Wang X. Vehicle Noise and Vibration Refinement

Подождите немного. Документ загружается.

Mid- and high-frequency problems in vehicle noise 163

7.6.1 Deﬁ ning the system model

There are also three steps in deﬁ ning the system model:

1. Divide the system into physical components of suitable size and combine

the natural modes of each component into groups (subsystems) with

similar characteristics.

2. Deﬁ ne the physical coupling between subsystems.

3. Deﬁ ne the form of excitation to the system.

7.6.2 Evaluating the subsystem parameters

After deﬁ ning the system model, there are four steps in evaluating the

subsystem parameters:

1. Mode count, modal density and group velocity

2. Damping loss factor

3. Coupling loss factors to connected subsystems

4. Input power from the external sources of excitation

where the deﬁ nition and evaluation method of the damping loss factor have

been given in Chapter 3, and are not duplicated in this chapter.

7.6.3 Evaluating the response variables

After deﬁ ning the system model and evaluating the subsystem parameters,

the response variables can be evaluated by

[D]{ε} = {〈P

〉} (7.85)

where [D] is a symmetric matrix of damping and coupling loss factors. The

modal energy vector {ε} can be found from

{ε} = [D]

−1

{〈P

〉} (7.86)

7.7 Evaluation of the statistical energy analysis

subsystem parameters

There are three main SEA parameters for the SEA wave approach:

1. Input power from external excitation – input power as shown in

Fig. 7.12(a)

2. Energy storage capacity of the reverberant ﬁ eld – modal density/group

velocity as shown in Fig. 7.12(b)

3. Energy transmission from the reverberant ﬁ eld to direct ﬁ elds of

adjacent subsystems – coupling loss factor as shown in Fig. 7.12(c).

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

164 Vehicle noise and vibration reﬁ nement

In order to calculate the SEA subsystem parameters using a wave approach,

we need to know the following:

• Types of waves that can propagate in a given subsystem, which depends

on subsystem dimension, cross-section, etc.

• The dispersion properties of each wave type.

7.7.1 Modal density and group velocity

for one-dimensional wave propagation

The exponential displacement ﬁ eld along the one-dimensional waveguide

is assumed to be

u = U(x, y)e

ikz

iωt

(7.87)

From the equation of motion for the assumed displacement ﬁ eld, the roots

of the homogeneous equation can be solved for values of U and k that

satisfy the homogeneous equation at a given frequency ω. Each root rep-

resents a wave type of the waveguide; roots with real k represent propagat-

ing waves, while roots with imaginary k represent evanescent waves.

U reﬂ ects wave type and k represents wave number in rad/m, which is

given by

== =

22ππ

(7.88)

where λ is wavelength in m, c

is phase velocity in m/s, and the group veloc-

ity c

(m/s) is deﬁ ned as

∂

(7.89)

which was mentioned in Equation 7.13. The group velocity is the speed at

which energy propagates by a wave. The dispersive relation is given by

External

excitation

(a) (b) (c)

7.12 Three main SEA parameters.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

Mid- and high-frequency problems in vehicle noise 165

curves or plots which show how the wave number varies with frequency for

each wave type where the group and phase velocity are related to the local

and global slopes of the dispersion curve at a given frequency. If group

velocity varies with frequency then the wave is said to be dispersive as

shown in Fig. 7.13. For a one-dimensional waveguide of length L, one mode

is expected on average for every increment in wave number of π/L radians,

which is shown in Fig. 7.14 which reﬂ ects wave mode duality.

Modal density is deﬁ ned as the expected number of modes per unit fre-

quency, which is given by:

()

∂

()

∂

()

=π

(7.90)

where the mode count N(ω) = kL/π is deﬁ ned as the expected number of

modes below frequency ω. Equation 7.90 gives

(

)

(7.91)

This means that the modal energy E/n is equal to the energy density π(E/L)

multiplied by the speed c

at which energy ﬂ ows, which is the incident

∂w

∂k

7.13 Dispersion curves.

4p/L

3p/L

2p/L

p/L

7.14 Wave mode duality.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

166 Vehicle noise and vibration reﬁ nement

power. From a wave viewpoint, the modal density relates the incident

power in a reverberant ﬁ eld to the total energy contained within the rever-

berant ﬁ eld.

7.7.2 Modal density and group velocity

for two-dimensional wave propagation

The exponential displacement ﬁ eld for plates/shells which support wave

propagation in two dimensions is assumed to be

uUz

()

eee

(7.92)

Wave types are calculated in the same way as for a one-dimensional wave-

guide, but now U(z) describes the displacement ﬁ eld through the thickness;

the wave number and wave velocities become vector quantities.

