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On the modal analysis of non-conservative linear systems
Journal of Sound and Vibration (1987) 112(1), 69-96
O N THE M O D A L ANALYSIS OF N O N - C O N S E R V A T I V E LINEAR SYSTEMS D. E. NEWLAND University Engineering Department, Trumpington Street, Cambridge CB2 IPZ, England
(Received 27 November 1985, and in revised form 22 January 1986) This paper applies the results of classical eigenvalue theory to the analysis of a nonconservative vibratory system. General results are obtained for the response of a multiple degree-of-freedom system which is represented by a set of linear second-order differential equations with constant coefficients. For the case when the eigenvalues of the system are distinct, these results confirm those obtained by Fawzy and Bishop [1] by a different method. The latter results are restricted to distinct eigenvalues, but the more general analysis presented here allows the case of multiple eigenvalues to be included. The physical interpretation of complex normal co-ordinate vectors is discussed and some simple numerical examples illustrate the results. The approach described here lends itself to numerical computation using computer library programs for eigenvalue determination.
1. INTRODUCTION The dynamic behaviour of a mechanical or structural vibratory system is usually studied by one of two methods: the mode superposition method or the direct integration method (see, for example, the book by Bathe [2]). The mode superposition method involves calculating analytically the response o f each mode separately and then summing the response of all the modes of interest to obtain the overall response. The direct integration method involves computing the response of the system by step-by-step numerical integration. A choice of which method to use depends on what is to be achieved and the numerical effectiveness of the two methods in a specific case. However for many problems, the mode superposition method offers greater insight into the dynamic behaviour and parameter dependence of the system being studied. This paper is about the mode superposition method. For passive structural systems, in which the damping terms are small and essentially dissipative, approximate methods of modal analysis are usually used. The viscous damping coefficients are assumed to be proportional to mass or stiffness terms in such a way that the normal modes of the undamped system are preserved in the presence of damping. Then the solution can be expressed in terms o f the normal co-ordinates of the undamped system, which can be found by well-tried methods and whose interpretation is well understood (for example, the book by Bishop et al. [3]). But for many practical problems, the assumption that the undamped normal mode shapes are preserved in the presence of damping terms is not valid. Some examples are problems of vehicle dynamics, aerodynamic flutter and rotor whirling, and there are many others. This work has arisen from a number of practical investigations of such systems, and is concerned with the modal analysis of any general dynamical system that can be represented by a set of linear second-order differential equations with constant coefficients. 69 0022-460x/87/010069+28 $03.00/0 9 1987 Academic Press Inc. (London) Limited
70
D.E.
NEWLAND
In previous papers Fawzy and Bishop [1] and Wahed and Bishop [4] have examined the behaviour o f such a system, described by the matrix equation
m~+~+~=f.
(1)
When there are n degrees-of-freedom, m, c and k are real matrices of order n x n and x and f are column vectors of order n. To include any general system, whether conservative or non-conservative, there are no restrictions on the m, c and k matrices, each o f which may be any square array of real numbers. The input or excitation of the system is the vector f and its output or response is given by the vector x. Two general expressions were developed in references [l, 4], one for the steady-state solution x when there is harmonic excitation at a single frequency to, and the other for the general response x when the elements of the vector f are each different arbitrary functions o f time. The method o f obtaining these expressions involved using eigencolumn and eigenrow matrices which are defined as follows. IfA~ (i= 1 to 2n) are the eigenvalues of the system, which are assumed to be distinct, then an eigencolumn vector tr ~ satisfies the equation
[A~m+A,c+k]o'(~
(i=1,2 .... ,2n)
(2)
and an eigenrow vector ~.(o satisfies the equation
~rt~
(i = 1, 2 , . . . , 2 n ) .
(3)
The eigencolumn matrix Z is an n x 2n matrix each of whose columns is an eigencolumn vector; the eigenrow matrix H is a 2n x n matrix each of whose rows is an eigenrow vector. In general, the elements of Z and / / a r e complex numbers. The purpose o f the present paper is (i) to show that the two general results referred to above may be obtained by an alternative and, it is believed, a simpler method by using a first-order formulation of the equations o f motion, (ii) to show how these results may be extended to the case of multiple eigenvalues, and (iii) to discuss the physical nature of the "normal co-ordinates'" which characterize the natural modes of non-conservative oscillatory systems. In addition the analysis which will be described is in a form which lends itself to numerical computation by using computer library programs for eigenvalue determination. These standard library programs have been developed for a range o f problems, including the analysis of linear control systems, and generally incorporate eigenvector sub-programs which generate normal co-ordinates in the form to be defined below. The use of a first-order formulation for vibration problems is not new (see, for example, the paper by Woodcock, [5]) and there are different methods of organizing the equations, but in order to apply the results of algebraic eigenvalue theory, which includes the case of multiple eigenvalues, it is necessary to express the n second-order equations (1) as 2n coupled first-order equations in the form :~ = a z + F,
(4)
where the system matrix A is a real matrix of order 2n • and z and F are column vectors of order 2n. When the eigenvalues of A are distinct, the solution is developed in terms o f t h e system's eigencolumn matrix U, which is of order 2n • 2n and which satisfies the equation
AU
=
U[diag (A,)].
(5)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
It will Fawzy matrix The
71
be shown that, subject to scaling the eigenvector columns in the same way, the and Bishop n x 2n matrix X is identical with the top half of U and their 211 x n product Ilm is identical with the right-hand half o f U -~. co-ordinate transformation
(6)
q = U-lz
allows the set of 2n coupled first-order equations (4) to be rewritten as an equivalent set of 2n uncoupled first-order equations and defines the normal co-ordinate vector q, of order 2n. In the general case, the elements of q are complex. The solution vectors become real (for real excitation) only after the inverse transformation has been effected to revert to the (coupled) physical co-ordinates of the system. When there are multiple eigenvalues, it may not be possible to find a co-ordinate transformation which will uncouple all the 2n first-order equations. It can be shown from eigenvalue theory that at least one equation can be uncoupled (that is to say written in terms o f a single co-ordinate) but that some or all of the remaining equations may remain coupled. However each such coupled equation involves two co-ordinates only. Therefore the general solution for arbitrary excitation may be obtained by solving the uncoupled equation first, and then substituting this result into the next equation and solving that, continuing until solutions for all the co-ordinates have been obtained. 2. EIGENSOLUTIONS
It is well-known (see, for example, the book by Wilkinson, [6]) that a system o f n simultaneous differential equations of order r with constant coefficients may be reduced to a system of nr first-order equations by introducing r ( n - 1) new variables. If, in the present case, the mass matrix m is non-singular, this reduction may be made as follows. One defines a new vector ), of order I1 so that j,=.~
(7)
and another new vector z of order 2n such that
z:[:] In the terminology of control theory, the latter is called the "state vector" and in a typical mechanical system its first n elements give the displacements and its second n elements give the velocities at the n co-ordinates of the system. By premultiplying both sides of (1) by m -t one obtains s + m-l cYc+ m-~ "kx=m-~f
(9)
which, on substituting from equation (7) gives j',=-m-lkx-m-lcy+ m-tf
(10)
Then using the definition of z in equation (8), gives the state equation
i=.4z+F,
(11)
where A=
_~ll _ m - ~
and
F=
.
(12, 13)
72
D.E.
NEWLAND
When there is no excitation, F = 0, and then the free motion of the system has the form z ( t ) = Z e A',
(14)
where, by substitution into equation (11), ,~z = A z
(15)
which is a statement of the fundamental algebraic eigenproblem [6]. The general theory (see, for example, the book by Gantmacher, [7]) shows that equation (15) has a non-trivial solution for z only if det [A - AI] = 0
(16)
and that the determinant in equation (16) gives a polynomial in A for the eigenvalues Ai of the problem. When the order of the matrix A is 2n x2n, the order of the characteristic polynomial obtained from equation (16) will be 2n and so there will be 2n eigenvalues Ai, i = 1,2 . . . . ,2n. Because A is a real matrix, the coefficients of the characteristic polynomial will be real, so that if A = a +jfl is an eigenvalue, then so must A = a -j/3 be an eigenvalue. Therefore if they are complex, the eigenvalues occur in complex conjugate pairs. Usually in practical problems the different Ai will be distinct, but this may not be the case and there may be one or more multiple eigenvalues. Each multiple eigenvalue may be repeated any number of times (up to 2n). An example is the torsional vibration of rotating machinery, when the eigenvalue A = 0 may be repeated several times (see example 5 following). When the eigenvalues are all distinct, equation (15) will yield a different solution vector zl for each eigenvalue A~. These are the eigenvectors of the problem and the 2n x 2n matrix each of whose columns is an eigenvector is called the eigencolumn matrix, here denoted by U. Then, from equation (15), after reversing the order of the equation, A U = U[diag (A,)]
(17)
U - t A U = [diag (A,)].
(18)
or, after premultiplying U -I,
Each column of U is arbitrary to the extent of a (different) scaling factor because, although the ratios o f the complex amplitudes of each co-ordinate are determined by the equations of motion, the absolute values of these amplitudes are determined only by the initial conditions. The general theory of eigenvalue analysis [7] shows that, if the eigenvalues of A are distinct, then there is always a non-singular matrix U for which equation (18) is true. When there are multiple eigenvalues, this may still be so, but it is not necessarily the case. If independent eigenvectors can be found for the multiple eigenvalues, then equation (18) will be true. However usually it is not possible to reduce A to diagonal form by a similarity transformation when A has multiple eigenvalues. Then the best that can be done is to reduce A to its Jordan canonical form, which is almost diagonal. The case of multiple eigenvalues and the Jordan form will be discussed later in this paper but, for the present, it is assumed that the eigenvalues are distinct so that equation (18) is true. 3. NORMAL CO-ORDINATES For distinct eigenvalues, it is always possible to define a new set of co-ordinates which allow the 2n coupled equations ( l l ) to be written as an equivalent set of 2n uncoupled
NON-CONSERVATIVE SYSTEM M O D A L ANALYS1S
73
equations. Let a new state vector be defined by the transformation
q = U-Iz
(19)
and substitute q for z in equation (11) to obtain
U(I= AUq+ F.
(20)
After premultiplying both sides of (20) by U -~, one finds
(! = U-~AUq + U-'F
(21)
which with equation (18) becomes ~j= [diag (A,)]q+ U-~E
(22)
The 2n separate equations represented by equation (22) are then uncoupled from each other and one has 2n separate first-order linear differential equations for each of the co-ordinates which are the elements of the 2n vector q. Since the matrix U -~ can be obtained from equation (17) and F is defined by equation (13), one can now calculate the time history of the response q for any specified excitation f The response in the physical co-ordinates z can then be regained from q by the inverse of equation (19). In the traditional terminology of vibration theory, the elements of q are called the normal co-ordinates of the system (11). 4. STEADY STATE ttARMONIC VIBRATION Consider the case when the nth order force vector f in equation (1) describes forced excitation at the single frequency to so that
f = f( t)= fo ~J~
(23)
where fo is the vector of force amplitudes, which are in general complex. Then, from equation (13), F = F ( t ) = ~m ---i 0 ~.o] e i~
(24,
and the state equation (20) becomes = [diag (Ai)]q +
Lm-'foJ
(25)
Consider the solution reached asymptotically as t ~ ~ on the assumption that the system is passive and that all the starting transients die away with increasing time. For this to be the case, the real parts of all the eigenvalues ,~ must be negative. Under that condition, the steady state solution of (25) has the form q = q(') = qo e i'~
(26)
where qo is the vector, of order 2n, of the complex amplitudes of the response of the normal co-ordinates. Substituting equation (26) into equation (25) gives
Jwq~
(A')]q~ U - ' f m 'O-_fot"
(27)
74
D.E.
NEWLAND
which can be solved for qo to give qo =
J
[ r e - ' f oJ"
If the steady state response in the physical co-ordinates z is represented by z = z ( t ) = Zo~,o,
(29)
then, combining equations (19), (28) and (29) yields =
V[diag
J
u-'l--Lm-'foJ ~
(30)
which is a general expression for the amplitudes of the steady state response of system (1) to harmonic excitation at frequency to. The first n elements o f ~ are the (complex) amplitudes of x; the second n elements of Zo are the (complex) amplitudes of .L 5. FORCED TRANSIENT VIBRATION For the general case of forced vibration, the excitation consists of a set of arbitrary functions of time so that f=f(t) (31) and, from equation (13), 0
F = F(I) = [-~_q ~(-~)--].
