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Linear analysis of transient vibration
J. Sound Vib. (1969) 9 (2), 313-337
LINEAR ANALYSIS OF TRANSIENT VIBRATION R. E. D. BISHOV, A. G. PARKr~SONAND J. W. PENDERED Department of Mechanical Engineering, University CollegeLondon, Gower Street, London, IV.C.1, England
(Received 27 May 1968) An elementary explanation is given of Duhamel's integral and Fourier and Laplace transform techniques in linear vibration analysis. In this, three types of receptances are described, the relationships between them are explained and their application illustrated by means of simple examples. The ways in which the theory is extended so as to apply to aeroelastic systems is also explained. Aeroelastic systems represent an important class of active systems, and it is probably in this direction that transient analysis has found its most important applications to date. 1. INTRODUCTION In practice, the analysis of transient vibration can be very difficult. Quite apart from the usual problems that beset the analyst in the process of idealization for the purpose of analysis, there are the difficulties (a) of adequately defining the excitation and (b) of working out the response. The first of these is a matter of physics, while the second is solely one of mathematical complication. Modern computing techniques have largely drawn the sting from the latter and the new techniques of measurement which are steadily improving our physical understanding are thus beginning to make this form of analysis a practical proposition--at least for the initial period following sudden application of force. It is clear that such analysis can be most important, as when aircraft designs are checked for landing and for gust loading; indeed some aircraft designs are considerably influenced by the results of gust analysis. Yet it is arguable that transient oscillation, as a subject of general study, has been somewhat neglected by vibration analysts in the past. It may be thought opportune therefore to attempt to set down the basic theory of transient loading in quite simple terms. It is also felt appropriate that the application of receptance theory to transient analysis should be reviewed. Here, three types of receptances that are in current use are explained, and the relationships that exist between them are developed. The adjective "transient" is usually applied to deterministic (as opposed to random) excitation that is a non-periodic function of time alone. The response is thus a non-periodic forced vibration, and (if the system concerned is a passive one) no question of self-excitation arises. It is not to be imagined, however, that transient excitation is relevant only to passive systems. In recent years, transient analysis has probably found most application in the determination of the response of a flexible aircraft to a gust. As an illustration of the application to active systems, the paper therefore includes an exposition of the principles of gust analysis. This article is therefore essentially one of explanation and its purpose is to collect information and to present it in a form that it is hoped will be found quite straightforward. Its subject matter falls naturally into five parts. These are (i) an introduction to harmonic and transient receptances, (ii) the use of Duhamel's integral and Fourier transform techniques in transient analysis, 313
314
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
(iii) the relationships between the various types of receptance, (iv) the derivation of transient receptances using Laplace transform techniques, (v) the application of transient receptances to aeroelastic (i.e. active) systems. 2. HARMONIC AND TRANSIENT RECEPTANCES A "receptance" is defined as the response at a generalized co-ordinate to a unit excitation applied either at the same or some other generalized co-ordinate. The responses to a harmonic excitation of unit amplitude, for example, are the familiar harmonic receptances.t In this paper we shall examine the response of a linear system to two other types of unit excitation and introduce two other forms of receptance. The three types of receptance will be explained in terms of an example. Consider the idealized system, shown in Figure 1, composed of rigid masses Ml and M2 connected by a light spring of stiffness k and a dashpot whose damping coefficient is b. If the
I / / / I / / / / / / / / /
/
/
Figure 1. displacements of the masses from pre-selected equilibrium positions are qt and q2, the equations of motion under the influence of a force Q~ (which "corresponds to" ql in the Lagrangian sense) are M1 ql + b~l + kql - b~2 - kq2 = Q1,
(1)
--bql - kqt + e 2 q 2 + bq2 + kq2 = 0. 2.1. HARMONIC RECEPTANCES
Suppose that the applied force Q1 in the system of Figure 1 is harmonic, having the amplitude ~t and frequency to. In terms of the complex exponential representation, we may write QI = ~a e lwt, and it should be noted that we assign no particular significance to the instant t -- 0. If this value of Ql is placed on the right-hand side of the first of equations (1), and we set out to find the steady-state harmonic solutions of those equations, then the trial solutions ql = I/'ti e f~°t, q2 = htt2e ~°~t, (2) are appropriate. When substituted into the equations of motion they give the pair o f algebraic equations (k - M1 to 2 + itob) WI - (k + itob) W2 = ~1,
- ( k + i, ob) ~gl + (~ - M2 ,o2 + i,ob) ~
= O.
(3)
These algebraic equations may be solved for WI and W2 so that the solutions of equations (1) may be written in the form qi --- el 1(£O)~1 et°~t, q2 = °~21(t°) ~1 ettOt, (4) t Because harmonic receptances have received a great deal more attention in the literature than the other types, they are usually referred to simply as "receptances". Here, the adjective harmonic will be necessary to
distinguish them from other forms.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
315
where ~11(03) is the direct harmonic receptance at ql and ~2~(03) is the cross harmonic receptance between qt and q2. In fact in the problem under discussion we have k - 342 032 + i03b oq ~(03) = (k - M~ 032 + i03b) (k - M2 032 + i03b) - (k + i03b)2' k + i03b a2t(03) ----(k - Ma to2 + i03b) (k - M2 032 Jv i03b) - (k + i03b)2"
(5)
Already the simple system of Figure 1 is seen to produce somewhat unwieldy results. It is worthwhile to simplify them as far as possible and to express them in forms which can be most easily applied later in the paper. We therefore note that they can be rewritten as C~ll(W)
M , + M 2 1 [ , - -1, ,~ M 2 /,M , ] ~ "[- if: + i¢.0~2_3r.O2 ,
~z2'(03)
,[, M, + M2 (ig)2
1 ] (~:+ i03)2 + r/2 ,
(6)
where =
a/[4Ml M2 k(Ml + 342) - b2(M1 + M2) z] 2M1 Mz
and
(7) b(Ml + Mz) 2M1 Mz
(It is assumed here that the system of Figure 1 possesses less than critical damping, so that is real.)
Figure 2. If Mz is made infinitely large, then the two harmonic receptances collapse to the forms appropriate for the system of Figure 2; ~21 -+ 0 and ~i~ takes the form, 1 ~11(03) = k - M1 032 + i03b
(8)
or
,1 (03)
1/M1 (v + i03)z + ~z
(9)
where = ~/4kM1 2Ml
b2 '
b v = 2M~"
(10)
The various uses to which harmonic receptances may be put and the various forms that they may take will not be discussed here, as they are examined at length in references 1 and 2. Nor will their usefulness in transient analysis be described at this stage as this can conveniently be left until later in the present paper. At this point we merely note that a harmonic receptance is, in effect, a quantity giving the amplitude and phase of a steady-state response to a harmonic excitation of unit amplitude, and as such can be easily measured experimentally.
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The technique of deriving the receptances from equations like (3) can usefully be generalized for more complicated systems. This is because the technique involves only algebraic manipulation and does not require the solution of differential equations. In fact, all that is really needed is an application of Cramer's rule [1, 2]. There are other methods of arriving at harmonic receptances, and some of them are set out in the form of rules in references 1 and 2. But, generally speaking, the algebraic technique is the most useful in practice. Many of the properties of harmonic receptances are discussed in reference 1. It is quite easy to show, for instance, that they obey the reciprocity rule, ~rs(to) = c~sr(o~).
(11)
This is readily illustrated in terms of the system shown in Figure 1 by simply interchanging the subscripts in the expression for a2t(to) in equation (5). 2.2. INDICIALRECEPTANCES We return now to the system of Figure 1, and to equations (1). Suppose that the force Q~(t) is suddenly applied at the instant t = 0, as indicated in Figure 3; the system having been at rest in the position qi = 0 = q2 for all t < 0. The force Ql(t) can conveniently be written
:It 0
"-'~!
Figure 3. symbolically as Ql(t) = ~1 l(t), where, once again, ~1 is the amplitude of the excitation and the function l(t) is defined as l(t) =
{~
t<0 t >/0"
(12)
The motion is governed by equation (1) with QI replaced by ~l for t i> 0. When the initial conditions ql = 0 = ~ 1 =q2 =q2 att=0 are imposed upon the general solution, expressions may be found for the displacements at q~ and q2. These expressions may be written in the form ql = ~ll(t).~l, q2 = ~211(t).~1,
(13)
where all(t) is the direct indicial receptance and ct121(t) is the cross indicial receptance. These receptances can easily be shown to have the forms ~ll(t)
t2(Mi ~- M2)
t2
k(M1
+ M2)2[
MIME
(14)
[1 _ e_~,(cosBt + ~sin
where the symbols ~: and ~/are defined in equations (7). It should be noted that the presence of the function l(t) in expressions (14) confirms that we are only concerned with response of the system for t t> O.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
317
If the value of Mz is taken as infinity, the (only) indicial receptance of the anchored system of Figure 2 is found; namely, e -vt cos ~t + ~sin ~t
-
(15)
The "rigid body" term in the indicial receptance is thus absent now. It will be shown later how the indicial receptances may be used with the Duhamel integral in transient analysis. At this stage it is merely to be noted that they are the direct and cross responses to a "unit step excitation". We shall now briefly mention certain of their properties. Generally speaking, the method that has been suggested here to find indicial receptances (that is, direct solution of the appropriate differential equation) is too cumbersome to be of value, whereas step-by-step integration techniques have been used successfully for many years. Other methods exist, however, and two of them are explained in reference 3. They are based on (a) Fourier integral and (b) operational techniques. These both require prior knowledge of the harmonic receptances of the system which, as we have previously noted, are relatively easy to determine numerically or experimentally. The first method is discussed later, as also is the use of Laplace transform techniques in transient analysis. It will be shown that an explicit expression can be found for an indicial receptance ~,~,(t) in terms of the corresponding harmonic receptance ~,s(oJ). The relationship is such as to preserve the property of reciprocity [see equation (11)], so that
o:~( t ) =: ~],( t ).
(16)
This result can easily be illustrated in terms of the system shown in Figure 1 by simply changing the subscripts in the expression for c~t(t ) in equation (14). 2.3. IMPULSIVERECEPTANCES Returning yet again to equations (1) and Figure 1, suppose now that an impulse is applied to the mass M~ at the instant t = 0. That is to say, the curve of Q~(t) is that shown in Figure 4 where the vertical arrow implies a curve having a peak at t = 0 of infinite height, zero width along the time axis and an area t/i~ beneath it. To express this symbolically, we may write
Ql(t) = ~1 3(0,
(17)
where 3(0 is the so-called delta function.
1
Figure 4.
The motion is now governed by equations (1) with Q~ replaced by 0. These are the equations of free motion and, to their general solution, we must apply the initial conditions q~=0=q2=0z
and 0 1 = ~
att=0.
(18)
In this way, expressions may be found for the displacements at qt and qz for t >I 0. These expressions are of the form ql = - ~ ( t ) . ~ ,
q2 = ~ ( t ) . tb,,
(19)
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R. E, D. BISHOP, A. G. PARKINSON AND J. W . PENDERED
where u~l(t) is the direct impulsive receptance and ~2~l(t) is the cross impulsive receptance. For the system of Figure 1, these receptances can be written as,
t a ? l ( t ) = ~/, + - - M:
(,
M2 e-~_t sin ~qt .t l(t), + M2)"q)
"q M ~ ( M I
~ , ( t ) = Ml + ME
(2o)
e-~t sin*/t t l(t). (M, + M2) */1
Again we have no interest in the behaviour of these functions for t < 0. If M2 is taken as infinity, we arrive at the impulsive receptance at ql of the system shown in Figure 2. This is
(,)= ( m e -
s,n
(21)
where ~ and v have the values given in equation (10). Again the "rigid body" term in the impulsive receptance is absent. Later on we shall describe how the impulsive receptances may be used with the Duhamel integral in transient analysis. Here we merely note that an impulsive receptance is the response to a unit impulse excitation. For a given system it would not be a practical proposition to find impulsive receptances in the manner just indicated. Some headway can be made by systematizing the process of solving the differential equations, but this process is inescapably cumbersome [1]. In general it is better to use an approach which does not require direct solution of the differential equations (see, for example, the use of Laplace transform techniques). One such method of finding an impulsive receptance is first to derive the appropriate indicial receptance and then to differentiate it. This method of deriving an impulsive receptance will become clear when we see how impulsive and indicial receptances are related to each other. We may also note that the impulsive receptance and the corresponding harmonic receptance are a Fourier transform pair. This relationship does not easily lend itself to the analytical derivation of impulsive receptances, but, it should be remembered that harmonic receptances are much easier to measure experimentally than impulsive receptances. The property of reciprocity is exhibited by impulsive receptances; that is to say, ~5(t) = =5(t).
