Ch05 more SDOF transient

Applied Structural and Mechanical Vibrations Theory, Methods and Measuring Instrumentation 5 More SDOF—transient response and approximate methods 5 1 Introduction The harmonic excitation considered so.

of the individual responses So, in principle, the complications seem to be more

of a mathematical nature rather than a physical nature, and such a statement

of the problem does not seem to add anything substantial to the understanding

of the behaviour of a linear SDOF system under the action of a complex excitingload However, things are not so simple; a number of different approaches andtechniques are available to deal with this problem and the choice is partly asubjective matter and partly dictated by the complexity of the situation, thefinal results that one wants to achieve and the mathematical tractability of thecalculations by means of analytical or computer-based methods

The first and main distinction can be made between:

• time-domain techniques

• frequency-domain techniques.

As the name itself implies, the first approach relies on the manipulation ofthe functions involved (generally the loading and the response functions) inthe domain of time as the independent variable of interest The important

concepts are ultimately the impulse response function and the convolution,

or Duhamel’s, integral.

On the other hand, the second technique is based on the powerful tool ofmathematical transforms: the manipulations are made in the domain of anappropriate independent variable (frequency, for example, hence the name)and then, if necessary, the result is transformed back to the domain of theoriginal variable

The two approaches, as one might expect, are strictly connected and theresult does not depend on the particular technique adopted for the problem athand Unfortunately, except for simple cases, both techniques involve evaluations

of integrals that are not always easy to solve and their practical applicationmust often rely heavily on numerical methods which, in their turn, require therelevant functions to be ‘sampled’ at regular intervals of the independent variable.This ‘sampling’ procedure introduces further complications that belong to the

specific field of digital signal analysis, but they cannot be ignored when

measurements are taken and computations via electronic instrumentation areperformed Their basic aspects will be dealt in future chapters

Until a few decades ago the computations involved in frequency-domaintechniques were no less than those in a direct evaluation of the discreteconvolution in the time domain The development of a special algorithmcalled the fast Fourier transform [1] has completely changed this situation,cutting down computational time of orders of magnitude and makingfrequency techniques more effective

Both the convolution integral and the transform methods apply whenlinearity holds; for nonlinear systems recourse must be made to a directnumerical integration of the equations of motion, a technique which,obviously, applies to linear systems as well

When the predominant frequency of vibration is the most importantparameter and the system is relatively complex, the Rayleigh ‘energy method’and other techniques with a similar approach turn out to be useful to obtainsuch a parameter The simplest application represents a multiple- (or infinite-)degree-of-freedom system as a ‘generalized’ SDOF system after an educatedguess of the vibration pattern has been made The method is approximate (but

so are numerical methods, and in general are much more time consuming) andits accuracy depends on how well the estimated vibration pattern matches thetrue one Its utility lies in the fact that even a crude but reasonable guess oftenresults in a frequency estimate which is good enough for most practical purposes

5.2 Time domain—impulse response, step response and

convolution integral

Let us refer back to Fig 4.7 The SDOF system considered so far is a particularcase of the situation that this figure illustrates, i.e a single-input single-output

linear system where the output x(t) and the input f(t) (written as a force for

simplicity, but it need not necessarily be so) are related through a lineardifferential equation of the general form

(5.1)

The coefficients a i and b j (i=1, 2, 3, ···, n; j=1, 2, 3, ···, r), that is the parameters

of the problem, may also be functions of time, but in general we shall consideronly cases when they are constants On physical grounds, this assumptionmeans that the system’s parameters (mass, stiffness and damping) do notchange, or change very slowly, during the time of occurrence of the vibrationphenomenon This is the case, for example, for our spring—mass-damperSDOF system whose equation of motion is eq (4.13), which is just a particularcase of eq (5.1)

Very common sources of excitation are transient phenomena and mechanical shocks, both of which are obviously nonperiodic and are

characterized by an energy release of short duration and sudden occurrence.Broadly speaking, we can define a mechanical shock as a transmission ofenergy to a system which takes place in a short time compared with thenatural period of oscillation of the system, while transient phenomena maylast for several periods of vibration of the system

