derivatives obtained in the following section.
3.3 Modal Energy Derivatives to Parameter Variation

The SEA response derivatives are derived with the chain rule by using (1.3).

armagmfi (3'7”
BE,- ae, an, Be,-

The derivatives of modal energy to design parameters are found by taking
derivatives on both sides of (1.1) with respect to the design parameters. Since the input
power vector is independent of the design parameters, the partial derivative on the right
hand side is zero. Two terms remain on the left hand side. Rearranging this result yields

the derivatives of modal energy in matrix form.

29

.33.-
8x,"

1 aNe

-———N'
Txme

(3.8)

Expanding partial derivative of matrix N with respect to each design

parameters, x,,,, in (3.8) through partial fraction expansion yields the results below.

aeN-1 8N can—ii 8N e877,: aNe 871i
—— — — — — —— (3.9)
073cm: N[,2I,2>,_3nij ”ax +§18m ”8): +21% exmd
(SN 5’ n,, k=m=iorkl=m=j
where TzIfN—km-I = -n,-.k=i,m=j0rk=j,m=i
nij "U 0, otherwise
(917,-: 817— 0 , otherwise
P0 ' 0 0 0
0 0 0 0
K
3N O 0 771' + z 771,: “771141 “771K
— = j=i+l
8n,- 0 "' 0 — 711.141 771,141 0' " 0
0 .
E 0
.0 0 _niK 0 "'0 711K _

 

 

The beauty of the elegant forms of the SEA coefficient matrix N derivatives to
SEA parameters is that they are easy to implement in computer algorithms.
The derivative of the modal energy vector, e, with respect to the element power

inputs, P, is found by observing that the partial derivative of the power input vector, P, is

non-zero in the I'm row only .

 

70‘
a“: 1 N‘llri (3.10)
8P,- Zflfk .

LO;

 

 

The partial derivatives of 171,, 17,-, and n, are dependent on the specific element

3O
definition equations.

3.4 Application of SP Sensitivity to Damping

Analysis of SP sensitivity to damping yields a quantitative evaluation of the effects
of damping on SP in an SEA model acoustic space. Damping dissipates energy in each
SEA model element. Damping is typically quantified in SEA models through each
element's internal loss factor, 17,. Increasing the damping will increase the energy
dissipation, decrease the SEA energy response and reduce the SP in the interior. Without
loss of generality, it is assumed that

1. the SEA model has K elements,

2. a single input source drives the first element of the SEA model,

3. the SEA model is irreducible (at least one path from each element to any other

element in the model),

4. the Km element is an acoustic space, and

5. the design parameters, xm, are internal loss factors, 17",.

The design goal is to find the internal loss factors which have the greatest effect on
the SPL in the last acoustic element. The goal requires the evaluation of sensitivity of

SPL to internal loss factors. Since the modal density is independent of internal loss
factors, 53" = 0, the sensitivity of SP (3.6) can be written as:

N!
53f = 201,55: (3.11)
i=1

The derivative of the interior modal energy, eKIX, f j), to the mth internal loss
factor, 17", (x, f j), can be found by taking i=m and selecting the last entry in the vector
saw in (3.9).

aeK("’fj)
anm(x9fj)

where N2" is the Km row and mth column entry in the N"1 .

-N}Innm(x, f,- )e,,, (x, f,) (3.12)

31
Note that one of the above derivatives exists for each of the internal loss factors in the

SEA model.
The modal energy response can be expressed under the above five assumptions for
(1 . 1) as

p1(fj) —1
2717f} ml

 

em(x,f,-)= form: 1,2,--~,K (3.13)

The sensitivities of the Km modal energy response, eKIX, f j), to any internal loss

factor, nm (x, f j), can be written as:

53::=-N21n(x.f.)nm(x,f.-JN;‘1(”J/(Na(1143/4040) (314)

The rankings of the SP sensitivities in the interior to internal loss factors are useful

for evaluating acoustic design strategies. SP sensitivities can be obtained by substituting

(3.14) into (3.6) noting that the sensitivity of modal density to any internal loss factor is

zero because modal density definitions are all independent of internal loss factor. The

ranking of the SP sensitivities is determined by sorting the sensitivities of the modal

energy to different internal loss factors. This sorting is typically frequency dependent.

The sensitivities of the modal energy responses for a one-input SEA model are only
associated with the entries in N "1 and invariant with the input power level.

The sensitivities of any modal energy responses to the source element internal loss

factor, n], is derived by letting m = l in (3.14).

sf;l = -nl(x,f,-)nl(x,f,-)Nf11(x,fj) (3.15)
The importance of (3.15) is that the sensitivities of all elements in our assumed SEA
model topology to the source internal loss factor, 171(x, f,), are equal. The change of the
source element internal loss factor will proportionally change the energy levels in all
elements. Since the all energy levels are associated linearly by the SEA equations and the
all other elements get their energy through the paths connected to the powered element,

the sensitivities of all elements to the internal loss factor are the same.

32
3.5 Additional Design Examples

Two examples are used here to demonstrate the sensitivity analysis. The first
example is a simple 3-element SEA model (Figures 1.1 and 1.2). We will illustrate the
SPL sensitivity analysis in space 2 to damping parameters in detail. All parameters are
listed in Appendix A. The second example is derived from a model of a production
vehicle and more complex in structure (Figure 2.2 and 2.3). The model includes 11
elements, 15 couplings and 140 parameters. Although there are too many parameters to be
listed here, the second example demonstrates the utility of SP sensitivity analysis for
vehicle interior acoustic design.

The goal for the first example is to evaluate which element damping has the

greatest effect on SP in element 3. Since modal density is independent of damping

parameters, the term 5:3 = O and (3.6) becomes:
NI
5;?” = 211,53 (3.16)
r=1

The modal energy sensitivity frequency components in Space 2 with respect to
internal loss factors are evaluated in (3.14) by taking K=3 and m=1, 2 ,3. Internal loss
factors are characterized differently in acoustic and structural elements (Lyon, 1975).
lntemal loss factor for an acoustic space is inversely proportional to acoustic reverberation
time. Internal loss factor for a structural element is directly proportional to damping ratio.
The derivatives of internal loss factors with respect to reverberation times in acoustic
spaces and plate are computed from (A4, Appendix A).

The absolute values of modal energy sensitivity frequency components in space 2
to reverberation times in acoustic elements 1 and 3 and damping ratio in the structural

element 2 are plotted in Figure 3.1.

33

 

 

 

 

 

 

 

 

 

4 I
-0. ....

500 1000 15.00 2000 2500 3000 35.00 4000 4500 5000
Third Octave Band Center Frequency (Hz.)

Figure 3.1 Modal Energy Sensitivities in Space 2 to Damping Parameters

Result examination in Figure 3.1 shows that the reverberation time in both acoustic
spaces have the largest sensitivities over all frequency bands. The reverberation time in
space 2 is less than the reverberation time in space 1 and has a larger effect on the
responses in the space 2. The acoustic space 2 responses have the least sensitivity to the
plate damping because the flanking path carries more power into space 2 than the plate.
Since the reverberation times in space 2 and 1 have the greatest and second greatest and
plate damping has the least sensitivities respectively over all frequency bands, from the
modal energy frequency component sensitivities one can get some conclusion of effects of
the design parameters on SP without computing SP sensitivity in (3.16). The conclusion
is that decreasing reverberation time through acoustic treatment is the most efficient way
to reduce the SP in space 2 and that increasing damping in the plate has the least effect on
the SP in space 2.

The second example is an automotive vehicle (Figures 2.2 and 2.3). Note that,

34
although right and left frame rail are symmetric in the model and have identical geometric

parameters, the measured responses are not the same for the two rails. Therefore the
internal loss factors are not identified the same and this will lead to different SP
sensitivities. The SP sensitivities of the interior acoustic responses to the internal loss

factors are computed based on the SEA model.

 

101 ........... ; ........... Z ........... ........... ; ........... ; ___________

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Third Octave Band Center Frequency (Hz.)

Figure 3.2 Sensitivity Rankings for Interior Modal Energies

Interior modal energy sensitivity frequency components to internal loss factor
variations are calculated by using (3.14) on the elements of the existing full scale
automobile model shown in Figure 3.2. The over car and under car volumes are not part

of the vehicle design, their sensitivities are excluded. The sensitivity of the vehicle

35
INTERIOR's internal loss factor has the largest magnitude over all frequencies. This

result arises from the diagonal element, N72,“ which determines this sensitivity and
confirms the diagonally dominant nature of N"(Hodges, et a1 1987). Figure 3.2 also
shows that the FLOORPAN, ENGCOMP and FOD have the next largest sensitivity
values. This ranking indicates that these four design areas are the next most important to
the acoustic performance of the vehicle. BODYPAN and HOOD have higher sensitivity
than LRAIL and RRAIL over lower frequencies, however, have lower sensitivity over
higher frequencies. Ranking all sensitivities of the interior SP to damping parameters of
elements provides information as to the most effective direction for acoustic design
changes. For the elements whose sensitivities have cross over these frequency ranges, one
has to evaluate SP sensitivities in (3.6) for each of the parameters to determine which

parameter has larger effect on the SP because one needs to evaluate both nodal energy

sensitivity and the normalized SP spectrum aj.