(22)
Later on it will be shown that the relationships that exist between the three types ofreceptance all preserve this property--a property tha~ is most easily demonstrated for the harmonic receptances. 3. TRANSIENT ANALYSIS: DUHAMEL'S INTEGRAL TECHNIQUES The principle of superposition may be applied to any system whose motion is described by linear differential equations. Now the systems under discussion are not merely linear; they have constant coefficients. This means that the response at some instant to a step function of force or to an impulsive force applied at a time ~- depends only on the elapsed time t - ~and not on t and ~- separately. In view of this we may use the principle of superposition to obtain the response to a force which is an arbitrary function of time, using the mathematical process of convolution. 3.1. DUHAMEL'S INTEGRAL WITH INDICIAL RECEPTANCES Let some force Qs(t), corresponding to the sth generalized co-ordinate of the system, commence to act at the instant t = 0. The function may be represented roughly by a series
L I N E A R ANALYSIS OF T R A N S I E N T V I B R A T I O N
319
of steps, acting at the instants t = 0, d~-, 2A~-, 3A~-..... as shown in Figure 5(a). The response at the rth generalized co-ordinate to all these step inputs is
q,(t) - ct~(t) Q~(O)+ S', ~ ( t - r) AQ,
(23)
A'r
where the sum is reckoned over all the steps. The sketches in Figure 5(b) and (c) will make the meaning clear. This result may be written in the form
q,(t) - ~(t) Q~(O)+ ~ o:~(t- z)-~--~z~Ar
(24)
so that, in the limit, as the steps are taken indefinitely small, t
q,(t) =
Q,(O) + |
(25)
0
This expression is known as Duhamel's integral. (a)
//Y/Y/C/Y/Y/,/YA~YY/A" :i' os co) "~ I I i r
0
,'t
(b)
t 0 rT
~;--
(c)
0
~t
r
Figure 5. So far, we have not considered the duration of the excitation to be finite. If, however, we are concerned with an excitation pulse of duration T(see, for example, Figure 6), then equation (25) should be replaced by
q,(t) = ~ ( t ) Qs(O)+ f ~ , ( t - z ) - ~ ~d r - ~s(t- T)Qs(T).
(26)
qtl
0
Note that, if t < T, the third term in expression (26) vanishes and, if t > T, the integrands of equations (25) and (26) are zero for T < ~- <~ t. In many cases Q,(T) = 0---that is, the pulse does not end abruptly--and consequently expressions (25) and (26) are identical. By way of illustration, suppose that a force Ql(t) = A. tiT acts on the system of Figure 1 during the interval 0 ~< t ~< T as indicated in Figure 6. The response at q2 is given by the expression t
q2(t)
=
- ~211(t- T ) A , f cc211(t- ~') ~d~" A 0
(27)
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R. E. D. BISHOP, A. G. PARKINSON AND J. W. PENDERED
A
0
T
t
Figure 6. where the indicial receptance is the second of those given in equations (14). That is, A q2(t)
T.2(Mt
(t - - r ) 2 l(t
+ m2)
- r)dr
-
Id
o t
o
[ (t - T) 2 - A [2(m~ + m2)
M , 3/2
k(m~ + m2) 2 x
x[1-e-~(t-r'(cos~l(t-T)+~sin~7(t-T))]}l(t-T).
(28)
In evaluating integrals such as those of expressions (28) and (36), it should be remembered that l(t - r) = 1 for 0 ~
=_At
!_'
TI6(M, + M2) M}M22~/
u, M2
k(M1 + 3/2) 2 t + [2~t,_
-'~k2(Ml-}-M2)3[,r]
il
~2
.
'~* e-e'c°s~Tt)+ I -~--0e-"slnr/t]} '
(29)
and for t f> T, 3t - 2T qz(t)= A T 6 ( M I + M2)
A. k(~+M~M2) ~ e-~t-r' [cos ~7(t- T ) + ~sin ~/(t- T ) ] +
M~M~ + A~,/~2(M~ + Mz)-3 ~/ e -et
1 - -~ sin ~t - 2 cos ~Tt -
-e-~"-r) [(1-~2)sin~(t-T)-2~cos~(t-T)]}.
(30)
Finally, returning to the general problem, it should be stressed that if more than one force Q,(t) is applied to the system, the result (25) may still be made to give the total response through the principle of superposition. Thus, if the number of degrees of freedom is n,
q,(t) = ~ , s=l
~s(t) Q~(O) +
~s(t - r)
dr - ~s(t - T) Q~(T) .
(31)
0
3.2. DUHAMEL'S INTEGRAL WITH IMPULSIVE RECEPTANCES
An alternative form of Duhamel's integral may be used, in which impulsive receptances are employed. Going back to the function Qs(t) of arbitrary form (Figure 5), we now divide
LINEAR ANALYSIS OF TRANSIENT
~it)(a)~
VIBRATION
321
f
0 ' r o,(i) (b)
H
0
"r
D-!
i'o'°'A a r V
k./ "-~
~t
Figure 7. up the area under the curve into vertical strips, each of which represents a discrete impulse of magnitude Q~(t).At (see Figure 7). The response at qr to all the impulses applied in the interval 0 ~<~- ~< t is qr(t) -- Y, =~,s(t- T) Q~(T) Ar, (32) /Jr
so that, in the limit, as AT -+ 0, t
q,(t) -----f =~(t - T) Q~(T) dr.
(33)
0
This result may be used instead of that employed in equation (27) to find the response at q2 when the force Q,(t) of Figure 6 is applied to the system of Figure 1. That is, t
q2(t) =
=~2~(t- T)A.~.d%
0 < t ~< T
(34)
=~21(t--T)A.~.d%
t I> T,
(35)
0
and T
q2(t) =
f 0
where the impulsive receptance is that of equations (20). Thus for t >1 T, T
q2(t)
AT.(M] +1 M2) f (t - T) Tl(t -- T) dT -0 T
A 1 f T. e -~<'-~) sin ~/(t - T) l(t -- T) d% T" (MI + M2)7} 0
which may be shown to be equivalent to the result of equation (30). 22
(36)
322
R.E.D.
BISHOP, A. G. PARKINSON AND J. W. PENDERED
To return to the general result (33) we note again that, if more than one force Q,(t) is applied to the system, the total response may be found from the principle of superposition. That is to say, t
q~(t) = ~ f u~(t - ~) Q~(~')d~',
(37)
s=l
n being the number of degrees of freedom possessed by the system.
4. TRANSIENT ANALYSIS: FOURIER TRANSFORM TECHNIQUES The Fourier integral theorem states that any f u n c t i o n f ( t ) satisfying the condition
f If(t)ldt <
(38)
--co
may be expressed in the form
f(t) =
1 ; F(to) e ~'°tdto
(39)
--oo
where oo
F(to) = f f(t) e-''°t dt.
(40)
--o0
That is to say, the f u n c t i o n f ( t ) is associated with a function F(to), which is known as its Fourier transform. A proof of the theorem is given in reference 3. L e t f ( t ) be identified with a generalized force Q~(t), which is a function of time. In this event it is clear that the quantity to must be in the nature of a frequency. And equation (39) shows that the transient force Qs(t) can be thought of as being built up from an infinite number of harmonic components, that of frequency to being of amplitude F(to)dto/2rr--or Q~(to)dto/2cr, as we shall now call it. The harmonic response at qr due to the excitation Qs(to)2zrdtoe t°'t .is ~,~(to) Q~(to) dto e ~ t 2rr The total response at qr is thus
q,(t) = ~
~,~(to)Q~(to)e '°~tdto
(41)
--oo
or
q,(t)=l f ~,~(to){_f Q~(t)e-t°'tdt} e~°'tdto.
(42)
--oo
If more than one generalized force Q,(t) acts on the system, it is necessary to add solutions of the type (42) by summing the results over all s.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
323
To illustrate the use of equation (42), we return to our previous problem in which the system of Figure I is excited by the force Ql(t) of Figure 6. Noting that the condition (38) is satisfied, and using the harmonic receptance given in equation (6), we find that, for t ~>T, 1
q2(t) = ~
1
1
M1 ~- M2 (i~)2
1
. t
(~ + io))2 + ~/2
A~e-
~,~t
dt
e ~°~ do).
(43)
By expanding the second term of the harmonic receptance further into partial fractions, and evaluating the inner integral, this integrand can be expressed in terms of components whose integrals are given in Appendix 1. On evaluation, this yields a result equivalent to that of equation (30). 5. RELATIONSHIPS BETWEEN THE VARIOUS RECEPTANCES 5.1. INDICIAL AND IMPULSIVE RECEPTANCES The response at some generalized co-ordinate q, to a pulse Q,(t) of the form shown in Figure 8(a) may be expressed in terms of the cross indicial receptance =,~(t). It is [from Figure 8(b)] 1
1
1
q" = -~t ~drs(t) -- - ~ c%(t -- At).
(44)
If we now take the interval A t to be a smaller and smaller quantity we find that
lim[~l,~(t)--e~__~,(t--At!] d l q , = A=~0L t = = ~ [=,~(t)].
(45)
But the process of letting At --~ 0 is also that of letting Q,(t) become an impulse function 8(0 of the type shown in Figure 4. Therefore, a d c%(t) = ~-~[~,s(t)].
(46)
This result may easily be illustrated with any of the corresponding indicial and impulsive receptances that have already been derived, whether direct or cross. Thus it can be seen that, for the system of Figure 1, %8 l ( t ) = ~d[ % ll( t ) ]
and
d l ~2t(t)=~[c~21(t)] ,
while a similar result to the first of these applies to the system of Figure 2.
0 At
"--~t
0 ~._7t
----~t
os(t) (b}
Figure 8. The result (46) is generally sufficient for practical purposes of vibration analysis, but a special case should perhaps be mentioned. The individual transient receptances of a linear system are usually well-behaved continuous functions of time so that the numerator of the
324
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
quantity in square brackets in equation (45) tends to zero as At ~ 0 for all values of t. In analysis of the type under discussion, however, there is a special case in which this is not so. It occurs when there is a discontinuous step in ~,~,(t) at t = 0, so that e,t~(0) ¢ 0 while, of course, ~,I~(t) = 0 for t < 0. In this special circumstance equation (45) remains valid, except when t = 0. For this particular instant the numerator in the square brackets of that equation remains finite while the denominator is made indefinitely small in the limiting process. Thus the response at t = 0 may be written as
q" = l~mo[el~(O) - °~l~(O ' A -t At) ] = lim [ ~ ( 0 ) ] = c~(O) lim Q,(t) = ~,~s(O)8(0. •at-,oL At J ,J,-,o
(47)
Remembering that 8(0 = 0 for all t ¢ 0 and that the transient receptances have the form
o~l,,(t) = a~,(t) l(t), ee~(t) = a~,~(t) l(t),
(48)
results (46) and (47) can be combined into one general result that ~,~,(t) = e~,(0) 8(0 + l(t) d [a~s(t)].
(49)
Thus, at any time other than t = 0, only the second term in equation (49) matters and then it is equal to (d/dr) [e,~,(t)]. If there is a discontinuity in e¢,(t) at t = 0, such that ~ ( 0 ) ~ 0, then only the first term in equation (49) is significant at that instant. Such a case as this would arise if Ml -----0 = b in the system of Figure 1 ; the step force then being applied at the end of a massless spring. In that event the spring would experience a sudden distortion at the instant t = 0. It is not our purpose, however, to pursue this point any further, partly because it is clearly of limited importance and only arises in easily identified circumstances and partly because it raises difficulties of a purely mathematical nature. 5.2. IMPULSIVE AND HARMONIC RECEPTANCES
Impulsive and harmonic receptances are related to each other in a very simple way. This can easily be shown by reference to the result (42). In that equation, if Q,(t) = 8(t) then, by definition, q,(t) = ~,~,(t). That is to say,
1
~,,(co)
3(t) e -"°t dt e u°' do).
(50)
--cO
Here, the inner integral has the value unity since 3(t) is everywhere zero except at t = 0, at which value e -~°'t has the value unity. Therefore,
1/
e,,(t) -- ~
e,~(o))e u°' do).
(51)
By comparing equations (39) and (51) it will be seen that e,,(o)) is merely the Fourier transform of e,~(t). The inverse relationship is that of equation (40), so that e,~(to) = ; e,~,(t) e -u°' dt. --cO
(52)
LINEAR ANALYSIS OF TRANSIENT VIBRATION
325
It is, of course, no surprise that ~,(t) may possess a Fourier transform, since the requirement (38) is clearly met by any impulsive receptance of any passive linear dissipative system that is anchored. It is not clear, however, that an unanchored system of the type shown in Figure 1 or an undamped system will satisfy the requirement, and indeed they do not do so directly. It can be shown, though, that the results (51) and (52) can still be retained providing that suitable limiting processes are employed. As an example, consider the unanchored (though damped) system of Figure 1, for which a~2z(t) does not satisfy requirement (38). Since e -13' a~t(t),
[3 > 0
satisfies the condition, we use the artifice of first evaluating the appropriate integral and then letting fl be vanishingly small so that (53)
a21(to) = lim J e -~, a2az(t)e-i°~t dt ~-~0 -oo
or, here,
2,(to) =lim f /3.o
t
M l + M2
e-¢t sin r/t / -w~t (M-~l~ M~-2)~Jl(t)e dt.
