9.5 MECHANICAL IMPEDANCE AND MOBILITY

Back

There are many cases in which the velocity of the mass is an important

vibration function to be controlled, instead of the displacement. In this

case, the mechanical impedance ZM may be utilized. The mechanical impedance

gives a measure of how strongly the system resists applied forces (or

moments). The mechanical impedance is defined as the ratio of the applied

force to the resulting velocity of the system:

ZM ј

FрtЮ

vрtЮ

(9-82)

Let us take the displacement of the system from Eq. (9-73) or (9-74)

and take the derivative with respect to time to obtain the velocity of the

system:

vрtЮ ј

dyрtЮ

dt ј j!ymax e jр!t_        Ю (9-83)

We note that we may write j ј e j_=2, so Eq. (9-83) may be written in the

following alternative form:

vрtЮ ј !ymax e jр!t_   ю_=2Ю ј !ymax e jр!t__Ю ј vmax e jр!t__Ю (9-84)

The quantity _ is related to the displacement phase angle           as follows:

_ ј         _ _=2 (radians) ј          _ 908 (degrees) (9-85)

The complex representation of the mechanical impedance may be

obtained by combining Eqs (9-84) and (9-82):

ZM ј

Fo e j_

!ymax ј jZMj e j_ (9-86)

424 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

The magnitude of the mechanical impedance may be expressed in terms of

the magnification factor (MF) by using Eq. (9-78), defining the factor:

jZMj ј

Fo

!ymax ј

KSрFo=KSЮ

!ymax ј

KS

!MF

(9-87)

If we introduce the expression for the magnification factor from Eq.

(9-81) into Eq. (9-87), we obtain the following result for the magnitude of

the mechanical impedance:

jZMj ј рKS=!ЮЅр1 _ r2Ю2 ю р2_rЮ2_1=2 (9-88)

The expression may be further simplified by using the expression for the

undamped natural frequency, Eq. (9-2), !2

n ј KS=M, and the damping factor

relationship, Eq. (9-25), 2_r ј !RM=KS.

jZMj ј RMf1 ю р1=2_Ю2Ѕr _ р1=rЮ_2g1=2 (9-89a)

The expression may also be written in terms of the mechanical quality factor

QM by using Eq. (9-46):

jZMj ј RMf1 ю Q2

MЅr _ р1=rЮ_2g1=2 (9-89b)

The variation of the mechanical impedance in the various limiting

regions may be noted. First, in the low-frequency region рr _ 1Ю, we note

that r is negligible compared with р1=rЮ, and р1=rЮ is much larger than unity.

In this region, the magnitude of the mechanical impedance approaches the

following limiting value:

jZMj

RM

2_r ј

KS

! ј

KS

2_f

(9-90)

For low frequencies of vibration (or, for practical purposes, when r < 0:16),

the motion is governed by the stiffness or the spring constant of the system.

Thus, Region I рr < 0:16Ю could be denoted as the stiffness-controlled region

of vibration. The mechanical impedance is inversely proportional to the

frequency in this region.

Secondly, for the frequency region around the undamped natural frequency

рr _ 1), the term Ѕr _ р1=rЮ_ is small. In this region, the magnitude of

the mechanical impedance approaches the following limiting value:

jZMj  RM (9-91)

For frequencies around the undamped natural (or, for practical purposes,

when Ѕ1 _ 0:22__ < r < Ѕ1 ю 0:22__Ю, the motion is governed by the damping

of the system. Thus, Region II could be denoted as the damping-controlled

region of vibration.

Vibration Isolation for Noise Control 425

Copyright © 2003 Marcel Dekker, Inc.

Finally, in the high-frequency region рr_1Ю, we note that р1=rЮ is

negligible compared with r. In this region, the magnitude of the mechanical

impedance approaches the following limiting value:

jZMj

RMr

2_ ј !M ј 2_fM (9-92)

For high frequencies of vibration (or, for practical purposes, when r > 6),

the motion is governed by the inertia or the mass of the system. Thus,

Region III рr > 6Ю could be denoted by the mass-controlled region of vibration.

The mechanical impedance is directly proportional to the frequency in

this region.

When analyzing electromechanical systems (combinations of electric

and mechanical components), it is convenient to use the mechanical admittance

YM, which is defined by the following expression:

YM ј

vрtЮ

FрtЮ ј

1

ZM ј jYMj e_j_ (9-93)

The mechanical admittance is also called the mechanical mobility of the

system. The magnitude of the mechanical admittance or mobility may be

expressed in terms of the magnification factor MF by using Eq. (9-78):

jYMj ј

vmax

Fo ј

!ymax

Fo ј

!ymax

KSрFo=KSЮ ј

!MF

KS

(9-94)

If we introduce the expression for the magnification factor from Eq.

(9-81) into Eq. (9-94), we obtain the following result for the mechanical

admittance or mobility:

jYMj ј

r!n

KSЅр1_r2Ю2 юр2_rЮ2_1=2 (9-95)

The undamped natural frequency may be eliminated from the expression as

follows:

!n

KS ј рKS=MЮ1=2

KS ј

1

рKSMЮ1=2 ј

2

RM;cr ј

2_

RM

(9-96)

jYMj ј

2r

RM;crЅр1_r2Ю2 юр2_rЮ2_1=2 ј

2_r

RMЅр1_r2Ю2 юр2_rЮ2_1=2 (9-97)

Aplot of the mobility as a function of the frequency ratio is shown in Fig. 9-

6.

Example 9-4. Determine the magnitude of the maximum velocity for the

system given in Example 9-3.

426 Chapter 9

Copyright © 2003 Marcel Dekker, Inc.

The magnitude of the mobility for the system is given by Eq. (9-94):

jYMj ј

vmax

Fo ј

2_fMF

KS ј р2_Юр10Юр0:3287Ю

р50,000Ю ј 0:413_10_3 m/N-s

jYMj ј 0:413mm=N-s р0:0723 in=lbf -sec)

The maximum amplitude of the velocity for the system is as follows:

vmax ј jYMjFo ј р0:413Юр40Ю ј 16:52mm=s р0:650 in=secЮ

Навигация