2.4 Mechanical Impedance Approach

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Any type of force or motion variable may be used as input and output variables in defining a system

transfer function. In vibration studies, three particular choices are widely used. The corresponding

Frequency-Domain Analysis 2-25

© 2005 by Taylor & Francis Group, LLC

frequency transfer functions are named impedance functions, mobility functions, and transmissibility

functions. These are described in the present section and in the subsequent section, and their use is

illustrated.

Through variables (force) and across variables (velocity), when expressed in the frequency domain

(as Fourier spectra), are used in defining the two important frequency transfer functions: mechanical

impedance and mobility. In the case of impedance functions, velocity is considered the input variable and

the force is the output variable; whereas in the case of mobility functions, the converse applies. Specifically,

M ¼

1

Z ð2:82Þ

It is clear that mobility ðMÞ is the inverse of impedance ðZÞ: Either transfer function may be used in a

given problem. One can define several other versions of frequency transfer functions that might be useful

in the modeling and analysis of mechanical systems. Some of the relatively common ones are listed in

Table 2.2.

Note that in the frequency domain, since acceleration¼ jv £ velocity; and displacement ¼

velocity/ðjvÞ; the alternative types of transfer functions as defined in Table 2.2, are related to mechanical

impedance and mobility through a factor of jv; specifically,

dynamic inertia ¼ force/acceleration ¼ impedance/ðjvÞ

acceleration ¼ acceleration/force ¼ mobility £ ðjvÞ

dynamic stiffness ¼ force/displacement ¼ impedance £ jv

receptance ¼ displacement ¼ mobility/ðjvÞ

In these definitions, the variable (force, acceleration, and displacement) should be interpreted as the

corresponding Fourier spectra.

The time-domain constitutive relations for the mass, spring, and the damper elements are well known.

The corresponding transfer relations are obtained by replacing the derivative operator d=dt by the Laplace

operator s: The frequency transfer functions are obtained by substituting jv or j2p for s: These results are

derived below.

Mass Element:

m

dv

dt ¼ f

In the frequency domain,

mjvv ¼ f or

f

v ¼ mjv

Hence,

Zm ¼ mjv ð2:83aÞ

and

Mm ¼

1

mjv ð2:83bÞ

TABLE 2.2 Definitions of Useful Mechanical Transfer Functions

Transfer Function Definition (in the Frequency Domain)

Dynamic stiffness Force/displacement

Receptance, dynamic flexibility, or compliance Displacement/force

Impedance ðZÞ Force/velocity

Mobility ðMÞ Velocity/force

Dynamic inertia Force/acceleration

Accelerance Acceleration/force

Force transmissibility ðTf Þ Transmitted force/applied force

Motion transmissibility ðTm Þ Transmitted velocity/applied velocity

2-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Spring Element:

f ¼ kx or

df

dt ¼ kv

In the frequency domain,

jvf ¼ kv or

f

v ¼

k

jv

Hence,

Zk ¼

k

jv ð2:84aÞ

and

Mk ¼

jv

k ð2:84bÞ

Damper Element:

f ¼ bv or

f

v ¼ b

Then,

Zb ¼ b ð2.85aÞ

or

Mb ¼

1

b ð2.85bÞ

These results are summarized in Table 2.3.

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