2.4.1 Interconnection Laws

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Any general impedance element or a mobility element may be interpreted as a two-port element in which,

under steady conditions, energy (or power) transfer into the device takes place at the input port and

energy (or power) transfer out of the device takes place at the output port. Each port of a two-port

element has a through variable, such as force or current, and an across variable, such as velocity or voltage,

associated with it. Through variables are called flux variables, and across variable are called potential

variables. Through variables are not always the same as flow variables (velocity and current). Similarly,

across variables are not the same as effort variables (force and voltage). For example, force is an effort

variable, but it is also a through variable. Similarly, velocity is a flow variable and is also an across

variable. The concept of effort and flow variables is useful in giving unified definitions for electrical and

mechanical impedance. However, in component interconnecting and circuit analysis, mechanical

impedance is not analogous to electrical impedance. The definition of mechanical impedance is

force/velocity in the frequency domain. This is a ratio of (through variable)/(across variable);

whereas electrical impedance, defined as voltage/current in the frequency domain, is a ratio of

TABLE 2.3 Impedance and Mobility Functions of Basic Mechanical Elements

Element Frequency Transfer Function (Set s ¼ jv ¼ j2p f )

Impedance Mobility Receptance

Mass,m Zm ¼ ms Mm ¼

1

ms

Rm ¼

1

ms2

Spring,k Zk ¼

k

s

Mk ¼

s

k

Rk ¼

1

k

Damper,b Zb ¼b Mb ¼

1

b

Rb ¼

1

bs

Frequency-Domain Analysis 2-27

© 2005 by Taylor & Francis Group, LLC

(across variable)/(through variable). Since both force and voltage are “effort” variables, and velocity and

current are “flow” variables, it is then convenient to use the definition

Impedance ðelectrical or mechanicalÞ ¼

Effort

Flow

In other words, impedance measures how much effort is needed to drive a system at unity flow.

Nevertheless, this definition does not particularly help us to analyze interconnected systems with

mechanical impedance, because mechanical impedance cannot be manipulated using the rules for

electrical impedance. For example, if two electric components are connected in series, the current

(through variable) will be the same for both components, and the voltage (across variable) will be

additive. Accordingly, the impedance of a series-connected electrical system is just the sum of the

impedances of the individual components. Now consider two mechanical components connected in

series. Here the force (through variable) will be the same for both components, and velocity (across

variable) will be additive. Hence, it is mobility, not impedance, that is additive in the case of seriesconnected

mechanical components. It can be concluded that, in circuit analysis, mobility behaves like

electrical impedance and mechanical impedance behaves like electrical admittance. Hence, the

“generalized series element” is electrical impedance or mechanical mobility, and the “generalized

parallel element” is electrical admittance or mechanical impedance. The corresponding interconnection

laws are summarized in Table 2.4.

Now, two examples are given to demonstrate the use of impedance and mobility methods in

frequency-domain problems.

Example 2.5

Consider the simple oscillator shown in Figure 2.9(a). A schematic mechanical circuit is given in

Figure 2.9(b). Note here that in this circuit, the broken line from the mass to the ground represents

how the inertia force of the mass is felt by, or virtually transmitted to, the ground. This is the case

because the net force that generates the acceleration in the mass (i.e., the inertia force) has to be

transmitted to the ground at the reference point of the force source. This is the same reference with

respect to which the velocity of the mass is expressed. If the input is the force f ðtÞ; the source

element is a force source. The corresponding response is the velocity v; and in this situation, the

transfer function V ð f Þ=Fð f Þ is a mobility function. On the other hand, if the input is the velocity

vðtÞ; the source element is a velocity source. Then, f is the output, and the transfer function

Fð f Þ=V ð f Þ is an impedance function.

TABLE 2.4 Interconnection Laws for Impedance and Mobility

Series Connections Parallel Connections

M1 M2

f

v1 v1

v

Z1 Z2

M2

M1

f

Z2

Z1

v

f1

f2

v ¼ v1 þ v2 f ¼ f1 þ f2

v

f ¼

v1

f þ

v2

f

f

v ¼

f1

v ¼

f2

v

M ¼ M1 þ M2 Z ¼ Z1 þ Z2

1

Z ¼

1

Z1 þ

1

Z2

1

M ¼

1

M1 þ

1

M2

2-28 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Suppose that a known forcing function is applied to this system (with zero initial conditions) using a

force source, and the velocity is measured. Now, if we were to move the mass exactly at this

predetermined velocity (using a velocity source), the force generated at the source will be identical to the

originally applied force. In other words, mobility is the reciprocal (inverse) of impedance, as noted

earlier. This reciprocity should be intuitively clear because we are dealing with the same system and same

initial conditions. Owing to this property, we may use either the impedance representation or the

mobility representation, depending on whether the elements are connected in parallel or in series, and

irrespective of whether the input is a force or a velocity. Once the transfer function is determined in one

form, its reciprocal gives the other form.

