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1.6 Forced Response

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The free response of a vibratory system is the response to some initial excitation and in the absence of any

subsequent forcing input. This corresponds to the “natural” response of the system. Mathematically, it is

the homogeneous solution, because it is obtained by solving the homogeneous equation of the system

(i.e., without the input terms). The natural response, the free response, and the homogeneous solution

are synonymous, in the absence of a forcing input to the system.

The forced response of a dynamic (vibratory) system is the response of the system to a forcing input.

When there is a forcing excitation (i.e., an input) on a system, the equation of motion will be

nonhomogeneous (i.e., the right-hand side will not be zero). Then, the total solution (total response T)

will be given by the sum of the homogeneous solution ðHÞ and the particular integral ðPÞ; subject to the

system initial conditions. This may be determined by the mathematical solution of the equation of

motion. The total response can be separated into the terms that depend on the initial conditions ðXÞ and

the terms that depend on the forcing excitation ðFÞ: This is in fact the physical interpretation of the total

solution. Note that X is called the “free response,” “initial-condition response,” or the “zero-input

0.5 1.0

t(s)

0.20

0.10

0.00

−0.10

−0.20

x(m)

0.5 1.0

t(s)

0.20

0.10

0.00

−0.10

−0.20

x(m)

0.5 1.0

t(s)

0.20

0.10

0.00

−0.10

−0.20

x(m)

ζ < 1

ζ = 1

ζ > 1

FIGURE 1.16 Free response of a damped oscillator: (a) underdamped; (b) critically damped; (c) overdamped.

Time-Domain Analysis 1-27

response.” The term F is called the “forced response” or the “zero-initial-condition response” or the

“zero-state response.” In general H is not identical to X and P is not identical to F: But, when there is no

forcing excitation (no input), by definition H and X will be identical. Furthermore, under steady-state

conditions the homogeneous part or initial-condition response will die down (assuming that there is

some damping, and the system is stable). Then, F will become equal to P: Note that, even when the initial

conditions are zero, F and P may not be identical because F may contain a natural response component

that is excited by the forcing input. This component will die out with time, however.

The total response will depend on the natural characteristics of the system (as for the free response)

and also on the nature of the forcing excitation. Mathematically, then, the total response will be

determined by both the homogeneous solution and the particular solution. The complete solution will

require a knowledge of the input (forcing excitation) and the initial conditions.

The behavior of a dynamic system when subjected to a certain forcing excitation may be studied by

analyzing a model of the system. This is commonly known as system-response analysis. System response

may be studied either in the time domain, where the independent variable of the system response is time,

or in the frequency domain, where the independent variable of the system response is frequency. Timedomain

analysis and frequency-domain analysis are equivalent. Variables in the two domains are

connected through the Fourier (integral) transform. The preference of one domain over the other

depends on such factors as the nature of the excitation input, the type of the available analytical model,

the time duration of interest, and the quantities that need to be determined. The frequency-domain

analysis will be addressed in Chapter 2.

1.6.1 Impulse-Response Function

Principle of superposition. Consider a linear dynamic (vibratory) system. The principle of superposition

holds for a linear system. More specifically, if y1 is the system response to excitation u1ðtÞ and y2 is the

response to excitation u2ðtÞ; then ay1 þ by2 is the system response to input au1ðtÞ þ bu2ðtÞ for any

s-Plane

(Eigenvalue Plane)

B D

FIGURE 1.17 Dependence of free response (or stability) on the pole location. (A and B are stable; C is marginally

stable; D and E are unstable.)

1-28 Vibration and Shock Handbook

constants a and b and any excitation functions u1ðtÞ and u2ðtÞ: This is true for both time-variantparameter

linear systems and constant-parameter linear systems.