(32)
Substituting for F from equation (32) into equation (22), one now has to find the solution of the 2n uncoupled equations represented by t~=[diag(A,)]q+U-'[- m - i0f ( t )~ "
(33)
Consider the ith row of equation (33), q, = A,q, + ~,,(t),
(34)
where ~ ( t ) represents the ith row of
u-,r I Lm-'f(b)J" A general solution of this first-order linear difterentiai equation may be found by using the integrating factor e -A,', because multiplying both sides of equation (34) by this factor and rearranging the terms gives (d/dt)(e-A,'q,) = e-~.',;b,(t),
.(35)
e-~.'qi = J e-~,'qS,(t) d t + Ni,
(36)
which has the general solution
where N~ is an arbitrary constant of integration. The solution of equation (36) is therefore q, =
eX,'fd e-X.'th,(t)dt q-e~.'Nj
(37)
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
75
and there is a similar result for each o f the 2n rows of equation (33). If one defines a 2n x 1 vector of the constants of integration, PC, then the 2n separate equations (37) may be written as the single matrix equation q = [diag
(e')][f
[diag
Also, if one defines a new matrix
(e-a")]U-'~mOf(-t)~ dt+N].
E(t) of order 2n •
(38)
such that
E ( t ) = [diag (e~,')],
(39)
then, by using equations (32) and (39), equation (38) may be written more briefly as
q=E(t)[f E-'(t)U-'F(t)dt+N] and, from equation (19), the state vector z = z(t) is given by z =UE(t)[I E-'(t)U-tF(t)dt+N].
(40)
(41)
This final equation gives a general solution o f the n degree-of-freedom system (1) when subjected to arbitrary forcing functions. Here z is the state vector of order 2n x 1 defined by equation (8), U is the 2n x 2 n eigencolumn matrix defined by equation (17), is the 2n x 2 n diagonal matrix of integrating factors defined by equation (39), is the excitation vector o f order 2n x 1 defined by equation (32) and N is the 2n x I vector of arbitrary constants of integration which have to be found from the initial conditions of the problem.
E(t) F(t)
6. COMPARISON WITH THE RESULTS OF FAWZY AND BISHOP [1] Equations (30) and (41) of this paper are in agreement with the corresponding equations (44), (70) and (71) of Fawzy and Bishop [1], who derived them by a different method. In order to demonstrate that the results derived here and those in Fawzy and Bishop are the same, it has to be shown how the n x 2n eigencolumn matrix ~; used by Fawzy and Bishop relates to the 2n x 2n eigencolumn matrix U used in this paper and how the 2n x n eigenrow matrix H used by Fawzy and Bishop relates to the inverse eigencolumn matrix U -I.
Consider the ith column of U. This satisfies equation (17) and, after substituting for A from equation (12) into equation (17), one has -m-~k
I
If now the column vector u (i) is partitioned into its top and bottom halves, r
.(i)
-]
uO) = I _ ~ _ o~__[
l . (,)
/'
(43)
L ==b o t t o m . J
where ,'top " (o and "hot,ore ' (i) are column vectors of order n, then equation (42) gives ~(0 = . H t(1) ~ bottom opltl
(44)
.") ~ i. - - I'~11- ~n, .(to O p - - i_l l - ~ i . ~.(o b o t l o m = 14bottomtl
(45)
and
76
D.E.
NEWLAND
By eliminating Ubouo m(1) between equations (44) and (45), rearranging terms, and premultiplying by m -~, one finds [A~m+A,e+t:l.(~ =0, 9 ~ JMIO
i=1,2,.
p
9 9
,2n.
(46)
By comparing equation (46) with equation (2), it can be seen that, subject to the same scaling, uo) _ o.(O, top-
i = 1, 2,.
9 9 ~I
2n,
(47)
and so, if the 2n x 2 n eigencolumn matrix U is partitioned into a top and bottom half so that U = t.Ubot,omJ
(48)
it has been demonstrated that, subject to scaling, U,o,, =
(49)
v.
Also, because of equation (44), one has the result that Ubo,,om= U,op[diag (A~)].
(50)
Now consider the ith row of U -~. Let V=
(51)
U-'
and denote the ith row of V by v(~ Postmultiplying both sides of equation (18) by U -~ and using equation (51) gives VA = [diag (A,)] V.
(52)
After substituting for A from equation (12), the equation for the ith row of V is, from equation (52),
o.Eo
-m-'k
I
,:]
-m-_ T
= Aiv <~
(53)
If the row vector v t~ is partitioned into left-hand and right-hand halves, so that v(') = [ vl~t where
IIl v~ig)ht],
(54)
' o) a n d '-'right 9 (i) are row vectors of order n, then equation (53) gives "'left _,(i) 9~n~h,.~ .- .] .b
_- A ,,a, I)(i)
(55)
and (i) _.(i)
_-t~_..(O
v,~f, ",ight. . . . .
",'-',iSh,-
(56)
By eliminating vl~, between equations (55) and (56) and rearranging terms, one obtains the result that v(o right m -Ir.~ L A i m + A , c + k ] =0,
i = 1 , 2 . . . . . 2n.
(57)
Comparing equation (57) with equation (3) shows that, subject to scaling, t,(i) ~-I rightS,r,
~--
n ('),
i = 1, 2 . . . . ,2n,
(58)
and that, if the matrix V is partitioned into left-hand and right-hand halves so that V = [ Vj,f, i
V,~h,],
(59)
NON-CONSERVATIVE SYSTEM MODAL ANALYSIS
77
then it can be concluded that, subject to corresponding rows having the same scaling?, Eigh, = lira.
(60)
In order to compare the results in this paper with those of Fawzy and Bishop, where the n second order equations which define the system are given as (61)
A ~ + B:I+ Cq = Q ( t ) ,
one has to write A=m,
B=c,
C=k,
Q=f
and
q=x.
(62)
In this paper q has been used to denote a vector of order 2n for the normal co-ordinates, whereas Fawzy and Bishop used q as a vector of order n for the physical (displacement) co-ordinates which have been called x here. The two principal results (30) and (41) can now be shown to lead to the same results as equations (44) and (70) and (71) of Fawzy and Bishop. Two new symbols are needed. These are the 2n x 2n matrices defined by A=[diag(Ai)]
and
ll=
diag
.
(63, 64)
Now, by using equations (7), (8), (48), (59) and (64), equation (30) may be written as Ix~ = i[--UL~ Ubo,omLJ -1D[ V~'r
V'h'] f m O'j'0t"
(65)
On account of the zeros in the top half of the column vector on the r.h.s, of equation (65), one obtains [~o]
=[__U,_oP__l fi~v, ght[m-,,o] L U~o,,om.I
(66,
and so the nth order vector giving the amplitudes of the physical co-ordinates is given by Xo= Utopfl V,~sh,[m-~fo].
(67)
In the notation of Fawzy and Bishop, =fo
(68)
and, together with the last of equations (62), and equations (49) and (60), one obtains the result that, for steady state harmonic excitation at frequency to,
q( t)= ~ I I H ~ ~,o, (69) which confirms equation (44) of the reference. Equation (41) of this paper becomes after substituting from equations (7), (8), (32), (51) and (59),
[x-] =[--U•176 Ubo,,omJ
LJ
Vngh']~n--~(t)] dt+N ]
(70,
and, with equations (49), (50), (59), (60), (63) and the last two of equations (62), this becomes
[_:_]
+
which agrees with equations (70) and (71) of Fawzy and Bishop. i See the Appendix.
,7,,
78
D. E. N E W L A N D
7. NUMERICAL EXAMPLES Two numerical examples of the use of normal co-ordinates to solve problems of forced vibration are as follows. The examples are the same as those given in the paper by Fawzy and Bishop [1] because they allow successive stages of the different methods of calculation to be compared. 7.1. E X A M P L E 1
The first example concerns the steady state harmonic response of the single degree-offreedom system ~ + 2.'~+ 2x = 5 ei'. (72) From equation (12), the A matrix is
From equation (16), the eigenvalues At.2 are the solutions of I for which det [-__~ - 2 1- A] = 0
(74)
A,.2 = -1 :i:j.
(75)
so that The column eigenvectors are the solutions of equation (15) for ;t = - 1 +j and 3. = - I - j , and these are u(') =
[ , ] a.d.<2,=[ 1 ] -1 +j
- 1 -j
(76)
so that U=
[
1 -l+j
1 ] -1-j
(77)
and its inverse is
v_, [0.5-j0.5 -jo.5] k0.5+j0.5
(78)
j0.5J"
The normal co-ordinates are defined by equation (19), so that in this example
[<,,]:ro.~-Jo.~ -Jo.~ar.
(79)
The steady state harmonic response is then given by equation (30), which becomes
[.o] [,
:fo = - l + j - 1 - j
,][ o:
jto+_l-j
o
1 /L0"S+j0"5 jto+l+j
j0"SJl.SJ"
<80,
Putting to = 1, from equation (72), and working out the matrix products gives
~o for the vector of amplitudes.
L 2+j J
(81)
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
79
7.2. E X A M P L E 2 The second example is a non-conservative two degree-of-freedom system for which, with the notation of equation (1),
:],
(82)
and
(83) The initial conditions are that, at t = 0,
(84)
Z ~
From equation (12), the system matrix is
o0 ,_,~] 0 0 1 6.
A=
4
0
0 6
01
(85)
-3
The eigenvalues may be found from equation (16) and give -1 0 0 0
[diag(A~)]=
0 0 i] -2 0 . 0 -3 0 0
(86)
The eigencolumn vectors are obtained by solving equation (15) for each eigenvalue and this leads to the eigencolumn matrix U=
-1
-5/6
-13/15
1
-2
-3
1
5/3
13/5
-
6
(87)
-2/31
and its inverse
2.. 7,.0.35] V = U -I =
7 -45/14 11/35
6 -15/7 12/35
-2 15/14 8/35
15/7[" 9/35J
(88)
By comparing the top two rows of (87) with the Fawzy and Bishop eigencolumn matrix v in their equation (86), which is [21-210 2"=1/210 _21 175
225 _195
_
61] ,
(89)
80
D.E. NEWLAND
one sees that, except for ditIerent scaling factors for the columns, equation (49) is satisfied. The Fawzy and Bishop eigenrow matrix is given in their equation (87), and using their values gives
[i-!JEi1[i !1
Hm=
=
(90)
which is the same as the right-hand half of U -1 in equation (88) (except for the different scaling of the individual rows), thus agreeing with equation (60). From equation (39), E(t)=
[e~176176 l 0
e -~'
0
0
0
0
e -3t
0
0
0
0
e 4'
(91)
and, from equation (32), for t > O,
0
~
o
':'
(92)
In order to evaluate the general result (41), one needs
~<')=
I'Y,Yl
<,>0)
(93,
L~/~a J and, using E - ' obtained from equation (91), [ 7/10C 1 / -2 e 2' /
E-lVF(t)=18tll5/lae3t[
(t>0).
(94)
L8/35 e-4'J
Evaluating the integrals then gives
E-tVF(t)dt=
63/5 e'Ct-1) ] -9e2'(2t-1) / 15/7 e3'(3t- 1) /
(t>0)
(95)
-9/35 e-4'(4t + 1)J and, after substituting this result into equation (41), and making use of equations (7) and
(8),
Z= 2, 2z
='E /
u [ 15/7e3,(3t_l)+N3 L-9/35 e -4 (4t + 1)'+ N4
(t>0).
(96)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
81
Substituting for E from equation (91) and working out the matrix product gives x2
/-9(2t-
1)+ N2 e-:' /
= Ui15]7(3t_l)+N3e_3, [ Yr
(t > 0).
(97)
[-9/35(4t+l)+N4e"J
Lastly, substituting for U from equation (85) and multiplying out the remaining matrix product gives, for t > 0,
x2 21 =
22
7-3t-N~e-'-5/6NIe-2'-13/15N3e-3'-l/6N4e - Ni e - ' - 2 N 2 e -2' - 3 N 3
e-3'q-4N4
"'
e 4'
-3+Nle-'+5/3N2e-Z'+13/5Nze-3'-2/3N4e
"
(98)
4'
For the initial conditions specified in equations (84), which are the same initial conditions as used by Fawzy and Bishop, all the arbitrary constants o f integration are zero, N, = N 2 = N3 = N4 = 0
(99)
and hence the complete solution is
ixl[ x2 =
7-3t
(t>0).
(100)
,~2
Note that the initial conditions are chosen to ensure that all the exponential terms in equation (98) are zero. These include the increasing exponential term e4'. Any small change in the specified initial conditions would mean that this term would eventually dominate the solution. 8. SINGULAR AND DEFECTIVE MATRICES The general solutions (30) for steady state harmonic vibration and (41) for forced transient vibration depend on the mass matrix m being non-singular and on the eigenvalues Ai being distinct. The requirement that m is non-singular is necessary if the n second-order equations (1) are to be written as an equivalent set of 2n first-order equations (1 I) by using equation (12). Note that the same condition that m should be non-singular applies also to the analysis of Fawzy and Bishop [1] because their eigencolumn and eigenrow matrices have to be normalized to satisfy equation (40) of the reference, which is
X A H = m -I,
(101)
so that m -~ must exist. The significance of this condition may be seen by returning to equation (1) and, for the case of free motion when f = 0, substituting the solution vector
x(t) = x e ~'
(102)
which is the upper half of z(t) in equation (14). This gives [Aim + Ac+ k]x = 0
(103)
82
D.E.