Table 3.1 Interior SP Sensitivities

 

 

ENGINE LEFT RIGHT HOOD BODY FRONT FLOOR INTERIOR
COMPARTMENT RAIL RAIL PANEL of PANEL
DASH
0.356 1.92x10-2 7.86x10-3 1.39x10’2 1.36x10'3 0.337 0.640 0.954

 

 

 

 

 

 

 

 

 

 

The interior SP sensitivities over the full frequency range to element damping
ratios and reverberation times are listed in Table 3.1. The table shows that the
reverberation time in the interior has the maximum sensitivity and the second is the
damping ratio in the floor panel and that the right frame rail and body panel have the least
sensitivities. From viewpoint of design, the most efficient ways to reduce the noise levels
in the vehicle interior from the engine surface vibration are to decrease the reverberation
time in the interior by acoustic treatment and increase the damping ratio in the floor panel.

The most inefficient ways are to increase the damping in the right frame rail and body

36
panels. The floor panel has the second greatest SP sensitivities and the acoustic treatment

to the interior may be much expensive. Then increasing the floor panel damping to reduce
interior SP may be chosen as a design alternative. SP sensitivities for front of dash and
engine compartment are in the same order. This also gives design alternative to balance
the cost effective in reducing vehicle interior noise.
3.6 Conclusions

Methodology developed in this paper extends traditional Statistical Energy
Analysis beyond prediction of mean-rms vibro-acoustic response. The methodology uses
SEA response sensitivity in each frequency band to derive SP sensitivity. SP sensitivity
analysis relates the variability of SP to model design parameter variability in the SEA
equations. This methodology is important for acoustic design of aircraft, ship and
automotive vehicles because it enables users to qualitatively and quantitatively determine
and rank the variations of vehicle interior SP with respect to changes of design parameters.
The methodology identifies the most and least sensitive design parameters to SP in an
SEA model. The methodology is powerful design tool for identifying the most effective

areas for change in acoustic design.

Chapter 4 Putting Statistics into SEA"

4.1 Introduction

Statistical Energy Analysis (SEA) is of particular interest to the automotive
industry because it can predict sound and vibration levels at higher frequencies than
differential equation based analytical methods. SEA has been successfully employed by
the aerospace industry (Lyon, R.H., 1975, Jacobs, E.W., et. al., 1989) and the maritime
industry (Jenson, J .O., 1979) The analytical foundation of SEA is the concept that the
vibration and sound energy of dynamic systems is shared equally between modes over
narrow band of frequency (Lyon, R.H., 1975). For automotive vehicles, this concept
makes SEA most accurate at frequencies above BOO-500 Hz. where the number of modes
in octave bands is high enough to allow averaging of the energy between the modes. As
an example, a typical automotive interior has approximately 27 modes per Hz. at 5,000
Hz. Differential equation methods like the Finite Element Method have difficulty
maintaining accuracy due to the same high modal density at these frequencies
(Remington, PJ. and Manning, J .E., 1975, Lu, L.K.H., et. al., 1983).

Design of quality vehicles requires analytical methods that can accurately predict
both the expected performance of designs and the variance of vehicle performance about
the expected performance. Design analysis is needed for two purposes, the evaluation of
existing vehicle designs and the development of specifications for new designs. Existing

vehicle designs are evaluated to predict typical vehicle vibration and acoustic response

 

’ The Chapter is based on the paper titled "Putting Statistics into The Statistical Energy Analysis of
Automotive Vehicles" in Vibration Isolation, Acoustics and Damping in Mechanical Systems, DE-VOL 62
ASME Book 000823

37

38
and the probability that a vehicle will perform within specified performance specifications.

During vehicle design development, subsystem specifications are used to develop
predictions of expected system performance and variance. Both expected value and
variance of SEA models are required to predict confidence levels in both classes of design
analysis.

Current SEA modeling software provides RMS response predictions for such an
SEA automotive vehicle model. Early SEA development by Lyons (1975) provided
variance analysis for a few simple example systems and the response variance for
randomly varying resonant frequencies was studied by DeJong (1985), however, no
general approach to the variance analysis problem for design parameters is currently
available in literature. The work discussed here extends SEA to include variance
calculations.
4.2 SEA Response Variance Analysis

Response variance analysis gives the statistical distribution of predicted SEA
energy levels about their mean values. Conventional SEA predicts the expected (RMS)
values of SEA energies in the form of frequency spectra. Random distributions of
physical parameter values lead to variance of predicted SEA energies about their expected
frequency spectra values. These variances in SEA responses are implicitly, non-linearly,
associated with variances of physical parameters via the SEA equation (1.1). This implicit
variance relationship is derived through non linear definitions of the SEA parameters such
as modal density, internal loss factor and coupling loss factor. SEA response statistics are
dependent on the statistics of the model’s physical parameters in these definitions. We
will derive a linear approximate variance expression for SEA responses about mean-rms
spectra in terms of physical parameters through a Taylor expansion.

The SEA response deviation about its mean-rms energy can be approximated by

first order Taylor expansion over the M physical parameters and K SEA element inputs.

39

Mi:

~.

BE- _ K aE- — (4,1)
l __ l P _ P
13x". ;:— [xm xm]+k2=:l aPk if [ k k]

 

 

TM: $7"

 

 

"6|
'5| "

The partial derivatives of the ith element's energy response above are evaluated at the
nominal physical parameters x = E and nominal power inputs P = P.

Taking the variance (Clifford, A.A., 1973, Oh, H.L., 1987) of (4.1) yields the
variance of SEA responses, 0203,), in terms of the variances of M physical parameters and

K input powers.

2

M BB. K

02(54): 2 -a—'— 02(xm)+ 2
m:] xm ;: k=1

—'— 02(Pk) (4.2)

"6|”

 

 

where 02(Ei) is the variance of total energy response in the ith element,

2 th
6 (xm) is the variance of the m physical parameter, and

62(1) k) is the variance of the kth power input
The SEA response derivatives are derived in Section 3.3.
4.3 Exampr and Monte Carlo Validation

Two examples and a Monte Carlo modeling procedure are used here to validate the
variance analysis. A set of two thousand (2,000) Gaussian distributed parameter values
are first generated to simulate a production process. The mean and variances of these
parameters are used in the variance analysis to predict the mean and variance of SEA
response energies. The two thousand parameter values are then used in two thousand
separate SEA analyses to compute 2,000 actual SEA responses. The mean and variance of
these actual SEA responses are then compared with the variance analysis predictions of
mean and variance.

The first example is a simple conceptual model (Figure 1.2) to illustrate the details

of the analytical development and Monte Carlo test verification. This SEA model has the

40
topology similar to that shown in Figure 1.1. Acoustic power is input to the first cube and

5,000 Hz. is taken as the analysis band center frequency. In the variance analysis, the
plate thickness h is chosen to be non stationary, x1 = h and we calculate the variance of

total energy response in the second acoustic space, 6(E3). A Gaussian distribution of

3

2,000 plate thicknesses with mean value of 7.5 x 10' m and standard deviation of 5% of

the mean, 6(xl) = 3.73 x10.4 (meters), is generated from a MATLAB software call. Note
that the actual standard deviation of the data sample generated does not exactly equal the
requested value for the sample. The histogram of 2,000 Gaussian distributed plate
thicknesses is shown in Figure 4.1.

The variance of the second acoustic space response (4.2) is dependent only on the

variance of the plate thickness because the variances of the input powers are zero.

02(53) = -— 02(xl) (4.3)

The derivative of total energy response with respect to x1 includes only the first
term in (3.7a) because modal density in the second acoustic space is independent of the

plate thickness.

4.4

3x1 3 3x1 ( )
The derivative of the SEA coefficient matrix N with respect to plate thickness is
needed in order to compute the derivative of modal response with respect to plate

thickness. Since the plate thickness is associated only with the plate modal density, n2,

coupling loss factors 1112 and 1123 (3.9) includes only three terms.

 

3N _ 3N 8712 + 8N 31112 + 8N 81123

—— — (4.5)
8x1 8"2 8"I 81112 8x1 at123 8x1
The terms, 1%, 93]— and _B_N_ , can be obtained by using (3.9). The derivatives
8"2 81112 81123

with respect to x, = h at 5,000 Hz of modal density n2 , as well as coupling loss factors

41
11,2 and 1123 can be derived from their definitions (Al, Appendix A).

 

 

 

 

    
 

 

     

 

 

 

 

 

 

3g _ _ ~54 (4 6)
3x1 hZCI '
C 1
38123 :_ PAIR AIR 1 1+ 1 (4.7)
1 1 ” 800]: l——
IOOOOPALURh2(l—m)2 ( 400,1)
81112 :_B_ "21123 zafiflg+flgan23 (4.8)
3x1 3x1 n1 3x1 n1 n1 3x,
Th bo a] ' ' 853 —40 10‘4 d E —15 10"7
ea vean YSlsglVCSa—M-xl=h— . x an o( 3)—. x .
P = P
30° I I I I I
250 ........ ; ......... ; ........ l: ........ g ....... ..
Pm: Thickness . hussian Probability
E200 ....... H. gem. ....stvath ...... ..
E 150 ...................... r E
s t
g 100 ..................... L i
i” so. ................ ;
on 6.5 7 5 l

 

 

111ickness(meters x 10.3)

Figure 4.1 Plate Thickness Histogram and Corresponding Gaussian Probability
Density
Monte Carlo verification of the above variance analysis is started by computing
2,000 SEA model responses from 2,000 separate models, one for each of the 2,000
thicknesses used in the SEA model. The standard deviation of the 2,000 values for total

energy response in the second cube, EjMC, from the Monte Carlo test is

0(E3MC) = 1.4 x10"7 . The histogram for the 2,000 total energy responses computed for

the second acoustic space is plotted in Figure 4.2 along with the predicted distribution

42
(solid line). In order to further compare the analytical results with the Monte Carlo

results, mean value and variance for the actual model responses in the second acoustic

space , E3MC, were used to plot the Gaussian probability density function (broken line) for

the 2,000 model analyses.