An impulse disturbance, or shock loading, may be for example a ‘hammerblow’: a force of large magnitude which acts for a very short time.Mathematically, the Dirac delta function (Chapter 2) can be used to representsuch a disturbance as

(5.2)

where has the dimensions newton-seconds and describes an impulse (timeintegral of the force) of magnitude

(5.3)

One generally speaks of unit impulse when

From Newton’s second law fdt=mdv, assuming the system at rest before

the application of the impulse, the result on our system will be a suddenchange in velocity equal to without an appreciable change in itsdisplacement Physically, it is the same as applying to the free system the

initial conditions x(0)=0 and The response can thus be written(eq (4.8))

(5.4)for an undamped system, and (eq (4.27))

(5.5)

for a damped system In both cases it is convenient to write the response as

(5.6)

where h(t) is called the unit impulse response (some authors also call it the

weighting function) and is given by

(5.7a)

for the undamped and damped case, respectively Equations (5.7a) represent

the response to an impulse applied at time t=0; if the impulse is applied at

(5.7b)

for and zero for since the change of variable from t to is

geometrically a simple translation of the coordinate axes to the right by anamount seconds In practice, an impact of duration ∆t of the order of

10–3 s (and ∆t is short compared to the system’s period) is acommon occurrence in vibration testing of structures In these cases, theconsiderations above apply

Figure 5.1 illustrates a graph of for a damped system with unitmass, damping ratio and damped natural frequency

Example 5.1 Let us consider the response of an undamped system to an

impulse of a constant force f0 that acts for the short (compared to the system’s

period) interval of time 0<t<t1 We assume the system to be initially at restand we have

(5.8)

During the ‘forced vibration era’ (0<t<t1) the response of the system is given

by eq (4.63) where β=0 (because ω=0) and the particular integral is given

(5.12)

and the approximations above hold for (or, better, ) By notingthat and that the impulse has a value from eq (5.11) weobtain in the limit

(5.13)

which is, as expected, the impulse response of the undamped system (eq(5.4)) It is left to the reader to determine how considerations similar to theones above lead to eq (5.5) for a damped system

A general transient loading such as the one shown in Fig 5.2 can beregarded as a superposition of impulses; each impulse is applied at time

and has a magnitude given by f( ) ∆ , with varying along the time axis

(shaded area in Fig 5.2)

Mathematically we can write

The response to the impulse of eq (5.14) is

and the total response from time to time is obtained by summing

the effects of all the impulses up to the instant t, i.e.

(5.16)Passing to the limit of the summation becomes an integral and theresponse from to is finally given by

that its evaluation is performed through multiplication of f( ) by each

incremental shift in As the present time t varies the impulse response scans the excitation function, producing a weighted sum of past inputs

up to the present instant; so, in terms of the superposition principle the

system’s response x(t) may be interpreted as being the weighted superposition

of past input f( ) values ‘weighted’ or multiplied by In other words,

to find the response x(t0) for some t=t0, we form the function and weshift it to ; the area of the product yields x(t0).

In the foregoing discussion we have assumed that the force f(t) starts at t=0; since this may not be case and f(t) can extend to negative values of t,

we can write eq (5.17) in the more general form

(5.20)

In order for a constant-parameter linear system to be physically realizable

(causal), it is necessary that it responds only to past inputs; this requires

(5.21)

In fact, if we recall that is the response to an impulse applied at time (i.e ), for (i.e ) there is no response because noimpulse has been applied This justifies eq (5.21) and allows us to extend theupper limit of integration in eq (5.20) from without changingthe result; we can then write

(5.22)

We note in passing that a physical system is always causal if the

independent variable is the real time t; however, not all physical systems are

causal (for example, if the independent variable is a space variable).Consider now, in eq (5.20), the change of variable obtained by defining where can be interpreted as the time delay between the occurrence

of the input and the instant when its result is calculated; we get

where the minus sign in the second integral comes from the fact that

Changing over the limits of integration to dispense with the minussign we get

(5.23)

which is an alternative form of the convolution integral

Another alternative form can be obtained either by putting in eq(5.20) or by noting—in the second integral of eq (5.23)—that for since there is no response before the impulse occurs: this yields