Unit impulse. A unit pulse of width Dt starting at time t ¼ t is shown in Figure 1.18(a). Its area is

unity. A unit impulse is the limiting case of a unit pulse when Dt ! 0: Unit impulse acting at time t ¼ t

is denoted by dðt 2 tÞ and is graphically represented as in Figure 1.18(b). In mathematical analysis, this

is known as the Dirac delta function. It is mathematically defined by the two conditions:

dðt 2 tÞ ¼

0 for t – t

1 for t ¼ t

(

ð1:88Þ

and

ð1

dðt 2 tÞdt ¼ 1 ð1:89Þ

The Dirac delta function has the following well-known and useful properties:

ð1

f ðtÞdðt 2 tÞdt ¼ f ðtÞ ð1:90Þ

and

ð1

dnf ðtÞ

dtn dðt 2 tÞdt ¼

dnf ðtÞ

dtn

􀀈 􀀈 􀀈 􀀈

t¼t ð1:91Þ

for any well-behaved time function f ðtÞ:

Impulse-response function. The system response (output) to a unit-impulse excitation (input) acted

at time t ¼ 0 is known as the impulse-response function and is denoted by hðtÞ:

1.6.2 Forced Response

The system output to an arbitrary input may be expressed in terms of its impulse-response function. This

is the essence of the impulse-response approach to determining the forced response of a dynamic system.

Without loss of generality, assume that the system input uðtÞ starts at t ¼ 0; that is,

uðtÞ ¼ 0 for t , 0 ð1:92Þ

For a physically realizable system, the response does not depend on the future values of the input.

Consequently,

yðtÞ ¼ 0 for t , 0 ð1:93Þ

Δt

(a) τ

u(t)

0 τ t (b)

u(t)

t + Δt 0 t

FIGURE 1.18 Illustrations of (a) unit pulse and (b) unit impulse.

Time-Domain Analysis 1-29

and

hðtÞ ¼ 0 for t , 0 ð1:94Þ

where yðtÞ is the response of the system to any general excitation uðtÞ:

Furthermore, if the system is a constant-parameter system, then the response does not depend on the

time origin used for the input. Mathematically, this is stated as follows: if the response to input uðtÞ

satisfying Equation 1.92 is yðtÞ; which satisfies Equation 1.93, then the response to input uðt 2 tÞ; which

satisfies,

uðt 2 tÞ ¼ 0 for t , t ð1:95Þ

is yðt 2 tÞ; and it satisfies

yðt 2 tÞ ¼ 0 for t , t ð1:96Þ

This situation is illustrated in Figure 1.19. It follows that the delayed-impulse input dðt 2 tÞ; having time

delay t; produces the delayed response hðt 2 tÞ:

Convolution integral. A given input uðtÞ can be

divided approximately into a series of pulses of

width Dt and magnitude uðtÞ Dt: In Figure 1.20,

as Dt ! 0; the pulse shown by the shaded area

becomes an impulse acting at t ¼ t; having the

magnitude ut dt: The value of this impulse is

given by dðt 2 tÞuðtÞdt: In a linear, constantparameter

system, it produces the response hðt 2

tÞuðtÞdt: By integrating over the entire time

duration of the input uðtÞ; the overall response

yðtÞ is obtained as

yðtÞ ¼

ð1

hðt 2 tÞuðtÞdt ð1:97Þ

u(t)

0 t

y(t)

0 t

u(t−τ)

0 t

y(t−τ)

0 t

FIGURE 1.19 Response to a delayed input.

u(t)

0 t

Area = u(t).Δt

t t+Δt

FIGURE 1.20 A general input treated as a continuous

series of impulses.

1-30 Vibration and Shock Handbook

Equation 1.97 is known as the convolution integral. This is in fact the forced response, under zero initial

conditions.

In view of Equation 1.94, it follows that hðt 2 tÞ ¼ 0 for t . t: Consequently, the upper limit of

integration in Equation 1.97 could be made equal to t without affecting the result. Similarly, in view of

