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Approximate decoupling of the equations of motion of damped linear systems
Journal of Sound and Vibration (1990) 136(l), 5 l-64
APPROXIMATE DECOUPLING OF THE EQUATIONS OF MOTION OF DAMPED LINEAR SYSTEMS S. M. SHAHRUZ Department of Electrical Engineering and Computer Sciences, and the Electronics Research Laboratory, University of California, Berkeley, California 94720, U.S.A. (Received 27 September 1988, and in revised form 18 April 1989)
One common procedure in solving a normalized damped linear system with non-zero off-diagonal damping elements is to replace the normalized damping matrix by a selected diagonal matrix. The extent of approximation introduced by this method of decoupling the system is evaluated, and tight upper bounds on the norm of errors are derived. Moreover, when the normalized damping matrix is diagonally dominant, it is shown that decoupling the system by neglecting the off-diagonal elements indeed minimizes the error upper bound. The results expounded in this paper are applicable to any damped linear dynamical system the coefficient matrices of which are symmetric and positive definite. 1. INTRODUCTION The equations of motion can be written as
of an n-degree-of-freedom
Mjl+Ci+Kx=f(t),
x(O) = x0,
linear system under external
i(0) = i”,
tso,
excitation
(1.1)
where the mass matrix M, the damping matrix C and the stiffness matrix K are n x n real matrices; the displacement vector x(t) and the external excitation f(t) are ndimensional vectors for all t 2 0. For strictly passive systems, the matrices M, C and K are symmetric and positive definite. Let U denote the n x n modal matrix (see, e.g., references [ 1,2]) corresponding to the matrix the columns of which are system (1.1). The modal matrix is an n x n non-singular the eigenvectors of the symmetric generalized eigenvalue problem Ku’” = &Q’),
(1.2)
where W: > 0, and uci), i = 1, . . . , n, are the eigenvalues (undamped natural frequencies of the system) and the corresponding eigenvectors, respectively. The modal matrix is commonly orthonormalized according to U*MU = Z, (UT denotes the transpose of U, and Z,, denotes the n x n identity matrix), and hence satisfies UTKU = diag (a:, . . . , co’,) = 0’. It is well known that by the linear change of co-ordinates x(t) = Uq( t), for all t z 0, the system (1.1) can be written in the normalized form G+&j+fFq=g(t),
q(0) = UTMx,,
$0)
= UTM&,
t 3 0,
(1.3)
where c = UTCU, and for all t 30,-g(t) = U’f(t), and q is the vector of normalized co-ordinates. The symmetric matrix C is called the normalized dampins matrix. The normalized damping matrix 6 is not diagonal in general; when C is not diagonal, the system (1.1) is said to be non-classically damped. If the damping matrix C is a linear combination of the mass and stiffness matrices, then e is diagonal; this is a sufficient condition under which c is diagonal, and was originally given by Lord Rayleigh 51 0022-460X/90/010051 +14%03.00/O @ 1990 Academic Press Limited
52
S. M. SHAHRUZ
[3, p. 1301. The necessary and sufficient condition under given by Caughey [4] and Caughey and O’Kelly [5].
which
C is diagonal
has been
When e is diagonal, system (1.3) is a set of n decoupled second order differential equations, which can be solved for q conveniently. Then, the solution of system (1.1) is obtained by x(t) = Uq(t), for all f 2 0. When e is not diagonal, system (1.1) may be solved for x either by direct numerical integration or by the method proposed by Foss [6]; these methods, although quite accurate, are involved. Several authors have studied non-classically damped linear systems, as well as proposed solutions for these systems by approximate techniques. For instance, Cronin [7] obtained an approximate solution for a non-classically damped system under harmonic excitation by perturbation technique. Chung and Lee [8] applied a perturbation technique to obtain the eigensolutions of damped systems with weakly non-classical damping. Prater and Singh [9], and Nair and Singh [lo] developed several indices to determine quantitatively the extent of nonproportional damping in discrete vibratory systems. Yae and Inman [ 111 obtained bounds on the response of non-classically underdamped systems. Nicholson [12] gave upper bounds for the response of a non-classically damped system under impulsive loads and step loads. An approximate technique for solving system (1.3) is to replace C by a diagonal matrix C,, and hence decouple equations (1.3). The solution of the decoupled equations leads to an approximate solution for system (1.1); this solution would be close to the exact solution of system (1.1) if a reasonable C, were to be chosen. Several approaches to et al. [13], and Cronin [7]; choosing C, have been given by Meirovitch [I], Thomson most of these approaches are quite involved. The simplest approach to choosing C, is to neglect the off-diagonal elements of (?’and take the remaining diagonal matrix for C,, whereby the system is decoupled. Intuitively, the solution of such an approximately dec_oupled system is close to the exact solution of system (1.3) if the off-diagonal elements of C are small compared to its diagonal elements. The extent of approximation introduced by replacing c with a selected C, for the linear systems, the approximately decoupled system of which has only underdamped modes, has been reported in reference [14]. In this paper, we examine the errors introduced by approximate decoupling when C is replaced by a selected diagonal matrix C,, and tight upper bounds on the norm of errors are obtained by analytical methods. The outline of the paper is as follows. In section 2, C, is taken to be the diagonal matrix obtained by omitting the off-diagonal elements of 6. An expression for the error in the solution of the approximately decoupled system is derived, and a relation between the norm of error and the elements of e is obtained; by this relation a rigorous proof is furnished for the fact that if the off-diagonal elements of E are small compared to its diagonal elements, then the approximately decoupled system obtained by neglecting the off-diagonal elements of d in system (1.3) has a solution close to the exact solution of system (1.3). In section 3, C, is assumed to be any diagonal matrix, and an upper bound on the norm of error is again derived. In section 4, it is shown that if e is diagonally dominant, then among all diagonal matrices C,, the one that minimizes the error upper bound is simply the diagonal matrix obtained by neglecting the off-diagonal elements of C. Examples in section 5 illustrate the application of the theory developed herein. There are two appendices in which the proofs of several expressions are given. The results in this paper are an extension of those in reference [14] for the systems the approximately decoupled system of which can have underdamped, critically damped, or overdamped modes.