(5.24)

The equality between the two integrals is obvious and in the last expression

we have used the common symbol * to indicate the convolution betweentwo functions Equation (5.24) is the most general form of the convolutionintegral for our purposes

An analogous line of reasoning, starting from eq (5.17) leads to

(5.25)

which, with eq (5.24), shows the symmetry of the convolution integral in

the excitation f(t) and the impulse response h(t) Simplicity suggests that the

simpler of the two functions be shifted in performing the calculations

A constant-parameter linear system is stable if any possible bounded input

produces a bounded output Since

(5.26)

if the input is bounded, i.e there exists some finite constant K such that

(5.27)then

(5.28)

It follows that if the impulse response function is absolutely integrable, i.e

(5.29)

then the output will be bounded and the system is stable Note that x(t) does

not need to satisfy

which is necessary in the classical Fourier transform (frequency response)approach

A final comment to show the important property of frequency preservation

in linear systems Let h(t) be the impulse response of a constant-parameter linear system; for an arbitrary, well behaved input f(t) the nth derivative of the output x(t) is given by

(5.31)

If we assume a sinusoidal input of the general form weobtain, for example, for its second derivative Insertingthis result in eq (5.31) we get

(5.32)

so that x(t) must be sinusoidal with the same frequency as f (t) It follows

that a constant-parameter linear system cannot cause any frequencytranslation, but can only modify the amplitude and phase of an appliedinput

Example 5.2 Let us now determine the response of a damped system to the step function given by eq (5.33), which is often called the Heaviside function

when f0=1 In real situations this could model some kind of machine operation

or a car running over a surface that changes level abruptly (i.e a curb)

(5.33)

For assuming that the system starts from rest, the response can bewritten as the convolution integral

(5.34)which can be easily calculated if we remember that

Since and after some manipulations we get

The initial conditions x(0)=0 and (0)=0 give for the constants

which, after substitution in eq (5.37) and a little algebra, give exactly thesolution of eq (5.35) Figure 5.3 is a graph of eq (5.35) with: f0=50 N, k=1000

For an undamped system eq (5.37) becomes

(5.38)

The response to a unit step input (sometimes called indicial response) and

the unit step function (Heaviside) itself are also very important in linearvibration theory and are often indicated with a symbol of their own: as wedid in Chapter 2, we will write θ(t) for the Heaviside function applied at t=0 while we will use the symbol s(t) for the response to a Heaviside input We

can write symbolically eq (4.13) for our SDOF linear system as

(5.39)

where D is the linear differential operator

(5.40)Following this notation, we can then write the symbolic relations

(5.41)

and since—in the sense of distributions (Chapter 2)—it can be shown that

it is left to the reader to determine the following important relations betweenthe impulse response and the Heaviside response

(5.42)

which can be verified in eqs (5.7a), (5.35) and (5.38) (see also eq (5.34))

Fig 5.3 Response to step function.

Two more examples will now show further applications of what has beensaid up to this point Both examples refer to an undamped SDOF system,since this system is often considered—for comparison purposes—in the

analysis of shock loading In these cases, the excitation is short compared to

the natural period of the structure and as a result, the response is notsignificantly influenced by the presence of damping We point out that theresponse of structures to shock is vital in design, particularly preliminarydesign, to select the system’s parameters in a manner that limits within aspecified range a certain response quantity—e.g maximum responseamplitude, maximum stress etc.—in order to prevent undesired vibration or

damage of the structure The concept of shock (or response) spectrum has

been devised to deal with these problems and, due to its importance, it will

be considered separately in the following section

Example 5.3 The first example is the response to a rectangular pulse of

duration t1 and amplitude f0 This problem has already been considered inExample 5.1 (eqs (5.10) and (5.11)), but now we proceed from eq (5.11) bysubstituting the appropriate initial conditions (5.12) to obtain an explicit

expression for the ‘free vibration era’ t>t1 The substitution gives

which, after some manipulations, yields the response for t>t1

(5.43)