NEWLAND
which has a non-trivial solution for x if and only if det [ A 2 m + A c + k ] =0.
(104)
Expanding the determinant leads to a polynomial of order 2n in A of the form a2nA 2n Jr" O ~ 2 n _ l A 2 n - I "1" 9 9 9 q" a l A "1" a 0 : 0 ,
(105)
where, from equation (104), a2n = det [m].
(106)
When a2, -~0, then at least one of the roots A of equation (105) approaches infinity. This means that, in the limiting case when a2, = 0, one at least of the eigenvalues does not exist and so equations (30) and (41) break down. Since, from equation (106), a 2 , = 0 when the mass matrix m is singular, the solution given fails when m is singular. In all the engineering problems known to the author for which m has been found to be singular, the reason has been zero mass associated with one of the physical co-ordinates x~. In that case there is no difficulty in dealing with the defective second-order equation as a separate first-order equation. The remaining (n - 1) second-order equations reduce to 2 ( n - 1 ) first-order equations without difficulty and the extra first-order equation is then added to give an A matrix in equation (4) of order (2n - 1) x (2n - 1). This calculation is illustrated in.the following example 3. When it has been found, the A matrix may turn out to have multiple eigenvalues. In that case, it may not be possible to find an eigencolumn matrix U which satisfies equation (18). When it is possible to satisfy equation (18), the columns of U for a repeated eigenvalue have an indeterminancy. Suppose, for example, that z~ and z2 are two columns of U for the same eigenvalue A~. Then, from equation (15), one has AIzI=Azl
and
A~z2--Az2.
(107, 108)
If z3 is ;., new vector defined by z3 = azl + flz2,
(109)
where a and fl are arbitrary multipliers, then, from equations (107) and (108) one must have Zzz~ = Az3.
(110)
Apart from the individual scaling of columns, the eigenvector matrix U is unique when the eigenvalues are distinct. When there are multiple eigenvalues, and equation (18) is still satisfied, then U is no longer unique. But provided that any U can be found which satisfies equation (18), then the results expressed by equations (30) and (41) apply without alteration. One can next consider the case when there are multiple eigenvalues and a matrix U to satisfy equation (18) cannot be found. Then the coupled fi rst-order equations in equation ( l l ) cannot be written in an uncoupled form (22), and so the general results that have been derived from equation (22) are no longer valid. Instead offinding a similarity transformation by a matrix U which reduces A to diagonal form, as in equation (18), it can be shown (see, for example, the book by Gantmacher, [7]) that a matrix W may be found which reduces A to its so-called Jordan form. This may be described as a nearly diagonal matrix and it is the most compact form to which A may be reduced by a similarity transformation. For example, if A has order 2n x 2n, and the first eigenvalue At is repeated once, then it may still be possible to reduce A to diagonal form, but, if not, then it will be possible always to find a non-singular matrix
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
83
W for which Ai 0
1 A~
I Ix \
x,.\A 3 \,.
W-lAW=J=
zeros
N
zeros (111)
~4 NN
N
\
x ~'2n
In that case there will be 2n - 1 independent eigenvectors. These are columns 1 and 3 to 2n of W. Column 2 satisfies the equation A w t2) = w~ + Alw (2~
(I 12)
and is not an eigenvector of A. Instead, it is called a principal vector of A. A n eigenvector is a special case of a principal vector and it m a y be shown (for example, [7]) that the principal vectors of a matrix are independent, although they are not unique. Thus all the columns of W satisfying equation (111) are independent vectors, but wtz) is not unique because if wt2) satisfies (112) then so does wt2)+aw") where a is an arbitrary multiplier. If equation (111) is true, instead of equation (19), one defines normal co.ordinates by the equation q= W-Iz (113)
and then equation (22) becomes
:l=dq+ W-~F,
(114)
where d is the Jordan matrix defined by equation (111). The steady state response to harmonic excitation is then, corresponding to equation (30), %= W [ j t o / - J ] - ~ W - ' [~n-_0Tfo].
(115)
For the forced transient response, when F is given by equation (32), the first row of equation (114) is t)~ = A~q, + q2+ ff~(t),
(116)
where ~b~(t) is the first row of W-~F, and the second row of equation (114) is t12= A~q2+ ~b2(t),
(117)
where qSz(t) is the second row of W-~E From equation (37), one knows that the solution of equation (117) is q2= e~'t I e-X"q~2(t) d t + e X " N2.
(118)
Hence equation (38), with W -~ replacing U -~ and with Az = A~, is true for rows 2 to 2n but is not true for row 1. Row 1 must be replaced by the q~(t) obtained by solving equation (116) with q2 from equation (118) being used. The state vector of the physical co-ordinates must then be recovered from q by using equation (113), and this expression replaces equation (41). When an eigenvalue is repeated more than once, or there is more than one multiple eigenvalue, the Jordan matrix is more complicated than shown in equation (111). The
84
D.E. NEWLAND
eigenvalues always lie along its diagonal and there may or may not be elements equal to unity lying on the first super-diagonal line. For example, for the case when 2n = 6 and the A matrix has eigenvalues An, At, An, A4, As, A6, then if there are six independent eigenvectors, matrix A is similar to a Jordan matrix of the form
J=
At 0
0 At
0 0
r t 0
0
0
At o
I
o~
n I
t-
zeros (119)
".A4 "-.
zeros
""3,
``.~ ,
if there are five independent eigenvectors, then A is similar to a Jordan matrix o f t h e form
J=
At 0
0 hj
0 I
0
0
At 0..
rI
o
zeros
I
~;"
|
II
ZerOS
(120)
~``
"" ~ .., ~ 5
I
~ "- ~.
"" "-
~6 I
and if there are four independent eigenvectors, then A is similar to a Jordan matrix of the form
a =
An 0
I At
0 r I 1 0
0
0
At
I
I
zeros
J
(121)
"``
"'..An''
zeros
"~
s --. ~"
I
"``
I
A6
If As=A4, then d may be as in equations (119), (120) or (121) or it may have any of these forms with the submatrix
replaced by
in which case there will be five, four or three independent eigenvectors respectively. For problems of forced transient vibration, when the forcing function F in equation (114) is an arbitrarily chosen function of time, the solution procedure is to solve each of the equations in equation (114), starling with the equation given by its last row. This will ahvays be an uncoupled equation because there is no super-diagonal element associated with the last row. Each other row of equation (114) is an equation with no more than two co-ordinates. For row r, a solution for q,(t) is possible as soon as q,+~(t) has been found; the whole solution vector q(t) can be determined by progressive substitution from one row into the next higher row until the equations for all the rows have been solved.
NON-CONSERVATIVE
SYSTEM
MOI)AL
85
ANALYSIS
The Jordan matrix with elements of unity in the super-diagonal is called the upper~ Jordan form and a similar matrix may be found with elements of unity in its sub-diagonal. Every matrix A is similar to an upper and a lower Jordan matrix and the choice of which Jordan form to use is arbitrary. Ifthe lower-Jordan form is used, the calculations described above have to be modified accordingly. In a particular problem with multiple eigenvalues, the form of the Jordan matrix has to be determined by finding the number of independent eigenvectors corresponding to each multiple eigenvalue. There do not appear to be any general rules for determining the number of independent eigenvectors a priori. Each case has to be considered individually. These calculations are illustrated by three further numerical examples in the next section. 9. FURTHER NUMERICAL EXAMPLES 9.1. EXAMPLE 3 This example is the response to steady state harmonic excitation of a non-conservative system with a singular mass matrix and repeated eigenvalues. With the notation of equation
(1) m=[l 0
~],
c=[20
11],
k=[l 0
~]
and
f = [ ~ ] e j'.
(124,125)
Because m is singular, equation (12) cannot be used to obtain a 4 x 4 A matrix. There is one second-order equation 5il + 22~ + 22 + xt + 2x2 = 0
(126)
.~2+ x2 = 2 e i'.
(127)
and one first-order equation
The latter may be used to eliminate -'/2 between equations (126) and (127) to give .#i+2Yq+Xl+X2
=
--2 e 3t.
(128)
If, corresponding to equations (7) and (8), one defines
ix]
z = x2 21 then the 3 x 3 A matrix is A=
0 0 -1
0 -1 -1
(129)
l]
0 -2
(130)
and this has the repeated eigenvalues AI=-I,
A2=-l,
(131)
There is one eigenvector only, which is (132)
86
D. E. N E W L A N D
and so one has to find a 3 x 3 W matrix for which
W-IAW=J
=
o1
Ai
1
0
.
(133)
AI
After premultiplying both sides of equation (133) by W, the second columns give
A w (2) =
The solution of
W(2)
- -
w(2)
(134)
.
from equation (134) is
W(2) --
(135) 1-
where 6 is arbitrary. The third columns of equation (133) give Aw(3) = w( 2 ) - w(3)
(136)
which, with equation (135), gives
w~
-1
(137)
,
where e is also arbitrary. Hence the W matrix is
W=
1
0 -
-1
.
(138)
6-e
1-6
If the second and third columns of (138) are normalized so that their first elements are unity, then 1
1
0
0
-1
0
W=
1
-
(139)
and this W has the inverse W-' =
1
.
(140)
-1 From equations (127) and (128), one sees that F is a column vector of order 3 for which
F =
ej .
(141)
NON-CONSERVATIVE
SYSTEM MODAL
o_,][
Hence, corresponding to equation (115), one has Zo=
[!1 1j[j+l 1 01[i -
0 -1
0
0
0
0
jto+l 0
87
ANALYSIS
-1
joJ+l
1
1
-1
0
(142)
.
-
Substituting to = 1 and evaluating the inverse matrix gives [ i Zo= -
1 1][0"5-j0"5-j0"5-0"25-j0"25 0 -1 0 0.5-j0.5 -jO.5 0 0 0 0 0.5-j0.5
-1
0
1
1
2
-1
0
-2 (143)
and multiplying out the matrices, one obtains finally Zo=
0"5+j1"5 1-j -1"5+j0"5
(144)
as the vector of amplitudes of the steady state response. Since equation (133) is true whatever the values of the arbitrary constants ,5 and e in equation (138), one will obtain the same results if different values of ,5 and e are chosen when determining W. 9.2. E X A M P L E 4 The second example here is a calculation for the forced transient vibration of a system with a repeated eigenvalue. This is the system represented by m=[lo
i]'
c=[21
~],
k=[20
21]
(145)
and
f=[~]
fort~>0,
f=[00]
fort<0.
(146, 147)
Then, from equation (12),
A= _
0 0 -1 -1 -1
-1
(148) -
and the system's eigenvalues are Al=--l,
A2=-l,
A3=-I-j,
A4=-I+j.
(149)
There is only one eigenvector for A~= -1 which is
W (I)
(150)
88
D.E.
NEWLAND
and then equation (112) gives
A w (2) _-
[:i]
(151)
_ w (2),
where A is given by equation (148). Hence w(2) is
--~J
w (2) =
(152)
I-8 ' - 1 -(5
where ,5 is arbitrary. For the case when 8 = 0, one has
(153)
W (2) =
After finding the eigenvectors for A3.4= - 1 + j from equation (111), one has
1 IV=
0
1
-1
0
-0.6+j0.2
-0.6-j0.2
-1
1
-l+j
-1-j
[
0.4-j0.8
0"4+j0"8
J
1
-1
1
] (154)
which has the inverse
W_ ~=
0.5-jl.5 L0.5+jl.5
Also, from equation (13),
-3 0.5-jl.5 0.5+jl.5
"Ji
-1 -0.5-j -0.5+j
. -0.5-j -0.5+jJ
for t~>0,
(155)
(156)
and so, from equation (114),
q=
0 0
-1 0
0
0
0 -l+j 0
W_ I 0
q+
0 , -
(157)
-1 -jJ
where W -~ is given by equation (155). The resulting equations for the elements of q are, for t~>0, tll= --ql+q2,
t12 = - - q 2 - - 5 ,
t~3 = ( - - I + j ) q 3 ,
t~4 = ( - - 1 - - j ) q 4 .
(158)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
89
Solving the second, third and fourth of equations (158) gives q2 = - 5 + N2 e-',
q3 = N3 e (-~+i',
q4 = N4 e (-~-m
(159)
and, combining the first of equations (159) with the first of equations (158) gives qt = - 5 + N~ e - l + N2t e -t.