It is helpful to view the response of a panel in wave number space where

the wave number space description is obtained by taking the two-

dimensional Fourier transform of the physical displacement ﬁ eld. The wave

number indicates the number of ‘wiggles’ per unit distance in a given direc-

tion, which is extremely useful for understanding wave propagation and

acoustic radiation. For example, inﬁ nite isotropic panel resonant wave

numbers lie on a circle in wave number space as shown in Fig. 7.3, where

the number of modes in a given band is proportional to the area contained

between two dispersion curves in the wave number space.

When boundary impedance is random, averaged modal densities for two-

and three-dimensional wave propagation are given by [9]

(

)

2π

(7.93)

(

)

(7.94)

respectively, where A is the surface area of the two-dimensional structure

and V is the volume of the three-dimensional acoustic cavity where bound-

ary corrections are sometimes applicable.

For a three-dimensional room with rigid walls, the averaged modal

density is given by [9]:

LLH

LL LH LH

xy x y x y

(

)

(

)

()

12 1 2

ππ

++ H

(7.95)

where L

and L

are the length and width of the room, H is the depth of

the room, c is the speed of sound in air, f is the frequency in Hz and ω is

the radial frequency in rad/s.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

Mid- and high-frequency problems in vehicle noise 167

In summary, energy storage in the SEA wave approach is represented

by modal density, which is related to group velocity and dimensions of a

subsystem/wave ﬁ eld. The SEA parameters for the wave ﬁ elds can be found

from dispersion curves.

7.7.3 Coupling loss factor

The coupling loss factor describes the rate at which energy ﬂ ows from the

reverberant ﬁ eld of the ith subsystem to the direct ﬁ eld of the jth subsystem

per unit energy in the reverberant ﬁ eld of the ith subsystem, which is

expressed as

coup,

(7.96)

According to the SEA wave approach assumption shown in Fig. 7.11(a),

that is neglecting direct ﬁ eld transmission between two junctions connected

to the same subsystem, or assuming that transmission is incoherent via the

reverberant ﬁ eld, the coupling loss factor for each junction can be calcu-

lated in isolation. This implies that the SEA ensemble average response

does not capture transmission in which coherent phase information is

propagated across several subsystems; localization, wave number ﬁ ltering

effects and global modes are not revealed in the SEA ensemble average

predictions.

In order to calculate the coupling loss factor, the semi-inﬁ nite impedance

to a junction has to be added to the direct ﬁ elds of each subsystem, and the

junction has to be loaded by diffuse reverberant loading with a particular

type of forcing. The diffuse ﬁ eld wave transmission coefﬁ cient from subsys-

tem i to subsystem j is calculated when a diffuse reverberant ﬁ eld is incident

upon the connection. For line and area connections, wave transmission

calculation is often simpliﬁ ed by assuming that junctions are planar and/or

large compared with a wavelength.

For pointed connected subsystems, direct ﬁ eld impedances of all subsys-

tems connected to a junction are assembled ﬁ rst, that is the 6 × 6 dynamic

stiffness matrix for connection degrees of freedom, and this process is

repeated and looped over the excited wave ﬁ eld/subsystem. The force on

the junction due to the diffuse incident wave ﬁ eld (with unit energy density)

in the excited subsystem is then found. This force is then applied to the

junction and the input power to the direct ﬁ elds of each of the receiving

subsystems is calculated. The columns of the coupling loss factor matrix are

calculated and the process is repeated.

For line-connected subsystems, the maximum and minimum trace-wave

numbers in the excited subsystem are ﬁ rst sought, then a set of incident

trace-wave numbers are chosen to describe the diffuse ﬁ eld, and the process

loops over all trace-wave numbers. For each trace-number, direct ﬁ eld

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

168 Vehicle noise and vibration reﬁ nement

impedances of all subsystems connected to the junction are assembled into

a 6 × 6 matrix for connection degrees of freedom. The force on the junction

due to an incident trace-wave number (with unit energy density) is then

sought in the excited subsystem. This force to the junction is then applied

and the input power to the direct ﬁ elds of each of the receiving subsystems

is calculated, the results being averaged over all incident trace-wave

numbers.

A formulation is available for predicting diffuse ﬁ eld transmission

through pointed and line-connected subsystems such as plates/shells, etc.,

which can include lumped masses/isolators, in-line beams, etc.; subsystem

impedances can be computed from local wave impedances (for example,

line wave impedance) which are related to dispersion properties of waves.

Coupling loss factors can be related to the matrix of subsystem wave trans-

mission coefﬁ cients from which the coupling loss factor matrix is obtained

for a given point or line junction. This matrix describes scattering/coupling/

transmission between different wave ﬁ elds.