Alternatively, the rectangular pulse f(t) can be seen as the superposition

of two step functions, i.e where f1(t) is given by eq (5.8),

f2(t) is a step of magnitude –f0 applied at t=t1 and the response can bedetermined accordingly as a superposition of the two responses Needless tosay, the results are again eq (5.10) for the ‘forced vibration era’ and eq (5.43)for the ‘free vibration era’ The reader is invited to draw a graph of the

response for different values of t1

Example 5.4 In this second example we consider the response of an

undamped system to a half-sine shock excitation, that is, to an excitationfunction of the form

(5.44)

During the ‘forced vibration era’ the exciting function is asinusoid with circular frequency In order to calculate the

response, we notice that f(t) can be written in phasor form as if weagree to take the imaginary part as described in Chapter 1 (recall thatthe form of the exponential and the choice of the real or imaginary part

is a matter of convenience, it is only important to be consistent) Theresponse is then given by

and the integral can be easily solved using again

calculation of the definite integral in brackets yields

Taking the imaginary part of the expression above we get the system’sresponse for which is

(5.45)

The response in the ‘free vibration era’ (t>t1) is obtained from eq (4.8) with initial conditions that are determined by the state of the system at t=t1:this is

(5.46)where now

(5.47)

Substitution of eqs (5.47) into eq (5.46)—considering that —yields

(5.48)

Figures 5.4(a) and 5.4(b) illustrate how the same undamped SDOF system

responds to two different half-sine excitations

Fig 5.4 Undamped SDOF system—half-sine response (a)

(b)

Both the excitation and the response are shown in the two graphs even if themeasurement units are obviously different (newtons for the excitation functionand metres for the response displacement function).

maximum value plotted as a function of t1/T or, alternatively, as a function

of ωn /ω when the quantity ω (frequency of the exciting pulse) can be defined.

Such curves, extensively used in engineering practice, are called response

or shock spectra Several kinds of maxima are important, but in general the so-called maximax response is considered, which is the maximum of the

response attained at any time following the onset of the shock pulse

Mathematically speaking, given an exciting force f(t) of duration t1, wewant to determine the quantity

(5.49)

Let us consider a rectangular pulse of amplitude f0 and duration t1 Theresponse to such excitation is given by eqs (5.10) and (5.43) in the ‘forced’and ‘free’ vibration era, respectively The maximum value of response will

be attained in either the first or second era, depending on the value of t1:

when t<t1, equating the derivative of eq (5.10) to zero gives the condition

which is verified at a time t m given by provided that

(5.50)

where n is an integer and we have taken n=1 because we are interested in the

first maximum and we know that Substitution of t m in eq (5.10)gives the maximum value

and this result can be stated by saying that the maximum value of the ratio

of dynamic response to static deformation (maximum dynamic magnification factor) is equal to 2.

In the free vibration era the response is given by eq (5.43) Equating itsderivative to zero gives the condition

which—considering again the first maximum—is verified at a time t m given by

The response spectrum for a rectangular pulse excitation is then obtained

by combining eqs (5.51) and (5.53) and is shown in Fig 5.5

We turn now to the half-sine excitation considered in Example 5.4 Theresponse to this kind of pulse is given by eqs (5.45) and (5.48) for the forcedand free vibration era, respectively and, once again, the maximum value can

be attained for or t>t1, depending on the value of t1 The two casesare illustrated in Figs 5.4(a) and 5.4(b)

Let us suppose that the maximum response occurs in the forced vibrationera; equating the derivative of eq (5.45) to zero gives

which is satisfied at a time t m determined by

(5.54)

The first maximum is attained by considering the minus sign and n=1 in

eq (5.54) and we get

(5.55)

provided that which means (or ) if we remember that

and hence Substitution of eq (5.55) in eq(5.45) yields after some manipulation

(5.56a)

or, defining

(5.56b)

So, for example, if it follows and the maximum value

of the dynamic magnification factor D is given by

Fig 5.5 Rectangular pulse—shock spectrum.

If the maximum occurs for t>t1, equating the derivative of eq (5.48) tozero leads to

which is satisfied at the time t m given by

(5.57a) (5.57b)

provided that i.e (or) and we have considered the

first maximum (n=1 and the minus sign) passing from eq (5.57a) to eq (5.57b) Substitution of the explicit expression for t m in eq (5.48) yields for themaximum value of displacement

(5.58a)

or, as a function of

(5.58b)for example, if the maximum value of D is found to be

The shock spectrum for a half-sine pulse excitation is shown in Fig 5.6,

where D max is plotted as a function of t1/T.