Equation 1.92, the lower limit of integration in Equation 1.97 could be made 21. Furthermore, by

introducing the change of variable t ! t 2 t; an alternative version of the convolution integral is

obtained. Several valid versions of the convolution integral (or response equation) for a linear, constantparameter

system are given below:

yðtÞ ¼

ð1

hðtÞuðt 2 tÞdt ð1:97aÞ

yðtÞ ¼

ð1

hðt 2 tÞuðtÞdt ð1:97bÞ

yðtÞ ¼

ð1

hðtÞuðt 2 tÞdt ð1:97cÞ

yðtÞ ¼

ðt

hðt 2 tÞuðtÞdt ð1:97dÞ

yðtÞ ¼

ðt

hðtÞuðt 2 tÞdt ð1:97eÞ

yðtÞ ¼

ðt

hðt 2 tÞuðtÞdt ð1:97f Þ

yðtÞ ¼

ðt

hðtÞuðt 2 tÞdt ð1:97gÞ

In fact, the lower limit of integration in the convolution integral could be any value satisfying t # 0; and

the upper limit could be any value satisfying t $ t: The use of a particular pair of integration limits depends

on whether the functions hðtÞ and uðtÞ implicitly satisfy the conditions given by Equation 1.93 and

Equation 1.94, or whether these conditions have to be imposed on them by means of the proper integration

limits. It should be noted that the two versions given by Equation 1.97f and Equation 1.97g take these

conditions into account explicitly and therefore are valid for all inputs and impulse-response functions.

It should be emphasized that the response given by the convolution integral assumes a zero initial state,

and is known as the zero-state response, because the impulse response itself assumes a zero initial state.

As we have stated, this is not necessarily equal to the “particular solution” in mathematical analysis.

Also, as t increases ðt ! 1Þ; this solution approaches the steady-state response denoted by yss; which is

typically the particular solution. The impulse response of a system is the inverse Laplace transform of the

transfer function. Hence, it can be determined using Laplace transform techniques. This aspect will be

addressed in Chapter 3. Some useful concepts of forced response are summarized in Box 1.3.

1.6.3 Response to a Support Motion

An important consideration in vibration analysis and in the testing of machinery and equipment is the

response to a support motion. To illustrate the method of analysis, consider the linear, single-DoF system

consisting of mass m; spring constant k; and damping constant b; subjected to support motion

(displacement) uðtÞ: Vertical and horizontal configurations of this system are shown in Figure 1.21.

Both configurations possess the same equation of motion, provided the support motion uðtÞ and the

mass response (displacement) y are measured from the fixed points that correspond to the initial, staticequilibrium

position of the system. In the vertical configuration, the compressive force in the spring

exactly balances the weight of the mass when it is in static equilibrium. In the horizontal configuration,

the spring is unstretched when in static equilibrium. It may be easily verified that the equation of motion

Time-Domain Analysis 1-31

Box 1.3

CONCEPTS OF FORCED RESPONSE

Total Response ðTÞ ¼ Homogeneous Solution ðHÞ þ Particular Integral ðPÞ

¼ Free Response ðXÞ þ Forced Response ðFÞ

¼ Initial-Condition Response ðXÞ þ Zero-Initial-Condition Response ðFÞ

¼ Zero-Input Response ðXÞ þ Zero-State Response ðFÞ

Note: In general, H – X and P – F

With no input (no forcing excitation), by definition, H ; X

At steady state, F becomes equal to P:

Convolution Integral: Response y ¼

Ðt

0 hðt 2 tÞuðtÞdt ¼

Ðt

0 hðtÞuðt 2 tÞdt; where

u ¼ excitation (input) and h ¼ impulse-response function (response to a unit-impulse input).

Damped Simple Oscillator: y€ þ 2zvny_ þv2

ny ¼ v2

nuðtÞ

Poles ðeigenvaluesÞ l1; l2 ¼

2zvn ^

ffiffiffiffiffiffiffiffi

z2 2 1

vn for z $ 1

2zvn ^ jvd for z a 1

vn ¼ undamped natural frequency, vd ¼ damped natural frequency, z ¼ damping ratio.