(54)
--oO
This gives ct2z(to) = lim{
-1
1
}
~-~o (M1 + ~)2)(to -- ifl) + (Mr + M2) [(to - ifl - i~)z _ 72]
(55)
which reduces to the expression for aEl(to) given in equations (6). Condition (38) must also be satisfied in performing the transformation from harmonic to impulsive receptance. Thus, again referring to the system of Figure 1, we consider a621(/) = lira
t3-,o 2rr __ M , + M2 ,(to 7 ifl)2 + (to _
ifl -
i~)2 _ ~=j
dto.
(56)
--co
This gives the impulsive receptance ~21(t) of equations (20). Undamped systems may be approached by first considering the corresponding damped system and applying a limiting process in which the damping is reduced to zero. Thus, for the undamped (though anchored) system corresponding to Figure 2 with b = 0 = v, ~ e1-
all(to)=l!m_.
~t-s m ~ t l ( t ) e - ' ° ' t d t
(57)
--co
and cc~,(t) = lim 1 ; M I [(v + ito)2 1 + ~2] e'c°t . dto. ,,~o 2~r
(58)
--00
These expressions yield the values for al~(to) and a~(t) given in equations (8) and (21), respectively, with b = 0. 5.3. HARMONIC AND INDICIAL RECEPTANCES
It is true that, having established the connections of the last two sections, we have effectively shown how harmonic and indicial receptances are related to each other. It is instructive,
326
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
nonetheless, to approach the matter directly. We may use the Fourier integral technique of the preceding section. But whereas the force Qs(t) = 8(0 obeys the requirement of equation (38), the force Qs(t) = l(t) does not. It is therefore not possible to use equation (42) directly. The difficulty may be overcome by investigating the response to a transient excitation
Q,(t) = e -/3'. l(t)
(59)
where fl > 0 and then letting fl ~ 0. The Fourier transform of Q,(t) is now Q~(to) =
;
Q~(t)e-t'°tdt=
--oo
;
1 e-t~+t'°)tdt-fl+i~.
60)
0
That is, the excitation (59) may be expressed in the form
Q,(t) =
1 ;,
.e~"~t 8 - - ~ i dto.
(61)
--00
If we wished merely to find a frequency expression for the function Q,(t) = l(t) we could in fact do so by evaluating the integral l ( t ) = - 2rr I lim ~-.0 f ~ e'°" do.
(62)
~o0
A method of doing this which avoids the arrival at an improper integral is to separate the real and imaginary parts of the integrand of expression (62) and then to use the substitution to/fl = x, finally setting fl = 0. The resulting expression turns out to be 1 1 ~ sintot l(t)=~+~ J oa do,
(63)
~oa
the so-called Dirichlet integral. Our purpose is not to find a frequency expression for l(t), however, but to use the result (60) to find ~,l,(t). By definition of the indicial receptance, equation (41) now gives ~s(t) =-~-1 lim f c%(to)e'~°'dto. zrr ~--,o fl + ia,
(64)
--00
Thus for the system of Figure 1, we have
~ll(t)= 2~r/3-,0 1 lim f
1
{ 1
M, + M2 (i~)2
1
/
(~ + ito)2 + 72j ~
e '°'*
do
(65)
which yields the value of =2tl(t) given in equations (14). If the system is unanchored or undamped, further limiting processes may be necessary. 6. TRANSIENT ANALYSIS: LAPLACE TRANSFORM TECHNIQUES The Fourier integral theorem defined by equations (39) and (40) is only directly applicable to functions which satisfy condition (38). For this reason various limiting processes have been adopted in forming some of the Fourier transforms. Again, the technique cannot be
LINEAR ANALYSIS OF TRANSIENT VIBRATION
327
used immediately for vibrating systems that are unstable. There is, however, an alternative process--that of the Laplace transform--which avoids these difficulties. The Laplace transform is a valuable tool in determining the solutions of differential equations which are required to satisfy certain initial conditions. The transient response of vibrating systems is an example of problems of this class. Unfortunately, however, the Laplace transform, unlike the Fourier integral, has no simple physical interpretation. The Laplace transform F(p) of a functionf(t) is defined as
F(p) = f f ( t ) e -pt dt
(66)
o
where p = cr +/co is chosen such that the real number o is large enough to ensure that the integral (66) is convergent. The functionf (t) must satisfy certain continuity conditions (see, for example, reference 4), but we shall not examine them here, apart from noting that l(t) and 3(0 are suitable functions. The inverse transform may be expressed in the form y+loo
The integration in equation (67) is performed in the complex p plane along the line p = ~,, where ~, is a real number greater than the real parts of the poles (or singularities) of F(p) (see Appendix 2 and Figure 9). Many inverse transforms can, of course, be found directly from standard references. :s / PI / p :- ~+i~ //
I
c!
o-- 7"
Figure 9. By means of the Laplace transform it is possible to reduce differential equations of motion to algebraic forms and then derive the corresponding indicial and impulsive receptances. The process of reduction is a standard one [5] and we shall illustrate first an important property of such a reduction. Consider, for example, the transform of a generalized velocity ~)~(t), say. Applying the definition (66) we have o0
co
[- ¢)l(t) e -pt dt = [ql(t) e-Pt]~ - [ ql(t) (-p) 0
e -or
dt,
0 co
=p f ql(t)e-P' dt - q~(O),
(68)
0
where ql(0) denotes the initial value of q~(t) at t = 0. Thus the Laplace transform of gl~(t) can be written simply in terms of the transform of ql(t).
328
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BISHOP, A. G. P A R K I N S O N
AND
J. W . P E N D E R E D
Likewise the Laplace transform of the generalizedacceleration~'~(t)is co
co
(0",(t) e-pt dt --p f ~x(t)e -pt dt- 0,(0), 0
0 co =p2
f ql(t) e-p' dt -
~,(0) -pq,(O),
(69)
0
where ~1(0) represents the value of~l(t) at t = 0. The transform of~l(t) can also therefore be expressed easily in terms of the transform of q~(t). We will illustrate the complete process by calculating the indicial and impulsive receptances of the system shown in Figure 1. Consider first the indicial receptances. The Laplace transform of the excitation Ql(t) = ~1 l(t) is
f ~ll(t)e-l"dt=~fe-'~dt=~Jp. 0
(70)
0
If we form the Laplace transform of both sides of the first of equations (1) we have
(Mlp2 +bp+k) f ql(t)e-'tdt-(bp+k) 0
q2(t)e-'tdt 0
= 1 ~l + MI ~(0) + (Mp + b)q~(O) P
bq2(0).
(71)
But here ql(t) and q2(t) have the form (13), so that ifAl l(P) and A,ll(p) represent respectively the Laplace transforms of e I ~(t) and e,! ~(t), equation (71) can be rewritten as (nl
p2 + bp + k)Aft(p) - (bp + k)Al~(p) = 1 ~
+ M~ 0t (0) + (M~ p + b)ql (0) - bq2(0). (72)
Of course, strictly, A~s(p) is the Laplace transform of al,s(t) [see equations (48)] for
Al,,(p) =
al~(t)l(t)e-'t dt,
e~(t)e-'tdt= 0
0
co
= f ars(t) e-'t dt.
(73)
0
The Laplace transform of the second of equations (1) can be found in a similar way. Both transformed equations can be combined in the algebraic matrix form
[Ap2+Bp+C][A~I(P)][A~1(p)J= [10/P] + A4(0) + (AP + B)q(0)'
(74)
where q(0) = rq~(0)] Lq2(0)J,
dl(0) = [*,(0) 1
(75)
L~2(0)J
and
A=[M,
02],
B=[_bb
-~],
C=[.,_kk - k ]
(76)
are the inertia, damping and stiffness matrices, respectively. The transform procedure is, of course, easily applicable to systems with any number of degrees of freedom and a matrix
LINEAR ANALYSIS OF TRANSIENT VIBRATION
329
equation similar to expression (74) is obtained. For a system with n degrees of freedom the various matrices are of order n. We are concerned here with a system which is initially at rest so that q(0) and q(0) are both null column vectors in equation (74). In these circumstances the following results are obtained for the transformed receptances:
A 11(P)
M2p 2 + bp + k p A (p) ' (77)
A~(p) - bp + k pA(p)' where
A(p) = lAp z + Bp + C I = Mx Mzp2(p + ~ - i~l)(p + ~ + i~1).
(78)
The quantities ~ and ~ are introduced in equations (7). The inversion of the Laplace transforms (77) is explained in Appendix 2. The results obtained for the indicial receptances are in complete agreement with those of equations (14). The impulsive receptances are calculated in a similar manner. Thus for the system of Figure 1 described by equations (1), if Ql(t)= q)l ~(t) and ql = 0 = q2 =01 =0z for t ~<0, then
ql = o~l(t) ~1 = a~l(t) l(t) ~1, l q2 ~ , ( t ) ~ l =a~,(t) l ( t ) ~ l . J
(79)
The transformed equations have the form
[Ap2+Bp+C][A~I(P)]LA~I(p)J= [10]
(80)
where A~I, A~ and 1 are the Laplace transforms ofa~l, a~ and 3(0, respectively. Aqt and A~I are calculated in a manner similar to that of equation (73). The transformed receptances have the form
A~i(p) = Mzp 2 + bp + k A(p) ' bp + k A~(p) A(p) '
(81)
and the inverse Laplace transforms of these expressions yield the impulsive receptances (20). The properties of the transient receptances which are discussed earlier in this paper can all be established through the results of the present section. For example, from the matrix equations (74) and (80) with q(0) = 0 = 0(0) one can see that for any system the transformed receptances are related in the following manner:
A~(p) =pA~,(p), = p f a~,(t) e -p' dt, o co
= [ - e - " a~s(t)]~° +
f o
e- pt~d [a~s(t)l dt.
(82)
330
R. E. D. BISHOP, A. G. PARK1NSON AND J. W. PENDERED
Therefore,
A~s(p) =
pt d
a,ls(0) +
e- ~ [a~s(t)] dt.
(83)
0
Thus by forming the inverse Laplace transforms of both sides of equation (83) and remembering that the transform of 8(0 is 1 we have ~,a~(t) = ~,~,(0)8(0 + l(t)~t [a~(t)].
(84)
This relationship between the impulsive and indicial receptances is in agreement with that derived earlier, when the physical significance of the first term on the right-hand side of equation (84) was explained. It is also possible to establish the expressions relating transient and harmonic receptances through the Laplace transform solution, but this will not be discussed here. 7. AEROELASTIC SYSTEMS By contrast with passive systems, we are concerned here with systems in which the exciting force (or at least part of it) depends upon the response. In these circumstances self-excitation is possible. The equation of motion of such a system may be expressed as A/i + Bd/+ Cq = Q(t) + F.
(85)
Here, A, B and C are the inertia, damping and stiffness matrices of the system, q(t) is a column matrix of generalized co-ordinates, Q(t) is the column matrix representing the applied forces and F is the column matrix of impressed forces due to the response. Aeroelastic systems represent an important class of active systems and it is to their analysis that techniques of transient analysis have been applied most frequently. Such systems differ from those considered earlier in this paper in one important aspect. So far we have discussed methods of representing the transient response of a system due to applied external forces. We shall see that, in part, determination of aerodynamic response entails solving the inverse problem--that is, the calculation of changes in aerodynamic loading due to arbitrary displacements, modifications in geometry, or any sudden change in flight conditions from one steady state to another. The purpose of this study lies in the calculation of loads, bending moments, shearing forces, etc., acting in an aircraft structure during rapidly changing conditions of flight. In this section we are concerned with an aircraft flying into a vertical gust, as it is this problem of transient aerodynamics which has received most attention in the past. If it could validly be assumed that aerodynamic forces act instantaneously--i.e, that the loads at any instant correspond to the current configuration and motion (as in piston theory) --then problems on transient loading would be quite straightforward. In particular, the vector F would depend at any instant on the prevailing vectors ii, ¢1and q. This is well illustrated in a paper by Huntley [6]. Unfortunately this assumption is not strictly valid. Vortices require time to form, circulation takes time to adjust itsdf to changes of incidence, and so on. As this clearly implies, the term F is made much more complicated as a consequence. At any instant t, the vector F depends on q(t) for all values of~" less than t. (In the special case of a "rigid body" motion of translation, the dependency is upon q(~-) for all values of ~- less than t.) This delay in the adjustment of aerodynamic forces also means that Q(t) is made more complex in the particular problem of gust loading where the delayed adjustments accompany the varying depth of penetration of the aircraft into the gust front.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
331
In order to simplify the problem, we first note that, in the matrix F, the object is to express the generalized forces in terms of generalized displacements (or generalized velocities for rigid body modes other than those of pitch and yaw). Equation (85) may therefore be written as n separate equations of which the rth is
2=1
[a~ + b~s + C,~q~]=~ F~ + Qr(t). s~l
(86)
Here, Fr~ is the force at the rth generalized co-ordinate due to the deformation in the sth co-ordinate. This deformation is a function of time dependent in part upon Qr(t). It is convenient to start by discussing the impressed force F~ due to an elementary response. The force F,~ due to an arbitrary response can then be derived from this. In general there are two practical ways of realizing elementary responses, namely the Fourier and Indicial techniques. These are considered separately below. 7.1. FOURIER TECHNIQUE
Consider a harmonic displacement (or velocity if appropriate) of the form
qs(t) = t/t~e'°~'.