At resonance, i.e when (or), the expressions given byeqs (5.56a or 5.56b) and (5.58a or 5.58b) become indeterminate but the value of

D max can be obtained by calculating the limit for and using L’Hospital’srule in either one of the above equations It is not difficult to show that

Following a similar line of reasoning, shock spectra for other impulsiveloading conditions can be worked out (e.g Harris [2], Jacobsen [3]).The calculations become more and more laborious, especially if damping

is taken into account, and other techniques are available (for example theLaplace transform, which will be considered later, and the plane-phase orother graphical methods), but often one has to make use of a computer for

an extensive investigation of the problem

However, the following general conclusions can be drawn for step-typeand pulse-type excitations, where damping is of relatively less importanceunless the system is highly damped

The excitation—a function of time only—may be a force applied to themass (as in the examples above) or a ground motion acting on the springanchorage The ground motion, in its turn, can be in the form of displacement,velocity or acceleration All these cases are mathematically similar and thegeneral equation

(5.59)

can be used, where v and ξ are the response and the excitation, respectively,

in the appropriate form For example, if y(t) is the ground displacement with

respect to a fixed frame of reference, the equation of motion is (4.55) with

c=0, i.e.

Fig 5.6 Half-sine pulse—shock spectrum.

which, differentiating twice with respect to time, can be rearranged to

and can be treated as a second-order equation in when the ground

acceleration ÿ(t) is given Note the formal analogy with eq (5.59) where v=

and

For step-type excitations x max occurs after the step has risen to its full

value (t1 being the rise time) and the extreme values of D max are 1 and 2

As the ratio t1/T approaches zero, D max approaches the upper limit of 2;

when t1/T approaches infinity (low rise time compared to the system’s natural period), the step loses its character of dynamic excitation and D max approaches the lower limit of 1, which means that x max tends to the static

value f0/k.

The reader is invited to verify that for some shapes of the step rise time,

D max is equal to unity at certain finite values of t1/T: for example, for a step with constant slope rise, D max=1 when Different values

are obtained for different shapes of the slope, but the minimum value of t1/

T for which this occurs is 1 When D max=1, the amplitude of motion with

respect to the final value of the excitation as a base (the so-called residual response) is zero and this fact is sometimes exploited in the practical design

situation in order to achieve the smallest possible residual response

For pulse-type excitations, when the ratio of pulse duration to system

natural period t1/T is less than 1/2, the shape of certain types of pulses of

equal area (equal impulse) is of secondary significance in determining themaxima of the system’s response If the pulse shape has littlesignificance in almost all cases; only when must the pulse shape

be considered carefully Furthermore, the maximum response usually occursfor and D max has values between 1.5 and 1.8 If the pulse has

a vertical rise, D max attains asymptotically the value of 2 and in the particular

case of rectangular pulse D max=2 for

Again, the reader is invited to verify that the residual response amplitude

is zero for certain values of t1/T and if the pulse has a vertical rise or a vertical

decay (but not both) there are no zero values except at In the case

of a rectangular pulse, the residual response is zero for As

an example, Fig 5.7 shows the response of the undamped SDOF to a

rectangular pulse of duration t1=4 s in the case of

Note that Fig 5.7 illustrates a particular case; for different values of

t1/T the oscillation of the system continues after the end of the pulse with

amplitude characteristics that depend on the value of t1/T considered in the

specific case under study

For several shapes of pulse the minimum value of t1/T for zero residual

response is 1 and occurs for an exciting rectangular pulse; for a sine pulsethe lowest value for zero residual response is and for a symmetricaltriangular pulse is

transforms

Fourier analysis is a mathematical technique that deals with the addition ofseveral harmonic oscillations to form a resultant and with the oppositeproblem Any branch of physical sciences in which harmonically varyingquantities play a part makes use of the theory of Fourier analysis at somestage of its theoretical development The subject of engineering vibrations is

no exception, and we have already given a review of the fundamentalmathematical aspects in Chapter 2 We move on from there, recalling some

of these fundamentals when needed in the course of our investigation Withrespect to Chapter 2, small differences in notation will be noted in a fewcases They have been adopted to suit our present needs and, in any case,are irrelevant to the essence of the discussion as long as consistency ismaintained from the beginning to the end of the specific argument beingdiscussed

Fig 5.7 Rectangular pulse response .