Note: vd ¼

ffiffiffiffiffiffiffiffi

1 2 z2

vn:

Impulse-Response Function (Zero Initial Conditions):

hðtÞ ¼

ffiffivffiffinffiffiffiffi

1 2 z2

p expð2zvntÞsin vdt for z a 1

ffiffiffiffiffiffiffiffi

z2 2 1

p ½exp l1t 2 exp l2t􀀉 for z s 1

nt expð2vntÞ for z ¼ 1

8>>>>><

>>>>>:

Unit Step Response (Zero Initial Conditions):

ystepðtÞ ¼

1 2

1 ffiffiffiffiffiffiffiffi

1 2 z2

p expð2zvntÞsinðvdt þ fÞ for z a 1

1 2

ffiffiffiffiffiffiffiffi

z2 2 1

vn ½l1 exp l2t 2 l2 exp l1t􀀉 for z s 1

1 2 ðvnt þ 1Þexpð2vntÞ for z ¼ 1

8>>>>>><

>>>>>>:

with

cos f ¼ z

Note:

Impulse Response ¼

dt ðStep ResponseÞ

1-32 Vibration and Shock Handbook

is given by

my€ þ by_ þ ky ¼ kuðtÞ þ bu_ ðtÞ ð1:98Þ

in which ð_Þ ¼ d=dt and ð€Þ ¼ d2=dt2: The two parameters vn and z are undamped natural frequency and

damping ratio, respectively, given by vn ¼

ffiffiffiffiffi

k=m p and 2zvn ¼ b=m; as usual. This results in the equivalent

equation of motion:

y€ þ 2zvny_ þv2

n y ¼ v2

nuðtÞ þ 2zvnu_ ðtÞ ð1:99Þ

There are several ways to determine the response y from Equation 1.99 once the excitation function uðtÞ is

specified. The procedure used below is to first solve the modified equation:

y€ þ 2zvny_ þv2

ny ¼ v2

nuðtÞ ð1:100Þ

This can be identified as the equation of motion of the single-DoF system shown in Figure 1.15. Once this

response is known, the response of the system (Equation 1.99) is obtained by applying the principle of

superposition.

1.6.3.1 Impulse Response

Many important characteristics of a system can be studied by analyzing the system response to an

impulse or a step-input excitation. Such characteristics include system stability, speed of response, time

constants, damping properties, and natural frequencies. In this way, a knowledge of the system response

to an arbitrary excitation is gained. A unit impulse or a unit step are baseline inputs or test inputs.

Responses to such inputs can also serve as the basis for system comparison. In particular, it is usually

possible to determine the degree of nonlinearity in a system by exciting it at two input intensity levels,

separately, and checking whether proportionality is retained at the output; or by applying a harmonic

excitation and checking whether limit cycles are encountered by the response.

The response of the system (Equation 1.100) to a unit impulse uðtÞ ¼ dðtÞ may be conveniently

determined by the Laplace transform approach. Here, we will use a time-domain approach, instead. First,

integrate Equation 1.100 over the almost zero time interval from t ¼ 02 to t ¼ 0þ: We obtain

y_ð0þÞ ¼ y_ð02Þ 2 2zvn½yð0þÞ 2 yð02Þ􀀉 2 v2

ð0þ

y dt þ v2

ð0þ

uðtÞdt ð1:101Þ

Suppose that the system starts from rest. Hence, yð02Þ ¼ 0 and y_ð02Þ ¼ 0: Also, when an impulse is

applied over an infinitesimal time period ½02; 0þ􀀉 the system will not be able to move through a finite

distance during that period. Hence, yð0þÞ ¼ 0 as well, and furthermore, the integral of y on the righthand

side of Equation 1.101 will also be zero. Now, by the definition of a unit impulse, the integral of u

k b

y(t)

u(t)

Static u(t) y(t)

Equilibrium Level

Static

Equilibrium Level

Equipment

Support

(a) (b)

FIGURE 1.21 A system subjected to support motion: (a) vertical configuration; (b) horizontal configuration.