(87)
This will give rise to a harmonic impressed force which can be expressed as F,s = ~p v 2 stz,s(o,) ~'~ e '~', where z,,(¢o) = s, l = p= V=
(88)
the inverse harmonic aerodynamic receptance,t reference area and length, respectively, air density, aircraft velocity.
The force due to an arbitrary displacement can now be built up using a Fourier transform technique. There is, however, a difficulty in that, although displacements in the distortion modes of a flexible aircraft satisfy condition (38), velocities in rigid body translational modes do not. To avoid this difficulty one should use acceleration co-ordinates, all of which obey condition (38). The acceleration response can then be calculated using the Fourier technique outlined here and the results integrated to yield the velocities and displacements. This is fully explained by Mitchell [7] but, for simplicity, we will confine our remarks here to displacement co-ordinates. Using the relations (39) and (40), therefore, an arbitrary displacement q,(t) can be expressed as an infinite array of harmonic components, that of frequency to being of amplitude
l { ~_fqs(t)e-i°~t dt)d¢o. The impressed force due to this component is therefore
21r
{_J q~(t)e-U°~dtd°JeU°t
t In aerodynamic analyses, one would normally work with non-dimensionalgeneralizedco-ordinates and receptances so that the terms in equation (85) have dimensions(force) x (length). However, in order to keep the discussion here in general terms, we will not enter into the question of non-dimensional aerodynamic equations. We merely note, therefore, that the receptances have dimensions appropriate to those of the other terms in equation (88).
332
R.E.D.
BISHOP, A. G. PARKINSON AND J. W. PENDERED
and that due to the complete response is
-
2~r
zr,(o~)
qs(t) e-'~t dt
e '~t dco.
(89)
The advantage of this approach lies in the fact that harmonic aerodynamic derivatives are relatively easy to measure and calculate. It will be noted that the method requires that these be known for all frequencies. It now remains to discuss the term Q,(t), the externally applied gust excitation in the absence of response. It is again convenient to reduce an arbitrary gust into an infinite array of harmonic components. Thus, we first consider the aircraft to fly through an infinite expanse of harmonic downwash having a velocity amplitude w and wavelength I. The aerodynamic generalized force in the rth mode arising from this may be expressed as Q,(t) =
½pV z slz,(t2)-~e w lea,,
(90)
where
g2 = 2~rV/A, z,(g2) = the appropriate complex aerodynamic inverse receptance. This value for Q,(t) together with the value for F,~(t) from expression (89), may now be substituted into equation (86) to give the response of the aeroelastic system to a harmonic gust. Thus, the rth equation becomes s=l
[ a ~ + b,~gl~+ c,~q~] =½pV2sl[~=~ _£z,~(eo){_f q~(t)e-'°"dt}e'°~rdco+z,(g2)vem' ].
(91)
Since the system is assumed to be linear, these equations may be solved to give the responses in the form
q,(t) =
W ~t
r(~2).~,e
.
(92)
The response to a gust having an arbitrary velocity profile w(t) (as seen by the aircraft) can now be derived. Assuming that w(t) satisfies relation (38) (or that a suitable limiting process can be applied) and applying the results of the Fourier transformation defined in (39) and (40), we see that the harmonic component of w(t) at frequency O has amplitude
The response in the rth mode due to this is therefore 1 y( 27r
e -t~t dt e ~ t dD,
and that due to the complete profile is
q,(t) = ~1
Y(12) ~o0
w(t)__t~t dt e t~t dO. -ff-
(93)
L I N E A R ANALYSIS OF T R A N S I E N T V I B R A T I O N
7.2.
333
INDICIAL TECHNIQUE
The second approach to this problem derives from the consideration of step distortions of the form
qs(t) =
Ws l(t)
(94)
rather than the harmonic displacements considered above. The impressed force which accompanies such a displacement can be expressed as
F,~(t) = z~(t) 7t~.
(95)
It is convenient to distinguish between those rigid body modes for which so-called "velocity co-ordinates" are relevant, and all other modes. For the former translation modes, forces are related to generalized velocities. For all other modes the forces are related to generalized displacements. The terms zl,~(t) can then be expressed as
z],(t) = ½pV 2 slz;'s K,(,~)(t) =
(s represents translation modes), 2 ,
= ½pV slz,~ Klt,~)(t)
(96)
(s represents all other modes), where
Kl(,s)(t)
= growth factor which approaches unity as t tends to ~,
z~s, z)'s = constant factors of appropriate dimensions corresponding to steady-state conditions at t = o,.]The evaluation of the Kj(,~)(t) is a matter of indicial aerodynamics. This can loosely be referred to as the Wagner problem, since Wagner gave the first solution of this type. The impressed force due to an arbitrary disturbance can now be derived using Duhamel's integral in the "indicial" (as opposed to the "impulsive") form. Thus, using the result of equation (25), and assuming that the initial displacement is zero (so that the first term on the right-hand side of equation (25) is zero), the impressed force can be written as t
0
=½pV2sl(zr"siKl(rs)(t-z)q's(T)d'r}
(97)
s = translation modes,
=½pV 2 sl z,~ K,(,s)(t-,)Os(r)d o
s = other modes. The summation term on the right-hand side of equation (86) can therefore be expressed as
½pV2sl
~
~=translatlon modes
z;'~ f Kl,.s)(t-~')~/s(1")dr 0
+
]~
s=other modes
Z;s f K,,.~)(t--:)O~(.)d. 0
t These factors are in fact the zero-frequency inverse harmonic receptances.
334
R.E.D.
BISHOP, A. G . P A R K I N S O N A N D J. W . P E N D E R E D
The term Q~(t) arising from the aerodynamic loading in the absence of response may also be approached by considering a step gust rather than a harmonic gust. Thus, we consider the aircraft flying into a vertical fronted up-gust of velocity w. In evaluating the aerodynamic force arising from such a gust, account must be taken of gradual development of aerodynamic forces and the gradual penetration of the aircraft into the gust front. The force is therefore expressed as W Q,(t) = ½pV2 sl-# Q,, K2(o(t),
(98)
where Q', = constant factor of appropriate dimensions corresponding to the steady state as t --~ oo (i.e. as the aircraft becomes more deeply immersed), K2(o(t) = a growth factor which approaches unity as t tends to o0. The evaluation of K2(o(t) is a second problem of indicial aerodynamics. Where, as is usual, it is assumed that an aircraft flies normally into a vertical-fronted up-gust (so that the only relevant translation mode is that of heave), this may loosely be referred to as the Kiissner problem, in honour of the man who first solved an elementary case. The complete response of the aircraft following a step gust is therefore given by n equations such as
[a~s(~+ b~,Os+ C,~q~] t
=½pV 2sl
X s=translation modes
t
zr", f Kl(r,)(t-~')(is('r)dr + ~ s=other modes
0
+ --~Q; K2(o(t)).
z~s f K,(,,)(t-*)O~(*)dr + 0
(99)
These n equations may be solved using step-by-step methods to give the variations in the n generalized co-ordinates in the form
qr(t) = Ylr(t) w.
(100)
Once the functions Y](t) have been evaluated, the response to any gust may be derived by a further application of Duhamel's integral. Thus, using the result of equation (25), the response to a gust w(t) [as seen by the aircraft and assuming w(t) = 0 at t = 0] is given by
q,(t)=f
Y l , ( t - r ) ~ r dr.
(101)
0
REFERENCES 1. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 2. R.E.D. BISHOP,G. M. L. GLADWELLand S. MICrlAELSON1965 The Matrix Analysis of Vibration. Cambridge: Cambridge University Press. 3. T. VONK/~RM/~Nand M. A. BlOT1940MathematicalMethods in Engineering. New York: McGrawHill. 4. W. R. LE PAGE 1961 Complex Variables and the Laplace Transform for Engineers. New York: McGraw-Hill.
335
L I N E A R A N A L Y S I S OF T R A N S I E N T V I B R A T I O N
5. H. S. CARSLAWand J. C. JAEGER1963 OperationalMethods in Applied Mathematics. New York: Dover Publications Inc. 6. E. HUNTLEY1961 ARC 23, 508, R.A.E. Tech. Note Aero. 2771. Calculations of the response of a flexible slender wing aircraft to discrete vertical gusts. 7. C. G. B. MITCHELL1968 A R C R & M N o . 3498. Calculation of the response of a flexible aircraft to harmonic and discrete gusts by a transform method. APPENDIX 1 TABLE 1 Standard Fourier Transformations co
g(t) = f f(to)el°~td~o
f(oJ)
(a + iw)-" n>0 (a + ioJ)-" (b + ion)-I n > -1
2~r[F(n)] -1 t"-I e-°,
t I> 0
0
t<0
27r[-P(n)]-x (a - b)-" e -bt 7(n, at - bt)
t >~0
0
t<0
F(n) = ; e-Ss "-1 ds o
(= (n - 1)! ifn is an integer)
Gt
7(n, ~t) = f e-Ss"-1 ds 0
APPENDIX 2: CALCULATION OF INVERSE LAPLACE TRANSFORMS F r o m the definition (67) the inverse transforms of expressions (77) are determined by the integrals y+l~
l(t) Ct]l(/) = 2~ri
(' J
M 2 p 2 + bp + k p , . pd(p) e ap,
--
(A.1)
b p + k e P , dp. pA(p)
(A.2)
y --i oo "F+lco
l(t) ~l'(t) = 2~ri
f ),-l~o
Here the zeros o f p A ( p ) and hence the poles of the integrands are at p = 0, - ~ ± iT, where and ~ are positive, that is, at the points 0, P~ and Pz in the complex p plane of Figure 9. Thus 7 can be any positive real number and the integrals (A.1) and (A.2) are performed along the line through the point p = y parallel to the imaginary axis (the broken line in Figure 9). There are several ways of evaluating the integrals (A.1) and (A.2). For example, the integrands can be expanded as a series of partial fractions and the inverse transform of each fraction determined separately from tables of Laplace transforms. The transient receptances
336
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
are, however, often more easily determined by contour integration in the complex plane. In some respects, this is merely an alternative way of forming partial fractions. Consider the contour ABCA in Figure 9, where BCA is a semicircle of diameter 2R --AB. As All(p) and A~,~(p) are 0(p -3) and 0(p-4), respectively, it may he shown (see, for example, reference 5) that, for t > 0, lim f A]l(p)ePtdp=O=lim R-->eo BCA
f Ai,,(p)ePtdp.
(A.3)
R~oo B~A
Thus from equations (A. 1) and (A.2)
l(t) f
.l,(t) =~-~
All(p)ePtdp,
r/ l(t) t" ~t211(t)= ~ | A~l(p) ept
(A.4)
dp,
td
/7
where/7 is the limit of the closed contour ABCA as R -+ oo, Expressions of the form (A.4) can be integrated by a standard procedure using the calculus of residues. Thus ~] 1(0 is equal to the sum of the residues of A~t(p)e pt at the poles p = 0, - ~ 4- i contained within the contour/7 (see reference 4). But AIl(p)e pt has no poles outside 11, so that the contour/-/may be replaced by any dosed curve containing the points p = 0, - ~ 4- in. Integrating expressions (A.4) in this way the transient receptances are determined as, /M2(-~ + in) 2 + b(-~ + in) + ket_~+,n), + b(-~ - i n ) + g l M2(-~ - in) 3 (-2in)
+ M 2 ( - ~ - in) 2 +
k e~_e_ln),
+
I]
Lrd2/ (M2£Z-+bP--+ k ) e ' / l(t), + 2! [@2 ~Ml M2(p + ~ - in)(p + ~ + in)/J,,-d
(A.5)
, . . _ [ b(-~___~+in_._)+___k c-~+,,7,, b(-~ - in) + k eC_e_tn,,+ =2~(t) - I M ~Mz(-~ + in) 32in e ~ M~ M2(-~ - in) x (-2in)
~1rd2 / (bp+k)eP' }] } 1(,). + 2! [dp ~tm~ m=(p + ~ -- in) (p + # + in) ,=0
(A.6)
These expressions for the indicial receptances reduce, after some manipulation, to equations (14). Note that, as pA(p) has a multiple root at p = 0, a special procedure is adopted in equations (A.5) and (A.6) in calculating the residues at p = 0. This is explained in general terms in reference 5. To illustrate one of the alternative methods of calculating inverse Laplace transforms we will consider the second of equations (81) and form the associated impulsive receptance ~28t(t). From this equation the transform A~21(p)can be expressed in terms of partial fractions as follows:
A~(p) = bp + k A(p)
bp + k M~ M2p2(p + ~ + in)(p + ~ - in)'
111
1]
--Ml +ME ,O2 (p+~:)2+n2 .
(A.7)
LINEAR ANALYSISOF TRANSIENTVIBRATION
337
The inverse transforms of these partial fractions, however, are standard forms, which may be found in any appropriate textbook. Thus we can deduce from reference 5 that M1
1
t
(A.8)
This result can be seen to agree with the second of equations (20), when it is remembered that a~zl(t) --- a2~x(t)l(t).
