9.6 TRANSMISSIBILITY

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One of the important factors in design for vibration isolation is reduction of

the force transmitted to the base or support of the system. Generally, the

objective of vibration isolation is to reduce the transmitted force to an

acceptable value. The force transmitted to the base is equal to the sum of

the spring force and the damper force, as illustrated in Fig. 9-7:

FTрtЮ ј FS юFd ј KSyрtЮюRMvрtЮ (9-98)

The displacement is given by Eq. (9-73), and the velocity is given by Eq. (9-

83):

Vibration Isolation for Noise Control 427

FIGURE 9-6 Plot of the dimensionless mechanical mobility jYMj _ Rm;cr ј RM;crvmax=Fo vs. frequency ratio r ј f =fn for an SDOF spirng–mass–damper system

with various values of the damping ratio _ ј RM=RM;cr.

Copyright © 2003 Marcel Dekker, Inc.

FTрtЮ ј KSymax ejр!t_         Ю юj!RMymax ejр!t_ Ю (9-99)

The transmissibility (Tr) is defined as the ratio of the transmitted force

to the applied force for the system.

Tr ј

FTрtЮ

FрtЮ ј

KSymax

Fo

1юj

!RM

KS

_ _

e_j       (9-100)

If we use Eq. (9-76) for the frequency ratio, Eq. (9-78) for the magnification

factor, and Eq. (9-96), we may write the transmissibility in the following

form:

Tr ј MFЅ1юjр2_rЮ_ e_j        ј jTrj e_jр       __Ю (9-101)

The magnitude of the transmissibility may be found from the real and

imaginary parts of Eq. (9-101):

jTrj ј MFЅ1юр2_rЮ2_1=2 (9-102)

The transmissibility may be written in the following fromby substituting for

the magnification factor from Eq. (9-81):

jTrj ј

1юр2_rЮ2

р1_r2Ю2 юр2_rЮ2

" #1=2

(9-103)

A plot of this function is shown in Fig. 9-8. The quantity _ is the damping

ratio and r ј f =fn is the frequency ratio.

The transmissibility may also be expressed in ‘‘level’’ form. The transmissibility

level LTr is defined by the following expression:

LTr ј 20log10 jTrj (9-104)

428 Chapter 9

FIGURE 9-7 The force transmitted to the foundation is the sum of the spring force

and the damper force.

Copyright © 2003 Marcel Dekker, Inc.

We note that the transmissibility level may be positive (if jTrj > 1) or negative

(if jTrj < 1).

The phase angle between the transmitted force and the applied force

may be determined as follows. The angle _ may be found from the real and

imaginary components of the term in brackets in Eq. (9-101) or,

tan _ ј Im=Re.

tan _ ј 2_r (9-105)

The phase angle is р     _ _Ю.

There are several observations that we may make from Fig. 9-8. First,

the transmissibility approaches unity as the frequency ratio becomes small

(less than about 0.2) for all values of the damping ratio. This means that, if

we wish to reduce the force transmitted, the natural frequency of the system

should not be large compared with the forcing frequency.

Vibration Isolation for Noise Control 429

FIGURE 9-8 Plot of the transmissibility Tr ј FT=Fo vs. frequency ratio r ј f =fn for

an SDOF spring–mass–damper system with various values of the damping ratio

_ ј RM=RM;cr.

Copyright © 2003 Marcel Dekker, Inc.

Secondly, if the frequency ratio r is less than

ffiffiffi

p2 (i.e., 1.414), the

transmissibility is always greater than unity for all values of the damping

ratio. In this range of frequencies, the effect of damping is to reduce the

transmitted forces below that which would occur with zero damping; however,

the transmitted force is still larger than the exciting or applied force.

Finally, the transmissibility is always less than unity for the frequency

ratio larger than 1.414. The effect of damping is to increase the transmitted

force above that which would occur with zero damping; however, the transmitted

force is smaller than the applied force, because some of the effort is

expended in accelerating the mass at the higher frequencies. This means

that, if we wish to isolate the foundation from the vibrating mass, we should

select a spring constant for the support system such that fn < 0:707f, or

r > 1:414, and use as little damping as is practical.

For the case of zero damping р_ ј 0Ю, the magnitude of the transmissibility

reduces to the following expression:

jTrj ј

1

jr2 _ 1j

(9-106)

In many cases of vibration isolation design, we need to determine the

frequency ratio required to achieve a given transmissibility. Using Eq. (9-

103), we may write the following:

Tr2Ѕр1 _ r2Ю2 ю р2_rЮ2_ ј 1 ю р2_rЮ2 (9-107)

This expression may be simplified as follows:

r4 _ 2_r2 _

1 _ Tr2

Tr2 ј 0 (9-108)

The quantity _ is defined by the following expression:

_ ј 1 ю

2_2р1 _ Tr2Ю

Tr2 (9-109)

For a known value of the damping ratio _, the frequency ratio r required to

achieve a given value of the transmissibility Tr may be determined. Because

Eq. (9-108) is a quadratic equation in r2, two solutions for r2 are obtained;

however, only the positive solution has a physical meaning.

Example 9-5. A machine has a mass of 50 kg (110.2 lbm). The damping

ratio for the support system is _ ј 0:10. The driving force acting on the

mass has a maximum amplitude of 5.00kN (1124 lbf ), and the frequency

of the driving force is 35 Hz. Determine the spring constant of the support

such that the transmissibility is 0.020 or the transmissibility level is _34 dB.

The value of the parameter _ may be found from Eq. (9-109):

430 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

_ ј 1 ю р2Юр0:10Ю2р1 _ 0:0202Ю

р0:020Ю2 ј 1 ю 49:98 ј 50:98

The required frequency ratio may be found by solving Eq. (9-108):

рr2Ю2 _ р2Юр50:98Юr2 _ р1 _ 0:0202Ю=р0:020Ю2 ј 0

r2 ј 50:98 ю Ѕр50:98Ю2 ю 2499_1=2 ј 50:98 ю 71:40 ј 122:38

r ј р122:38Ю1=2 ј 11:06 ј f =fn

The required undamped natural frequency may be determined:

fn ј

35

11:06 ј 3:164 Hz ј рKS=MЮ1=2

2_

The required spring constant for the support system may now be found:

KS ј Ѕр2_Юр3:164Ю_2р50Ю ј 19,759N=m ј 19:76kN=m р112:8 lbf=inЮ

The magnitude of the transmitted force may be calculated from the

definition of the transmissibility:

FT ј FojTrj ј р5000Юр0:020Ю ј 100N р22:5 lbf Ю

The phase angle between the transmitted force and the applied force may be

found from Eqs (9-80) and (9-105):

tan        ј р2Юр0:10Юр11:06Ю

1 _ р11:06Ю2 ј _0:01823

             ј _0:0182 rad ј _1:048

tan _ ј р2Юр0:10Юр11:06Ю ј 2:213

_ ј 1:146 rad ј 65:688

The phase angle between the transmitted and applied forces is as follows:

р          ј __Ю ј _1:048 _ 65:688 ј _66:728 ј _1:164 rad

Note that, because of the small damping ratio, the phase angle              between

the displacement and the applied force is almost zero.

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