250 . fi

 

: : : . ' 4r. 2,000 Model Gaussian
I. . WHY-MW . . . . .5 ....... .z ......

 

  

“8’

150.......E ....... :. ..... ....I.... .. .. ....... I ......

oooooooooooooooooooo

Number of Energy Levels
3
'-

M
O
I

 

 

 

 

 

 

 

 

 

 

 

 

 

0 A : : : :7 7
3.207 3.208 3.209 3.21 3.211 3.212 3.213 3.214 3.215 3.216
Space 2 Acoustic Energy (Joules x 104)

Figure 4.2 Gaussian Analytical Probability Density Function (PDF) Prediction for
Acoustic Energy in Space #2 Compared with Histogram and PDF
for 2,000 Model Monte Carlo Test Results

The variance analysis probability density function result (solid line) agrees very

well with the probability density function computed for the 2,000 SEA model Monte
Carlo test. The linear analytical standard deviation prediction has an error that is less than
5% of the actual standard deviation for the 2,000 models computed from Monte Carlo test
results. This error can be partially attributed to the population of random physical
parameter used in Monte Carlo tests. Figure 4.1 shows that the thickness population is not
large enough (2,000 models) for the random generator used in the computation to
approach a Gaussian distribution accurately. Increasing the number of models generated
in the Monte Carlo tests would have required a prohibitive amount of computational

effort. The remaining error can be attributed to the effects of non-linearities in the

parameters on predicted variance and will be discussed in the second example.

43
The second example is an automotive vehicle SEA model (Figure 2.2 and 2.3). To

demonstrate the variance analysis, the floor panel thickness is chosen to be non stationary
and analysis of the vehicle interior sound level will be conducted at a band center
frequency of 5,000 Hz. Two thousand (2,000) of the Gaussianly distributed floor panel
thicknesses are generated with mean 7.50><10_4 (m) and 5% standard deviation
0.38x104(m). The floor panel thickness histogram and associated ideal Gaussian

probability density function are presented in Figure 4.3. The mean and variance of the

 

    

       

3m T E X ‘l I

250.5uu-oooé eeeeeeee S IIIIIII =.I—I|Ioon-:\Ieoooooos ooooooo n
a ' ' 7‘ 3‘ Gaussian Probébility
é) 2moaooooooto-uuucou:oc alo:bcl 0.. mew ...... I
.2 Floor Panel Thickness: 2 '
E 150 ..... H1539” ..................
O z '
E 100 ...............
3
Z

50 .......................

0 r. o .7 7

6 6.5 7 7.1 8 8.5 0

 

 

 

 

 

 

 

 

Floor Panel Thickness (meters xio‘ )
Figure 4.3 Floor Panel Thickness Histogram and Probability Density Function

floor panel data generated are then used in the variance analysis to compute predictions of
mean and variance of the SEA energy responses. These predictions used formulas
developed in this paper and evaluated the derivative of the interior total energy responses
with respect to the floor panel thickness at the nominal system parameters. The predicted
variance of the interior total energy response at 5,000 Hz. from floor panel thickness
variance is 2.84 x10”12 (Joules). After computing 2,000 of SEA total energy responses
in the interior at 5,000 Hz by our Monte Carlo procedure, actual model interior acoustic
energy responses were found to have a variance of 2.79 x 10—12 (Joules). The predicted
and actual variance differ by less than 2%. The analytical variance prediction requires
only about 1/300 of the computational effort (flops) of that required to compute the 2,000

models responses. The histogram of the interior total energy responses and the Gaussian

44
probability density functions from Monte Carlo test and analytical variance analysis at

5,000 Hz. are shown in the Figure 4.4. The results show that the linear variance

approximations are effective to a complicated automotive vehicle SEA model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

250 I I I I
2,000 Model Gaussian : I "~ I 5
Probability Density 5 , ' g
a, 200 g g J" Analytical Gaussian
.9. .................. E ................... E.... .r..p--~'"1:‘.... .Pfébébifitynéfiéity...uu
E} 150 ........................................ .
O - .
L5 2,000Model I .
“5 Energy Histogram ‘ V
E 100 ................. . . ............................. .
E I
3
Z 50. ........................... 7 r i ...................... -
A
O '__, : : "_
6 6.5 7 7.5 * 5 8 8.5

Interior Acoustic Energy (in Joules x10'll )

Figure 4.4 Histogram and Gaussian Probability Densities from Analytical and
Monte Carlo Tests for Standard Deviation 5% of Floor Panel
Thickness at 5,000 Hz.

The effect of nonlinearities in the SEA coefficient computations on the accuracy of
the Taylor expansion was evaluated by running 8 different Monte Carlo validations for
differing floor panel variances ( of mean value). 2,000 of the physical parameter are
generated from the Gaussian distributions for each of the standard deviation choices, 2,000
total interior energy responses (in Joules) are computed for each of the cases and the
standard deviations for the responses are computed from Monte Carlo test runs and the
analytical standard deviations from linear variance analysis are computed for each of the

cases in Figure 4.5.

45

 

 

 

 

-Il
12“,,on
.2
g” ......................................... I .......
>-. . . . .
”1.2 IIIla-0000:!COIIIIOII:OOIOIOIOII.~ IIIIIIII . ....... q
Lg I Monte'CarloTest I .
U 1 ............................................. ..
'3 I I O I
§0.8 ............................ ,. ................... .4
< I
80_6 IIIIIIIIIIIIIIIIIIIIIIIIIIII .. oooooooooooooooooooo d
g Dmitri/mane:
£04 ooooooooooooooooooooooooooo WWI'W ooooooooooooooo I
"o‘ .
$0.2pooI-I. s\uuon.ooooudooooaonuoo:a ooooooooooooooooooo u
:3 . . . .
.0. 0 . . . .
a 5 IO 15 20 5

Pheroentage Floor Panel Thiclmess Standard Deviation (%)

Figure 4.5 Comparison of Standard Deviations of Response Energy in the Vehicle
Interior at 5,000 Hz. from Analytical Prediction and Monte Carlo
Test due to Deviation in Floor Panel Thickness
The linear variance analysis prediction for a standard deviation of 15% of floor
panel thickness has only a 6% error relative to the actual Monte Carlo test results which
corresponds to only about 0.7% of the mean vehicle interior sound energy of 7.33 x 10’”
Joules computed for the models. Note that a 5-15% floor panel thickness standard
deviation may well be greater than acceptable in any actual production process and is used
here for discussion purposes only. The variance linear approximation is quite effective
and robust to physical parameter variations.
4.4 Conclusions
The analytical variance theory put forth in this paper can accurately predict the
SEA model response statistics, the linear approximation is shown to be robust over a
broad range of the physical parameter variations tested and the effectiveness of the
analytical algorithm is validated by two examples. The algorithm can be applied to
evaluate the robustness of existing vehicle designs to variations of design parameters and
to develop specifications for new designs. In particular the algorithm can also be used in
the early design stages for a prototype. The accurately predicted variances of SEA model

with expected values can be used to predict confidence levels in the design analyses.

46
With those features and applications, the variance theory on SEA vibro-acoustic responses

provides a powerful design tool for quality automotive vehicle designs. Designers can
acquire quantitative information on the relationship between the design SEA responses
and the design physical parameters affecting the mean-rms performance and the variability
in the design performance caused by the variability in the design physical parameters.
This will allow designers to develop specifications to achieve optimal and/or ultimate

robust designs.

Chapter 5 Probability Distribution of SEA Responses"

5.1 Introduction

Quality acoustic designs for aircraft, ships, and automotive vehicles require the
probability analysis of extreme vibro-acoustic response values. Probability distributions
of SEA responses determine the response variances and confidence levels needed to
evaluate the probability of excessive acoustic noise and structural vibration (Lyon, 1987).
Study of variance and confidence level of SEA responses was initiated by Lyon (1967,
1975, 1987). In his study, Gamma response distributions were chosen to compute
confidence levels because they have the desirable feature of strictly positive values. SEA
response variance analysis for automotive vehicle was investigated by DeJong (1985). An
exceptionally large experimental study (156 production cars) of vibro-acoustic response
statistics was conducted by Kompella and Bernhard (1993).

SEA response distributions in frequency bands are proven to be Gaussian for
infinite number of design parameters in this work. Sound and velocity are directly
proportional to SEA responses, their distributions are also proven to be Gaussian. In
engineering applications, the number of design parameters is always finite for SEA
models. A Monte Carlo test and a Statistical Hypothesis test on a simple 3 element SEA
model show that Gaussian distributions are good approximations for responses of an SEA

model with a finite number of design parameters.