23
LINEAR ANALYSIS OF TRANSIENT VIBRATION R. E. D. BISHOV, A. G. PARKr~SONAND J. W. PENDERED Department of Mechanical Engineering, University CollegeLondon, Gower Street, London, IV.C.1, England
(Received 27 May 1968) An elementary explanation is given of Duhamel's integral and Fourier and Laplace transform techniques in linear vibration analysis. In this, three types of receptances are described, the relationships between them are explained and their application illustrated by means of simple examples. The ways in which the theory is extended so as to apply to aeroelastic systems is also explained. Aeroelastic systems represent an important class of active systems, and it is probably in this direction that transient analysis has found its most important applications to date. 1. INTRODUCTION In practice, the analysis of transient vibration can be very difficult. Quite apart from the usual problems that beset the analyst in the process of idealization for the purpose of analysis, there are the difficulties (a) of adequately defining the excitation and (b) of working out the response. The first of these is a matter of physics, while the second is solely one of mathematical complication. Modern computing techniques have largely drawn the sting from the latter and the new techniques of measurement which are steadily improving our physical understanding are thus beginning to make this form of analysis a practical proposition--at least for the initial period following sudden application of force. It is clear that such analysis can be most important, as when aircraft designs are checked for landing and for gust loading; indeed some aircraft designs are considerably influenced by the results of gust analysis. Yet it is arguable that transient oscillation, as a subject of general study, has been somewhat neglected by vibration analysts in the past. It may be thought opportune therefore to attempt to set down the basic theory of transient loading in quite simple terms. It is also felt appropriate that the application of receptance theory to transient analysis should be reviewed. Here, three types of receptances that are in current use are explained, and the relationships that exist between them are developed. The adjective "transient" is usually applied to deterministic (as opposed to random) excitation that is a non-periodic function of time alone. The response is thus a non-periodic forced vibration, and (if the system concerned is a passive one) no question of self-excitation arises. It is not to be imagined, however, that transient excitation is relevant only to passive systems. In recent years, transient analysis has probably found most application in the determination of the response of a flexible aircraft to a gust. As an illustration of the application to active systems, the paper therefore includes an exposition of the principles of gust analysis. This article is therefore essentially one of explanation and its purpose is to collect information and to present it in a form that it is hoped will be found quite straightforward. Its subject matter falls naturally into five parts. These are (i) an introduction to harmonic and transient receptances, (ii) the use of Duhamel's integral and Fourier transform techniques in transient analysis, 313
314
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
(iii) the relationships between the various types of receptance, (iv) the derivation of transient receptances using Laplace transform techniques, (v) the application of transient receptances to aeroelastic (i.e. active) systems. 2. HARMONIC AND TRANSIENT RECEPTANCES A "receptance" is defined as the response at a generalized co-ordinate to a unit excitation applied either at the same or some other generalized co-ordinate. The responses to a harmonic excitation of unit amplitude, for example, are the familiar harmonic receptances.t In this paper we shall examine the response of a linear system to two other types of unit excitation and introduce two other forms of receptance. The three types of receptance will be explained in terms of an example. Consider the idealized system, shown in Figure 1, composed of rigid masses Ml and M2 connected by a light spring of stiffness k and a dashpot whose damping coefficient is b. If the
I / / / I / / / / / / / / /
/
/
Figure 1. displacements of the masses from pre-selected equilibrium positions are qt and q2, the equations of motion under the influence of a force Q~ (which "corresponds to" ql in the Lagrangian sense) are M1 ql + b~l + kql - b~2 - kq2 = Q1,
(1)
--bql - kqt + e 2 q 2 + bq2 + kq2 = 0. 2.1. HARMONIC RECEPTANCES
Suppose that the applied force Q1 in the system of Figure 1 is harmonic, having the amplitude ~t and frequency to. In terms of the complex exponential representation, we may write QI = ~a e lwt, and it should be noted that we assign no particular significance to the instant t -- 0. If this value of Ql is placed on the right-hand side of the first of equations (1), and we set out to find the steady-state harmonic solutions of those equations, then the trial solutions ql = I/'ti e f~°t, q2 = htt2e ~°~t, (2) are appropriate. When substituted into the equations of motion they give the pair o f algebraic equations (k - M1 to 2 + itob) WI - (k + itob) W2 = ~1,
- ( k + i, ob) ~gl + (~ - M2 ,o2 + i,ob) ~
= O.
(3)
These algebraic equations may be solved for WI and W2 so that the solutions of equations (1) may be written in the form qi --- el 1(£O)~1 et°~t, q2 = °~21(t°) ~1 ettOt, (4) t Because harmonic receptances have received a great deal more attention in the literature than the other types, they are usually referred to simply as "receptances". Here, the adjective harmonic will be necessary to
distinguish them from other forms.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
315
where ~11(03) is the direct harmonic receptance at ql and ~2~(03) is the cross harmonic receptance between qt and q2. In fact in the problem under discussion we have k - 342 032 + i03b oq ~(03) = (k - M~ 032 + i03b) (k - M2 032 + i03b) - (k + i03b)2' k + i03b a2t(03) ----(k - Ma to2 + i03b) (k - M2 032 Jv i03b) - (k + i03b)2"
(5)
Already the simple system of Figure 1 is seen to produce somewhat unwieldy results. It is worthwhile to simplify them as far as possible and to express them in forms which can be most easily applied later in the paper. We therefore note that they can be rewritten as C~ll(W)
M , + M 2 1 [ , - -1, ,~ M 2 /,M , ] ~ "[- if: + i¢.0~2_3r.O2 ,
~z2'(03)
,[, M, + M2 (ig)2
1 ] (~:+ i03)2 + r/2 ,
(6)
where =
a/[4Ml M2 k(Ml + 342) - b2(M1 + M2) z] 2M1 Mz
and
(7) b(Ml + Mz) 2M1 Mz
(It is assumed here that the system of Figure 1 possesses less than critical damping, so that is real.)
Figure 2. If Mz is made infinitely large, then the two harmonic receptances collapse to the forms appropriate for the system of Figure 2; ~21 -+ 0 and ~i~ takes the form, 1 ~11(03) = k - M1 032 + i03b
(8)
or
,1 (03)
1/M1 (v + i03)z + ~z
(9)
where = ~/4kM1 2Ml
b2 '
b v = 2M~"
(10)
The various uses to which harmonic receptances may be put and the various forms that they may take will not be discussed here, as they are examined at length in references 1 and 2. Nor will their usefulness in transient analysis be described at this stage as this can conveniently be left until later in the present paper. At this point we merely note that a harmonic receptance is, in effect, a quantity giving the amplitude and phase of a steady-state response to a harmonic excitation of unit amplitude, and as such can be easily measured experimentally.
316
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
The technique of deriving the receptances from equations like (3) can usefully be generalized for more complicated systems. This is because the technique involves only algebraic manipulation and does not require the solution of differential equations. In fact, all that is really needed is an application of Cramer's rule [1, 2]. There are other methods of arriving at harmonic receptances, and some of them are set out in the form of rules in references 1 and 2. But, generally speaking, the algebraic technique is the most useful in practice. Many of the properties of harmonic receptances are discussed in reference 1. It is quite easy to show, for instance, that they obey the reciprocity rule, ~rs(to) = c~sr(o~).
(11)
This is readily illustrated in terms of the system shown in Figure 1 by simply interchanging the subscripts in the expression for a2t(to) in equation (5). 2.2. INDICIALRECEPTANCES We return now to the system of Figure 1, and to equations (1). Suppose that the force Q~(t) is suddenly applied at the instant t = 0, as indicated in Figure 3; the system having been at rest in the position qi = 0 = q2 for all t < 0. The force Ql(t) can conveniently be written
:It 0
"-'~!
Figure 3. symbolically as Ql(t) = ~1 l(t), where, once again, ~1 is the amplitude of the excitation and the function l(t) is defined as l(t) =
{~
t<0 t >/0"
(12)
The motion is governed by equation (1) with QI replaced by ~l for t i> 0. When the initial conditions ql = 0 = ~ 1 =q2 =q2 att=0 are imposed upon the general solution, expressions may be found for the displacements at q~ and q2. These expressions may be written in the form ql = ~ll(t).~l, q2 = ~211(t).~1,
(13)
where all(t) is the direct indicial receptance and ct121(t) is the cross indicial receptance. These receptances can easily be shown to have the forms ~ll(t)
t2(Mi ~- M2)
t2
k(M1
+ M2)2[
MIME
(14)
[1 _ e_~,(cosBt + ~sin
where the symbols ~: and ~/are defined in equations (7). It should be noted that the presence of the function l(t) in expressions (14) confirms that we are only concerned with response of the system for t t> O.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
317
If the value of Mz is taken as infinity, the (only) indicial receptance of the anchored system of Figure 2 is found; namely, e -vt cos ~t + ~sin ~t
-
(15)
The "rigid body" term in the indicial receptance is thus absent now. It will be shown later how the indicial receptances may be used with the Duhamel integral in transient analysis. At this stage it is merely to be noted that they are the direct and cross responses to a "unit step excitation". We shall now briefly mention certain of their properties. Generally speaking, the method that has been suggested here to find indicial receptances (that is, direct solution of the appropriate differential equation) is too cumbersome to be of value, whereas step-by-step integration techniques have been used successfully for many years. Other methods exist, however, and two of them are explained in reference 3. They are based on (a) Fourier integral and (b) operational techniques. These both require prior knowledge of the harmonic receptances of the system which, as we have previously noted, are relatively easy to determine numerically or experimentally. The first method is discussed later, as also is the use of Laplace transform techniques in transient analysis. It will be shown that an explicit expression can be found for an indicial receptance ~,~,(t) in terms of the corresponding harmonic receptance ~,s(oJ). The relationship is such as to preserve the property of reciprocity [see equation (11)], so that
o:~( t ) =: ~],( t ).
(16)
This result can easily be illustrated in terms of the system shown in Figure 1 by simply changing the subscripts in the expression for c~t(t ) in equation (14). 2.3. IMPULSIVERECEPTANCES Returning yet again to equations (1) and Figure 1, suppose now that an impulse is applied to the mass M~ at the instant t = 0. That is to say, the curve of Q~(t) is that shown in Figure 4 where the vertical arrow implies a curve having a peak at t = 0 of infinite height, zero width along the time axis and an area t/i~ beneath it. To express this symbolically, we may write
Ql(t) = ~1 3(0,
(17)
where 3(0 is the so-called delta function.
1
Figure 4.
The motion is now governed by equations (1) with Q~ replaced by 0. These are the equations of free motion and, to their general solution, we must apply the initial conditions q~=0=q2=0z
and 0 1 = ~
att=0.
(18)
In this way, expressions may be found for the displacements at qt and qz for t >I 0. These expressions are of the form ql = - ~ ( t ) . ~ ,
q2 = ~ ( t ) . tb,,
(19)
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R. E, D. BISHOP, A. G. PARKINSON AND J. W . PENDERED
where u~l(t) is the direct impulsive receptance and ~2~l(t) is the cross impulsive receptance. For the system of Figure 1, these receptances can be written as,
t a ? l ( t ) = ~/, + - - M:
(,
M2 e-~_t sin ~qt .t l(t), + M2)"q)
"q M ~ ( M I
~ , ( t ) = Ml + ME
(2o)
e-~t sin*/t t l(t). (M, + M2) */1
Again we have no interest in the behaviour of these functions for t < 0. If M2 is taken as infinity, we arrive at the impulsive receptance at ql of the system shown in Figure 2. This is
(,)= ( m e -
s,n
(21)
where ~ and v have the values given in equation (10). Again the "rigid body" term in the impulsive receptance is absent. Later on we shall describe how the impulsive receptances may be used with the Duhamel integral in transient analysis. Here we merely note that an impulsive receptance is the response to a unit impulse excitation. For a given system it would not be a practical proposition to find impulsive receptances in the manner just indicated. Some headway can be made by systematizing the process of solving the differential equations, but this process is inescapably cumbersome [1]. In general it is better to use an approach which does not require direct solution of the differential equations (see, for example, the use of Laplace transform techniques). One such method of finding an impulsive receptance is first to derive the appropriate indicial receptance and then to differentiate it. This method of deriving an impulsive receptance will become clear when we see how impulsive and indicial receptances are related to each other. We may also note that the impulsive receptance and the corresponding harmonic receptance are a Fourier transform pair. This relationship does not easily lend itself to the analytical derivation of impulsive receptances, but, it should be remembered that harmonic receptances are much easier to measure experimentally than impulsive receptances. The property of reciprocity is exhibited by impulsive receptances; that is to say, ~5(t) = =5(t).