Time-Domain Analysis 1-33

on the right-hand side of Equation 1.101 will be unity. Hence, we have y_ð0þÞ ¼ v2

n: It follows that as soon

as a unit impulse is applied to the system (Equation 1.100) the initial conditions will become

yð0þÞ ¼ 0 and y_ð0þÞ ¼ v2

n ð1:102Þ

Also, beyond t ¼ 0þ the excitation uðtÞ ¼ 0; according to the definition of an impulse. Then, the impulse

response of the system (Equation 1.100) is obtained by its homogeneous solution (as carried out before,

under free response), but with the initial conditions given in Equation 1.102. The three cases of damping

ratio (z , 1; z . 1; and z ¼ 1) should be considered separately. We obtain the following results:

yimpulseðtÞ ¼ hðtÞ ¼

ffiffivffiffinffiffiffiffi

1 2 z2

p expð2zvntÞsin vdt for z , 1 ð1:103aÞ

yimpulseðtÞ ¼ hðtÞ ¼

ffiffiffiffiffiffiffiffi

z2 2 1

p ½exp l1t 2 exp l2t􀀉 for z . 1 ð1:103bÞ

yimpulseðtÞ ¼ hðtÞ ¼ v2

nt expð2vn tÞ for z ¼ 1 ð1:103cÞ

An explanation concerning the dimensions of hðtÞ is appropriate here. Note that yðtÞ has the same

dimensions as uðtÞ: Since hðtÞ is the response to a unit impulse, dðtÞ; it follows that they have the same

dimensions. The magnitude of dðtÞ is represented by a unit area in the uðtÞ versus t plane. Consequently,

dðtÞ has the dimensions of (1/time) or (frequency). Clearly then, hðtÞ also has the dimensions of (1/time)

or (frequency).

1.6.3.2 The Riddle of Zero Initial Conditions

For a second-order system, zero initial conditions correspond to yð0Þ ¼ 0 and y_ð0Þ ¼ 0: It is clear

from Equations 1.103 that hð0Þ ¼ 0; but h_ ð0Þ – 0; which appears to violate the zero-initialconditions

assumption. This situation is characteristic in a system response to an impulse and its

derivatives. This may be explained as follows. When an impulse is applied to a system at rest (zero

initial state), the highest derivative of the system differential equation momentarily becomes infinity.

FIGURE 1.22 Impulse-response functions of a damped oscillator.

1-34 Vibration and Shock Handbook

As a result, the next lower derivative becomes finite (nonzero) at t ¼ 0þ: The remaining lower

derivatives maintain their zero values at that instant. When an impulse is applied to the system

given by Equation 1.100, for example, the acceleration y€ðtÞ becomes infinity, and the velocity y_ðtÞ

takes a nonzero (finite) value shortly after its application ðt ¼ 0þÞ: The displacement yðtÞ; however,

would not have sufficient time to change at t ¼ 0þ: The impulse input is therefore equivalent to a

velocity initial condition, in this case. This initial condition is determined by using the integrated

version (Equation 1.101) of the system (Equation 1.100), as has been done.

The impulse-response functions given by Equations 1.103 are plotted in Figure 1.22 for some

representative values of the damping ratio. It should be noted that for 0 , z , 1 the angular frequency of

damped vibrations is vd; the damped natural frequency, which is smaller than the undamped natural

frequency vn:

1.6.3.3 Step Response

A unit-step excitation is defined by

uðtÞ ¼

1 for t . 0

0 for t # 0

(

ð1:104Þ

Unit-impulse excitation dðtÞ may be interpreted as the time derivative of uðtÞ;

dðtÞ ¼

duðtÞ

dt ð1:105Þ

Note that Equation 1.105 re-establishes the fact that for a nondimensional uðtÞ; the dimension of dðtÞ is

(time)21. Then, since a unit step is the integral of a unit impulse, the step response can be obtained

directly as the integral of the impulse response; thus,

ystepðtÞ ¼

ðt

hðtÞdt ð1:106Þ

This result also follows from the convolution integral (Equation 1.97g) because, for a delayed unit step,

we have

uðt 2 tÞ ¼

1 for t , t

0 for t $ t

(

ð1:107Þ

Thus, by integrating Equations 1.103 with zero initial conditions, the following results are obtained for

step response:

ystepðtÞ ¼ 1 2

1 ffiffiffiffiffiffiffiffi

1 2 z2

p expð2zvntÞsinðvdt þ fÞ for z , 1 ð1:108aÞ

ystep ¼ 1 2

ffiffiffiffiffiffiffiffi

1 2 z2

vn ½l1 exp l2t 2 l2 exp l1t􀀉 for z . 1 ð1:108bÞ

ystep ¼ 1 2 ðvnt þ 1Þexpð2vntÞ for z ¼ 1 ð1:108cÞ

with

cos f ¼ z ð1:109Þ

The step responses given by Equations 1.108 are plotted in Figure 1.23 for several values of damping

ratio.