For area-connected subsystems, resonant energy transmission is gov-

erned by the radiation efﬁ ciency of resonant modes, while non-resonant

energy transmission is governed by the mass law and can be described by

additional coupling loss factors. For example, the coupling loss factor

between a cavity and a plate is given by [9]

ρσ

ωρ

(7.97)

where ρ

is the mass density of air, ρ is the mass density of the plate, h

the thickness of the plate, and σ is the radiation efﬁ ciency of the plate,

which is given by [9]

=∗ <

(

)

−

(

)

(7.98)

=<<

(

)

ffff,

(7.99)

=<<

()

−

()

018

10 0 25

fff f

(7.100)

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

−

()

−

018 4

025

ffff.,<<

()

(7.101)

⎛

⎝

⎜

⎞

⎠

⎟

()

−

fff f

(7.102)

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

Mid- and high-frequency problems in vehicle noise 169

where the critical frequency is given by [10]

Lt t

18.

(7.103)

where

−

()

ρν

(7.104)

and the critical wavelength is given by

(7.105)

The perimeter of the plate is given by

P = 2(L

+ L

) (7.106)

The radiation area is given by

S = L

(7.107)

Frequency limits for the radiation ratio are given by

=−

⎛

⎝

⎜

⎞

⎠

⎟

(7.108)

(

)

100

(7.109)

The radiation efﬁ ciency can also be calculated from the Leppington–Heron

radiation efﬁ ciency [2].

7.7.4 Transfer matrices and insert loss of a trim lay-up

It is assumed that there are three characteristic wave types in poroelastic

materials: the ﬂ uid standing wave, the solid standing wave, and a shear wave

of both solid and ﬂ uid. Based on inertial/stiffness coupling, Biot equations

provide a simple model (semi-empirical) way to describe the three wave

types in terms of various measurable properties of a poroelastic material

for which the response of a poroelastic material can be solved from known

properties, domain and boundary conditions.

Noise control treatment is often encountered by planar lay-ups of poro-

elastic material where the transfer matrix method provides an efﬁ cient

numerical method for computing wave propagation through such lay-ups.

The method assumes the trim lay-ups are inﬁ nite in the lateral direction

and homogeneous, and the response across each layer is sought analytically

based on transfer properties/matrices for a given layer.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

170 Vehicle noise and vibration reﬁ nement

7.7.5 Leaks

Acoustic leakage paths are signiﬁ cant for acoustic energy transmission.

Typical leaks arise because of access holes/pass-throughs, grillages, gaps,

imperfect coverage, etc. It is important to account for leakage in the SEA

model, which can be represented by additional coupling loss factors at area

junctions.

7.7.6 Energy inputs

There are ﬁ ve types of energy inputs for the SEA wave approach: direct

ﬁ eld impedance and input power, distributed random loading, turbulent

boundary layer loading, diffuse acoustic ﬁ eld loading and propagating wave

ﬁ eld loading.

Input power from point force excitation applied to point/line/area-type

connections can be computed from the direct ﬁ eld impedance, which is

identical to calculations performed when calculating coupling loss factors.

Input power from distributed pressure loads is calculated from the auto-

spectrum and cross-spectrum. The auto-spectrum function of turbulent

boundary layer loading depends on the boundary layer thickness and ﬂ ow

condition, and is typically obtained empirically from wind tunnel tests. The

cross-spectrum of turbulent boundary layer loading contains information

about spatial correlation structures within a ﬂ ow in a Corcos model. For

convection ﬂ ows, different characteristic decay lengths in the ﬂ ow and

cross-ﬂ ow directions are observed. The decay coefﬁ cients and convection

wavelengths are important parameters.

Loading from diffuse acoustic ﬁ eld excitation can be calculated by the

use of the diffuse ﬁ eld reciprocity principle. Spatial correlation in a diffuse

ﬁ eld is given by a sine function. Input power from diffuse acoustic ﬁ eld

excitation is proportional to radiation efﬁ ciency, in principle.

Input power from propagating wave ﬁ eld excitation can be calculated

from modal joint acceptance or wave number space integrals in which trace

velocities are derived from components of acoustic wave numbers and cor-

relation decays are generally low.

7.8 Hybrid deterministic and the statistical energy

analysis approach

A hybrid analysis method was developed in which the various components

of a complex vibro-acoustic system can be modelled either deterministically

or statistically [3–6]. The coupling between these two types of component

is effected by using a diffuse ﬁ eld reciprocity relation, which relates the

cross-spectrum of the forces at the boundary of a statistical component to

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

Mid- and high-frequency problems in vehicle noise 171

the vibration energy level of the component. This method is most appli-

cable in the mid-frequency ranges.