(22)
Later on it will be shown that the relationships that exist between the three types ofreceptance all preserve this property--a property tha~ is most easily demonstrated for the harmonic receptances. 3. TRANSIENT ANALYSIS: DUHAMEL'S INTEGRAL TECHNIQUES The principle of superposition may be applied to any system whose motion is described by linear differential equations. Now the systems under discussion are not merely linear; they have constant coefficients. This means that the response at some instant to a step function of force or to an impulsive force applied at a time ~- depends only on the elapsed time t - ~and not on t and ~- separately. In view of this we may use the principle of superposition to obtain the response to a force which is an arbitrary function of time, using the mathematical process of convolution. 3.1. DUHAMEL'S INTEGRAL WITH INDICIAL RECEPTANCES Let some force Qs(t), corresponding to the sth generalized co-ordinate of the system, commence to act at the instant t = 0. The function may be represented roughly by a series
L I N E A R ANALYSIS OF T R A N S I E N T V I B R A T I O N
319
of steps, acting at the instants t = 0, d~-, 2A~-, 3A~-..... as shown in Figure 5(a). The response at the rth generalized co-ordinate to all these step inputs is
q,(t) - ct~(t) Q~(O)+ S', ~ ( t - r) AQ,
(23)
A'r
where the sum is reckoned over all the steps. The sketches in Figure 5(b) and (c) will make the meaning clear. This result may be written in the form
q,(t) - ~(t) Q~(O)+ ~ o:~(t- z)-~--~z~Ar
(24)
so that, in the limit, as the steps are taken indefinitely small, t
q,(t) =
Q,(O) + |
(25)
0
This expression is known as Duhamel's integral. (a)
//Y/Y/C/Y/Y/,/YA~YY/A" :i' os co) "~ I I i r
0
,'t
(b)
t 0 rT
~;--
(c)
0
~t
r
Figure 5. So far, we have not considered the duration of the excitation to be finite. If, however, we are concerned with an excitation pulse of duration T(see, for example, Figure 6), then equation (25) should be replaced by
q,(t) = ~ ( t ) Qs(O)+ f ~ , ( t - z ) - ~ ~d r - ~s(t- T)Qs(T).
(26)
qtl
0
Note that, if t < T, the third term in expression (26) vanishes and, if t > T, the integrands of equations (25) and (26) are zero for T < ~- <~ t. In many cases Q,(T) = 0---that is, the pulse does not end abruptly--and consequently expressions (25) and (26) are identical. By way of illustration, suppose that a force Ql(t) = A. tiT acts on the system of Figure 1 during the interval 0 ~< t ~< T as indicated in Figure 6. The response at q2 is given by the expression t
q2(t)
=
- ~211(t- T ) A , f cc211(t- ~') ~d~" A 0
(27)
320
R. E. D. BISHOP, A. G. PARKINSON AND J. W. PENDERED
A
0
T
t
Figure 6. where the indicial receptance is the second of those given in equations (14). That is, A q2(t)
T.2(Mt
(t - - r ) 2 l(t
+ m2)
- r)dr
-
Id
o t
o
[ (t - T) 2 - A [2(m~ + m2)
M , 3/2
k(m~ + m2) 2 x
x[1-e-~(t-r'(cos~l(t-T)+~sin~7(t-T))]}l(t-T).
(28)
In evaluating integrals such as those of expressions (28) and (36), it should be remembered that l(t - r) = 1 for 0 ~
=_At
!_'
TI6(M, + M2) M}M22~/
u, M2
k(M1 + 3/2) 2 t + [2~t,_
-'~k2(Ml-}-M2)3[,r]
il
~2
.
'~* e-e'c°s~Tt)+ I -~--0e-"slnr/t]} '
(29)
and for t f> T, 3t - 2T qz(t)= A T 6 ( M I + M2)
A. k(~+M~M2) ~ e-~t-r' [cos ~7(t- T ) + ~sin ~/(t- T ) ] +
M~M~ + A~,/~2(M~ + Mz)-3 ~/ e -et
1 - -~ sin ~t - 2 cos ~Tt -
-e-~"-r) [(1-~2)sin~(t-T)-2~cos~(t-T)]}.
(30)
Finally, returning to the general problem, it should be stressed that if more than one force Q,(t) is applied to the system, the result (25) may still be made to give the total response through the principle of superposition. Thus, if the number of degrees of freedom is n,
q,(t) = ~ , s=l
~s(t) Q~(O) +
~s(t - r)
dr - ~s(t - T) Q~(T) .
(31)
0
3.2. DUHAMEL'S INTEGRAL WITH IMPULSIVE RECEPTANCES
An alternative form of Duhamel's integral may be used, in which impulsive receptances are employed. Going back to the function Qs(t) of arbitrary form (Figure 5), we now divide
LINEAR ANALYSIS OF TRANSIENT
~it)(a)~
VIBRATION
321
f
0 ' r o,(i) (b)
H
0
"r
D-!
i'o'°'A a r V
k./ "-~
~t
Figure 7. up the area under the curve into vertical strips, each of which represents a discrete impulse of magnitude Q~(t).At (see Figure 7). The response at qr to all the impulses applied in the interval 0 ~<~- ~< t is qr(t) -- Y, =~,s(t- T) Q~(T) Ar, (32) /Jr
so that, in the limit, as AT -+ 0, t
q,(t) -----f =~(t - T) Q~(T) dr.
(33)
0
This result may be used instead of that employed in equation (27) to find the response at q2 when the force Q,(t) of Figure 6 is applied to the system of Figure 1. That is, t
q2(t) =
=~2~(t- T)A.~.d%
0 < t ~< T
(34)
=~21(t--T)A.~.d%
t I> T,
(35)
0
and T
q2(t) =
f 0
where the impulsive receptance is that of equations (20). Thus for t >1 T, T
q2(t)
AT.(M] +1 M2) f (t - T) Tl(t -- T) dT -0 T
A 1 f T. e -~<'-~) sin ~/(t - T) l(t -- T) d% T" (MI + M2)7} 0
which may be shown to be equivalent to the result of equation (30). 22
(36)
322
R.E.D.
BISHOP, A. G. PARKINSON AND J. W. PENDERED
To return to the general result (33) we note again that, if more than one force Q,(t) is applied to the system, the total response may be found from the principle of superposition. That is to say, t
q~(t) = ~ f u~(t - ~) Q~(~')d~',
(37)
s=l
n being the number of degrees of freedom possessed by the system.
4. TRANSIENT ANALYSIS: FOURIER TRANSFORM TECHNIQUES The Fourier integral theorem states that any f u n c t i o n f ( t ) satisfying the condition
f If(t)ldt <
(38)
--co
may be expressed in the form
f(t) =
1 ; F(to) e ~'°tdto
(39)
--oo
where oo
F(to) = f f(t) e-''°t dt.
(40)
--o0
That is to say, the f u n c t i o n f ( t ) is associated with a function F(to), which is known as its Fourier transform. A proof of the theorem is given in reference 3. L e t f ( t ) be identified with a generalized force Q~(t), which is a function of time. In this event it is clear that the quantity to must be in the nature of a frequency. And equation (39) shows that the transient force Qs(t) can be thought of as being built up from an infinite number of harmonic components, that of frequency to being of amplitude F(to)dto/2rr--or Q~(to)dto/2cr, as we shall now call it. The harmonic response at qr due to the excitation Qs(to)2zrdtoe t°'t .is ~,~(to) Q~(to) dto e ~ t 2rr The total response at qr is thus
q,(t) = ~
~,~(to)Q~(to)e '°~tdto
(41)
--oo
or
q,(t)=l f ~,~(to){_f Q~(t)e-t°'tdt} e~°'tdto.
(42)
--oo
If more than one generalized force Q,(t) acts on the system, it is necessary to add solutions of the type (42) by summing the results over all s.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
323
To illustrate the use of equation (42), we return to our previous problem in which the system of Figure I is excited by the force Ql(t) of Figure 6. Noting that the condition (38) is satisfied, and using the harmonic receptance given in equation (6), we find that, for t ~>T, 1
q2(t) = ~
1
1
M1 ~- M2 (i~)2
1
. t
(~ + io))2 + ~/2
A~e-
~,~t
dt
e ~°~ do).
(43)
By expanding the second term of the harmonic receptance further into partial fractions, and evaluating the inner integral, this integrand can be expressed in terms of components whose integrals are given in Appendix 1. On evaluation, this yields a result equivalent to that of equation (30). 5. RELATIONSHIPS BETWEEN THE VARIOUS RECEPTANCES 5.1. INDICIAL AND IMPULSIVE RECEPTANCES The response at some generalized co-ordinate q, to a pulse Q,(t) of the form shown in Figure 8(a) may be expressed in terms of the cross indicial receptance =,~(t). It is [from Figure 8(b)] 1
1
1
q" = -~t ~drs(t) -- - ~ c%(t -- At).
(44)
If we now take the interval A t to be a smaller and smaller quantity we find that
lim[~l,~(t)--e~__~,(t--At!] d l q , = A=~0L t = = ~ [=,~(t)].
(45)
But the process of letting At --~ 0 is also that of letting Q,(t) become an impulse function 8(0 of the type shown in Figure 4. Therefore, a d c%(t) = ~-~[~,s(t)].
(46)
This result may easily be illustrated with any of the corresponding indicial and impulsive receptances that have already been derived, whether direct or cross. Thus it can be seen that, for the system of Figure 1, %8 l ( t ) = ~d[ % ll( t ) ]
and
d l ~2t(t)=~[c~21(t)] ,
while a similar result to the first of these applies to the system of Figure 2.
0 At
"--~t
0 ~._7t
----~t
os(t) (b}
Figure 8. The result (46) is generally sufficient for practical purposes of vibration analysis, but a special case should perhaps be mentioned. The individual transient receptances of a linear system are usually well-behaved continuous functions of time so that the numerator of the
324
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
quantity in square brackets in equation (45) tends to zero as At ~ 0 for all values of t. In analysis of the type under discussion, however, there is a special case in which this is not so. It occurs when there is a discontinuous step in ~,~,(t) at t = 0, so that e,t~(0) ¢ 0 while, of course, ~,I~(t) = 0 for t < 0. In this special circumstance equation (45) remains valid, except when t = 0. For this particular instant the numerator in the square brackets of that equation remains finite while the denominator is made indefinitely small in the limiting process. Thus the response at t = 0 may be written as
q" = l~mo[el~(O) - °~l~(O ' A -t At) ] = lim [ ~ ( 0 ) ] = c~(O) lim Q,(t) = ~,~s(O)8(0. •at-,oL At J ,J,-,o
(47)
Remembering that 8(0 = 0 for all t ¢ 0 and that the transient receptances have the form
o~l,,(t) = a~,(t) l(t), ee~(t) = a~,~(t) l(t),
(48)
results (46) and (47) can be combined into one general result that ~,~,(t) = e~,(0) 8(0 + l(t) d [a~s(t)].
(49)
Thus, at any time other than t = 0, only the second term in equation (49) matters and then it is equal to (d/dr) [e,~,(t)]. If there is a discontinuity in e¢,(t) at t = 0, such that ~ ( 0 ) ~ 0, then only the first term in equation (49) is significant at that instant. Such a case as this would arise if Ml -----0 = b in the system of Figure 1 ; the step force then being applied at the end of a massless spring. In that event the spring would experience a sudden distortion at the instant t = 0. It is not our purpose, however, to pursue this point any further, partly because it is clearly of limited importance and only arises in easily identified circumstances and partly because it raises difficulties of a purely mathematical nature. 5.2. IMPULSIVE AND HARMONIC RECEPTANCES
Impulsive and harmonic receptances are related to each other in a very simple way. This can easily be shown by reference to the result (42). In that equation, if Q,(t) = 8(t) then, by definition, q,(t) = ~,~,(t). That is to say,
1
~,,(co)
3(t) e -"°t dt e u°' do).
(50)
--cO
Here, the inner integral has the value unity since 3(t) is everywhere zero except at t = 0, at which value e -~°'t has the value unity. Therefore,
1/
e,,(t) -- ~
e,~(o))e u°' do).
(51)
By comparing equations (39) and (51) it will be seen that e,,(o)) is merely the Fourier transform of e,~(t). The inverse relationship is that of equation (40), so that e,~(to) = ; e,~,(t) e -u°' dt. --cO
(52)
LINEAR ANALYSIS OF TRANSIENT VIBRATION
325
It is, of course, no surprise that ~,(t) may possess a Fourier transform, since the requirement (38) is clearly met by any impulsive receptance of any passive linear dissipative system that is anchored. It is not clear, however, that an unanchored system of the type shown in Figure 1 or an undamped system will satisfy the requirement, and indeed they do not do so directly. It can be shown, though, that the results (51) and (52) can still be retained providing that suitable limiting processes are employed. As an example, consider the unanchored (though damped) system of Figure 1, for which a~2z(t) does not satisfy requirement (38). Since e -13' a~t(t),
[3 > 0
satisfies the condition, we use the artifice of first evaluating the appropriate integral and then letting fl be vanishingly small so that (53)
a21(to) = lim J e -~, a2az(t)e-i°~t dt ~-~0 -oo
or, here,
2,(to) =lim f /3.o
t
M l + M2
e-¢t sin r/t / -w~t (M-~l~ M~-2)~Jl(t)e dt.
(54)
--oO
This gives ct2z(to) = lim{
-1
1
}
~-~o (M1 + ~)2)(to -- ifl) + (Mr + M2) [(to - ifl - i~)z _ 72]
(55)
which reduces to the expression for aEl(to) given in equations (6). Condition (38) must also be satisfied in performing the transformation from harmonic to impulsive receptance. Thus, again referring to the system of Figure 1, we consider a621(/) = lira
t3-,o 2rr __ M , + M2 ,(to 7 ifl)2 + (to _
ifl -
i~)2 _ ~=j
dto.