 

* This Chapter is based on the paper with the same title to be submitted to Journal of Vibration and
Acoustics.

47

48

5.2 Statistical Distribution of SEA Responses

Distributions of SEA responses originate from randomness of design parameters.
The design parameters have arbitrary distributions. The design parameters are used to
compute SEA responses and the randomness from the design parameters passes to the
SEA responses through SEA equations. The SEA response randomness is characterized
by SEA response distribution. The SEA response distributions can be used for vibro-
acoustic design purposes by computing probability of excessive vibro-acoustic response
and response confidence levels.

The Gaussian distribution derivations for SEA responses have been done by the
following steps. Step 1 makes all necessary assumptions. Step 2 expands SEA responses
by Taylor expansions. Step 3 shows Lyapunov condition in The Central Limit Theorem
for SEA responses is satisfied under these assumptions. Step 4 checks these assumptions
in Step 1 for SEA responses and points out situations for both holding and violating these
assumptions.

Step 1: Assumptions. (1): SEA response derivatives with respect to SEA
parameters and SEA parameter derivatives to design parameters are bounded above. (2):
Third order absolute moments of all design parameters are bounded above. (3): there is
no single statistically dominant variable.

Step 2: SEA Expansions. The SEA responses are non-linearly and implicitly
related to design parameters. The probability distribution derivation of SEA responses

requires SEA response for the ith element at a single frequency, fk, be linearly and

explicitly expressed in terms of design parameters by Taylor Expansions.

 

._ NX
Eu. z EiJc + 2
P=F (5.1)

where Ek is SEA response expected value evaluated at nominal parameters

49

 

N1 is the number of design parameters.
83 . . . .
75—"- 18 SEA response derivatives With respect to design parameters
xm
3E,- k _ 8E,- k _
ym _ a ’ X=- xm and ym : a , X=i xm
xm P=P xm P=P

 

 

The partial derivatives above are evaluated at the nominal design parameters x = i and
nominal power inputs P = P .

Step 3: The Central Limit Theorem (CLT) and Lyapunov Condition. CLT
can be invoked to derive the distributions of SEA energy responses which are sums

consisting of a number of independent random variables with arbitrary distributions (5.1).
The CLT states (Cramér, pg 215, 1963): let y1,y2,...,y,, be independent variables with

arbitrary distributions, finite expected values, and variances 62(yi). If a Lyapunov

Condition:

 

w. 1/ 2M3(Iym—7ml) / Evian—>0 as ...... (5.2)

m=l m=l
where M3 (lym — ml) is the third order absolute central moment of ym and
n is the number of the variables
I! n
is satisfied, the random variable, 2 = 2 y,- 262 (y) tends asymptotically to a Gaussian

i=1

distribution.
The third order absolute moment M3(|y,,,—y,,,|) and variance 62(J’m) of

intermediate variable ym about its mean in, in (5.1) are needed to check the Lyapunov
condition for SEA responses. The third order absolute central moment of ym for SEA
responses can be expressed in terms of the third order absolute moments of xm about its

mean in, using the linearity property of the moments.

8E”,

ax,"

 

3
E] M3(|xm — imD (5.3)

P=P

M3(lym _ yml) =[

 

The variance 02(ym) of y," can be expressed in terms of variance 02(xm) of xm.

50

2
- — ..:] 62(xm) (5.4)

 

The Lyapunov Condition for SEA energy responses can be obtained by substituting (5.3)

 

 

and (5.4) into (5.2).
3 /’ 2
Nx aEi k _ .- Nx aEi k 2
(0 = 2 —' F; M xm - xm / —’— x; O m 5.5
N; "1:1 [ axm P=F] 3(l )/ m2=l l7 axm P=F:l (x ) ( )

 

 

where N x is the number of random design parameters in SEA model

If the upper bound of (ON): in (5.5) approaches zero as N, goes to infinity, (5.5)

will satisfy the Lyapunov Condition (5.2) for SEA energy responses. The Lyapunov
Condition (5.5) is determined by three quantities, SEA response derivatives, 8E“, /8xm ,

design parameter third order absolute central moments M3(|xm—fm|) and

variances,(}'2 (xm ). The absolute values of SEA response derivatives under assumption (1)

are bounded above and therefore have a maximal value.

...{_aEz-.k}-m..
.. _

The absolute values of SEA response derivatives under assumption (1) are bounded below

3E“,
8x,"

 

x:

,, m=1,2,---,N,r (5.6)
1):?

 

 

 

away from zero and have a minimal value obtained by ignoring the trivial case of zero-

SEA parameter derivatives.

0<min 2% -mm —
8x _

An upper bound for the Lyapunov Condition (5.5) can be obtained by replacing the

, m=1,2,---,Nx (5.7)

 

 

 

derivatives in the numerator with the maximal value and replacing the derivatives in the

denominator with the minimal value.

51

05, N. _ , 35,. N.
(0N. :max{max‘k}‘[m 2:] M3(lxm _xml)/mm){—aT’k} mz=102(xm)
C'méi/ith 3(,n|x —xm|) )/,’N 2:02“
whereCmm - max{a—axk}/mi nlraka;

The bounds of the variances and moments can be used to further simplify the

 

 

 

(5.8)

upper bound of (0N: (5.8). The third order absolute moments of design parameters,

M3(|xm - TM), are bounded above under assumption (2). Therefore the third absolute

central moments have a maximal value. Zero variance variables are not random hence the
zero-variance variables can be excluded from the moment and variance calculation. For
random variables, the variances are bounded below away from zero and have a non-zero
minimal value. The upper bound for the Lyapunov Condition (5.8) can be simplified by
replacing the summation for the third absolute central moments in the numerator with the
product of the moment maximal value and the number of the design parameters. The
summation for the variances in the denominator is replaced with the product of the non-

zero variance minimal value and the number of the design parameters.

 

 

m”. _<. Cm Vmflax{M3(|xm — J?,,,|)}NJr /\/n’n’;n{02(xm )}Nx
= DM/VN—x

where Dmm = Cmm {/mflax{M3(|xm — fm|)}/\Finin{62

The Lyapunov Condition for SEA energy response in (5.9) has a constant Dmm In

(5.9)

 

 

the numerator and N x in the denominator. Letting N x go to infinity, the constant term in
the numerator remains unchanged and the denominator, Q/ 1v,r , goes to infinity, so The

Lyapunov Condition, (0N; —-> O. The Lyapunov condition is satisfied which guarantees

that probability distributions of SEA responses, EM, tend asymptotically to Gaussian

 

distributions with a mean of EM and a variance of Nx[i’k
m=l

52

Sup 4: Assumption Check on SEA Response Derivatives. The SEA modal

response derivatives with respect to SEA parameters, 38,31, /8x,,, , in the first term in (3.7a)

are the product of three terms: N71, aN/é’xm (xm =n,-,n,- and nij) and e (3.8). For a

strongly connected SEA model, N is positive definite, diagonally dominant and N—1
exists and is diagonally dominant (Hodges, et al, 1987, Huang and Radcliffe, 1993). All
entries in N are linear combinations of modal densities, internal loss factors and coupling
loss factors. These entries are finite for finite frequencies. The entries in N are bounded
by their diagonal entries and so are the entries in N "1. The entries in the derivatives are
sparse and functions of modal densities, internal loss factors and coupling loss factors
(3.9). For this reason, the entries in aN/axm are bounded for finite frequencies. The
modal energy response, e, is associated linearly to power inputs which drive the systems.
The modal energy response, e, is bounded over finite frequency bands for bounded power
inputs. The modal density derivatives with respect to SEA parameters are either 1 or zero.
Therefore, the SEA response derivatives with respect to SEA parameters in (3.7a) are
bounded.

Most SEA parameter derivatives with respect to design parameters exist and are
continuous in a closed interval in which the design parameters are defined and hence
bounded. Some SEA parameters, such as structural-acoustic coupling loss factor (Lyon,
1975), have jumps at a critical frequency and their derivative at that criticallfrequency
does not exist, however, the right-side and left-side derivatives exist and are bounded.
The analysis frequency band can be divided into two closed sub-intervals separated by the
critical frequency. The SEA parameter derivatives are continuous in the closed sub-
intervals and hence are bounded. One can conclude in general that SEA parameter
derivatives with respect to design parameters are piece-wise continuous in closed
intervals and bounded. Although bounds do not exist at critical frequencies, SEA
response derivatives are bounded in piece-wise closed intervals for finite frequencies.

Assumption (2) sets an upper bound for the third order absolute moments of design

53

parameters. The larger the upper bound the slower the Lyapunov Condition goes to zero.
This feature has significance in engineering applications. There is never an infinite
number of design parameters in any SEA engineering models. Gaussian distributions
derived here are only an approximation for SEA responses in engineering applications.
Small third order absolute moments mean small deviations among design parameters.
Small deviations from nominal design parameters will lead to good approximations to
Gaussian distributions. In other words, for small third order absolute moment upper
bounds, only a small number of random design parameters are needed to achieve good
approximations to Gaussian distributions. Assumption (3) guarantees that no single
random design parameter, xm, in (5.1) is statistically dominant. Otherwise, the SEA
response distributions will follow the distribution of the dominant random variable in
(5.1).
5.3 Distribution of Total Pressure and Velocity

Distributions of total mean squared (ms) pressure and velocity, p? and vi2 in (1.5),
will have a normal distribution if the ms pressure and velocity frequency components p5,,
and via have a normal distribution because the total pressure and velocity are linear

combinations of independent pressure and velocity frequency components, p5,, and vi”.