Note that, since a step input does not cause the highest derivative of the system equation to approach

infinity at t ¼ 0þ; the initial conditions that are required to solve the system equation remain unchanged at

t ¼ 0þ; provided that there are no derivative terms on the input side of the system equation. If there are

derivative terms in the input, then, for example, a step can become an impulse and the situation changes.

Now, the response of the system in Figure 1.21, when subjected to a unit step of support excitation

(see Equation 1.99), is obtained by using the principle of superposition, as the sum of the unit-step

Time-Domain Analysis 1-35

response and ð2z=vnÞ times the unit-impulse response of Equation 1.100. Thus, from Equation 1.103 and

Equation 1.108, we obtain the step response of the system in Figure 1.21 as

yðtÞ ¼ 1 2

expðffi2ffiffiffizffivffiffinffitÞ

1 2 z2

p ½sinðvdt þ fÞ 2 2z sin vdt􀀉 for z , 1 ð1:110aÞ

yðtÞ ¼ 1 þ

ffiffiffiffiffiffiffiffi

1 2 z2

vn ½l2 exp l2t 2 l1 exp l1t􀀉 for z . 1 ð1:110bÞ

yðtÞ ¼ 1 þ ðvnt 2 1Þexpð2vntÞ for z ¼ 1 ð1:110cÞ

1.6.3.4 Liebnitz’s Rule

The time derivative of an integral whose limits of integration are also functions of time may be obtained

using Liebnitz’s rule. It is expressed as

ðbðtÞ

aðtÞ

f ðt; tÞdt ¼ f ½bðtÞ; t􀀉

dbðtÞ

2 f ½aðtÞ; t􀀉

daðtÞ

dt þ

ðbðtÞ

aðtÞ

›f

›t ðt; tÞdt ð1:111Þ

By repeated application of Liebnitz’s rule to Equation 1.97g, we can determine the ith derivative of the

response variable; thus,

diyðtÞ

dti ¼ hðtÞ þ

dhðtÞ

dt þ · · · þ

di21hðtÞ

dti21

" #

uð0Þ þ hðtÞ þ

dhðtÞ

dt þ · · · þ

di22hðtÞ

dti22

" #

duð0Þ

þ · · · þ hðtÞ

di21uð0Þ

dti21 þ

ðt

hðtÞ

diuðt 2 tÞ

dti dt ð1:112Þ

From this result, it follows that the zero-state response to input ½diuðtÞ􀀉=dti is ½diyðtÞ􀀉=dti; provided that

all lower-order derivatives of uðtÞ vanish at t ¼ 0: This result verifies the fact that, for instance, the first

derivative of the unit-step response gives the impulse-response function.

It should be emphasized that the convolution integral (Equation 1.97) gives the forced response of a

system, assuming that the initial conditions are zero. For nonzero initial conditions, the homogeneous

FIGURE 1.23 Unit-step response of a damped simple oscillator.

1-36 Vibration and Shock Handbook

solution (e.g., Equation 1.54 or Equation 1.58) should be added to this zero-initial-condition response

and then the unknown constants should be evaluated by using the initial conditions. Care should be

exercised in the situation in which there is an initial velocity in the system and to this an impulsive

excitation is applied. In this case, one approach would be to first determine the velocity at t ¼ 0þ by

adding to the initial velocity at t ¼ 02; the velocity change in the system due to the impulse. The initial

displacement will not change, however, due to the impulse. Once the initial conditions at t ¼ 0þ are

determined in this manner, the complete solution can be obtained as usual.

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