(56)
--co
This gives the impulsive receptance ~21(t) of equations (20). Undamped systems may be approached by first considering the corresponding damped system and applying a limiting process in which the damping is reduced to zero. Thus, for the undamped (though anchored) system corresponding to Figure 2 with b = 0 = v, ~ e1-
all(to)=l!m_.
~t-s m ~ t l ( t ) e - ' ° ' t d t
(57)
--co
and cc~,(t) = lim 1 ; M I [(v + ito)2 1 + ~2] e'c°t . dto. ,,~o 2~r
(58)
--00
These expressions yield the values for al~(to) and a~(t) given in equations (8) and (21), respectively, with b = 0. 5.3. HARMONIC AND INDICIAL RECEPTANCES
It is true that, having established the connections of the last two sections, we have effectively shown how harmonic and indicial receptances are related to each other. It is instructive,
326
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
nonetheless, to approach the matter directly. We may use the Fourier integral technique of the preceding section. But whereas the force Qs(t) = 8(0 obeys the requirement of equation (38), the force Qs(t) = l(t) does not. It is therefore not possible to use equation (42) directly. The difficulty may be overcome by investigating the response to a transient excitation
Q,(t) = e -/3'. l(t)
(59)
where fl > 0 and then letting fl ~ 0. The Fourier transform of Q,(t) is now Q~(to) =
;
Q~(t)e-t'°tdt=
--oo
;
1 e-t~+t'°)tdt-fl+i~.
60)
0
That is, the excitation (59) may be expressed in the form
Q,(t) =
1 ;,
.e~"~t 8 - - ~ i dto.
(61)
--00
If we wished merely to find a frequency expression for the function Q,(t) = l(t) we could in fact do so by evaluating the integral l ( t ) = - 2rr I lim ~-.0 f ~ e'°" do.
(62)
~o0
A method of doing this which avoids the arrival at an improper integral is to separate the real and imaginary parts of the integrand of expression (62) and then to use the substitution to/fl = x, finally setting fl = 0. The resulting expression turns out to be 1 1 ~ sintot l(t)=~+~ J oa do,
(63)
~oa
the so-called Dirichlet integral. Our purpose is not to find a frequency expression for l(t), however, but to use the result (60) to find ~,l,(t). By definition of the indicial receptance, equation (41) now gives ~s(t) =-~-1 lim f c%(to)e'~°'dto. zrr ~--,o fl + ia,
(64)
--00
Thus for the system of Figure 1, we have
~ll(t)= 2~r/3-,0 1 lim f
1
{ 1
M, + M2 (i~)2
1
/
(~ + ito)2 + 72j ~
e '°'*
do
(65)
which yields the value of =2tl(t) given in equations (14). If the system is unanchored or undamped, further limiting processes may be necessary. 6. TRANSIENT ANALYSIS: LAPLACE TRANSFORM TECHNIQUES The Fourier integral theorem defined by equations (39) and (40) is only directly applicable to functions which satisfy condition (38). For this reason various limiting processes have been adopted in forming some of the Fourier transforms. Again, the technique cannot be
LINEAR ANALYSIS OF TRANSIENT VIBRATION
327
used immediately for vibrating systems that are unstable. There is, however, an alternative process--that of the Laplace transform--which avoids these difficulties. The Laplace transform is a valuable tool in determining the solutions of differential equations which are required to satisfy certain initial conditions. The transient response of vibrating systems is an example of problems of this class. Unfortunately, however, the Laplace transform, unlike the Fourier integral, has no simple physical interpretation. The Laplace transform F(p) of a functionf(t) is defined as
F(p) = f f ( t ) e -pt dt
(66)
o
where p = cr +/co is chosen such that the real number o is large enough to ensure that the integral (66) is convergent. The functionf (t) must satisfy certain continuity conditions (see, for example, reference 4), but we shall not examine them here, apart from noting that l(t) and 3(0 are suitable functions. The inverse transform may be expressed in the form y+loo
The integration in equation (67) is performed in the complex p plane along the line p = ~,, where ~, is a real number greater than the real parts of the poles (or singularities) of F(p) (see Appendix 2 and Figure 9). Many inverse transforms can, of course, be found directly from standard references. :s / PI / p :- ~+i~ //
I
c!
o-- 7"
Figure 9. By means of the Laplace transform it is possible to reduce differential equations of motion to algebraic forms and then derive the corresponding indicial and impulsive receptances. The process of reduction is a standard one [5] and we shall illustrate first an important property of such a reduction. Consider, for example, the transform of a generalized velocity ~)~(t), say. Applying the definition (66) we have o0
co
[- ¢)l(t) e -pt dt = [ql(t) e-Pt]~ - [ ql(t) (-p) 0
e -or
dt,
0 co
=p f ql(t)e-P' dt - q~(O),
(68)
0
where ql(0) denotes the initial value of q~(t) at t = 0. Thus the Laplace transform of gl~(t) can be written simply in terms of the transform of ql(t).
328
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BISHOP, A. G. P A R K I N S O N
AND
J. W . P E N D E R E D
Likewise the Laplace transform of the generalizedacceleration~'~(t)is co
co
(0",(t) e-pt dt --p f ~x(t)e -pt dt- 0,(0), 0
0 co =p2
f ql(t) e-p' dt -
~,(0) -pq,(O),
(69)
0
where ~1(0) represents the value of~l(t) at t = 0. The transform of~l(t) can also therefore be expressed easily in terms of the transform of q~(t). We will illustrate the complete process by calculating the indicial and impulsive receptances of the system shown in Figure 1. Consider first the indicial receptances. The Laplace transform of the excitation Ql(t) = ~1 l(t) is
f ~ll(t)e-l"dt=~fe-'~dt=~Jp. 0
(70)
0
If we form the Laplace transform of both sides of the first of equations (1) we have
(Mlp2 +bp+k) f ql(t)e-'tdt-(bp+k) 0
q2(t)e-'tdt 0
= 1 ~l + MI ~(0) + (Mp + b)q~(O) P
bq2(0).
(71)
But here ql(t) and q2(t) have the form (13), so that ifAl l(P) and A,ll(p) represent respectively the Laplace transforms of e I ~(t) and e,! ~(t), equation (71) can be rewritten as (nl
p2 + bp + k)Aft(p) - (bp + k)Al~(p) = 1 ~
+ M~ 0t (0) + (M~ p + b)ql (0) - bq2(0). (72)
Of course, strictly, A~s(p) is the Laplace transform of al,s(t) [see equations (48)] for
Al,,(p) =
al~(t)l(t)e-'t dt,
e~(t)e-'tdt= 0
0
co
= f ars(t) e-'t dt.
(73)
0
The Laplace transform of the second of equations (1) can be found in a similar way. Both transformed equations can be combined in the algebraic matrix form
[Ap2+Bp+C][A~I(P)][A~1(p)J= [10/P] + A4(0) + (AP + B)q(0)'
(74)
where q(0) = rq~(0)] Lq2(0)J,
dl(0) = [*,(0) 1
(75)
L~2(0)J
and
A=[M,
02],
B=[_bb
-~],
C=[.,_kk - k ]
(76)
are the inertia, damping and stiffness matrices, respectively. The transform procedure is, of course, easily applicable to systems with any number of degrees of freedom and a matrix
LINEAR ANALYSIS OF TRANSIENT VIBRATION
329
equation similar to expression (74) is obtained. For a system with n degrees of freedom the various matrices are of order n. We are concerned here with a system which is initially at rest so that q(0) and q(0) are both null column vectors in equation (74). In these circumstances the following results are obtained for the transformed receptances:
A 11(P)
M2p 2 + bp + k p A (p) ' (77)
A~(p) - bp + k pA(p)' where
A(p) = lAp z + Bp + C I = Mx Mzp2(p + ~ - i~l)(p + ~ + i~1).
(78)
The quantities ~ and ~ are introduced in equations (7). The inversion of the Laplace transforms (77) is explained in Appendix 2. The results obtained for the indicial receptances are in complete agreement with those of equations (14). The impulsive receptances are calculated in a similar manner. Thus for the system of Figure 1 described by equations (1), if Ql(t)= q)l ~(t) and ql = 0 = q2 =01 =0z for t ~<0, then
ql = o~l(t) ~1 = a~l(t) l(t) ~1, l q2 ~ , ( t ) ~ l =a~,(t) l ( t ) ~ l . J
(79)
The transformed equations have the form
[Ap2+Bp+C][A~I(P)]LA~I(p)J= [10]
(80)
where A~I, A~ and 1 are the Laplace transforms ofa~l, a~ and 3(0, respectively. Aqt and A~I are calculated in a manner similar to that of equation (73). The transformed receptances have the form
A~i(p) = Mzp 2 + bp + k A(p) ' bp + k A~(p) A(p) '
(81)
and the inverse Laplace transforms of these expressions yield the impulsive receptances (20). The properties of the transient receptances which are discussed earlier in this paper can all be established through the results of the present section. For example, from the matrix equations (74) and (80) with q(0) = 0 = 0(0) one can see that for any system the transformed receptances are related in the following manner:
A~(p) =pA~,(p), = p f a~,(t) e -p' dt, o co
= [ - e - " a~s(t)]~° +
f o
e- pt~d [a~s(t)l dt.
(82)
330
R. E. D. BISHOP, A. G. PARK1NSON AND J. W. PENDERED
Therefore,
A~s(p) =
pt d
a,ls(0) +
e- ~ [a~s(t)] dt.
(83)
0
Thus by forming the inverse Laplace transforms of both sides of equation (83) and remembering that the transform of 8(0 is 1 we have ~,a~(t) = ~,~,(0)8(0 + l(t)~t [a~(t)].
(84)
This relationship between the impulsive and indicial receptances is in agreement with that derived earlier, when the physical significance of the first term on the right-hand side of equation (84) was explained. It is also possible to establish the expressions relating transient and harmonic receptances through the Laplace transform solution, but this will not be discussed here. 7. AEROELASTIC SYSTEMS By contrast with passive systems, we are concerned here with systems in which the exciting force (or at least part of it) depends upon the response. In these circumstances self-excitation is possible. The equation of motion of such a system may be expressed as A/i + Bd/+ Cq = Q(t) + F.
(85)
Here, A, B and C are the inertia, damping and stiffness matrices of the system, q(t) is a column matrix of generalized co-ordinates, Q(t) is the column matrix representing the applied forces and F is the column matrix of impressed forces due to the response. Aeroelastic systems represent an important class of active systems and it is to their analysis that techniques of transient analysis have been applied most frequently. Such systems differ from those considered earlier in this paper in one important aspect. So far we have discussed methods of representing the transient response of a system due to applied external forces. We shall see that, in part, determination of aerodynamic response entails solving the inverse problem--that is, the calculation of changes in aerodynamic loading due to arbitrary displacements, modifications in geometry, or any sudden change in flight conditions from one steady state to another. The purpose of this study lies in the calculation of loads, bending moments, shearing forces, etc., acting in an aircraft structure during rapidly changing conditions of flight. In this section we are concerned with an aircraft flying into a vertical gust, as it is this problem of transient aerodynamics which has received most attention in the past. If it could validly be assumed that aerodynamic forces act instantaneously--i.e, that the loads at any instant correspond to the current configuration and motion (as in piston theory) --then problems on transient loading would be quite straightforward. In particular, the vector F would depend at any instant on the prevailing vectors ii, ¢1and q. This is well illustrated in a paper by Huntley [6]. Unfortunately this assumption is not strictly valid. Vortices require time to form, circulation takes time to adjust itsdf to changes of incidence, and so on. As this clearly implies, the term F is made much more complicated as a consequence. At any instant t, the vector F depends on q(t) for all values of~" less than t. (In the special case of a "rigid body" motion of translation, the dependency is upon q(~-) for all values of ~- less than t.) This delay in the adjustment of aerodynamic forces also means that Q(t) is made more complex in the particular problem of gust loading where the delayed adjustments accompany the varying depth of penetration of the aircraft into the gust front.
LINEAR ANALYSIS OF TRANSIENT VIBRATION
331
In order to simplify the problem, we first note that, in the matrix F, the object is to express the generalized forces in terms of generalized displacements (or generalized velocities for rigid body modes other than those of pitch and yaw). Equation (85) may therefore be written as n separate equations of which the rth is
2=1
[a~ + b~s + C,~q~]=~ F~ + Qr(t). s~l
(86)
Here, Fr~ is the force at the rth generalized co-ordinate due to the deformation in the sth co-ordinate. This deformation is a function of time dependent in part upon Qr(t). It is convenient to start by discussing the impressed force F~ due to an elementary response. The force F,~ due to an arbitrary response can then be derived from this. In general there are two practical ways of realizing elementary responses, namely the Fourier and Indicial techniques. These are considered separately below. 7.1. FOURIER TECHNIQUE
Consider a harmonic displacement (or velocity if appropriate) of the form
qs(t) = t/t~e'°~'.