The ms pressure and velocity frequency components are linearly associated with SEA
energy responses (1.4). The method used to derive the approximate distribution of SEA
energy responses in the previous section can be used to derive the distributions of the ms
sound pressure and velocity.

The distributions of the ms pressure and velocity frequency components require
the linear approximation of the ms pressure and velocity frequency components in terms
of the random design parameters. Taking the structural velocity frequency response

component as an example, , and assuming the first design parameter x1 = m, the element

mass, the second equation in (1.4) can be written as:

54

E-
v3k(fk,x) = f (5.10)

The Taylor expansion on (5.10) yields an approximation to the ms velocity.

2 ..:—.2 i x—i'k
Vi,k(fk'x) v'-k(fk’x)+[ Eraxm]

 

_ E- k _
F— [xm — x,,,]—[-"T]l,=i [xl — x1] (5.11)
x1 p=fi

Comparing the ms velocity derivatives (5.11) to the SEA energy response derivatives
(5.1), one finds that the ms velocity derivatives (5.11) can be obtained by dividing the

SEA energy response derivatives (5. 1) by a factor of f, , the nominal mass of the structural

. E:
element, and subtracting a term, 3:7" F; [x1 —3c'1], from the SEA energy response
1 P=P

derivative with respect to x1 2 mi. Both the factor and the additional term are bounded.
Therefore the derivatives of the ms structural velocity are bounded above and below away
from zero. The assumptions on the variances and the third order absolute central moments
of x are the same. The conclusions drawn from the bounded derivatives, variances and
moments on x from the previous section can be applied to (5.11). One can conclude that
the ms velocity frequency components, v3,“ have Gaussian distributions. The same
argument can also be applied to the ms pressure frequency components and they also have
Gaussian distributions as the number of design parameters go to infinity.

The distributions of the ms pressure and velocity integrated over all frequencies are
also Gaussian because they are the linear combinations of the ms pressure and velocity
frequency components. The linear combination of Gaussian distributions of ms pressure
and velocity frequency components yield Gaussian distributions for ms pressure and
velocity.

5.4 Monte Carlo Test

Monte Carlo analysis of a simple model (Figure 1.1 and 1.2) is used here to test the

theoretical results. The Gaussian distributions are only approximations for SEA responses

distributions. The reasons are that the Lyapunov Condition for SEA responses in

55

engineering applications will not be zero because there will never be a infinite number of
design parameters in any practical SEA model. The Lyapunov Condition is close to zero
for large numbers of design parameters in an SEA model and this feature will be
demonstrated in an SEA model example.

The SEA model example has 18 design parameters and 1 power input. Power
inputs are set inversely proportional to band center frequency: Pi = ff” x10"4 Watts.
Formulas used to calculate SEA parameters are from Lyon (Appendix A). Geometric
parameters (Table A1, Appendix A), such as lengths and thicknesses are assumed to have
Gaussian distributions with design values as means and 5% of the design values as their
standard deviations. All other parameters, such as mass densities, speeds of sound,
reverberation times, damping ratio, and input powers et al, are assumed to have uniform
distributions in intervals of i10% from their nominal values (see Appendix A). Third
octave Band center frequencies range from 125 Hz. to 10,000 Hz. in 20 bands. 2,000 SEA
models for each band are generated from these distributions and 2,000 total mean-squared
acoustic pressures and structural velocities are computed from the SEA response statistics.
The statistical population of SEA responses is used to verify the hypothetical Gaussian

distributions and analytical standard deviation computations of SEA responses.

0.95

 

   
 
    

Plate Velocity

0.9 252.30...
0.85 - Space 1 Sound Pressure

0.8 -

 

Space 2 Sound Pressure

 

0.75

 

0 5000 10000
Third Octave Band Center Frequency (Hz.)

Figure 5.1 Lyapunov Condition Terms for SEA Responses

56

The Lyapunov Condition terms for each analysis band are shown in Figure 5.1.
The maximum Lyapunov Condition terms are 0.832 at 10,000 Hz for space 1 sound
pressure, 0.810 at 5,000 Hz. for space 2 sound pressure and 0.916 at 125 Hz for plate
velocity in Figure 5.1. The Lyapunov Condition terms are non-zero for the finite number
of random design parameters in contrast with the theoretical infinite order analysis. The
finite order Lyapunov Condition terms are close to zero and the distribution from Monte
Carlo test will be approximately Gaussian because of (l) the finite number of design

parameters, N x =19, in the SEA model, (2) the finite number, ch = 2,000, of Monte

Carlo tests. Note that the Figure 5.1 shows that the Lyapunov Condition terms for

 

 

 

 

 

 

 

 

 

 

 

 

structural plate are
400 400 .
7”“
200 — -« 200 ~ -
0 0 ,
0.8 1 1.2 1.4 4 6 8 10

Figure 5.2 Space 1 ms Sound Pressure @
1,000 Hz. Histogram

 

 

 

 

 

 

 

(x104Pa2)
400
r“?
200 - 4
O .
1 2 3 4

Figure 5.4 Plate ms Velocity @ 1,000 Hz.

Histogram (><10‘4 mm2 /sec2)

Figure 5.3 Space 1 Total ms Sound
Pressure Histogram

 

 

 

 

 

 

(x10'3 Paz)
500 . f
:
0 .
0.5 1 1.5 2

Figure 5.5 Plate Total ms Velocity
Histogram( x10’1mm2 lsecz)

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

400 500 .
FR 77.
200— —
0 - - . , 0 .
0.5 1 3 4 5 6 7

Figure 5.6 Space 2 ms Sound Pressure @ Figure 5.7 Space 2 Total ms Sound
1,000 Hz. Histogram Pressure Histogram

(x104Pa2) (x10'3Pa2)

always larger than those for acoustic spaces. One can expect that the test data for spaces
closer to Gaussian distribution than those of the plate.

Histograms for ms acoustic pressure at 1,000 Hz and total ms acoustic pressure
and velocity for spaces and structural velocity for the plate are shown in Figures 5.2 to 5.7.
The bars represent the histograms from Monte Carlo test runs and the solid lines represent
the hypothetical Gaussian distribution with estimated means and standard deviation from
the Monte Carlo samples. The histograms show that the mean-squared acoustic sound
pressure and structural velocity distributions are well approximated by Gaussian.
5.5 Hypothesis Test

A Hypothesis test is used to examine the agreement between hypothetical Gaussian
distributions and distributions from the Monte Carlo tests of the SEA model. The
objective of the test is to find if deviations of hypothetical distributions from sample
distributions are small enough to accept the hypothesis. The deviation is quantified by the
normalized ms difference between cumulative frequency from the sample and from the
hypothesis. 12 will be zero if the samples come from the true hypothesis distribution.
R
12 = 21117,. — NM P(A,))’/Nm P(A,.) (5.12)

I

where Nm is the number of tests

58

A, is a non-overlapping data interval
P(A,.) hypothetical probability that data fall into the internal A,
F ,- is the number of data falling into A, from Monte Carlo test and
R is the number of interval A,
The 12 hypothesis test was introduced by Pearson (Cramér, 1963) who shown that,
when ch is large enough, statistical quantity 12 obeys xz-distribution of degree of
freedom R— r—l (r is the number of parameters estimated from the sample). Let 1?,

denote the (1% value of xz-distribution for degree of freedom R— 1. The probability

P = F(xz > 13,) will, for large N be approximately equal to 01%. The probability is

explained in two ways. First, if the value x2 > x3, is found, the samples show a significant
deviation from a hypothesis and the hypothesis shall be rejected at least until further
experience had made it plausible that the deviation was due to random fluctuations.
Second, the probability that the hypothesis is actually true and is falsely rejected is
precisely the probability P = P()(2 > xi). On the other hand, if value x2 S x: is found,
then distribution from Monte Carlo test samples has no significant deviation from the
hypothetical distribution and the hypothesis is accepted at the significant level a.

Hypothesis that SEA model response distributions are Gaussian is verified from

the 12 hypothesis test. The frequency F,- and response data interval A. are determined by
the Monte Carlo test histogram results of SEA responses. The median x,- of the interval
A, and the frequency F,- are used to estimate the mean and standard deviations of the
hypothetical Gaussian distribution by the maximum likelihood method. The estimated
expected mean and standard deviation are used to evaluate the Gaussian distribution and
to compute the probability P(A,-).

A typical test result for total ms sound pressure in Space 2 is listed in Table 5.1.
2,000 (ch = 2,000) total ms sound pressure in space 2 from Monte Carlo test are divided
into 15 intervals A. . in the second column. The mean and standard deviation for the ms

sound pressure are estimated from the medians x,- of intervals A; in the second column

 

59
and the observed frequency in the fifth column in Table 5.1.

