45.2 Power Flow Equations

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With the SEA method, we do not deal with specific characteristic modes (see Chapter 3 and Chapter 4) of

the analyzed structure. Instead, we consider the structural components as a set of equivalent vibrating

elements, and evaluate the vibration condition of the components as a macroscopic quantity averaged

statistically over the frequency band and space (by describing the energy). We assume that the vibration

modes within a given frequency band are distributed uniformly and are excited to the same degree.

Using the SEA method, we can formalize the relationships of power flows between subsystems, and by

solving these relationships, we can compute the energy stored in each subsystem. Next, the equations of

such basic power flow [4] are explained.

45.2.1 Power Flow Equations of a Two-Subsystem Structure

The power flow relationships of a structure

consisting of a two-subsystem structure are

shown in Figure 45.1. The equations for the

power flows between subsystem 1 and subsystem 2

under typical conditions are expressed as

Subsystem 1: Pi1 ¼ Pl1 þ Pl2 ð45:1Þ

Subsystem 2: Pi2 ¼ Pl2 þ P21 ð45:2Þ

where Pi1 is the input power to subsystem 1 from

outside, Pl1 is the internal power loss of subsystem

1, and P12 is the transmitted power from

subsystem 1 to subsystem 2.

The internal power loss, Pl1; is written as

Pl1 ¼ vh1E1 ð45:3Þ

where v is the central angular frequency in the band, E1 is the energy in the bandwidth Dv of subsystem 1,

and h1 is the internal loss factor (ILF).

The average modal energy Em1 in subsystem 1, and Em2 in subsystem 2, are given by

Em1 ¼

E1

N1

; Em2 ¼

E2

N2 ð45:4Þ

where N1 is number of modes in the bandwidth Dv of subsystem 1, and N2 is number of modes in the

bandwidth Dv of subsystem 2.

The transferred power, P12; between subsystems 1 and 2 is expressed as

P12 ¼ 2P21 ¼ P 012 2 P 021 ð45:5Þ

P 012 ¼ vh12E1 ¼ vh12N1Em1 ð45:6Þ

P 021 ¼ vh21E2 ¼ vh21N2Em2 ð45:7Þ

P21

Sub system1 Sub system 2

n2 E2

Pi1

P12

Pi2

Pl1 Pl2

n1 E1 h2 h1

h12

FIGURE 45.1 Power flow relationships between two

subsystems.

45-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

where h12 and h21 are the coupling loss factors (CLFs) from subsystem 1 to subsystem 2, and from

subsystem 2 to subsystem 1. They satisfy the reciprocity relationship h12n1 ¼ h21n2: Therefore,

transferred power, P12; becomes

P12 ¼ vh12N1ðEm1 2 Em2Þ ¼ vh12N1

E1

N1

2

E2

N2

􀀏 􀀐

ð45:8Þ

Consequently, the power flow equations (Equation 45.1 and Equation 45.2) can be expressed as

follows:

Pi1 ¼ vh1E1 þ vh12N1

E1

N1

2

E2

N2

􀀏 􀀐

ð45:9Þ

Pi2 ¼ vh2E2 þ vh21N2

E2

N2

2

E1

N1

􀀏 􀀐

ð45:10Þ

If the SEA parameters (i.e., the modal density, intrinsic loss factor, CLF, and input power) are given,

then each subsystem’s energy condition can be easily computed.

45.2.2 Power Flow Equations of a Multiple Subsystem Structure

By expanding the formulation in the previous section, it is possible to formalize the power flow

relationships of a structure composed of multiple subsystems in the same way. The power flow equation

for a structure composed of N subsystems is expressed by the following equation in the matrix form:

v

h1 þ

XN

i–1

h1i

􀁻 !

n1 2h12n1 · · · 2h1N n1

2h21n2 h2 þ

XN

i–2

h2i

􀁻 !

n2 · · · 2h2N n2

.. .

.. .

. .

. .. .

2hN1nn · · · · · · hN þ

NX21

i–N

hNi

􀁻 !

nN

2

66666666666666664

3

77777777777777775

£

E1=n1

E2=n2

.. .

EN =nN

2

66666664

3

77777775

¼

Pi1

Pi2

.. .

PiN

2

66666664

3

77777775

ð45:11Þ

From Equation 45.11, if the SEA parameters are given in the same way as for the structure of two

subsystems, then the energy equation of each subsystem can be obtained.

The average energy of a subsystem is expressed by the following equations by using the vibration

velocity and sound pressure:

E ¼ Mkv2l ð45:12Þ

E ¼

Mkp2l

Z2

0 ð45:13Þ

where M is the mass of the subsystem, kv2l is the average spatial square of the vibration velocity,

kp2l is the average spatial square of the sound pressure, and Z0 is the specific acoustic impedance

of air.

Accordingly, if each condition of component’s energy is determined from Equation 45.11, it is

possible to compute the vibration variable and the sound pressure with Equation 45.12 and

Equation 45.13.

Statistical Energy Analysis 45-3

© 2005 by Taylor & Francis Group, LLC

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