(87)
This will give rise to a harmonic impressed force which can be expressed as F,s = ~p v 2 stz,s(o,) ~'~ e '~', where z,,(¢o) = s, l = p= V=
(88)
the inverse harmonic aerodynamic receptance,t reference area and length, respectively, air density, aircraft velocity.
The force due to an arbitrary displacement can now be built up using a Fourier transform technique. There is, however, a difficulty in that, although displacements in the distortion modes of a flexible aircraft satisfy condition (38), velocities in rigid body translational modes do not. To avoid this difficulty one should use acceleration co-ordinates, all of which obey condition (38). The acceleration response can then be calculated using the Fourier technique outlined here and the results integrated to yield the velocities and displacements. This is fully explained by Mitchell [7] but, for simplicity, we will confine our remarks here to displacement co-ordinates. Using the relations (39) and (40), therefore, an arbitrary displacement q,(t) can be expressed as an infinite array of harmonic components, that of frequency to being of amplitude
l { ~_fqs(t)e-i°~t dt)d¢o. The impressed force due to this component is therefore
21r
{_J q~(t)e-U°~dtd°JeU°t
t In aerodynamic analyses, one would normally work with non-dimensionalgeneralizedco-ordinates and receptances so that the terms in equation (85) have dimensions(force) x (length). However, in order to keep the discussion here in general terms, we will not enter into the question of non-dimensional aerodynamic equations. We merely note, therefore, that the receptances have dimensions appropriate to those of the other terms in equation (88).
332
R.E.D.
BISHOP, A. G. PARKINSON AND J. W. PENDERED
and that due to the complete response is
-
2~r
zr,(o~)
qs(t) e-'~t dt
e '~t dco.
(89)
The advantage of this approach lies in the fact that harmonic aerodynamic derivatives are relatively easy to measure and calculate. It will be noted that the method requires that these be known for all frequencies. It now remains to discuss the term Q,(t), the externally applied gust excitation in the absence of response. It is again convenient to reduce an arbitrary gust into an infinite array of harmonic components. Thus, we first consider the aircraft to fly through an infinite expanse of harmonic downwash having a velocity amplitude w and wavelength I. The aerodynamic generalized force in the rth mode arising from this may be expressed as Q,(t) =
½pV z slz,(t2)-~e w lea,,
(90)
where
g2 = 2~rV/A, z,(g2) = the appropriate complex aerodynamic inverse receptance. This value for Q,(t) together with the value for F,~(t) from expression (89), may now be substituted into equation (86) to give the response of the aeroelastic system to a harmonic gust. Thus, the rth equation becomes s=l
[ a ~ + b,~gl~+ c,~q~] =½pV2sl[~=~ _£z,~(eo){_f q~(t)e-'°"dt}e'°~rdco+z,(g2)vem' ].
(91)
Since the system is assumed to be linear, these equations may be solved to give the responses in the form
q,(t) =
W ~t
r(~2).~,e
.
(92)
The response to a gust having an arbitrary velocity profile w(t) (as seen by the aircraft) can now be derived. Assuming that w(t) satisfies relation (38) (or that a suitable limiting process can be applied) and applying the results of the Fourier transformation defined in (39) and (40), we see that the harmonic component of w(t) at frequency O has amplitude
The response in the rth mode due to this is therefore 1 y( 27r
e -t~t dt e ~ t dD,
and that due to the complete profile is
q,(t) = ~1
Y(12) ~o0
w(t)__t~t dt e t~t dO. -ff-
(93)
L I N E A R ANALYSIS OF T R A N S I E N T V I B R A T I O N
7.2.
333
INDICIAL TECHNIQUE
The second approach to this problem derives from the consideration of step distortions of the form
qs(t) =
Ws l(t)
(94)
rather than the harmonic displacements considered above. The impressed force which accompanies such a displacement can be expressed as
F,~(t) = z~(t) 7t~.
(95)
It is convenient to distinguish between those rigid body modes for which so-called "velocity co-ordinates" are relevant, and all other modes. For the former translation modes, forces are related to generalized velocities. For all other modes the forces are related to generalized displacements. The terms zl,~(t) can then be expressed as
z],(t) = ½pV 2 slz;'s K,(,~)(t) =
(s represents translation modes), 2 ,
= ½pV slz,~ Klt,~)(t)
(96)
(s represents all other modes), where
Kl(,s)(t)
= growth factor which approaches unity as t tends to ~,
z~s, z)'s = constant factors of appropriate dimensions corresponding to steady-state conditions at t = o,.]The evaluation of the Kj(,~)(t) is a matter of indicial aerodynamics. This can loosely be referred to as the Wagner problem, since Wagner gave the first solution of this type. The impressed force due to an arbitrary disturbance can now be derived using Duhamel's integral in the "indicial" (as opposed to the "impulsive") form. Thus, using the result of equation (25), and assuming that the initial displacement is zero (so that the first term on the right-hand side of equation (25) is zero), the impressed force can be written as t
0
=½pV2sl(zr"siKl(rs)(t-z)q's(T)d'r}
(97)
s = translation modes,
=½pV 2 sl z,~ K,(,s)(t-,)Os(r)d o
s = other modes. The summation term on the right-hand side of equation (86) can therefore be expressed as
½pV2sl
~
~=translatlon modes
z;'~ f Kl,.s)(t-~')~/s(1")dr 0
+
]~
s=other modes
Z;s f K,,.~)(t--:)O~(.)d. 0
t These factors are in fact the zero-frequency inverse harmonic receptances.
334
R.E.D.
BISHOP, A. G . P A R K I N S O N A N D J. W . P E N D E R E D
The term Q~(t) arising from the aerodynamic loading in the absence of response may also be approached by considering a step gust rather than a harmonic gust. Thus, we consider the aircraft flying into a vertical fronted up-gust of velocity w. In evaluating the aerodynamic force arising from such a gust, account must be taken of gradual development of aerodynamic forces and the gradual penetration of the aircraft into the gust front. The force is therefore expressed as W Q,(t) = ½pV2 sl-# Q,, K2(o(t),
(98)
where Q', = constant factor of appropriate dimensions corresponding to the steady state as t --~ oo (i.e. as the aircraft becomes more deeply immersed), K2(o(t) = a growth factor which approaches unity as t tends to o0. The evaluation of K2(o(t) is a second problem of indicial aerodynamics. Where, as is usual, it is assumed that an aircraft flies normally into a vertical-fronted up-gust (so that the only relevant translation mode is that of heave), this may loosely be referred to as the Kiissner problem, in honour of the man who first solved an elementary case. The complete response of the aircraft following a step gust is therefore given by n equations such as
[a~s(~+ b~,Os+ C,~q~] t
=½pV 2sl
X s=translation modes
t
zr", f Kl(r,)(t-~')(is('r)dr + ~ s=other modes
0
+ --~Q; K2(o(t)).
z~s f K,(,,)(t-*)O~(*)dr + 0
(99)
These n equations may be solved using step-by-step methods to give the variations in the n generalized co-ordinates in the form
qr(t) = Ylr(t) w.
(100)
Once the functions Y](t) have been evaluated, the response to any gust may be derived by a further application of Duhamel's integral. Thus, using the result of equation (25), the response to a gust w(t) [as seen by the aircraft and assuming w(t) = 0 at t = 0] is given by
q,(t)=f
Y l , ( t - r ) ~ r dr.
(101)
0
REFERENCES 1. R. E. D. BISHOPand D. C. JOHNSON1960 The Mechanics of Vibration. Cambridge: Cambridge University Press. 2. R.E.D. BISHOP,G. M. L. GLADWELLand S. MICrlAELSON1965 The Matrix Analysis of Vibration. Cambridge: Cambridge University Press. 3. T. VONK/~RM/~Nand M. A. BlOT1940MathematicalMethods in Engineering. New York: McGrawHill. 4. W. R. LE PAGE 1961 Complex Variables and the Laplace Transform for Engineers. New York: McGraw-Hill.
335
L I N E A R A N A L Y S I S OF T R A N S I E N T V I B R A T I O N
5. H. S. CARSLAWand J. C. JAEGER1963 OperationalMethods in Applied Mathematics. New York: Dover Publications Inc. 6. E. HUNTLEY1961 ARC 23, 508, R.A.E. Tech. Note Aero. 2771. Calculations of the response of a flexible slender wing aircraft to discrete vertical gusts. 7. C. G. B. MITCHELL1968 A R C R & M N o . 3498. Calculation of the response of a flexible aircraft to harmonic and discrete gusts by a transform method. APPENDIX 1 TABLE 1 Standard Fourier Transformations co
g(t) = f f(to)el°~td~o
f(oJ)
(a + iw)-" n>0 (a + ioJ)-" (b + ion)-I n > -1
2~r[F(n)] -1 t"-I e-°,
t I> 0
0
t<0
27r[-P(n)]-x (a - b)-" e -bt 7(n, at - bt)
t >~0
0
t<0
F(n) = ; e-Ss "-1 ds o
(= (n - 1)! ifn is an integer)
Gt
7(n, ~t) = f e-Ss"-1 ds 0
APPENDIX 2: CALCULATION OF INVERSE LAPLACE TRANSFORMS F r o m the definition (67) the inverse transforms of expressions (77) are determined by the integrals y+l~
l(t) Ct]l(/) = 2~ri
(' J
M 2 p 2 + bp + k p , . pd(p) e ap,
--
(A.1)
b p + k e P , dp. pA(p)
(A.2)
y --i oo "F+lco
l(t) ~l'(t) = 2~ri
f ),-l~o
Here the zeros o f p A ( p ) and hence the poles of the integrands are at p = 0, - ~ ± iT, where and ~ are positive, that is, at the points 0, P~ and Pz in the complex p plane of Figure 9. Thus 7 can be any positive real number and the integrals (A.1) and (A.2) are performed along the line through the point p = y parallel to the imaginary axis (the broken line in Figure 9). There are several ways of evaluating the integrals (A.1) and (A.2). For example, the integrands can be expanded as a series of partial fractions and the inverse transform of each fraction determined separately from tables of Laplace transforms. The transient receptances
336
R . E . D . BISHOP, A. G. PARKINSON AND J. W. PENDERED
are, however, often more easily determined by contour integration in the complex plane. In some respects, this is merely an alternative way of forming partial fractions. Consider the contour ABCA in Figure 9, where BCA is a semicircle of diameter 2R --AB. As All(p) and A~,~(p) are 0(p -3) and 0(p-4), respectively, it may he shown (see, for example, reference 5) that, for t > 0, lim f A]l(p)ePtdp=O=lim R-->eo BCA
f Ai,,(p)ePtdp.
(A.3)
R~oo B~A
Thus from equations (A. 1) and (A.2)
l(t) f
.l,(t) =~-~
All(p)ePtdp,
r/ l(t) t" ~t211(t)= ~ | A~l(p) ept
(A.4)
dp,
td
/7
where/7 is the limit of the closed contour ABCA as R -+ oo, Expressions of the form (A.4) can be integrated by a standard procedure using the calculus of residues. Thus ~] 1(0 is equal to the sum of the residues of A~t(p)e pt at the poles p = 0, - ~ 4- i contained within the contour/7 (see reference 4). But AIl(p)e pt has no poles outside 11, so that the contour/-/may be replaced by any dosed curve containing the points p = 0, - ~ 4- in. Integrating expressions (A.4) in this way the transient receptances are determined as, /M2(-~ + in) 2 + b(-~ + in) + ket_~+,n), + b(-~ - i n ) + g l M2(-~ - in) 3 (-2in)
+ M 2 ( - ~ - in) 2 +
k e~_e_ln),
+
I]
Lrd2/ (M2£Z-+bP--+ k ) e ' / l(t), + 2! [@2 ~Ml M2(p + ~ - in)(p + ~ + in)/J,,-d
(A.5)
, . . _ [ b(-~___~+in_._)+___k c-~+,,7,, b(-~ - in) + k eC_e_tn,,+ =2~(t) - I M ~Mz(-~ + in) 32in e ~ M~ M2(-~ - in) x (-2in)
~1rd2 / (bp+k)eP' }] } 1(,). + 2! [dp ~tm~ m=(p + ~ -- in) (p + # + in) ,=0
(A.6)
These expressions for the indicial receptances reduce, after some manipulation, to equations (14). Note that, as pA(p) has a multiple root at p = 0, a special procedure is adopted in equations (A.5) and (A.6) in calculating the residues at p = 0. This is explained in general terms in reference 5. To illustrate one of the alternative methods of calculating inverse Laplace transforms we will consider the second of equations (81) and form the associated impulsive receptance ~28t(t). From this equation the transform A~21(p)can be expressed in terms of partial fractions as follows:
A~(p) = bp + k A(p)
bp + k M~ M2p2(p + ~ + in)(p + ~ - in)'
111
1]
--Ml +ME ,O2 (p+~:)2+n2 .
(A.7)
LINEAR ANALYSISOF TRANSIENTVIBRATION
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The inverse transforms of these partial fractions, however, are standard forms, which may be found in any appropriate textbook. Thus we can deduce from reference 5 that M1
1
t
(A.8)
This result can be seen to agree with the second of equations (20), when it is remembered that a~zl(t) --- a2~x(t)l(t).
23