 

 

l — 1 '5 -3 2
nzz—gFix, =4.909x10 (Pa)
1 1 ’ 15” (5.13)
-— _— . .2 _ — — “4
.S _ J2,000(,§1(F‘x‘ ) 2.00011) _ 4.257 x 10

The estimated mean and standard deviation are used to compute hypothetical probability

PM.) and hypothetical frequency ,NmCP(A,-), in the fourth column in Table 5.1. Since
Pearson Theorem is derived under the assumption that ch is large enough and chP(A,-)

should not be small (in general chP(A,-) should be larger than 10), the 11th to 15th row

in the fourth and fifth columns are combined to form the eleventh row in the sixth and

seventh column. The combined hypothetical and observed frequencies in the sixth and

seventh column are used to compute the contributions in each interval, A, , to the x2 in

Table 5.1 xz-Test for Space 2 Total ms Sound Pressure

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

interval] response probability hypothetical observed hypothetical observed contributiorl
interval
number Ai (x10’3) F(Ai) chP(A,') pi (combined) (combined) to X2
1 3654-3880 0.006171 12.342 7 12.34 7 2.312
2 3880-4. 107 0.02229 44.58 44 44.58 44 0.007546
3 4.107-4.334 0.05846 1 16.92 122 1 16.9 122 0.2207
4 4334-4561 0.1 176 235.18 231 235.2 231 0.07429
5 4561-4787 0.1798 359.6 387 359.6 387 2.088
6 4.787-5.014 0.2128 425.6 419 425.6 419 0.1023
7 5014-5241 0.1836 367.2 363 367.2 363 0.04804
8 5241-5468 0.1226 245.2 235 245.2 235 0.4243
9 5.468-5.694 0.06222 124.44 129 124.4 129 0.1671
10 5.694-5.921 0.02422 48.448 45 48.45 45 0.2454
11 5921-6. 148 0.006849 13.698 14 17.31 18 0.02739
12 6148-6375 0.001516 3.0322 1
13 6.3750-6.602 0.0002564 0.51288 1
14 6.6026828 3.1219E-05 0.062438 1
15 6828-7055 2.939lE-06 0.0058782 1
sum 0.9984 1997 2000 5.717

 

 

60

the last column. The normalized difference x2 is the summation Of all rows in the last
column, x2 = 5.717. There are R =11 intervals after combination and r=2 parameters
estimated. The degree of freedom for the 12 distribution is R— r— l=11-2-1=8.

Tabulated values (Cramér, 1963) gives x35 =7.344>x2=5.717. The inequality

indicates that the hypothesis is accepted at significant level 50%.

Table 5.2 Hypothesis Test Result Summary for Spaces

 

 

 

 

 

 

 

 

 

 

 

 

 

Space 1 Space 2
x2 x: (d.o.f) a xz x: (d.o.f) 01
500 Hz. 7.297 7.584 (11) 0.75 5.709 8.383 (7) 0.3
1,000 Hz. 9.894 12.55 (10) 0.25 7.855 8.343 (9) 0.5
5,000 Hz. I 9.268 9.342 (10) 0.5 8.493 10.66 (9) 0.3
Total ms SP (20 band) II 6.784 8.343 (9) 0.5 5.717 7.344 (8) 0.5

 

The hypothesis test result for spaces is summarized in Table 5.2. The results show
that the hypothesis that ms sound pressures in frequency bands and ms sound pressures
obey Gaussian distribution is accepted.

A typical hypothesis test for total ms structural velocity of the plate, composed by
20 frequency bands, is listed in Table 5.3. The computation procedure and table contents
are the same as Table 5.1. The normalized difference x2 = 43.33 and the hypothesis
should be rejected. However detailed examination of the contribution shows that the main
contributors to x2 (60%) come from the first and last row. In other words, these
contributions come from the neighborhood of 3-0' region. This phenomenon can be
illustrated by plotting the cumulative percent frequency of the total ms velocity of the
plate on a normal probability paper (Figure 5.8). The normal probability paper is designed
such that all cumulative percent frequency points from a normal distribution lie in a
straight line(Volk, 1969). The horizontal axis in Figure 5.8 is the response intervals, the

vertical axis is the cumulative percent frequency and the straight line is determined by two

CU]

Cd!

162

C11

61

points: 50% at mean value and 84.13%. It can be seen from Figure 5.8 that most
cumulative percent frequency points are close to the straight line and the points near two
edge sides are farther from the straight line than other points. Overall the total ms
velocity of the plate can be considered to have a Gaussian distribution. Therefore it is

reasonable to exclude the contributions from the first and last row without causing large

errors and compute xz = 18.22.

2
Table 5.3 X -Test for Total ms Structural Velocity of the Plate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

interval response probability hypothetical observed hypothetical observed Icontribution
number interval (x104) F(Ai) chP(A,') F.- (combined) (combined)! to 12
1 0870-0929 0.002778 5.560 1
2 0.9290989 0.01043 20.87 10 26.43 1 1 9.005
3 0989-1029 0.03067 61.34 51 61.34 51 1.743
4 1.049-1.108 0.07053 141.1 154 141.1 154 1.187
5 1.108-1.168 0.1269 253.8 274 253.8 274 1.608
6 1. 168-1228 0.1787 357.4 376 357.4 376 0.9680
7 1228-1288 0.1972 394.4 430 394.4 430 3.213
8 1.288-1.347 0.1702 340.4 308 340.4 308 3.084
9 1.347- 1 .407 0. 1 151 230.2 202 230.2 202 3.455
10 1.497-1.467 0.06087 121.7 1 15 121.7 1 15 0.3732
1 1 1.467-1.526 0.02521 50.42 39 50.42 39 2.587
12 1526-1586 0.008165 16.33 25 21.42 40 16.11
13 1.586-1.646 0.002072 4.143 9
14 1646-1 .705 0.00041 1 1 0.8221 3
15 1.705-1.765 6.381E-05 0.1276 3
sum 0.9993 1999 2000 43.33

 

 

 

 

62

1

 

° 0.901.02 1.14 1.26 1.38 1.50 1.62 1.74 x10'
11:1.25 n+§=137

Figure 5 .8 Normal Cumulative Graph of Total ms Velocity of Plate

Table 5.4 x2 Test for Structural Velocity of The Plate

II plate
2 x: (do. f ) a

 

 

 

 

 

X
500 Hz. 16.42 16.81 (6) 0.01
1,000 Hz. 17.45 19.68 (9) 0.02
5,000 Hz. 18.89 22.43 (6) 0.001

 

 

 

 

 

 

Total ms Vel. (20 band) 18.22 22.43 (6) 0.001

 

The same pattern in Figure 5.8 is shown by other data for the plate. Based on the
reasons above, eliminating the contributions from the vicinity of 3-0' region yields

hypothesis test result summary for the plate in Table 5.4. Note that the structure plate

 

63
failed to pass the hypothesis test. The reason for the failure is that the Lyapunov condition

terms for structures are larger than those for spaces.
5.6 Conclusions

Distributions of SEA responses are proven to be Gaussian regardless of
distributions of design parameters in this paper. A Monte Carlo test on a three element
SEA model was conducted and 2,000 SEA models are generated for each of 20 frequency
bands. A statistical hypothesis test was conducted based on the statistical population of
the SEA responses. Test results show that SEA responses and total acoustic pressure and
structural velocity from SEA obey approximately Gaussian distributions. The results also
show that the Gaussian distributions are better approximated to distributions of sound
pressures for acoustic spaces than those of the structural velocity for the plate. The
reasons are that plate element has a constant modal density (A1, Appendix A) and SEA
method works better for high modal density. The Gaussian distributions of SEA
responses can be applied to evaluate the confidence levels of the existing acoustic designs
and develop specifications for new designs. With these features and applications, the
Gaussian distributions for SEA responses provide a powerful design tool for quality

acoustic designs for aerospace, ship and automotive vehicles.

1.- 3.551-533“ ‘
' A

 

Chapter 6 Summary and Conclusions

6.1 Dissertation Summary

This dissertation develops a parameter identification algorithm for internal loss
factors. There are no analytical expressions available for damping and they must be
determined experimentally. lntemal loss factors in each frequency band can be uniquely
identified by solving inverse SEA equations given measure acoustic sound pressure and
structural velocities in an SEA model. The least square method is then used to determine
frequency-independent acoustic reverberation time and structural damping by minimizing
the ms errors between identified and computed internal loss factor in each frequency band.

This dissertation presents a sound pressure sensitivity methodology for SEA.
From the design point of view, SEA needs the power of not only computing the
variability of sound pressure level of an SEA model to variability of physical parameters
but also determining which design parameters cause the maximum sound pressure level
variation so that those design parameters can be selected which reduce the overall sound
pressure level with the least effort. The methodology discussed here allows quantitative
prediction of sound pressure sensitivity to variations in SEA design specification
parameters. and is able to determine the maximum sensitivity of the sound pressure levels
to the design parameters.

This dissertation develops a variance analysis method for SEA to extend the design

application of the SEA method through prediction of the variances of RMS responses of

vibro-acoustic structures and interior spaces from variances in SEA model physical

 

65
parameters. This analytical ability would allow variance analysis of sound and vibration

response to the variances of design specifications.

This dissertation shows that probability distributions of SEA acoustic pressure and
structural velocity responses are Gaussian. The Gaussian distributions characterize the
random features of the sound pressure and velocity and provides a means for predictions
of response variances and confidence levels.

6.2 Conclusions

The SEA methodology has been extended in this dissertation to include optimal
internal loss factor identification, sound pressure sensitivity analysis, SEA response
variance analysis and statistical distribution analysis. These new developments and
features increase the power of the SEA method as a tool for vibro-acoustic design and

modeling at audio frequencies.

