2.6 Receptance Method

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Receptance is another name for dynamic flexibility or compliance, and is given by the transfer function

(output displacement)/(input force) in the frequency domain (see Table 2.2). Also, as has been

mentioned previously, it is directly related to mobility through

Receptance ¼

Mobility

jv ð2:105Þ

and this relationship should be clear due to the fact that velocity ¼ jv £ displacement, with zero initial

conditions, in the frequency domain. Hence, the receptance functions for the basic elements mass ðmÞ;

spring ðkÞ; and damper ðbÞ can be derived from the mobility functions of these elements, as given in

Table 2.3. As a result, the interconnection laws given for mobility ðMÞ will be valid for receptance ðRÞ as

well. Specifically, for two receptance elements R1 and R2 connected in series, we have the combined

receptance

Series: R ¼ R1 þ R2 ð2:106Þ

because the displacement is additive and the force is common. For two receptance elements connected

in parallel, the combined receptance R is given by

Parallel:

1

R ¼

1

R1 þ

1

R2 ð2:107Þ

because the forces are additive and the displacement is common. The inverse of receptance is dynamic

stiffness.

2.6.1 Application of Receptance

The receptance method is widely used in the frequency-domain analysis of multi-DoF systems. This is

true particularly because the receptance of a multicomponent system can be expressed in a convenient

form in terms of the receptances of its constituent components. In deriving such relations, we use the

conditions of continuity (forces balance at points of interconnection, or nodes) and compatibility

(relative displacements in a loop add to zero). In fact, Equation 2.106 and Equation 2.107 are special cases

of receptance relations for multicomponent systems.

It should be clear from Table 2.3 that the receptance Rm of an inertia element ð21=ðv2mÞÞ; and the

receptance Rk of a spring element ð1=kÞ; are real quantities, unlike the corresponding mobility functions.

The receptance Rb of a damper ð1=ðbjvÞÞ is imaginary, however. It follows that receptance functions of

undamped systems are real, and we will have to deal with real quantities only in the receptance analysis

of undamped systems. This makes the analysis quite convenient. Also, since the displacement response

of an undamped system becomes infinite when excited by a harmonic force at its natural frequency, we

see that the receptance function of an undamped system goes to infinity (or its inverse becomes zero) at

its natural frequencies. This property can be utilized in determining an undamped natural frequency

(say, the fundamental natural frequency) of a system using the receptance method. In particular, the

characteristic equations for a system with two interconnected components are

Series:

1

R1 þ R2 ¼ 0 ð2:108Þ

Parallel: R1 þ R2 ¼ 0 ð2:109Þ

and their solutions will give the undamped natural frequencies of the combined system. Now, we will

consider two examples to illustrate the application of receptance techniques.

Undamped Simple Oscillator

Consider the simple oscillator shown in Figure 2.9, but assume that the damper is not present. As was

noted earlier, the mass and the spring elements are connected in parallel. Hence, the characteristic

Frequency-Domain Analysis 2-37

© 2005 by Taylor & Francis Group, LLC

equation of the undamped system is

Rm þ Rk ¼ 0 ð2:110Þ or

2

1

v2m þ

1

k ¼ 0 ð2:111Þ

or

2k þ v2m ¼ 0

whose positive solution is

v ¼ vn ¼

ffiffiffiffi

k

m

s

which gives the undamped natural frequency.

Dynamic Absorber

Dynamic absorbers are commonly used for vibration suppression in machinery over narrow frequency

ranges. Specifically, a dynamic absorber can “absorb” the vibration energy from the main system

(machine) at a specific frequency (the tuned frequency) and thereby completely balance the vibration

excitation in the system.

Consider a machine of equivalent mass M and equivalent stiffness K that is mounted on a rigid

foundation, as modeled in Figure 2.15(a). A dynamic absorber, which is a lightly damped oscillator, of

mass m and stiffness k is mounted on the machine. The damping is neglected in the model. The machine

receives a vibration excitation f ðtÞ; and the objective of the absorber is to counteract this excitation.

A schematic mechanical circuit of the system is shown in Figure 2.15(b). The overall system can be

considered to consist of two subsystems: the subsystem a, representing the machine, has M and K

connected in parallel with the excitation source; and the subsystem b; representing the vibrating absorber,

k

f (t)

(a) (b)

(c) (d)

f(t)

K

0

Source

m

k

M

Subsystem

b

Subsystem

a

Rb Ra f (t)

Source

RM f (t)

Source

Rk

Rm

RK

b a

K

Ground

m

M

Dynamic

Absorber

(Subsystem b)

Machine

(Subsystem a)

Excitation

FIGURE 2.15 (a) A machine with a vibration absorber; (b) schematic mechanical circuit; (c) component receptance

circuit; and (d) subsystem receptance circuit.

2-38 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

has m and k connected in series, as is clear from Figure 2.15(b). The corresponding receptance circuit,

indicating the two subsystems with receptances Ra and Rb; is shown in Figure 2.15(d).

Since M and K are connected in parallel, from Equation 2.107 we have

1

Ra ¼ 2v2M þ K ð2:112Þ

Since m and k are connected in series, from Equation 2.106 we have

Rb ¼ 2

1

v2m þ

1

k ð2:113Þ

Now, since the subsystems a and b are connected in parallel, from Equation 2.109, the characteristic

equation of the overall system is given by

Ra þ Rb ¼ 0 ð2:114Þ

Substitute Equations 2.112 and 2.113 in Equation 2.114. We obtain

1

2v2M þ K þ

1

2 v2m þ

1

k ¼ 0 ð2:115Þ

On simplification, after multiplying throughout by the common denominator, we obtain the

characteristic equation

mMv4 2 ðkM þ Km þ kmÞv2 þ kK ¼ 0 ð2:116Þ

This will give two positive roots for v; which are the two undamped natural frequencies of the system.

Typically, the natural frequency of the vibration absorber has to be tuned to the frequency of excitation in

order to achieve effective vibration suppression, as discussed in Chapter 12 (Enunciations 1 – 26).

Here, we have only considered direct receptance functions, where the considered excitation and

response are both for the same node. For more complex, multicomponent, multi-DoF systems, we will

need to consider cross receptance functions, where the response is considered at a node other than where

the excitation force is applied. Such situations are beyond the scope of the present, introductory material.

Some concepts of receptance are summarized in Box 2.3.

Box 2.3

CONCEPTS OF RECEPTION

Receptance; R

ðCompliance; Dynamic FlexibilityÞ ¼

Displacement

Force

Receptance ¼ Mobility=jv

Series Connection: R ¼ R1 þ ____________R2

Parallel Connection: 1=R ¼ 1=R1 þ 1=R2

Note: R is real for undamped systems.

Natural Frequency: R ! 1 (undamped case)

Characteristic Equation:

For system with series components: 1=

P

Ri ¼ 0

For system with parallel components:

P

1=Ri ¼ 0

Frequency-Domain Analysis 2-39

© 2005 by Taylor & Francis Group, LLC

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