45.3 Estimation of SEA Parameters

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To solve the power flow equations, it is necessary to determine the SEA parameters (i.e., the modal

density, ILF, CLF, and input power). In this subsection, a method is given for computing the SEA

parameters.

45.3.1 Modal Density

45.3.1.1 Structural Subsystem

Modal density is a key parameter for determining the dynamic characteristic of a structure. The number

of modes, N; included in the frequency band (for estimation), is a factor denoting how easily energy, in

transferring between subsystems, can be obtained. To determine N in the prescribed frequency band, it is

first necessary to determine the modal density nð f Þ; that is, the gradient of N in the frequency band.

The modal density of a structural subsystem is computed by using the following equation [4,5]:

nð f Þ ¼

dN

df ¼

1

f0 ¼

A

2t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12rð1 2 n 2Þ

E

s

ð45:14Þ

where A is the area of cross section, t is the thickness of the structural subsystem, r is the mass density,

n is the Poisson’s ratio, E is the Young’s modulus, and f0 is the fundamental natural frequency of the

structural subsystem.

45.3.1.2 Acoustic Subsystem

The modal density of an acoustic subsystem is determined by the following analytical equation [6]:

nðf Þ ¼

dN

df ¼

4pV

c3 f 2 ð45:15Þ

where c is the speed of sound propagation within the acoustic subsystem, and V denotes the volume of

the acoustic subsystem.

Modal density of the cavity in the low frequency band is deduced in a similar manner to that in the

two-dimensional space. Define the depth of the cavity by d; and the frequency of the standing wave in the

cavity by fd ¼ c=2d: If f , fd; then the modal density is assumed to be uniformly distributed, and is

estimated by

nðf Þ ¼

2pS

c2 f ð45:16Þ

where S is the area of the cavity.

If f . fd; modal density can be estimated using Equation 45.15, because the cavity is designated as

acoustically three dimensional.

45.3.2 Internal Loss Factor

45.3.2.1 Structural Subsystem

The ILF, h1; of a subsystem gives the loss ratio when the input power to the subsystem from the outside is

converted to kinetic energy of the subsystem. An excitation test to measure the damping ratio is

employed to estimate the ILF of the structural subsystem. There are several methods for estimating the

internal loss factor. The ILF applied in the SEA method is estimated by measuring input energy and

output energy simultaneously, or by measuring the attenuation ratio within a given period of time. Both

methods require the same setup to conduct an excitation test, and one is able to improve the

measurement precision by conducting both methods.

45-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

With the energy measuring methods mentioned above, the ILF can be estimated by

h ¼

ðf2

f1

ReðY ÞF2df

v0M

ðf2

f1

v2df

* + ð45:17Þ

where Y is the complex mobility at the driving point in the range of f1 to f2; F2 is the power spectrum of

the input vibration force, and v2 is the power spectrum of the response speed. In addition, k l denotes the

space average operator.

45.3.2.2 Acoustic Subsystem

The ILF of an acoustic subsystem is determined by [7]

h ¼

cSa􀀊

4Vv ð45:18Þ

where a􀀊 is the average acoustic absorption coefficient, V is the volume of the acoustic subsystem, and

S denotes the surface area. The acoustic absorption coefficient can be estimated by measuring the

reverberation time.

Both the ILF of the cavity in the low-frequency band and the modal density are deduced similar to that

in two-dimensional spaces. For f , fd; the ILF is estimated using

h ¼

cSpap

pvVc ð45:19Þ

where ap is the acoustic absorption coefficient in the cavity, Sp is the peripheral area of the cavity, and

Vc is the volume of the cavity.

For f . fd; the modal density can be estimated using Equation 45.18 because the cavity is taken as an

acoustic subsystem.

45.3.3 Coupling Loss Factor

45.3.3.1 Between Structural Subsystems

The CLF hij gives the loss ratio when power

transmits between two subsystems [4]. For

example, the CLF between two flat plates can be

estimated using

hij ¼

cgiLct

pvSi ð45:20Þ

where cgi is the group velocity of the bending

waves, Lc is the coupled length, Si is the surface

area, and tij is the energy transmission factor from

subsystem i to subsystem j. The transmission

factor varies with the type of coupling, for

example, I-type, L-type, or T-type shown in

Figure 45.2. In this section, we use energy

transmission efficiency of vertical incidence,

reported by Cremer [7].

I-type

L-type

T-type

(a)

(b)

(c)

FIGURE 45.2 Coupled type.

Statistical Energy Analysis 45-5

© 2005 by Taylor & Francis Group, LLC

45.3.3.2 Between a Structural Subsystem and an Acoustic Subsystem

Coupling power between a structural subsystem

and an acoustic subsystem is the power flow based

on resonance at transmission. The CLF between a

structural subsystem and an acoustic subsystem is

given by

hij ¼

Z0Scs

vMi ð45:21Þ

where Z0 is the specific acoustic impedance of air,

Sc is the surface area of coupling, s is the acoustical

radiation efficiency, and Mi is the mass of the

structural subsystem.

45.3.3.3 Between an Acoustic Subsystem

and a Cavity

Coupling power between an acoustic subsystem

and a cavity is power flow based on resonance in

transmission. The CLF between an acoustic

subsystem and a cavity is given by [8]

hsc ¼

csScstm

4vVs ð45:22Þ

where cs is the sound velocity in the acoustic space,

Vs is the volume in the acoustic space, tm is the

transmission factor at random incidence depending

on mass flow through the partition, and Scs is

the coupling area.

45.3.4 Input Power

45.3.4.1 Vibration Input Power

The vibration input power PiN is given by

PiN ¼ vMikv2

i l ð45:23Þ

where Mi is the equivalent mass, kv2

i l is the spatial

average of square of the vibration velocity, and v is

the central angular frequency.

45.3.4.2 Acoustical Input Power

The acoustical input power Ps is given by

Ps ¼

kp2lS2nð f Þ

4M

srad

c2

2pSf 2 ð45:24Þ

where kp2l is the square average of the input sound

pressure, S is the surface area of the component, M

is the mass of the component, nðf Þ is the modal

density, and srad is the sound radiation factor [2].

3

4

5

6

7

8

9

10

11

12

13

14

16 15

1 2 18 17

FIGURE 45.3 Modeling of the tractor cabin.

4

14 9 12

16 15 6 7 11

18 17 8 10 1

13

5

2

3

Pi4

Pl4

19

FIGURE 45.4 Power flows in the tractor cabin.

45-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

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