 

APPENDICES

 

APPENDIX A SEA EQUATIONS

(l) Modal Density Equations (Lyon, R.H., 1975):

 

 

Plate Structure:
n2 2 [3—A (A1)
hc,
where A is the area of the panel (m"2)
h is the thickness of the panel (m)
c, is longitudinal wave speed in the panel (In/s)
Acoustic Space:
2
4nf V flfAs Le
n = n = + + — (A2)
1 3 c3 2c2 80
where f is the analysis band center frequency (Hz.),
V is volume of the space (m"3)
As is the surface area of the space (m"2)
Le is the edge length of the space (m)
c is the speed of sound in air (m/s)
(2) Coupling Loss Factor Equations (Lyon, R.H., 1975):
From Plate to Acoustic Space:
'22,} . -1 f 1,2 f < f.
T51“ (—) 16.
pcA 71' A f c
7721:7733sz ‘ — 2 (A3)
,, I, _ .:..I ,
L f f > fc

 

Where p is fluid density (kg/m"3).

c is the speed of sound in the fluid (m/s),
66

 

67
A is the area of the panel (mA2),

P is the perimeter of the panel (m),

M p is the mass of the panel (kg)

f C = g; is the critical frequency (Hz.)

Ac 2 76- is the wavelength of sound at the critical frequency (In)
C

1 for simple edge supports
B = 2 for clamped edge supports

V2 for typical mounting conditions

2 C for structures

17. =1 2 2 (A4)
° for acoustic space

 

 

LTR k
where C is damping ratio of flat plate

TR is reverberation time in acoustic space

For the flanking path between Engine Compartment and Over Car Volume, a

constant coupling loss factor is assumed.

1726 = 0.0005 (A5)

Table A1 Geometric and Material Properties Used in Example 1

ave
1n
m
m

In ununum
Ol'l rme 1n
OI'l Ime 1n

actor

 

 

APPENDIX B GEO METRO MODEL DATA

B.1. GEO METRO Design Parameters

There are eight (8) elements in the EGO SEA Model. Front of Dash (plate) is
numbered as 1, Engine Compartment (acoustic volume) is numbered as 2, Hood (plate) is
numbered as 3, Under Car Volume (acoustic volume) is numbered as 4, Floor Panel
(plate) is numbered as 5, Over Car Volume (acoustic volume) is numbered as 6, Body

Panel (plate) is numbered as 7 and Interior (acoustic volume) is numbered as 8.

Table B1 List of Design Parameters for The GEO SEA Model Body Structures

 

 

 

 

 

 

element thickness surface area mass edge longitudinal length width Young's
number length wave speed modulus
(m) (m?) (kg) (m) (tn/S) (m) (m) (GPa)
Front 1 0.0077 0.6368 10 3.1496 5181.7 1.0414 0.5334 190
of Dash
Hood 3 0.0030 1.2876 40 4.4958 5181.7 1.3208 1.0541 190
Floor 5 0.0038 4.1 l 1 180 8.4836 5181.7 2.7432 1.4986 190
Panel
Body 7 0.0038 9.7711 340 53.996 5181.7 3.12 3.12 190
Panel

 

 

 

 

 

 

 

 

 

 

 

 

Note that the thicknesses for structural elements in the GEO SEA model are the
effective thickness which are obtained by multiplying the nominal thicknesses with an
effective factor. The effective factor is introduced to take care of local stiffener, add-on

structures and curvatures on plates. The effective factor here is taken as 2 to 5.

68

69

Table B2 List of Design Parameters for The GEO SEA Model Acoustic Spaces

 

 

 

 

 

element volume surface area! edge medium speed of
number length density sound
(m"3) (mAZ) (11)) (kg/m“) (m/s)
Engine 2 0.6368 10 3. 1496 1.244 344
compartment
under car 4 1.2876 40 4.4958 1.244 344
volume
over car 6 4.1 1 1 180 8.4836 1.244 344
volume
interior 8 9.771 1 340 53.996 1.244 344

 

 

 

 

 

 

 

 

 

B.2. GEO SEA Model Connectors

There are 16 connectors in the GEO SEA Model. There is a flanking path between
engine compartment and over car volume. The plate to acoustic space connectors are
listed in Table B3, the plate to plate connectors are list in Table B4 and the space to space

connector is listed in Table B5.

Table B3 List of Connectors between panels and spaces for The GEO SEA Model

element numbers Fixation [5
between a connector

1 ,2
l ,8
2,3
2,7
3,6
4,5
5,8
6,7
7,8

2
2
2
2
2
2
2
2
2

 

70

Table B4 List of Connectors between panels for The GEO SEA Model

element number connection angle connection
between a connector m

1 1.3462

1 1.0414

1 7 2.0
7 2.0828
7 6.985

 

Table B5 List of Space to Space Connector for The GEO SEA Model

 

 

element number coupling area sectional area
between a connector (mAZ) gmAZ)
I 2.4 | 90 | 6.985 |

 

B.3. GEO METRO Test Results
The body was excited by 0.9 V amplitude random noise signal from HP 35660A
Dynamic Signal Analyzer respectively. The accelerations for structural elements are in

dB referring to 1 g and SPL for acoustic spaces has standard pressure reference
Pref = 20uPa.

Table B6 List of Measured Acceleration and SPL of State 1 for GEO SEA Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

element number
1 2 3 4 5 6 7 8
third octave band dB dB dB dB dB dB dB dB
center frequency (Hz.)

500 -29.35 64.99 -53.94 57.89 -56.75 58.50 -61.02 66.66
630 -28.20 63.93 -52.54 60.07 -53.29 59.27 -58.69 64.46
800 -23.18 68.51 -45.92 62.61 -48.00 63.38 -56.96 67.52
1000 -16.95 75.52 -42.62 70.49 —43.48 71.49 -51.54 69.47
1250 ~15.58 77.69 -39.55 70.53 -40.45 71.85 -52.85 70.60
1600 -8.89 82.31 34.34 75.22 -31.54 75.76 -50.68 72.80
2000 -20. 14 71.29 48.66 64.25 —45.58 63.74 —61.72 60.04
2500 -30.72 68.41 -54.41 63.19 ~54.04 58.34 -63 .46 56.33
3150 ~32.65 82.91 -44.72 73.49 —49.40 70.55 -59.59 60.79
4000 -24.05 76.20 -50.78 69.01 -53.47 65.13 -61.81 58.53
5000 -26.55 80.82 -49.32 75.02 -54.51 69.96 -60.37 59.36

 

 

Table B7 List of Measured Acceleration and SPL of Operating State 2 for The GEO SEA

71

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model
element Dash Panel Interior
third octave band dB dB

center frequency (Hz.) (re 1 E)

500 -28. 13 68.13

630 -25.25 66.49

800 ~28.47 63.57

1000 -22.92 64.83

1250 -l9.53 67.46

1600 -l 1.61 70.26

2000 -26.40 59.88

2500 -34.29 52.96

3150 -34.79 51.28

4000 -32. 18 51.65

5000 -33.23 52.82

 

B.4. Model Predictions and Comparison

 

—I— predicted mean 0 measured mean

 

 

 

t'n
C?

8.
i
1
1
1
1
1
1
1
1
1
1
1
1
1
1

L:
c;
l

3;
1
!
i
!
!
!
!
!
!
1
1
l
l
l
l
I
I
l
i
i
i
i
I
I
!
!
!

Velocity in dB (re 1 m/s)
8 8
1 I
1
1
!
!
! .
1 1
l .
1 1
! !
! .
! !
l !
l
l l
l 1
l .
I l
l .
I l
l i
i i
1 i
i i
i i
i
1 1
. 1
1 1
1 1
I
1 1

 

l
j t i

500 630 800 1000 1250 1600 2000 2500 3150 4000

I
p—e
—e

11-

III-

Third Octave Band Center Frequency (Hz.)

Figure B1 GEO Front of Dash Velocity Prediction and Measurement

72

 

——o— predicted mean 0 measured mean I

 

r.--_._._._._.-...-.-.-.--..._._._.-...--_._._._..._._ ...........
measured srn SPL = 88 dB predicted ms SPL = 86 dB

é

3}
I
I
i
I
I
!
I
1
i
I
I
I
I
:
i
I
I
I
I
1
i
1
1
1
I
1
1
1
1
l
I

\1
o_

SPL in dB (re 2X10"(-5)Pa)
3. °<=f

 

u.
9.

 

Y

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000

3

Third Octave Band Center Frequency (Hz.)

Figure B2 GEO Engine Compartment SPL Prediction and Measurement

 

 

    
   

I —o_ predlcted mean . measured mean

A ___________________________________________________________
E -8 measured ms velocity = -94 dB (re 1 m/s)
Ft _ .............................................................
é predicted ms velocity = -91 dB (re lmls)
% -l ''''''''''''' --------------------------—-
5 -11--I ----- ~+ ------------------------------------------------------
.1?

5 -12 -------------------------------------- —-—-—---—- ......
o

> _13 ...............................................................

 

-1 I I ‘r I i I i 3 : ‘.‘ .
500 630 800 1000 1250 1600 2000 2500 3150 4000 5000

Third Octave Band Center Frequency (Hz.)