In the present example, the three elements are connected in parallel, as is clear from the mechanical

circuit shown in Figure 2.9(c). Hence, the impedance representation is appropriate. The overall

impedance function of the system is

Zð f Þ ¼

Fð f Þ

V ð f Þ ¼ Zm þ Zk þ Zb ¼ ms þ

k

s þ b

􀀈 􀀈 􀀈 􀀈

s¼j2p f ¼

ms2 þ bs þ k

s

􀀈 􀀈 􀀈 􀀈 􀀈

s¼j2p f ð2:86Þ

Then, the mobility function is

Mð f Þ ¼

V ð f Þ

Fð f Þ ¼

s

ms2 þ bs þ k

􀀈 􀀈 􀀈 􀀈

s¼j2p f ð2:87Þ

Note that, if in fact the input is the force, the mobility function will govern the system behavior. In this

case, the characteristic polynomial of the system is s2 þ bs þ k; which corresponds to a simple oscillator,

and accordingly the (dependent) velocity response of the system would be governed by this. If, on

the other hand, the input is the velocity, the impedance function will govern the system behavior. The

characteristic polynomial of the system, in this case, is s; which corresponds to a simple integrator. The

(dependent) force response of the system would be governed by an integrator-type behavior. To explore

v

f (t)

0

Zm Zk Zb

(c)

fs

k b

m

f (t)

v

Suspension

(a) fs

v

f (t)

0

(b)

FIGURE 2.9 (a) A ground-based mechanical oscillator; (b) schematic mechanical circuit; (c) impedance circuit.

Frequency-Domain Analysis 2-29

© 2005 by Taylor & Francis Group, LLC

this behavior further, suppose that the velocity source has a constant value. The inertia force will be zero.

The damping force will be constant. The spring force will increase linearly. Hence, the net force will have

an integration (linearly increasing) effect. If the velocity source provides a linearly increasing velocity

(constant acceleration), the inertia force will be constant, the damping force will increase linearly, and the

spring force will increase quadratically.

Example 2.6

Consider the system shown in Figure 2.10(a). In this example, the motion of the mass, m, is not

associated with an external force. The support motion, v; however, is associated with a force, f :

The schematic mechanical circuit representation shown in Figure 2.10(b), and the corresponding

impedance circuit shown in Figure 2.10(c), indicate that the spring and the damper are connected in

parallel and that the mass is connected in series with this pair. By impedance addition for parallel

elements and mobility addition for series elements, it follows that the mobility function is

V ð f Þ

Fð f Þ ¼ Mm þ

1

ðZk þ ZbÞ ¼

1

ms þ

1

ðk=s þ bÞ

􀀈 􀀈 􀀈 􀀈

s¼j2pf ¼

ms2 þ bs þ k

msðbs þ kÞ

􀀈 􀀈 􀀈 􀀈 􀀈

s¼j2pf ð2:88Þ

It follows that, when the support force is the input (force source) and the support velocity is the

output, the system characteristic polynomial is msðbs þ kÞ, which is known to be inherently unstable due

to the presence of a free integrator, and has a non-oscillatory transient response.

The impedance function that corresponds to support velocity input (velocity source) is the reciprocal

of the previous mobility function; thus,

Fð f Þ

V ð f Þ ¼

msðbs þ kÞ

ms2 þ bs þ k

􀀈 􀀈 􀀈 􀀈

s¼j2pf ð2:89Þ

v(t)

v(t) v(t)

vm

vm

vm

o

f f

k b

m

–

+

–

+

Zm

Zs, Ms

(a) (b)

(c)

FIGURE 2.10 (a) An oscillator with support motion; (b) schematic mechanical circuit; (c) impedance circuit.

2-30 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

and

Vmð f Þ

Fð f Þ ¼

1

ms

􀀈 􀀈 􀀈 􀀈

s¼j2pf ð2:90Þ

The resulting impedance function Fð f Þ=Vmð f Þ is not admissible and is physically non-realizable

because Vm cannot be an input as there is no associated force. This is confirmed by the fact that the

corresponding transfer function is a differentiator, which is not physically realizable. The mobility

function Vmð f Þ=Fð f Þ corresponds to a simple integrator. Physically, when a force, f , is applied to the

support, it is transmitted to the mass, unchanged, through the parallel spring – damper unit. Accordingly,

when f is constant, a constant acceleration is produced at the mass causing its velocity to increase linearly

(an integration behavior).

Maxwell’s principle of reciprocity can be demonstrated by noting that the mobility function

Vmð f Þ=Fð f Þ; obtained in this example, will be identical to the mobility function when the locations of

f and Vm are reversed (i.e., when a force, f , is applied to the mass m and the resulting motion, Vm, of the

support, which is not restrained by a force, is measured), with the same initial conditions. The reciprocity

property is valid for linear, constant-parameter systems in general and is particularly useful in vibration

analysis and the testing of multi-DoF systems; for example, to determine a transfer function that is

difficult to measure, by measuring its symmetrical counterpart in the transfer function matrix.

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