(160)
z= Wq,
(161)
Lastly, from equation (113),
from which the vector of the physical co-ordinates may be regained. If one takes a particular set of initial conditions, the arbitrary constants of integration may be determined and, for initial conditions that give, at t = 0,
z:E_],i
(162)
the final solution is, for t ~ 0,
Ix]
- 5 + e - ' ( l + t + 1 0 c o s t) ] 5 - e - ' ( l + t + 6 c o s t + 2 s i n t) I - e - r ( t + I0 cos t + 1 0 s i n t) / " e-t(t+4cost+8sint) J
X2
Z=
(163)
9.3. EXAMPLE5 The final example is a system with six eigenvalues, three of which are the same, and five (not four) independent eigenvectors. This is the torsional system shown in Figure 1. Co-axial wheels of moments of inertia I~, 12 and 13 are arranged so that wheels 1 and 2 are joined by an elastic shaft of torsional stiffness k and wheels 2 and 3 are joined by a fluid coupling with a torsional viscous damping coefficient c. This system represents approximately the behaviour of a primitive engine with a viscously-coupled vibration damper. The following parameter values are
1
c
5
I1
Figure 1. The torsional system.
90
D.E.
NEWLAND
to be used: /2 = 1 "5 kg m s,
It = 3 kg m 2,
13 = 4-5 kg m s,
k =200 Nm/rad,
c = 135/8 Nms/rad.
(164)
If xt, x2, x3 give the angular movements o f the three wheels from a static equilibrium position, then the equations of motion for the system when subjected to harmonic excitation at wheel 1 are given by equation (1) where
0 0] [0oi]
m=
k=
I2
0
0
13
-k 0
,
k 0
c=
0
c
0 ,
-
,
-c
f=
e j~
(165)
On substituting the numerical values and using equation (12), one finds that -
A=
0
0
0
1
0
0
0 0 -200/3 400/3 0
0 0 200/3 -400/3 0
0 0 0 0 0
0 0 0 0 0
1 0 0 -45/4 15/4
0 1 0 45/4 -15/4
(166)
and, from equation (16), the eigenvalues are At=A2=A3=0,
A4=-10,
A5.6= ( 5 / 2 ) ( - I + j x / ~ )
(167)
the units being 1/seconds. When the system is stationary, with no applied torque, angles xt = x2 may have any value and are independent of x3. Hence an eigenvector is (1, 1, a, 0, 0, 0) t, where a is an arbitrary real constant. Therefore there are infinitely many eigenvectors corresponding to the eigenvalue A = 0. But only two of these are independent since the rest can all be constructed by combining, for example, (1, 1, - 1 , 0, 0, 0) t and (1, 1, 1, 0, 0, 0) t in appropriate proportions. One thus concludes that the triple eigenvalue At = A2 = A3 = 0 has two independent eigenvectors, which one can take to be those given above. Therefore altogether there are five independent eigenvectors and the Jordan matrix has the form (120). Hence in this example
j=
0 0 0 0 0 0
0 0 0 0 0 0
0 1 0 0 0 0
0 0 0 -10 0 0
0 0 0 0 - (5/2)(1 + j , / ~ ) 0
0 0 0 0 0 - (5/2)(1 - j , . / ~ )
(168)
The three other independent eigenvectors can be found by solving (15) for each of A4, A5 and A6, so that one has five of the six columns of the W matrix. The missing column,
NON-CONSERVATIVE SYSTEM M O D A L ANALYSIS
91
which is the third column, can be found from, corresponding to equation (112),
Aw~3)=wC2)+ A3wt3) = wt2)
(169)
since 3.3 = 0 in this example. If one puts wt2) = (1, 1, 1, 0, 0, 0) t
(170)
wc3) = (/3,/3, % 1, 1, 1)',
(171)
then from equation (169)
where/3 and y are arbitrary constants. One can take/3 = 3' = 1 and put wt3)= (1, 1, 1, 1, 1, 1) t.
(172)
If, instead o f equation (170), one had put w~2)= (1, I, - 1 , 0, 0, 0) t,
(173)
then a solution for w~3) satisfying equation (169) could not have been found, so that it is important that the columns of W are ordered so that w~) = (1, 1, - 1 , 0, 0, 0) t
(174)
and w~2) is g i v e n b y equation (170). Then the full W matrix is
W=
1
1
1
1
1
1
1
2.5
-1
1
1
-1-5
0
0
1
-10
0
0
1
-25
0
0
1
15
1
-1-0625 -j0.8992 -0.3125 +j0.2997 -2-5 +j11.9896 13.4375 -j10.4909 -2.8125 -j4.4961
1
-1.0625 +j0.8992 -0.3125 -j0.2997 -2.5 -j11.9896 13.4375 +j10.4909 -2.8125 +j4.4961
(175)
and its inverse has been calculated to be
W-I=
0 0"3333 0 0"1667 0.25 -j0"1216 0"25 +j0"1216
0"5 0"1667 0 -0"1667 -0-25 +j0.1216 -0"25 -j0.1216
-0"5 0"5 0 0 0 0
0.0444 -0"3333 0.3333 -0.0167 -0"0139 -j0.0180 -0.0139 +j0.0180
0.0222 -0"0667-0.1667 - 0 ' 5 0.1667 0"5 -0.0208 0.0375 -0.0007 0.0146 +j0"0158 +j0.0022 -0.0007 0"0146 -j0"0158 -j0.0022
(176)
These values are given to four significant decimal figures. A note on their method of calculation is given at the end of the paper.
92
D.E. NEWLAND
The solution required is given by equation (115) for which one needs [jwl - d ] -~ which it is not difficult to show is given by
_=_1 joJ 0 0
o
o
1
1
joJ
(joJ) 2
0
[jwl-J]-I =
1
~
j~
o
o
o
0
0
0
0
0
0 (177)
1
0
0
0
0
0
0
0
0
0
0
0
0
jto+10
0
1
0
jw + ~(1 - j x / ~ )
1
0
jco +-~(1 +jx/~)
and then, evaluating equation (115), one can find the steady state amplitudes of x~, x2, x3 (the first three elements of Zo in equation (115)). For example, the first element xl is given by 0.0444 0"3333 Xl = k joJ 4 (jo~)2
0.0167 Ota+ 10)
(0.0139+j0.0180) (jw +~(1 - j 4 ' ~ ) )
(0.0139-j0.0180)],
(178)
which reduces to
0t~ + 15(jr~ + 133"3(Jt~ 500 (-~) x, - 0,o)5+ 15(j~,), + 2000,o)3+ 1500(j,o)~ .
(179)
10. INTERPRETATION OF COMPLEX EIGENVECTORS For free vibration, one has from equation (41) with F ( t ) = 0 , z = UE(t)N
(180)
or, substituting for E ( t ) from equation (39), z = U[diag (eX")]N,
(181)
where U is the eigencolumn matrix and N is a vector of arbitrary constants defining the initial conditions. At t =0, equation (181) gives z(t = 0 ) = UN.
(182)
Solving for N from equation (182) and substituting into equation (181) then gives z(t) = U[diag (e~,')]U-'z(t =0).
(183)
Alternatively, in terms of the normal co-ordinates defined by q= U-Iz
(184)
q(t) = [diag (eA,')]q(t = 0).
(185)
one has
NON-CONSERVATIVE
SYSTEM MODAL
ANALYSIS
93
Now let
(186)
q2,~(t).l and consider the case when
q~(t=O)--ql(O)
and
q2(t=O)=q3(t=O) . . . . .
q2.(t=O)=O.
(187,188)
Then, from equation (185),
ql(t)=eXltql(O ) and
q2(t)=q3(t) . . . . .
q2,(t)=0.
(189, 190)
When ~.~ = - a - j t o ,
ql(t) = e-=' e-J'Otq,(O)
(191)
which can be represented in an Argand diagram as aline which rotates about the origin in a clockwise direction at angular speed to, while decreasing in length with a time constant 1let (see Figure 2). If OQ~ is the line representing qt(t), then Q~ rotates clockwise about O while spiralling in towards O.
,,I- IIm I/
l
O
qs(0)
Re
[~....~--(ot]
,"'--b-\i (W,<,> ~
\ length qi (O) e'at= ql (O)e-tal~)O Figure 2. Complex normal co-ordinate behaviour for a system with damping.
It has already been remarked that, because A is a real matrix, complex eigenvalues must occur in complex conjugate pairs. It follows from equation (15), that if AI and A2 is a pair of complex conjugate eigenvalues with corresponding eigenvectors zl and z2, then the corresponding elements of zi and z2 will also be complex conjugate pairs. Since z~ becomes column u~ and z2 becomes column u2 of the eigenvector matrix /3, in this case the corresponding elements of the columns ui and u2 of U are complex conjugate pairs. In a physical case, the coordinates z(t) must always be real. The mode q~(t) corresponding to a complex eigenvalue Az cannot then be excited without a comparable excitation of mode q:(t) which corresponds to the complex conjugate eigenvalue As = AI*. Suppose that only modes ql(t) and q2(/) are excited. Then, from equation (19),
z(t) = Uq(t)
(192)
q3(t)=q4(t) ..... q2n(t)=O,
(193)
z( t)= u~q~(t)+ u2qe(t).
(194)
which, for
gives
94
D . E . NEWLAND
Then since z(t) is real and u2 = u*, one must have q2(t) = q,(t)*.
(195)
Hence the rotating line OQ~ in Figure 2 representing the oscillatory decay of mode q~(t) can occur only in the presence of a second line OQ2 representing co-ordinate q2(t) rotating in the opposite direction at the same speed to and satisfying the relation
OQ2 = OO*
(196)
as shown in Figure 3.
Im
Oz wt
clz(t}
~O
Re
q,(O 001=OO z
Figure 3. Showing the complex normal co-ordinate conjugate to that of Figure 2.
If, in equation (194), one puts
I-z,(,)1 (197)
/~i ~
tq,2. J
L u2..2 J
and also puts fftl = ,~ e-J~h,
ul2 ----lll*l = Pl e+J~':,
(198)
then
zl( t) = Pl e-J~"OQ~ + p~ e§
(199)
which one can write as z,(t) = o z , + oz2
(200)
o2 ~
x~4'1""""O
\
ZI
Figure 4. Behaviour of a real variable which derives from the complex conjugate normal co-ordinates of Figures 2 and 3.
NON-CONSERVATIVE SYSTEM MODAL ANALYSIS
95
as shown in Figure 4. The time history o f the changing real variable Zl(t) is therefore given by the sum of the projections onto the real axis of the Argand diagram of the rotating lines OZi and OZ2, or since
OZ2 = OZ*
(201)
by twice the projection onto the real axis of either OZi or OZ2. 11. NOTE ON NUMERICAL CALCULATIONS The first-order formulation for modal analysis described here, with equations in the form (4), allows readily available computer programs to be used to solve the eigenvalue problem (5). The real elements of the 2n x 2n system matrix A can be formed from the elements of the n x n mass, damping and stiffness matrices m, c and k by a sub-program which makes the calculation (12). Then there are a variety o f colnputer library programs to compute the eigenvalues and eigenvectors o f A. The 2n eigenvalues and the 2n x 2n eigenvector matrix U are the output from such a program. The eigenvector matrix U does not have to be normalized in any particular way. Its inverse U -~ can be calculated directly by using a library sub-program for the inversion of a complex matrix. When there are repeated eigenvalues, the first-order a p p r o a c h still allows the modal analysis method to be used, subject to an ad hoc exploration o f the eigenvectors associated with repeated eigenvalues in order to ensure that all the independent eigenvectors have been discovered and that the Jordan matrix has been formulated properly. REFERENCES
1.t I. FAWZY and R. E. D. BISHOP 1976 Proceedings of the Royal Society London A 352, 25-40. On the dynamics of linear non-conservative systems. 2. K.-J. BATHE 1982 Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall. See chapter 9. 3. R. E. D. BISHOP, G. M. L. GLADWELL and S. MICHAELSON 1965 The Matrix Analysis of Vibration. Cambridge University Press. 4.j- I. F. A. WAHED and R. E. D. BISHOP 1976 Journal of Mechanical Engineering Science 18, 6-10. On the equations governing the free and forced vibrations of a general non-conservative system. 5. D. L. WOODCOCK 1963 Aeronautical Quarterly 14, 45-62. On the interpretation of the vector plots of forced vibrations of a linear system with viscous damping. 6. J. H. WILKINSON 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.See chapter 1. 7. F. R. GANTMACHER 1959 The Theory of Matrices, Volume 1. New York: Chelsea APPENDIX: CONSISTENCY OF SCALING To confirm that the scaling of columns in equation (49) and of rows in equation (60) is consistent, equation (51) may be written in the form
r l[' --U-t~
L. UbottomJ
]
V,~ft I V"sht = I I
(202)
from which one sees that UbottomV~isht= I.
(203)
Utop[diag(A,)] V~ight= I
(204)
Hence, using equation (5) gives
1 Professor Bishop had the same co-author in references [1] and [4] in this list: Ibrahim Fawzy A. Wahed.
96
D.E.
NEWLAND
and, upon substituting from equations (49) and (60), this requires that, in the nomenclature of Fawzy and Bishop, ~YAHA
= I.
(205)
Equation (205) is the same as equation (40) in the paper by Fawzy and Bishop [1], thereby confirming that the scaling of ~ a n d / 7 needed to ensure that equations (49) and (60) are true is consistent with the scaling of these parameters that has been used by Fawzy and Bishop.
