9.3 DAMPING FACTORS

Back

In Sec. 9.2, we considered the damping as being produced by a linear viscous

damper, in which the force of the damper is directly proportional to the

relative velocity of the ends of the damper element. There are many other

types of damping or energy dissipation elements, and many of these elements

are nonlinear. It is usually possible to define an ‘‘equivalent damping

coefficient’’ or mechanical resistance for these elements, so the general

results that we have developed are valid.

In analogy with the Helmholtz resonator analysis, one alternative

measure of the effect of damping may be expressed through the mechanical

quality factor QM (Kinsler et al., 1982), which is defined in a form similar to

the acoustic quality factor in Eq. (8-48):

Vibration Isolation for Noise Control 413

Copyright © 2003 Marcel Dekker, Inc.

QM ј

!nM

RM ј

2_fnM

RM

(9-44)

From Eq. (9-24), we find:

!nM ј 1

2RM;cr (9-45)

Using this result, the mechanical quality factor may be written in terms of

the damping ratio _ ј RM=RM;cr:

QM ј

RM;cr

2RM ј

1

2_

(9-46)

The mechanical quality factor or damping ratio may be measured

experimentally by measuring the frequencies at which the power dissipated

in the damper element is one-half of the power dissipated at resonance. An

expression similar to Eq. (8-60) may be used to relate the half-power frequencies

р f1 and f2) to the mechanical quality factor:

f2 _ f1

fn ј

1

QM ј 2_ (9-47)

One of the oldest measures of damping in mechanical systems is the

logarithmic decrement _ (Thomson and Dahleh, 1998), which is defined by

414 Chapter 9

FIGURE 9-3 Vibratory motion for various values of the damping ratio _. The curves

are plotted for an initial displacement yр0Ю ј yo and an initial velocity vр0Ю ј 0.

Copyright © 2003 Marcel Dekker, Inc.

the natural logarithm of the ratio of the peak amplitudes N cycles apart,

divided by the number of cycles:

_ ј

1

N

ln

ymaxрtoЮ

ymaxрtNЮ

_ _

(9-48)

The quantities ymaxрtoЮ and ymaxрtNЮ are the maximum or peak amplitudes of

the motion at time to and tN, respectively. The quantity N is the number of

cycles during the time interval between to and tN:

N ј

!dрtN _ toЮ

2_

(9-49)

For small damping (or for _ _ 0:3Ю, Eq. (9-36) indicates that the undamped

natural frequency !n and the damped natural frequency !d are practically

equal, with less than a 5% error. With this approximation, Eq. (9-49) may

be written in the following form:

N ј

!nрtN _ toЮ

2_

(9-50)

At the peak amplitude, the cosine function in Eq. (9-41) is unity, so the

ratio of the peak amplitudes may be written in the following form, with the

help of Eq. (9-50):

yрtoЮ

yрtNЮ ј expЅ__!nрto _ tNЮ_ ј expЅ2__N_ (9-51)

Taking the natural logarithm of both sides of Eq. (9-51), we obtain the

following relationship:

ln

yрtoЮ

yрtNЮ

_ _

ј 2__N (9-52)

By comparing Eqs (9-48) and (9-52), we see that the logarithmic decrement

is directly related to the damping ratio:

_ ј 2__ (9-53)

The logarithmic decrement may be conveniently measured by displaying

the motion (on an oscilloscope, for example) and measuring the amplitude

ratio directly.

The peak amplitude ymax may be expressed in ‘‘level’’ form, where the

displacement level is defined by the following relationship:

Ld ј 20 log10р ymax=yref Ю (9-54)

The reference displacement is:

yref ј 10pm ј 10 _ 10_12 m

Vibration Isolation for Noise Control 415

Copyright © 2003 Marcel Dekker, Inc.

Another measure of the damping is the decay rate _ (Plunkett, 1959),

defined as the change in the peak displacement level with time, in units of

dB/s:

_ ј _

dLd

dt ј _20рlog10 eЮ рdymax=dtЮ

ymax

(9-55)

According to Eq. (9-41), the peak amplitude for the system with a

viscous damper is given by the following relationship:

ymaxрtЮ ј Ce__!nt (9-56)

Using this expression in Eq. (9-55), we find the following relationship for the

decay rate (dB/s):

_ ј р_20Юр0:43429Юр__!nЮ ј 8:6859_!n ј 54:575_ fn (9-57)

In analogy with the acoustic concepts presented in Chapter 7, the

reverberation time T60 may be defined as the time required for the displacement

level to decrease by 60 dB:

T60 ј

60 dB

_dB=s ј

1:0994

_ fn

(9-58)

Finally, the loss factor or energy dissipation factor _ (Ungar and

Kerwin, 1962) may be defined as the ratio of the average energy dissipated

per radian (energy dissipated per cycle Ediss divided by 2_) to the total

energy (kinetic energy plus potential energy) of the system Etot:

_ ј

Ediss=2_

Etot

(9-59)

The energy dissipated per cycle may be evaluated from the following

expression, involving an integration over one cycle of vibration:

Ediss ј

р

Fd dy (9-60)

For a viscous damper, the force is Fd ј RMvрtЮ and the displacement

dy ј vрtЮ dt. If we write the displacement in the form yрtЮ ј ymax cosр!tЮ,

where ymax is the peak amplitude, the velocity of the system may be evaluated

as follows:

vрtЮ ј

dyрtЮ

dt ј _!ymax sinр!tЮ (9-61)

Making the substitutions from Eq. (9-61) into Eq. (9-60), we obtain the

following integral:

416 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

Ediss ј RM!2y2

max

р2_=!

0

cos2р!tЮ dt (9-62)

The following result for the energy dissipated per cycle is obtained after

carrying out the integration:

Ediss ј _RM!y2

max (9-63)

The total energy stored in the system is equal to the potential energy

stored in the spring at the peak displacement (the point at which the kinetic

energy of the mass is zero):

Etot ј

р

FS dy ј KS

рymax

0

ydy ј 1

2KSy2

max (9-64)

If we make the substitutions from Eqs (9-63) and (9-64) into the expression

for the loss coefficient, Eq. (9-59), the following result is obtained:

_ ј

RM!

KS

(9-65)

The loss factor is often measured in systems involving freely decaying

vibrations, which occur at the resonant frequency, ! ј !n. We note from

Eqs (9-15) and (9-24) that:

2KS

!n ј 2M!n ј RM;cr (9-66)

Using this result in Eq. (9-65), we find that the loss factor and the damping

ratio are related by the following expression:

_ ј 2_ р9-67Ю

The relationships between the various measures of damping are summarized

in the following expression:

_ ј 1

2_ ј

1

2QM ј

_

2_ ј

D

54:575fn ј

1:0994

fnT60

(9-68)

Representative values for the loss factor _ for materials at room temperatures

are given in Table 9-1. In general, the loss factor is strongly temperature-

dependent, as illustrated in Table 9-2 for an acoustic absorbing foam

material.

Example 9-2. A machine has a mass of 50 kg (110.2 lbm). The spring constant

for the support is 49.35 kN/m (281.8 lbf /in) and the undamped natural

frequency is 5 Hz. It is desired to select the damping for the support such

that the damping ratio is 0.100. Determine the damping coefficient RM and

the other measures of damping capacity for the system.

Vibration Isolation for Noise Control 417

Copyright © 2003 Marcel Dekker, Inc.

418 Chapter 9

TABLE 9-1 Typical Values of the Loss Factor _ for

Materials at Room Temperature

Material _ Material _

Aluminum 0.0010 Masonary blocks 0.006

Brass; bronze 0.0010 Plaster 0.005

Brick 0.015 Plexiglas1 0.020

Concrete 0.015 Plywood 0.030

Copper 0.002 Sand (dry) 0.90

Cork 0.150 Steel; iron 0.0013

Glass 0.0013 Tin 0.002

Gypsum board 0.018 Wood, oak 0.008

Lead 0.015 Zinc 0.0003

Source: Beranek (1971).

TABLE 9-2 Variation of the

Loss Factor _ for a Typical

Acoustic Foam Material Used in

Composite Panelsa

Temperature, 8C Loss factor, _

0 0.087

5 0.116

10 0.168

15 0.28

20 0.48

25 0.72

30 0.79

35 0.66

40 0.47

45 0.30

50 0.19

aThe material has a density of 32 kg/

m3 (2 lbm=ft3). The temperature

range of application for the material

is between _408C (_408F) and

ю708C (1608F).

Copyright © 2003 Marcel Dekker, Inc.

The damping ratio is defined by Eq. (9-25). The damping coefficient

for a damping ratio of _ ј 0:100 is found as follows:

RM ј 2_рKSMЮ1=2 ј р2Юр0:100ЮЅр49:35Юр103Юр50Ю_1=2

RM ј 314:2 N-s/m

The mechanical quality factor is given by Eq. (9-46):

QM ј 1=2_ ј 1=Ѕр2Юр0:100Ю_ ј 5:00

The logarithmic decrement is given by Eq. (9-53):

_ ј 2__ ј р2_Юр0:100Ю ј 0:6283

The decay rate for the system is given by Eq. (9-57):

_ ј 54:575_ fn ј р54:575Юр0:100Юр5Ю ј 27:3dB=s

The reverberation time is given by Eq. (9-58):

T60 ј 60=_ ј р60Ю=р27:3Ю ј 2:20 sec

Finally, the loss factor is given by Eq. (9-67):

_ ј 2_ ј р2Юр0:100Ю ј 0:200

Навигация