5.3.1 Response to periodic excitation

A periodic signal is a particular type of deterministic (i.e that can be expressed

by an explicit mathematical relationship) signal that repeats itself in time

every T seconds; T is called the period and for all values of t

where n=1, 2, 3, ··· Provided that the fluctuations in f(t) are not too

pathological, a periodic function can be represented by a convergent series

of harmonic functions whose frequencies are integral multiples of a certainfundamental frequency

A periodic exciting function f(t) (written in the form of a force for our

present convenience) can be written as

establish the relation between eqs (5.60a) and (5.60b), and

give the relation between eq (5.60a) and the complex form

If all the Cs are real, then all the bs are zero, f(t) is real and an even function of t, i.e f(–t)=f(t); if (i.e all as and bs are real, as in our case), then f(t) is real but not necessarily symmetrical about t=0 We point

out that eq (5.60c) is not a phasor representation of the input signal (thesecond expression of eq (5.60b) is!), it is a different way of writing the periodic

function f(t) in its own right, where no convention of taking only the real or

imaginary part is implied

The conditions for the convergence of the series are extremely generaland cover the majority of engineering situations; one important restriction

to be kept in mind is that, when f(t) is discontinuous, the series gives the

average value of f(t) at the discontinuity The Fourier coefficients of the

series are given by (eq (2.6b) or (2.10))

(5.61)

for the expression of eq (5.60a) and

(5.62)

for the complex representation of eq (5.60c)

The mean (over one period) and mean-square values of f(t) and of its time

derivative can be expressed in terms of the Fourier coefficients as (Parseval’stheorem)

Under such an excitation, we have the equation of motion for our dampedSDOF system

Alternatively, if the excitation is written in the more compact form of eq

(5.60c), x2 too can be taken in the form of a series with summation from –

to + Since we have determined in Chapter 4 (eqs (4.48) and (4.92)) thatthe steady-state response of our system to a sinusoidal load is written

(5.68)

where H(ω0) is the complex frequency response function (receptance in this

case, see also subsequent sections) and is given by

(5.69)

the response to our periodic load can be written as

(5.70)

from which it follows that

(5.71)

The complementary function x1(t) must be added to eq (5.66) or (5.70) in

order to obtain the general solution Again (Chapter4),the complementary function dies out with time and contains two arbitraryconstants that must be determined by the initial conditions From thediscussion above we see that if the frequency of one of the harmonics in theexcitation is close to the system’s natural frequency ωn , its contribution to

the response will be relatively large (especially for light damping) and acondition of resonance may exist when for some value of n (it is clear from the context, but we note that the subscript n in ωn is for ‘natural’

frequency of the system and is not an integer n=1, 2, 3, ···)·

The undamped case can be easily obtained as a particular case of the

equations above with c=0.

5.3.2 Transform methods I: preliminary remarks

All the problems of transient response that we have considered up to thepresent can be solved by means of other methods by, so to speak, looking atthem from a different angle These techniques—which involve integraltransforms—may seem only mathematical complications when applied torelatively simple cases but their usefulness and power become clear whenmore complicated problems of nonperiodic excitations are encountered.Nonperiodic signals cannot be represented in terms of a fundamentalcomponent plus a sequence of harmonics, but must be considered as asuperposition of signals of all frequencies In other words, the series must begeneralized into an integral over all values of ω In Chapter 2 we introduced

the Fourier transform of a function y(t) as (eq (2.16))

(5.72)and the inverse Fourier transform (Fourier integral representation) as

(5.73)

Equation (5.73) represents the synthesis of y(t) from complex exponential harmonics while eq (5.72) represents analysis of y(t) into its frequency components, each frequency having for its amplitude the magnitude |Y( ω)|