O N THE M O D A L ANALYSIS OF N O N - C O N S E R V A T I V E LINEAR SYSTEMS D. E. NEWLAND University Engineering Department, Trumpington Street, Cambridge CB2 IPZ, England
(Received 27 November 1985, and in revised form 22 January 1986) This paper applies the results of classical eigenvalue theory to the analysis of a nonconservative vibratory system. General results are obtained for the response of a multiple degree-of-freedom system which is represented by a set of linear second-order differential equations with constant coefficients. For the case when the eigenvalues of the system are distinct, these results confirm those obtained by Fawzy and Bishop [1] by a different method. The latter results are restricted to distinct eigenvalues, but the more general analysis presented here allows the case of multiple eigenvalues to be included. The physical interpretation of complex normal co-ordinate vectors is discussed and some simple numerical examples illustrate the results. The approach described here lends itself to numerical computation using computer library programs for eigenvalue determination.
1. INTRODUCTION The dynamic behaviour of a mechanical or structural vibratory system is usually studied by one of two methods: the mode superposition method or the direct integration method (see, for example, the book by Bathe [2]). The mode superposition method involves calculating analytically the response o f each mode separately and then summing the response of all the modes of interest to obtain the overall response. The direct integration method involves computing the response of the system by step-by-step numerical integration. A choice of which method to use depends on what is to be achieved and the numerical effectiveness of the two methods in a specific case. However for many problems, the mode superposition method offers greater insight into the dynamic behaviour and parameter dependence of the system being studied. This paper is about the mode superposition method. For passive structural systems, in which the damping terms are small and essentially dissipative, approximate methods of modal analysis are usually used. The viscous damping coefficients are assumed to be proportional to mass or stiffness terms in such a way that the normal modes of the undamped system are preserved in the presence of damping. Then the solution can be expressed in terms o f the normal co-ordinates of the undamped system, which can be found by well-tried methods and whose interpretation is well understood (for example, the book by Bishop et al. [3]). But for many practical problems, the assumption that the undamped normal mode shapes are preserved in the presence of damping terms is not valid. Some examples are problems of vehicle dynamics, aerodynamic flutter and rotor whirling, and there are many others. This work has arisen from a number of practical investigations of such systems, and is concerned with the modal analysis of any general dynamical system that can be represented by a set of linear second-order differential equations with constant coefficients. 69 0022-460x/87/010069+28 $03.00/0 9 1987 Academic Press Inc. (London) Limited
70
D.E.
NEWLAND
In previous papers Fawzy and Bishop [1] and Wahed and Bishop [4] have examined the behaviour o f such a system, described by the matrix equation
m~+~+~=f.
(1)
When there are n degrees-of-freedom, m, c and k are real matrices of order n x n and x and f are column vectors of order n. To include any general system, whether conservative or non-conservative, there are no restrictions on the m, c and k matrices, each o f which may be any square array of real numbers. The input or excitation of the system is the vector f and its output or response is given by the vector x. Two general expressions were developed in references [l, 4], one for the steady-state solution x when there is harmonic excitation at a single frequency to, and the other for the general response x when the elements of the vector f are each different arbitrary functions o f time. The method o f obtaining these expressions involved using eigencolumn and eigenrow matrices which are defined as follows. IfA~ (i= 1 to 2n) are the eigenvalues of the system, which are assumed to be distinct, then an eigencolumn vector tr ~ satisfies the equation
[A~m+A,c+k]o'(~
(i=1,2 .... ,2n)
(2)
and an eigenrow vector ~.(o satisfies the equation
~rt~
(i = 1, 2 , . . . , 2 n ) .
(3)
The eigencolumn matrix Z is an n x 2n matrix each of whose columns is an eigencolumn vector; the eigenrow matrix H is a 2n x n matrix each of whose rows is an eigenrow vector. In general, the elements of Z and / / a r e complex numbers. The purpose o f the present paper is (i) to show that the two general results referred to above may be obtained by an alternative and, it is believed, a simpler method by using a first-order formulation of the equations o f motion, (ii) to show how these results may be extended to the case of multiple eigenvalues, and (iii) to discuss the physical nature of the "normal co-ordinates'" which characterize the natural modes of non-conservative oscillatory systems. In addition the analysis which will be described is in a form which lends itself to numerical computation by using computer library programs for eigenvalue determination. These standard library programs have been developed for a range o f problems, including the analysis of linear control systems, and generally incorporate eigenvector sub-programs which generate normal co-ordinates in the form to be defined below. The use of a first-order formulation for vibration problems is not new (see, for example, the paper by Woodcock, [5]) and there are different methods of organizing the equations, but in order to apply the results of algebraic eigenvalue theory, which includes the case of multiple eigenvalues, it is necessary to express the n second-order equations (1) as 2n coupled first-order equations in the form :~ = a z + F,
(4)
where the system matrix A is a real matrix of order 2n • and z and F are column vectors of order 2n. When the eigenvalues of A are distinct, the solution is developed in terms o f t h e system's eigencolumn matrix U, which is of order 2n • 2n and which satisfies the equation
AU
=
U[diag (A,)].
(5)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
It will Fawzy matrix The
71
be shown that, subject to scaling the eigenvector columns in the same way, the and Bishop n x 2n matrix X is identical with the top half of U and their 211 x n product Ilm is identical with the right-hand half o f U -~. co-ordinate transformation
(6)
q = U-lz
allows the set of 2n coupled first-order equations (4) to be rewritten as an equivalent set of 2n uncoupled first-order equations and defines the normal co-ordinate vector q, of order 2n. In the general case, the elements of q are complex. The solution vectors become real (for real excitation) only after the inverse transformation has been effected to revert to the (coupled) physical co-ordinates of the system. When there are multiple eigenvalues, it may not be possible to find a co-ordinate transformation which will uncouple all the 2n first-order equations. It can be shown from eigenvalue theory that at least one equation can be uncoupled (that is to say written in terms o f a single co-ordinate) but that some or all of the remaining equations may remain coupled. However each such coupled equation involves two co-ordinates only. Therefore the general solution for arbitrary excitation may be obtained by solving the uncoupled equation first, and then substituting this result into the next equation and solving that, continuing until solutions for all the co-ordinates have been obtained. 2. EIGENSOLUTIONS
It is well-known (see, for example, the book by Wilkinson, [6]) that a system o f n simultaneous differential equations of order r with constant coefficients may be reduced to a system of nr first-order equations by introducing r ( n - 1) new variables. If, in the present case, the mass matrix m is non-singular, this reduction may be made as follows. One defines a new vector ), of order I1 so that j,=.~
(7)
and another new vector z of order 2n such that
z:[:] In the terminology of control theory, the latter is called the "state vector" and in a typical mechanical system its first n elements give the displacements and its second n elements give the velocities at the n co-ordinates of the system. By premultiplying both sides of (1) by m -t one obtains s + m-l cYc+ m-~ "kx=m-~f
(9)
which, on substituting from equation (7) gives j',=-m-lkx-m-lcy+ m-tf
(10)
Then using the definition of z in equation (8), gives the state equation
i=.4z+F,
(11)
where A=
_~ll _ m - ~
and
F=
.
(12, 13)
72
D.E.
NEWLAND
When there is no excitation, F = 0, and then the free motion of the system has the form z ( t ) = Z e A',
(14)
where, by substitution into equation (11), ,~z = A z
(15)
which is a statement of the fundamental algebraic eigenproblem [6]. The general theory (see, for example, the book by Gantmacher, [7]) shows that equation (15) has a non-trivial solution for z only if det [A - AI] = 0
(16)
and that the determinant in equation (16) gives a polynomial in A for the eigenvalues Ai of the problem. When the order of the matrix A is 2n x2n, the order of the characteristic polynomial obtained from equation (16) will be 2n and so there will be 2n eigenvalues Ai, i = 1,2 . . . . ,2n. Because A is a real matrix, the coefficients of the characteristic polynomial will be real, so that if A = a +jfl is an eigenvalue, then so must A = a -j/3 be an eigenvalue. Therefore if they are complex, the eigenvalues occur in complex conjugate pairs. Usually in practical problems the different Ai will be distinct, but this may not be the case and there may be one or more multiple eigenvalues. Each multiple eigenvalue may be repeated any number of times (up to 2n). An example is the torsional vibration of rotating machinery, when the eigenvalue A = 0 may be repeated several times (see example 5 following). When the eigenvalues are all distinct, equation (15) will yield a different solution vector zl for each eigenvalue A~. These are the eigenvectors of the problem and the 2n x 2n matrix each of whose columns is an eigenvector is called the eigencolumn matrix, here denoted by U. Then, from equation (15), after reversing the order of the equation, A U = U[diag (A,)]
(17)
U - t A U = [diag (A,)].
(18)
or, after premultiplying U -I,
Each column of U is arbitrary to the extent of a (different) scaling factor because, although the ratios o f the complex amplitudes of each co-ordinate are determined by the equations of motion, the absolute values of these amplitudes are determined only by the initial conditions. The general theory of eigenvalue analysis [7] shows that, if the eigenvalues of A are distinct, then there is always a non-singular matrix U for which equation (18) is true. When there are multiple eigenvalues, this may still be so, but it is not necessarily the case. If independent eigenvectors can be found for the multiple eigenvalues, then equation (18) will be true. However usually it is not possible to reduce A to diagonal form by a similarity transformation when A has multiple eigenvalues. Then the best that can be done is to reduce A to its Jordan canonical form, which is almost diagonal. The case of multiple eigenvalues and the Jordan form will be discussed later in this paper but, for the present, it is assumed that the eigenvalues are distinct so that equation (18) is true. 3. NORMAL CO-ORDINATES For distinct eigenvalues, it is always possible to define a new set of co-ordinates which allow the 2n coupled equations ( l l ) to be written as an equivalent set of 2n uncoupled
NON-CONSERVATIVE SYSTEM M O D A L ANALYS1S
73
equations. Let a new state vector be defined by the transformation
q = U-Iz
(19)
and substitute q for z in equation (11) to obtain
U(I= AUq+ F.
(20)
After premultiplying both sides of (20) by U -~, one finds
(! = U-~AUq + U-'F
(21)
which with equation (18) becomes ~j= [diag (A,)]q+ U-~E
(22)
The 2n separate equations represented by equation (22) are then uncoupled from each other and one has 2n separate first-order linear differential equations for each of the co-ordinates which are the elements of the 2n vector q. Since the matrix U -~ can be obtained from equation (17) and F is defined by equation (13), one can now calculate the time history of the response q for any specified excitation f The response in the physical co-ordinates z can then be regained from q by the inverse of equation (19). In the traditional terminology of vibration theory, the elements of q are called the normal co-ordinates of the system (11). 4. STEADY STATE ttARMONIC VIBRATION Consider the case when the nth order force vector f in equation (1) describes forced excitation at the single frequency to so that
f = f( t)= fo ~J~
(23)
where fo is the vector of force amplitudes, which are in general complex. Then, from equation (13), F = F ( t ) = ~m ---i 0 ~.o] e i~
(24,
and the state equation (20) becomes = [diag (Ai)]q +
Lm-'foJ
(25)
Consider the solution reached asymptotically as t ~ ~ on the assumption that the system is passive and that all the starting transients die away with increasing time. For this to be the case, the real parts of all the eigenvalues ,~ must be negative. Under that condition, the steady state solution of (25) has the form q = q(') = qo e i'~
(26)
where qo is the vector, of order 2n, of the complex amplitudes of the response of the normal co-ordinates. Substituting equation (26) into equation (25) gives
Jwq~
(A')]q~ U - ' f m 'O-_fot"
(27)
74
D.E.
NEWLAND
which can be solved for qo to give qo =
J
[ r e - ' f oJ"
If the steady state response in the physical co-ordinates z is represented by z = z ( t ) = Zo~,o,
(29)
then, combining equations (19), (28) and (29) yields =
V[diag
J
u-'l--Lm-'foJ ~
(30)
which is a general expression for the amplitudes of the steady state response of system (1) to harmonic excitation at frequency to. The first n elements o f ~ are the (complex) amplitudes of x; the second n elements of Zo are the (complex) amplitudes of .L 5. FORCED TRANSIENT VIBRATION For the general case of forced vibration, the excitation consists of a set of arbitrary functions of time so that f=f(t) (31) and, from equation (13), 0
F = F(I) = [-~_q ~(-~)--].
(32)
Substituting for F from equation (32) into equation (22), one now has to find the solution of the 2n uncoupled equations represented by t~=[diag(A,)]q+U-'[- m - i0f ( t )~ "
(33)
Consider the ith row of equation (33), q, = A,q, + ~,,(t),
(34)
where ~ ( t ) represents the ith row of
u-,r I Lm-'f(b)J" A general solution of this first-order linear difterentiai equation may be found by using the integrating factor e -A,', because multiplying both sides of equation (34) by this factor and rearranging the terms gives (d/dt)(e-A,'q,) = e-~.',;b,(t),
.(35)
e-~.'qi = J e-~,'qS,(t) d t + Ni,
(36)
which has the general solution
where N~ is an arbitrary constant of integration. The solution of equation (36) is therefore q, =
eX,'fd e-X.'th,(t)dt q-e~.'Nj
(37)
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
75
and there is a similar result for each o f the 2n rows of equation (33). If one defines a 2n x 1 vector of the constants of integration, PC, then the 2n separate equations (37) may be written as the single matrix equation q = [diag
(e')][f
[diag
Also, if one defines a new matrix
(e-a")]U-'~mOf(-t)~ dt+N].