7.9 Application example

In order to illustrate the above approaches, vibro-acoustic analysis of a

plate–room cavity system has been conducted by using deterministic analy-

sis, statistical energy analysis (SEA) and the hybrid approach in which the

plate–room cavity system can be considered as a simpliﬁ ed model of a

vehicle cabin cavity coupled with a roof panel.

The plate–room cavity system is shown in Fig. 7.15 in which a sine excita-

tion force is acting at the centre of a plate. The aluminium plate is simply

supported and has a length of L

= 1.2 m, a width of L

= 1.35 m, a thick-

ness of h

= 5 mm and a damping loss factor of 0.03. The room is a rectan-

gular room with all the walls smooth, rigid and perfectly reﬂ ecting except

for the top wall of the thin aluminium plate. The room has dimensions of

length L

= 1.2 m, width L

= 1.35 m and height L

= 1.45 m, and a

damping loss factor of 0.03. The room size is close to a typical passenger

car cabin size.

The modal density for the plate is assumed to be n

(ω) = 0.0942/(2π)

s/rad, the modal density for the room cavity is calculated from Equation

7.95, and the coupling loss factor from the plate to the room cavity is cal-

culated from Equation 7.97. The radiation efﬁ ciency is calculated from

Equations 7.98–7.102. The speed of sound in air, c, is assumed to be

340 m/s; air mass density ρ

is 1.3 kg/m

and mass density ρ of the aluminium

plate is 2800 kg/m

. Equations 7.66 and 7.67 are applied to calculate mean

plate and room cavity energy for the SEA approach.

7.15 A plate–room system with a simply supported aluminium plate

on the top and an excitation at the centre.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2

172 Vehicle noise and vibration reﬁ nement

In order to validate the SEA mean energy estimations, a deterministic

analysis using modal expansion has been conducted in which coupling

effects of the solid structure and ﬂ uid cavity are considered. Multiple con-

stant bandwidth analysis was conducted on the results of the deterministic

analysis, in which mean values of the deterministic energy at frequency

sampling points in individual frequency bands were calculated at the central

frequencies of the frequency bands and compared with the SEA mean

value predictions. A hybrid solution was also sought by applying the deter-

ministic analysis to calculate the mean room cavity energy (master system–

room) and by applying the SEA equation to calculate mean plate energy

(subsystem–plate). In this way, the coupling effects of the solid structure

and the ﬂ uid cavity were considered in the SEA coupling loss factors.

The simulation results are shown in Plake VI which shows that the results

calculated by the three methods support one another. The SEA and the

hybrid methods tend to overestimate the mean response energy levels

in comparison with the discrete sample average/median values from the

deterministic results in the low to mid-frequencies.

In regard to the computational efﬁ ciency for the results in Plate VI

(between pages 114 and 115), the SEA computation took 0.5 minutes, the

deterministic computation took 10 hours and the hybrid computation took

2 hours. In the frequency range of 0–100 Hz, the deterministic method

showed modal information in detail; the SEA method does not provide any

modal information and has poor computational accuracy in this frequency

range. In the frequency range of 100–600 Hz, the deterministic method

showed more modal overlaps and structure variation-induced uncertainty,

and its computational speed slowed down; the hybrid method showed the

strong ﬂ uid–solid coupling mode peak and its computational speed was

much faster than that of the deterministic method in this frequency range;

and the SEA method does not provide any modal information but its com-

putational accuracy improved in this frequency range.

7.10 References

1. Lyon, R.H. and DeJong, R.G. (1995), Theory and Application of Statistical

Energy Analysis, Butterworth-Heinemann, Boston, MA.

2. Leppington, F.G., Broadbent, E.G. and Heron, K.H. (1982), The acoustic

radiation efﬁ ciency of rectangular panels, Proc. Roy. Soc. Lond., A382,

245–271.

3. Langley, R.S. and Cotoni, V. (2007), Response variance prediction for uncer-

tain vibro-acoustic systems using a hybrid deterministic–statistical method, J.

Acoustical Society of America, 122(6), 3445–3463.

4. Langley, R.S. and Bremner, P. (1999), A hybrid method for the vibration analy-

sis of complex structural–acoustic systems, J. Acoustical Society of America,

105(3), 1657–1671.

Copyrighted Material downloaded from Woodhead Publishing Online

Delivered by http://woodhead.metapress.com

ETH Zuerich (307-97-768)

Sunday, August 28, 2011 12:01:08 AM

IP Address: 129.132.208.2