Figure B3 GEO Hood Velocity Prediction and Measurement

 

73

 

 

—°— predicted mean 0 measured mean

 

 

 

 

 

1600 2000 2500 3150 4000 5000

500 I 630 800 1000 ' 1250
Third Octave Band Center Frequency (Hz.)

Figure B4 GEO Under Car Volume SPL Prediction and Measurement

 

 

—0— predicted mean in dB 0 measured mean in dB

 

 

 

’5? -3o-~--- ---------------------------------------------------------
E measured ms velocity = ~90 dB (re 1 m/s)
~ _901— ——————————————————————————————————————————————————————————
g ' predicted ms velocity = ~90 dB (re 1 m/s)
$400" "' '''''''''''''''''''''''''''''''''''''''
5-11 - --- ------------------------ - —------- -------------
“g-.. ---------------------------------------------- -- ---
E

-13 -—- -------------------------------------------------------

-14 . . 4. I I . . . I

 

500 630 800 1000 1250 1600 2000 2500 3150 4000
Third Octave Band Center Frequency (Hz.)

Figure B5 GEO Floor Panel Velocity Prediction and Measurement

74

 

 

 

 

 

 

 

 

 

 

   

 

 

 

{.5 —°— predicted mean 0 measured mean I
k
"a 90 - -------------------------------------------------------
2’ measure ms SPL = 80 dB predicted ms SPL = 81 dB
2 80‘
i5.
70'
35
5 60'
.5 50'
1.:
A: 40'
m
30 I I I I I I I f I I i
500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
Third Octave Band Center Frequency (Hz.)
Figure B6 GEO Over Car Volume SPL Prediction and Measurement
—°-— predicted mean 0 measured mean J
a 90 ——————————————————————————————————————————————————— I —————————
E - measured ms velocity = -102 dB (re 1 m/s)
fl ———————————————————————————————————————————————————————————
a '1 predicted ms velocity = -102 dB (re 1 m/s)
5 -1 1 '''''''''''''''''''''''''''
-12 ---------------------
.5
£143 --------- "‘"f‘"
g -1. ------------------------------------------------------------
> -15 --------------------------------------------------------------
-16(r ‘ . 4 . .

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000

Third Octave Band Center Frequency (Hz.)
Figure B7 GEO Body Panel Velocity Prediction and Measurement

 

75

 

 

     

 

—0— predicted mean 0 measured mean

a 90...... —————————————————————————————————————————————————————
{a measured ms SPL=78dB
2" 80‘”'-"'"-"""'“"'-"‘""""'""'"""'"""'. """""""""""
c pred1cted ms SPL: 78 dB
g 701- ----------------------------------
N

.11 ...—- m1—
% sow-- ................................................. ..
.5

A 40:11..“ ———————————————————————————————————————————————————————
m 30 I I I ‘ ‘ i L i 4

 

500 630 800 1000 1250 1600 2000 2500 3150 4000 5000
Third Octave Band Center Frequency (Hz.)

Figure B8 GEO Interior SPL Prediction and Measurement

BIBLIOGRAPHY

BIBLIOGRAPHY

Bies, DA. and Hamid, S., 1980, In Situ Determination of Loss and Coupling Loss
Factors by The Power Injection Method, Journal of Sound and Vibration,
Vol. 70, No. 2, pp.187-204,

Clarkson, BL. and Pope, R.J., 1981 Experimental Determination of Modal
Densities and Loss Factors of Flat Plates and Cylinders, Journal of Sound

and Vibration, Vol.. 77, No.4, pp. 535-549

Clifford, A.A., 1973, Multivariate Error Analysis, A handbook of Error

Propagation and Calculation in Many-Parameter Systems, Halsted Press.

Cramér, H., 1963, Mathematical Methods of Statistics , Princeton University Press,

Princeton

DeJong, R. G., 1985, "A Study of Vehicle Interior Noise Using Statistical Energy
Analysis, " SAE Paper No. 850960

Frank, P. M., 1978, Introduction to System Sensitivity Theory , Academic Press,

New York

Ghering, W.L. and Raj, D., 1987 "Comparison of Statistical Energy Analysis
Predictions with Experimental Results for Cylinder-Plate-Beam
Structures," Statistical Energy Analysis, Edited by Hsu, K.H. and Nefske,
DJ. and Akay, A., NCA-Vol.3, ASME, N.Y., NY.

76

 

77

Hodges, C.H., Nash, P., and Woodhouse, J., 1987, " Measurement of Coupling
Loss Factors by Matrix Fitting: An Investigation of Numerical Procedures,"

Applied Acoustics, Vol. 22, pp. 47—69

Huang, X.L. and Radcliffe, C.J., 1993, “Sensitivity of The Statistical Energy
Analysis for Automotive Vehicles”, Paper No. WAM93-18, The 4’h
Symposium on Advanced Automotive Technologies of the 1993 ASME
Winter Annual Meeting, New Orleans, Louisiana, USA

Jacobs, R.H., Yoerkie, C.A., Gintoli, P.J., and Manning, J.E., 1989, “Statistical
Energy Analysis Modeling of the Sikorsky ACAP Helicopter,” Analysis
Techniques in Acoustics Edited by Hsu, K.H., and Keltie, R.F., NCA-
Vol.9, ASME, N.Y., N.Y.

Jenson, J .0. , 1979, "Prediction of Noise in Ships by the Application of "Statistical
Energy Analysis," J. Acoustic. Soc. Am, Vol. 66(Sl), S4(A).

Kompella, MS and Bernhard, R.J., 1993, "Measurement of the Statistical
Variation of Structural-Acoustic Characteristics of Automotive Vehicles,"

Proceedings of the 1993 Noise and Vibration Conference , SAE P-264,

pp.65-82

Lu, L.K.H. and et. al., 1983, "Comparison of Statistical Energy Analysis and
Finite Element Analysis Vibration Prediction with Experimental Results,"

The Shock and Vibration Bulletin, May, No. 53, Part 4, pp. 145-154.

Lu, L. K. H., 1987, “Optimum Damping selection by Statistical Energy Analysis,”
Statistical Energy Analysis, Edited by Hsu, K. H., Nefske, DJ. and Aka,
A., NCA-Vol. 3, ASME, N.Y., N.Y.

 

78
Lyon, R.H. and Maidanik, G., 1962, “Power Flow between Linearly coupled

Oscillators,” J. Acoustic. Soc. Am, May, Vol. 34, No. 5, pp. 623-639.

Lyon, R.H., 1975, Statistical Energy Analysis of Dynamical Systems: Theory and
Applications, MIT Press, Cambridge, Mass.

Lyon, R.H., 1987, “The SEA Population Model - Do We Need a New One?,”
Statistical Energy Analysis, Edited by Hsu, K.H., Nefske, DJ. and Aka,
A., NCA-Vol.6, ASME, N.Y., N .Y.

Norton, MP. and Greenhalgh, R., 1986 "On the Estimation of Loss Factors in
Lightly Damped Pipeline Systems: Some Measurements Techniques and
Their Limitations," Journal of Sound and Vibration, Vol. 105, No. 3, pp.
397-423

Oh, H.L., 1987, “Variation Tolerant Design,” Quality: Design, Planning, and
Control, Edited by DeVor, R.E., and Kapoor, S.G, FED-Vol. 27, ASME,

N.Y., N.Y.

Powell, RE. and Quartararo, L.R., 1987, “Statistical Energy Analysis of
Transient Vibration ,” Statistical Energy Analysis, K. H. Hsu et al, ed.,

NCA-Vol. 3, ASME, N.Y., N.Y.

Radcliffe, C. J. and Huang, X. L., 1993, "Putting Statistics into the Statistical
Energy Analysis of Automotive Vehicles," Vibration Isolation, Acoustics,

and Damping in Mechanical Systems, DE-Vol. 62, CD. Johnson, et al, ed.,
ASME book 600823. pp. 51-60

Radcliffe, C. J. and Huang, X. L., 1993, "Identification of Internal Loss Factors
During Statistical Energy Analysis of Automotive Vehicles," Proceedings

of the 1993 Noise and Vibration Conference, SAE P—264, pp. 311-318.

 

79
Remington, R]. and Manning, J.E., 1975, "Comparison of Statistical Energy

Analysis Power Flow Predictions with an 'Exact' Calculation," J. Acoustic.

Soc. Am., Feb., Vol.57, No.2, pp. 374-379.

Smith, P.W. Jr., 1962, “Response and Radiation of Structural Modes Excited by
Sound,” J. Acoustic. Soc. Am., May, Vol. 34, No. 5, pp. 640-647.

Yoerkie, C.A., Moor, J.A., and Manning, J .E., 1983, " Development of Rotorcraft
Interior Noise Control Concepts - Phase 1: Definition Study," NASA

Contractor Report 166101

Lyon, R.H., 1967, "Statistical Analysis of Power Injection and Response in

Structures and Rooms," J. Acoust. Soc. Am. , 45, No. 3, pp. 545-565

Volk, W., 1969, Applied Statistics for Engineers, 2nd ed., McGraw-Hill Book

Company, New York

Vail, CR, 1972 "Dynamics Modeling of Automobile Structures from Test Data,"
System Identification of Vibrating Structures, Edited by Pilkey, W.D. and
Cohen, R., ASME, N.Y., N.Y.