E(t) of order 2n •
(38)
such that
E ( t ) = [diag (e~,')],
(39)
then, by using equations (32) and (39), equation (38) may be written more briefly as
q=E(t)[f E-'(t)U-'F(t)dt+N] and, from equation (19), the state vector z = z(t) is given by z =UE(t)[I E-'(t)U-tF(t)dt+N].
(40)
(41)
This final equation gives a general solution o f the n degree-of-freedom system (1) when subjected to arbitrary forcing functions. Here z is the state vector of order 2n x 1 defined by equation (8), U is the 2n x 2 n eigencolumn matrix defined by equation (17), is the 2n x 2 n diagonal matrix of integrating factors defined by equation (39), is the excitation vector o f order 2n x 1 defined by equation (32) and N is the 2n x I vector of arbitrary constants of integration which have to be found from the initial conditions of the problem.
E(t) F(t)
6. COMPARISON WITH THE RESULTS OF FAWZY AND BISHOP [1] Equations (30) and (41) of this paper are in agreement with the corresponding equations (44), (70) and (71) of Fawzy and Bishop [1], who derived them by a different method. In order to demonstrate that the results derived here and those in Fawzy and Bishop are the same, it has to be shown how the n x 2n eigencolumn matrix ~; used by Fawzy and Bishop relates to the 2n x 2n eigencolumn matrix U used in this paper and how the 2n x n eigenrow matrix H used by Fawzy and Bishop relates to the inverse eigencolumn matrix U -I.
Consider the ith column of U. This satisfies equation (17) and, after substituting for A from equation (12) into equation (17), one has -m-~k
I
If now the column vector u (i) is partitioned into its top and bottom halves, r
.(i)
-]
uO) = I _ ~ _ o~__[
l . (,)
/'
(43)
L ==b o t t o m . J
where ,'top " (o and "hot,ore ' (i) are column vectors of order n, then equation (42) gives ~(0 = . H t(1) ~ bottom opltl
(44)
.") ~ i. - - I'~11- ~n, .(to O p - - i_l l - ~ i . ~.(o b o t l o m = 14bottomtl
(45)
and
76
D.E.
NEWLAND
By eliminating Ubouo m(1) between equations (44) and (45), rearranging terms, and premultiplying by m -~, one finds [A~m+A,e+t:l.(~ =0, 9 ~ JMIO
i=1,2,.
p
9 9
,2n.
(46)
By comparing equation (46) with equation (2), it can be seen that, subject to the same scaling, uo) _ o.(O, top-
i = 1, 2,.
9 9 ~I
2n,
(47)
and so, if the 2n x 2 n eigencolumn matrix U is partitioned into a top and bottom half so that U = t.Ubot,omJ
(48)
it has been demonstrated that, subject to scaling, U,o,, =
(49)
v.
Also, because of equation (44), one has the result that Ubo,,om= U,op[diag (A~)].
(50)
Now consider the ith row of U -~. Let V=
(51)
U-'
and denote the ith row of V by v(~ Postmultiplying both sides of equation (18) by U -~ and using equation (51) gives VA = [diag (A,)] V.
(52)
After substituting for A from equation (12), the equation for the ith row of V is, from equation (52),
o.Eo
-m-'k
I
,:]
-m-_ T
= Aiv <~
(53)
If the row vector v t~ is partitioned into left-hand and right-hand halves, so that v(') = [ vl~t where
IIl v~ig)ht],
(54)
' o) a n d '-'right 9 (i) are row vectors of order n, then equation (53) gives "'left _,(i) 9~n~h,.~ .- .] .b
_- A ,,a, I)(i)
(55)
and (i) _.(i)
_-t~_..(O
v,~f, ",ight. . . . .
",'-',iSh,-
(56)
By eliminating vl~, between equations (55) and (56) and rearranging terms, one obtains the result that v(o right m -Ir.~ L A i m + A , c + k ] =0,
i = 1 , 2 . . . . . 2n.
(57)
Comparing equation (57) with equation (3) shows that, subject to scaling, t,(i) ~-I rightS,r,
~--
n ('),
i = 1, 2 . . . . ,2n,
(58)
and that, if the matrix V is partitioned into left-hand and right-hand halves so that V = [ Vj,f, i
V,~h,],
(59)
NON-CONSERVATIVE SYSTEM MODAL ANALYSIS
77
then it can be concluded that, subject to corresponding rows having the same scaling?, Eigh, = lira.
(60)
In order to compare the results in this paper with those of Fawzy and Bishop, where the n second order equations which define the system are given as (61)
A ~ + B:I+ Cq = Q ( t ) ,
one has to write A=m,
B=c,
C=k,
Q=f
and
q=x.
(62)
In this paper q has been used to denote a vector of order 2n for the normal co-ordinates, whereas Fawzy and Bishop used q as a vector of order n for the physical (displacement) co-ordinates which have been called x here. The two principal results (30) and (41) can now be shown to lead to the same results as equations (44) and (70) and (71) of Fawzy and Bishop. Two new symbols are needed. These are the 2n x 2n matrices defined by A=[diag(Ai)]
and
ll=
diag
.
(63, 64)
Now, by using equations (7), (8), (48), (59) and (64), equation (30) may be written as Ix~ = i[--UL~ Ubo,omLJ -1D[ V~'r
V'h'] f m O'j'0t"
(65)
On account of the zeros in the top half of the column vector on the r.h.s, of equation (65), one obtains [~o]
=[__U,_oP__l fi~v, ght[m-,,o] L U~o,,om.I
(66,
and so the nth order vector giving the amplitudes of the physical co-ordinates is given by Xo= Utopfl V,~sh,[m-~fo].
(67)
In the notation of Fawzy and Bishop, =fo
(68)
and, together with the last of equations (62), and equations (49) and (60), one obtains the result that, for steady state harmonic excitation at frequency to,
q( t)= ~ I I H ~ ~,o, (69) which confirms equation (44) of the reference. Equation (41) of this paper becomes after substituting from equations (7), (8), (32), (51) and (59),
[x-] =[--U•176 Ubo,,omJ
LJ
Vngh']~n--~(t)] dt+N ]
(70,
and, with equations (49), (50), (59), (60), (63) and the last two of equations (62), this becomes
[_:_]
+
which agrees with equations (70) and (71) of Fawzy and Bishop. i See the Appendix.
,7,,
78
D. E. N E W L A N D
7. NUMERICAL EXAMPLES Two numerical examples of the use of normal co-ordinates to solve problems of forced vibration are as follows. The examples are the same as those given in the paper by Fawzy and Bishop [1] because they allow successive stages of the different methods of calculation to be compared. 7.1. E X A M P L E 1
The first example concerns the steady state harmonic response of the single degree-offreedom system ~ + 2.'~+ 2x = 5 ei'. (72) From equation (12), the A matrix is
From equation (16), the eigenvalues At.2 are the solutions of I for which det [-__~ - 2 1- A] = 0
(74)
A,.2 = -1 :i:j.
(75)
so that The column eigenvectors are the solutions of equation (15) for ;t = - 1 +j and 3. = - I - j , and these are u(') =
[ , ] a.d.<2,=[ 1 ] -1 +j
- 1 -j
(76)
so that U=
[
1 -l+j
1 ] -1-j
(77)
and its inverse is
v_, [0.5-j0.5 -jo.5] k0.5+j0.5
(78)
j0.5J"
The normal co-ordinates are defined by equation (19), so that in this example
[<,,]:ro.~-Jo.~ -Jo.~ar.
(79)
The steady state harmonic response is then given by equation (30), which becomes
[.o] [,
:fo = - l + j - 1 - j
,][ o:
jto+_l-j
o
1 /L0"S+j0"5 jto+l+j
j0"SJl.SJ"
<80,
Putting to = 1, from equation (72), and working out the matrix products gives
~o for the vector of amplitudes.
L 2+j J
(81)
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
79
7.2. E X A M P L E 2 The second example is a non-conservative two degree-of-freedom system for which, with the notation of equation (1),
:],
(82)
and
(83) The initial conditions are that, at t = 0,
(84)
Z ~
From equation (12), the system matrix is
o0 ,_,~] 0 0 1 6.
A=
4
0
0 6
01
(85)
-3
The eigenvalues may be found from equation (16) and give -1 0 0 0
[diag(A~)]=
0 0 i] -2 0 . 0 -3 0 0
(86)
The eigencolumn vectors are obtained by solving equation (15) for each eigenvalue and this leads to the eigencolumn matrix U=
-1
-5/6
-13/15
1
-2
-3
1
5/3
13/5
-
6
(87)
-2/31
and its inverse
2.. 7,.0.35] V = U -I =
7 -45/14 11/35
6 -15/7 12/35
-2 15/14 8/35
15/7[" 9/35J
(88)
By comparing the top two rows of (87) with the Fawzy and Bishop eigencolumn matrix v in their equation (86), which is [21-210 2"=1/210 _21 175
225 _195
_
61] ,
(89)
80
D.E. NEWLAND
one sees that, except for ditIerent scaling factors for the columns, equation (49) is satisfied. The Fawzy and Bishop eigenrow matrix is given in their equation (87), and using their values gives
[i-!JEi1[i !1
Hm=
=
(90)
which is the same as the right-hand half of U -1 in equation (88) (except for the different scaling of the individual rows), thus agreeing with equation (60). From equation (39), E(t)=
[e~176176 l 0
e -~'
0
0
0
0
e -3t
0
0
0
0
e 4'
(91)
and, from equation (32), for t > O,
0
~
o
':'
(92)
In order to evaluate the general result (41), one needs
~<')=
I'Y,Yl
<,>0)
(93,
L~/~a J and, using E - ' obtained from equation (91), [ 7/10C 1 / -2 e 2' /
E-lVF(t)=18tll5/lae3t[
(t>0).
(94)
L8/35 e-4'J
Evaluating the integrals then gives
E-tVF(t)dt=
63/5 e'Ct-1) ] -9e2'(2t-1) / 15/7 e3'(3t- 1) /
(t>0)
(95)
-9/35 e-4'(4t + 1)J and, after substituting this result into equation (41), and making use of equations (7) and
(8),
Z= 2, 2z
='E /
u [ 15/7e3,(3t_l)+N3 L-9/35 e -4 (4t + 1)'+ N4
(t>0).
(96)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
81
Substituting for E from equation (91) and working out the matrix product gives x2
/-9(2t-
1)+ N2 e-:' /
= Ui15]7(3t_l)+N3e_3, [ Yr
(t > 0).
(97)
[-9/35(4t+l)+N4e"J
Lastly, substituting for U from equation (85) and multiplying out the remaining matrix product gives, for t > 0,
x2 21 =
22
7-3t-N~e-'-5/6NIe-2'-13/15N3e-3'-l/6N4e - Ni e - ' - 2 N 2 e -2' - 3 N 3
e-3'q-4N4
"'
e 4'
-3+Nle-'+5/3N2e-Z'+13/5Nze-3'-2/3N4e
"
(98)
4'
For the initial conditions specified in equations (84), which are the same initial conditions as used by Fawzy and Bishop, all the arbitrary constants o f integration are zero, N, = N 2 = N3 = N4 = 0
(99)
and hence the complete solution is
ixl[ x2 =
7-3t
(t>0).
(100)
,~2
Note that the initial conditions are chosen to ensure that all the exponential terms in equation (98) are zero. These include the increasing exponential term e4'. Any small change in the specified initial conditions would mean that this term would eventually dominate the solution. 8. SINGULAR AND DEFECTIVE MATRICES The general solutions (30) for steady state harmonic vibration and (41) for forced transient vibration depend on the mass matrix m being non-singular and on the eigenvalues Ai being distinct. The requirement that m is non-singular is necessary if the n second-order equations (1) are to be written as an equivalent set of 2n first-order equations (1 I) by using equation (12). Note that the same condition that m should be non-singular applies also to the analysis of Fawzy and Bishop [1] because their eigencolumn and eigenrow matrices have to be normalized to satisfy equation (40) of the reference, which is
X A H = m -I,
(101)
so that m -~ must exist. The significance of this condition may be seen by returning to equation (1) and, for the case of free motion when f = 0, substituting the solution vector
x(t) = x e ~'
(102)
which is the upper half of z(t) in equation (14). This gives [Aim + Ac+ k]x = 0
(103)
82
D.E.
NEWLAND
which has a non-trivial solution for x if and only if det [ A 2 m + A c + k ] =0.
(104)
Expanding the determinant leads to a polynomial of order 2n in A of the form a2nA 2n Jr" O ~ 2 n _ l A 2 n - I "1" 9 9 9 q" a l A "1" a 0 : 0 ,
(105)
where, from equation (104), a2n = det [m].
(106)
When a2, -~0, then at least one of the roots A of equation (105) approaches infinity. This means that, in the limiting case when a2, = 0, one at least of the eigenvalues does not exist and so equations (30) and (41) break down. Since, from equation (106), a 2 , = 0 when the mass matrix m is singular, the solution given fails when m is singular. In all the engineering problems known to the author for which m has been found to be singular, the reason has been zero mass associated with one of the physical co-ordinates x~. In that case there is no difficulty in dealing with the defective second-order equation as a separate first-order equation. The remaining (n - 1) second-order equations reduce to 2 ( n - 1 ) first-order equations without difficulty and the extra first-order equation is then added to give an A matrix in equation (4) of order (2n - 1) x (2n - 1). This calculation is illustrated in.the following example 3. When it has been found, the A matrix may turn out to have multiple eigenvalues. In that case, it may not be possible to find an eigencolumn matrix U which satisfies equation (18). When it is possible to satisfy equation (18), the columns of U for a repeated eigenvalue have an indeterminancy. Suppose, for example, that z~ and z2 are two columns of U for the same eigenvalue A~. Then, from equation (15), one has AIzI=Azl
and
A~z2--Az2.
(107, 108)
If z3 is ;., new vector defined by z3 = azl + flz2,
(109)
where a and fl are arbitrary multipliers, then, from equations (107) and (108) one must have Zzz~ = Az3.
(110)
Apart from the individual scaling of columns, the eigenvector matrix U is unique when the eigenvalues are distinct. When there are multiple eigenvalues, and equation (18) is still satisfied, then U is no longer unique. But provided that any U can be found which satisfies equation (18), then the results expressed by equations (30) and (41) apply without alteration. One can next consider the case when there are multiple eigenvalues and a matrix U to satisfy equation (18) cannot be found. Then the coupled fi rst-order equations in equation ( l l ) cannot be written in an uncoupled form (22), and so the general results that have been derived from equation (22) are no longer valid. Instead offinding a similarity transformation by a matrix U which reduces A to diagonal form, as in equation (18), it can be shown (see, for example, the book by Gantmacher, [7]) that a matrix W may be found which reduces A to its so-called Jordan form. This may be described as a nearly diagonal matrix and it is the most compact form to which A may be reduced by a similarity transformation. For example, if A has order 2n x 2n, and the first eigenvalue At is repeated once, then it may still be possible to reduce A to diagonal form, but, if not, then it will be possible always to find a non-singular matrix
NON-CONSERVATIVE
SYSTEM
MODAL
ANALYSIS
83
W for which Ai 0
1 A~
I Ix \
x,.\A 3 \,.
W-lAW=J=
zeros
N
zeros (111)
~4 NN
N
\
x ~'2n
In that case there will be 2n - 1 independent eigenvectors. These are columns 1 and 3 to 2n of W. Column 2 satisfies the equation A w t2) = w~ + Alw (2~
(I 12)
and is not an eigenvector of A. Instead, it is called a principal vector of A. A n eigenvector is a special case of a principal vector and it m a y be shown (for example, [7]) that the principal vectors of a matrix are independent, although they are not unique. Thus all the columns of W satisfying equation (111) are independent vectors, but wtz) is not unique because if wt2) satisfies (112) then so does wt2)+aw") where a is an arbitrary multiplier. If equation (111) is true, instead of equation (19), one defines normal co.ordinates by the equation q= W-Iz (113)
and then equation (22) becomes
:l=dq+ W-~F,
(114)
where d is the Jordan matrix defined by equation (111). The steady state response to harmonic excitation is then, corresponding to equation (30), %= W [ j t o / - J ] - ~ W - ' [~n-_0Tfo].
(115)
For the forced transient response, when F is given by equation (32), the first row of equation (114) is t)~ = A~q, + q2+ ff~(t),
(116)
where ~b~(t) is the first row of W-~F, and the second row of equation (114) is t12= A~q2+ ~b2(t),
(117)
where qSz(t) is the second row of W-~E From equation (37), one knows that the solution of equation (117) is q2= e~'t I e-X"q~2(t) d t + e X " N2.
(118)
Hence equation (38), with W -~ replacing U -~ and with Az = A~, is true for rows 2 to 2n but is not true for row 1. Row 1 must be replaced by the q~(t) obtained by solving equation (116) with q2 from equation (118) being used. The state vector of the physical co-ordinates must then be recovered from q by using equation (113), and this expression replaces equation (41). When an eigenvalue is repeated more than once, or there is more than one multiple eigenvalue, the Jordan matrix is more complicated than shown in equation (111). The
84
D.E. NEWLAND
eigenvalues always lie along its diagonal and there may or may not be elements equal to unity lying on the first super-diagonal line. For example, for the case when 2n = 6 and the A matrix has eigenvalues An, At, An, A4, As, A6, then if there are six independent eigenvectors, matrix A is similar to a Jordan matrix of the form
J=
At 0
0 At
0 0
r t 0
0
0
At o
I
o~
n I
t-
zeros (119)
".A4 "-.
zeros
""3,
``.~ ,
if there are five independent eigenvectors, then A is similar to a Jordan matrix o f t h e form
J=
At 0
0 hj
0 I
0
0
At 0..
rI
o
zeros
I
~;"
|
II
ZerOS
(120)
~``
"" ~ .., ~ 5
I
~ "- ~.
"" "-
~6 I
and if there are four independent eigenvectors, then A is similar to a Jordan matrix of the form
a =
An 0
I At
0 r I 1 0
0
0
At
I
I
zeros
J
(121)
"``
"'..An''
zeros
"~
s --. ~"
I
"``
I
A6
If As=A4, then d may be as in equations (119), (120) or (121) or it may have any of these forms with the submatrix
replaced by
in which case there will be five, four or three independent eigenvectors respectively. For problems of forced transient vibration, when the forcing function F in equation (114) is an arbitrarily chosen function of time, the solution procedure is to solve each of the equations in equation (114), starling with the equation given by its last row. This will ahvays be an uncoupled equation because there is no super-diagonal element associated with the last row. Each other row of equation (114) is an equation with no more than two co-ordinates. For row r, a solution for q,(t) is possible as soon as q,+~(t) has been found; the whole solution vector q(t) can be determined by progressive substitution from one row into the next higher row until the equations for all the rows have been solved.
NON-CONSERVATIVE
SYSTEM
MOI)AL
85
ANALYSIS
The Jordan matrix with elements of unity in the super-diagonal is called the upper~ Jordan form and a similar matrix may be found with elements of unity in its sub-diagonal. Every matrix A is similar to an upper and a lower Jordan matrix and the choice of which Jordan form to use is arbitrary. Ifthe lower-Jordan form is used, the calculations described above have to be modified accordingly. In a particular problem with multiple eigenvalues, the form of the Jordan matrix has to be determined by finding the number of independent eigenvectors corresponding to each multiple eigenvalue. There do not appear to be any general rules for determining the number of independent eigenvectors a priori. Each case has to be considered individually. These calculations are illustrated by three further numerical examples in the next section. 9. FURTHER NUMERICAL EXAMPLES 9.1. EXAMPLE 3 This example is the response to steady state harmonic excitation of a non-conservative system with a singular mass matrix and repeated eigenvalues. With the notation of equation
(1) m=[l 0
~],
c=[20
11],
k=[l 0
~]
and
f = [ ~ ] e j'.
(124,125)
Because m is singular, equation (12) cannot be used to obtain a 4 x 4 A matrix. There is one second-order equation 5il + 22~ + 22 + xt + 2x2 = 0
(126)
.~2+ x2 = 2 e i'.
(127)
and one first-order equation
The latter may be used to eliminate -'/2 between equations (126) and (127) to give .#i+2Yq+Xl+X2
=
--2 e 3t.
(128)
If, corresponding to equations (7) and (8), one defines
ix]
z = x2 21 then the 3 x 3 A matrix is A=
0 0 -1
0 -1 -1
(129)
l]
0 -2
(130)
and this has the repeated eigenvalues AI=-I,
A2=-l,
(131)
There is one eigenvector only, which is (132)
86
D. E. N E W L A N D
and so one has to find a 3 x 3 W matrix for which
W-IAW=J
=
o1
Ai
1
0
.
(133)
AI
After premultiplying both sides of equation (133) by W, the second columns give
A w (2) =
The solution of
W(2)
- -
w(2)
(134)
.
from equation (134) is
W(2) --
(135) 1-
where 6 is arbitrary. The third columns of equation (133) give Aw(3) = w( 2 ) - w(3)
(136)
which, with equation (135), gives
w~
-1
(137)
,
where e is also arbitrary. Hence the W matrix is
W=
1
0 -
-1
.
(138)
6-e
1-6
If the second and third columns of (138) are normalized so that their first elements are unity, then 1
1
0
0
-1
0
W=
1
-
(139)
and this W has the inverse W-' =
1
.
(140)
-1 From equations (127) and (128), one sees that F is a column vector of order 3 for which
F =
ej .
(141)
NON-CONSERVATIVE
SYSTEM MODAL
o_,][
Hence, corresponding to equation (115), one has Zo=
[!1 1j[j+l 1 01[i -
0 -1
0
0
0
0
jto+l 0
87
ANALYSIS
-1
joJ+l
1
1
-1
0
(142)
.
-
Substituting to = 1 and evaluating the inverse matrix gives [ i Zo= -
1 1][0"5-j0"5-j0"5-0"25-j0"25 0 -1 0 0.5-j0.5 -jO.5 0 0 0 0 0.5-j0.5
-1
0
1
1
2
-1
0
-2 (143)
and multiplying out the matrices, one obtains finally Zo=
0"5+j1"5 1-j -1"5+j0"5
(144)
as the vector of amplitudes of the steady state response. Since equation (133) is true whatever the values of the arbitrary constants ,5 and e in equation (138), one will obtain the same results if different values of ,5 and e are chosen when determining W. 9.2. E X A M P L E 4 The second example here is a calculation for the forced transient vibration of a system with a repeated eigenvalue. This is the system represented by m=[lo
i]'
c=[21
~],
k=[20
21]
(145)
and
f=[~]
fort~>0,
f=[00]
fort<0.
(146, 147)
Then, from equation (12),
A= _
0 0 -1 -1 -1
-1
(148) -
and the system's eigenvalues are Al=--l,
A2=-l,
A3=-I-j,
A4=-I+j.
(149)
There is only one eigenvector for A~= -1 which is
W (I)
(150)
88
D.E.
NEWLAND
and then equation (112) gives
A w (2) _-
[:i]
(151)
_ w (2),
where A is given by equation (148). Hence w(2) is
--~J
w (2) =
(152)
I-8 ' - 1 -(5
where ,5 is arbitrary. For the case when 8 = 0, one has
(153)
W (2) =
After finding the eigenvectors for A3.4= - 1 + j from equation (111), one has
1 IV=
0
1
-1
0
-0.6+j0.2
-0.6-j0.2
-1
1
-l+j
-1-j
[
0.4-j0.8
0"4+j0"8
J
1
-1
1
] (154)
which has the inverse
W_ ~=
0.5-jl.5 L0.5+jl.5
Also, from equation (13),
-3 0.5-jl.5 0.5+jl.5
"Ji
-1 -0.5-j -0.5+j
. -0.5-j -0.5+jJ
for t~>0,
(155)
(156)
and so, from equation (114),
q=
0 0
-1 0
0
0
0 -l+j 0
W_ I 0
q+
0 , -
(157)
-1 -jJ
where W -~ is given by equation (155). The resulting equations for the elements of q are, for t~>0, tll= --ql+q2,
t12 = - - q 2 - - 5 ,
t~3 = ( - - I + j ) q 3 ,
t~4 = ( - - 1 - - j ) q 4 .
(158)
N O N - C O N S E R V A T I V E SYSTEM M O D A L A N A L Y S I S
89
Solving the second, third and fourth of equations (158) gives q2 = - 5 + N2 e-',
q3 = N3 e (-~+i',
q4 = N4 e (-~-m
(159)
and, combining the first of equations (159) with the first of equations (158) gives qt = - 5 + N~ e - l + N2t e -t.
(160)
z= Wq,
(161)
Lastly, from equation (113),
from which the vector of the physical co-ordinates may be regained. If one takes a particular set of initial conditions, the arbitrary constants of integration may be determined and, for initial conditions that give, at t = 0,
z:E_],i
(162)
the final solution is, for t ~ 0,
Ix]
- 5 + e - ' ( l + t + 1 0 c o s t) ] 5 - e - ' ( l + t + 6 c o s t + 2 s i n t) I - e - r ( t + I0 cos t + 1 0 s i n t) / " e-t(t+4cost+8sint) J
X2
Z=
(163)
9.3. EXAMPLE5 The final example is a system with six eigenvalues, three of which are the same, and five (not four) independent eigenvectors. This is the torsional system shown in Figure 1. Co-axial wheels of moments of inertia I~, 12 and 13 are arranged so that wheels 1 and 2 are joined by an elastic shaft of torsional stiffness k and wheels 2 and 3 are joined by a fluid coupling with a torsional viscous damping coefficient c. This system represents approximately the behaviour of a primitive engine with a viscously-coupled vibration damper. The following parameter values are
1
c
5
I1
Figure 1. The torsional system.
90
D.E.
NEWLAND
to be used: /2 = 1 "5 kg m s,
It = 3 kg m 2,
13 = 4-5 kg m s,
k =200 Nm/rad,
c = 135/8 Nms/rad.
(164)
If xt, x2, x3 give the angular movements o f the three wheels from a static equilibrium position, then the equations of motion for the system when subjected to harmonic excitation at wheel 1 are given by equation (1) where
0 0] [0oi]
m=
k=
I2
0
0
13
-k 0
,
k 0
c=
0
c
0 ,
-
,
-c
f=
e j~
(165)
On substituting the numerical values and using equation (12), one finds that -
A=
0
0
0
1
0
0
0 0 -200/3 400/3 0
0 0 200/3 -400/3 0
0 0 0 0 0
0 0 0 0 0
1 0 0 -45/4 15/4
0 1 0 45/4 -15/4
(166)
and, from equation (16), the eigenvalues are At=A2=A3=0,
A4=-10,
A5.6= ( 5 / 2 ) ( - I + j x / ~ )
(167)
the units being 1/seconds. When the system is stationary, with no applied torque, angles xt = x2 may have any value and are independent of x3. Hence an eigenvector is (1, 1, a, 0, 0, 0) t, where a is an arbitrary real constant. Therefore there are infinitely many eigenvectors corresponding to the eigenvalue A = 0. But only two of these are independent since the rest can all be constructed by combining, for example, (1, 1, - 1 , 0, 0, 0) t and (1, 1, 1, 0, 0, 0) t in appropriate proportions. One thus concludes that the triple eigenvalue At = A2 = A3 = 0 has two independent eigenvectors, which one can take to be those given above. Therefore altogether there are five independent eigenvectors and the Jordan matrix has the form (120). Hence in this example
j=
0 0 0 0 0 0
0 0 0 0 0 0
0 1 0 0 0 0
0 0 0 -10 0 0
0 0 0 0 - (5/2)(1 + j , / ~ ) 0
0 0 0 0 0 - (5/2)(1 - j , . / ~ )
(168)
The three other independent eigenvectors can be found by solving (15) for each of A4, A5 and A6, so that one has five of the six columns of the W matrix. The missing column,
NON-CONSERVATIVE SYSTEM M O D A L ANALYSIS
91
which is the third column, can be found from, corresponding to equation (112),
Aw~3)=wC2)+ A3wt3) = wt2)
(169)
since 3.3 = 0 in this example. If one puts wt2) = (1, 1, 1, 0, 0, 0) t
(170)
wc3) = (/3,/3, % 1, 1, 1)',
(171)
then from equation (169)
where/3 and y are arbitrary constants. One can take/3 = 3' = 1 and put wt3)= (1, 1, 1, 1, 1, 1) t.
(172)
If, instead o f equation (170), one had put w~2)= (1, I, - 1 , 0, 0, 0) t,
(173)
then a solution for w~3) satisfying equation (169) could not have been found, so that it is important that the columns of W are ordered so that w~) = (1, 1, - 1 , 0, 0, 0) t
(174)
and w~2) is g i v e n b y equation (170). Then the full W matrix is
W=
1
1
1
1
1
1
1
2.5
-1
1
1
-1-5
0
0
1
-10
0
0
1
-25
0
0
1
15
1
-1-0625 -j0.8992 -0.3125 +j0.2997 -2-5 +j11.9896 13.4375 -j10.4909 -2.8125 -j4.4961
1
-1.0625 +j0.8992 -0.3125 -j0.2997 -2.5 -j11.9896 13.4375 +j10.4909 -2.8125 +j4.4961
(175)
and its inverse has been calculated to be
W-I=
0 0"3333 0 0"1667 0.25 -j0"1216 0"25 +j0"1216
0"5 0"1667 0 -0"1667 -0-25 +j0.1216 -0"25 -j0.1216
-0"5 0"5 0 0 0 0
0.0444 -0"3333 0.3333 -0.0167 -0"0139 -j0.0180 -0.0139 +j0.0180
0.0222 -0"0667-0.1667 - 0 ' 5 0.1667 0"5 -0.0208 0.0375 -0.0007 0.0146 +j0"0158 +j0.0022 -0.0007 0"0146 -j0"0158 -j0.0022
(176)
These values are given to four significant decimal figures. A note on their method of calculation is given at the end of the paper.
92
D.E. NEWLAND
The solution required is given by equation (115) for which one needs [jwl - d ] -~ which it is not difficult to show is given by
_=_1 joJ 0 0
o
o
1
1
joJ
(joJ) 2
0
[jwl-J]-I =
1
~
j~
o
o
o
0
0
0
0
0
0 (177)
1
0
0
0
0
0
0
0
0
0
0
0
0
jto+10
0
1
0
jw + ~(1 - j x / ~ )
1
0
jco +-~(1 +jx/~)
and then, evaluating equation (115), one can find the steady state amplitudes of x~, x2, x3 (the first three elements of Zo in equation (115)). For example, the first element xl is given by 0.0444 0"3333 Xl = k joJ 4 (jo~)2
0.0167 Ota+ 10)
(0.0139+j0.0180) (jw +~(1 - j 4 ' ~ ) )
(0.0139-j0.0180)],
(178)
which reduces to
0t~ + 15(jr~ + 133"3(Jt~ 500 (-~) x, - 0,o)5+ 15(j~,), + 2000,o)3+ 1500(j,o)~ .
(179)
10. INTERPRETATION OF COMPLEX EIGENVECTORS For free vibration, one has from equation (41) with F ( t ) = 0 , z = UE(t)N
(180)
or, substituting for E ( t ) from equation (39), z = U[diag (eX")]N,
(181)
where U is the eigencolumn matrix and N is a vector of arbitrary constants defining the initial conditions. At t =0, equation (181) gives z(t = 0 ) = UN.
(182)
Solving for N from equation (182) and substituting into equation (181) then gives z(t) = U[diag (e~,')]U-'z(t =0).
(183)
Alternatively, in terms of the normal co-ordinates defined by q= U-Iz
(184)
q(t) = [diag (eA,')]q(t = 0).
(185)
one has
NON-CONSERVATIVE
SYSTEM MODAL
ANALYSIS
93
Now let
(186)
q2,~(t).l and consider the case when
q~(t=O)--ql(O)
and
q2(t=O)=q3(t=O) . . . . .
q2.(t=O)=O.
(187,188)
Then, from equation (185),
ql(t)=eXltql(O ) and
q2(t)=q3(t) . . . . .
q2,(t)=0.
(189, 190)
When ~.~ = - a - j t o ,
ql(t) = e-=' e-J'Otq,(O)
(191)
which can be represented in an Argand diagram as aline which rotates about the origin in a clockwise direction at angular speed to, while decreasing in length with a time constant 1let (see Figure 2). If OQ~ is the line representing qt(t), then Q~ rotates clockwise about O while spiralling in towards O.
,,I- IIm I/
l
O
qs(0)
Re
[~....~--(ot]
,"'--b-\i (W,<,> ~
\ length qi (O) e'at= ql (O)e-tal~)O Figure 2. Complex normal co-ordinate behaviour for a system with damping.
It has already been remarked that, because A is a real matrix, complex eigenvalues must occur in complex conjugate pairs. It follows from equation (15), that if AI and A2 is a pair of complex conjugate eigenvalues with corresponding eigenvectors zl and z2, then the corresponding elements of zi and z2 will also be complex conjugate pairs. Since z~ becomes column u~ and z2 becomes column u2 of the eigenvector matrix /3, in this case the corresponding elements of the columns ui and u2 of U are complex conjugate pairs. In a physical case, the coordinates z(t) must always be real. The mode q~(t) corresponding to a complex eigenvalue Az cannot then be excited without a comparable excitation of mode q:(t) which corresponds to the complex conjugate eigenvalue As = AI*. Suppose that only modes ql(t) and q2(/) are excited. Then, from equation (19),
z(t) = Uq(t)
(192)
q3(t)=q4(t) ..... q2n(t)=O,
(193)
z( t)= u~q~(t)+ u2qe(t).
(194)
which, for
gives
94
D . E . NEWLAND
Then since z(t) is real and u2 = u*, one must have q2(t) = q,(t)*.
(195)
Hence the rotating line OQ~ in Figure 2 representing the oscillatory decay of mode q~(t) can occur only in the presence of a second line OQ2 representing co-ordinate q2(t) rotating in the opposite direction at the same speed to and satisfying the relation
OQ2 = OO*
(196)
as shown in Figure 3.
Im
Oz wt
clz(t}
~O
Re
q,(O 001=OO z
Figure 3. Showing the complex normal co-ordinate conjugate to that of Figure 2.
If, in equation (194), one puts
I-z,(,)1 (197)
/~i ~
tq,2. J
L u2..2 J
and also puts fftl = ,~ e-J~h,
ul2 ----lll*l = Pl e+J~':,
(198)
then
zl( t) = Pl e-J~"OQ~ + p~ e§
(199)
which one can write as z,(t) = o z , + oz2
(200)
o2 ~
x~4'1""""O
\
ZI
Figure 4. Behaviour of a real variable which derives from the complex conjugate normal co-ordinates of Figures 2 and 3.
NON-CONSERVATIVE SYSTEM MODAL ANALYSIS
95
as shown in Figure 4. The time history o f the changing real variable Zl(t) is therefore given by the sum of the projections onto the real axis of the Argand diagram of the rotating lines OZi and OZ2, or since
OZ2 = OZ*
(201)
by twice the projection onto the real axis of either OZi or OZ2. 11. NOTE ON NUMERICAL CALCULATIONS The first-order formulation for modal analysis described here, with equations in the form (4), allows readily available computer programs to be used to solve the eigenvalue problem (5). The real elements of the 2n x 2n system matrix A can be formed from the elements of the n x n mass, damping and stiffness matrices m, c and k by a sub-program which makes the calculation (12). Then there are a variety o f colnputer library programs to compute the eigenvalues and eigenvectors o f A. The 2n eigenvalues and the 2n x 2n eigenvector matrix U are the output from such a program. The eigenvector matrix U does not have to be normalized in any particular way. Its inverse U -~ can be calculated directly by using a library sub-program for the inversion of a complex matrix. When there are repeated eigenvalues, the first-order a p p r o a c h still allows the modal analysis method to be used, subject to an ad hoc exploration o f the eigenvectors associated with repeated eigenvalues in order to ensure that all the independent eigenvectors have been discovered and that the Jordan matrix has been formulated properly. REFERENCES
1.t I. FAWZY and R. E. D. BISHOP 1976 Proceedings of the Royal Society London A 352, 25-40. On the dynamics of linear non-conservative systems. 2. K.-J. BATHE 1982 Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall. See chapter 9. 3. R. E. D. BISHOP, G. M. L. GLADWELL and S. MICHAELSON 1965 The Matrix Analysis of Vibration. Cambridge University Press. 4.j- I. F. A. WAHED and R. E. D. BISHOP 1976 Journal of Mechanical Engineering Science 18, 6-10. On the equations governing the free and forced vibrations of a general non-conservative system. 5. D. L. WOODCOCK 1963 Aeronautical Quarterly 14, 45-62. On the interpretation of the vector plots of forced vibrations of a linear system with viscous damping. 6. J. H. WILKINSON 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press.See chapter 1. 7. F. R. GANTMACHER 1959 The Theory of Matrices, Volume 1. New York: Chelsea APPENDIX: CONSISTENCY OF SCALING To confirm that the scaling of columns in equation (49) and of rows in equation (60) is consistent, equation (51) may be written in the form
r l[' --U-t~
L. UbottomJ
]
V,~ft I V"sht = I I
(202)
from which one sees that UbottomV~isht= I.
(203)
Utop[diag(A,)] V~ight= I
(204)
Hence, using equation (5) gives
1 Professor Bishop had the same co-author in references [1] and [4] in this list: Ibrahim Fawzy A. Wahed.
96
D.E.
NEWLAND
and, upon substituting from equations (49) and (60), this requires that, in the nomenclature of Fawzy and Bishop, ~YAHA
= I.
(205)
Equation (205) is the same as equation (40) in the paper by Fawzy and Bishop [1], thereby confirming that the scaling of ~ a n d / 7 needed to ensure that equations (49) and (60) are true is consistent with the scaling of these parameters that has been used by Fawzy and Bishop.