2. SMALL
OFF-DIAGONAL
ELEMENTS
We consider the system ( 1.1) under bounded external excitation f(t), x0 = 0. Since the coefficient matrices M, C’ and K are positive definite,
and assume that the system (1.1)
APPROXIMATE
DECOUPLING
OF
LINEAR
53
SYSTEMS
with zero input is asymptotically stable (see, e.g., reference [15]). Moreover, it can be shown that the system (1.1) is bounded-input bounded-output stable (see, e.g., reference [16]). In the following, we use L,-norm of vectors defined by IIv]]= ,m,“lf,suf? ]Vi(t)l, . . ZS for ~(f)=[~,(f)... ~,(t)]~, t>O. We rewrite system (1.3) as ij+(ed+cJ4+.n’s=g(t),
q(0) = 0,
4(O) = iYkfx(),
t 2 0,
(2.1)
where ed + e, = e, cd = diag (25,~~). . . ,2&w,) is a diagonal matrix, the elements of which are the diagonal elements of 6;, and er = [&,I is an n x n symmetric matrix with zero diagonal elements. Note that by the positive definiteness of C, the damping ratios &>O, for all i=l,..., n. We decouple system (2.1) by neglecting & and denote the solution of the resulting equation by qa. Thus we have &7+Z;r4,+fi2%=g(t),
tso,
(2.2)
with qa(0) = q(0) = 0, and &(O) = Q(0). The ith approximate mode, denoted by qoi, is underdamped if 0 < ti < 1, critically damped if 5; = 1, and overdamped if & > 1. We denote the set of indices i, for which O< 4 < 1, & = 1, and b > 1, by I,, I, and I,, respectively. An analysis of the error introduced by approximate decoupling is now pursued. Subtracting equation (2.2) from equation (2.1), and denoting the n-dimensional vector of error q - qa by e, we obtain ?+Ej+f12e+er;,g=0.
(2.3)
Applying the Laplace transform to equation (2.3), and noting that e(0) = P(0) = 0, and q(0) = 0, we arrive at e^(s)=-s(I,s2+~~;ds+~2)-‘~~~(s),
(2.4)
where e^= Le is the Laplace transform of e. Let G(S) denote the n x n matrix &(S, = -s(Z,?+
d;ds+P)‘&
(2.5)
The ith row of I?(s) is
(2.6) Rewriting equation (2.4) as e^(S) = k(.s)g*( s), we can write the error in the time domain as I e(t)=(H*q)(t)=
50
H(r-T)q(T)dT,
rz0,
(2.7)
* where H(t) = L-‘(H(s)), t 20, and H*q denotes the convolution of the time functions H and q, The above expression for the approximation error e is exact. We define the linear operator fi by Rq = fJ*q,
(2.8)
where (fiq)(t) = (H*q)( t) for all t ~0, and is given by equation (2.7). From equations (2.7) and (2.8), we obtain llell = where ]jfi]] is the &-induced we obtain
Ilfiqll G11~11 11~11~
(2.9)
norm of n. By an identity given on p. 26 of reference [ 171,
54
S. M. SHAHRUZ
where for all i=l,..., off-diagonal elements
n, the quantity (+i is the sum on the ith row of e, i.e.,
of the absolute
values
u, = i 1q,
of the
(2.11)
IPI IfI
hi, is given by
and the function
hi,(t) = The function h,, has different the following three cases: (i) For 0~ & < 1,
s= + 25,w,s + w f
forms depending
-1 hii( t, = (* _
s
L-’
[f)1/2
’
t
20.
on the values
(2.12)
that 6 assumes.
e~‘*“~‘sin(Wi(l-,$)“‘t-~i)r
t30,
We have
(2.13)
where 4; = tan’
((1 - &)“2/&}.
(2.14)
(ii) For & = 1, h,,(t) = e-“1’ (1 - w,t), (iii)
t 3 0.
(2.15)
For 5, > 1, hii( t) =
-1 e-“t”‘l’sin h(w;(&f - 1)“‘t - +i), (8; - 1)“2
tzo,
(2.16)
where *,=In(&+([f-l)“2). We carry out the integration A), and denote
I,” Jh,,( T)/ d7, for the functions
max (T;
,
evaluated
(2.17)
on I,, I,. and I,,, respectively,
hji, given above (see Appendix
x lkb)l d7, J0
by h,, h,. and h,,. We have
u, 45, ew h,, = max I,, 25,~~ 1 -exp
(V(L))
O<&
(W(L))
(2.18)
’
where V(&)=-
5s (*_g)ldan
~,(1-g)“’ 5,
’
W(&) = -
GZi (1 -[f)“’
(2.19a, b)
Also h, = max -!?- (4ee’), I, 24% u, 45, exp (X(5,)) h,, = max I,. 2&W! Y(5,)
5,= 1,
’
5z> 1,
(2.20)
(2.21)
APPROXIMATE
DECOUPLING
OF
LINEAR
SYSTEMS
55
where X([.)=_&-(5i-I)“‘,n5’+(li-I)“2 _I 2(5f- I)“2 Therefore,
the norm in equation
(2.10) is IIfill = max {h,
Thus, by expression norm of q by
(2.22a, b)
y(5,)=&+(5f-l)“‘.
t;-(,$f_l)‘l”
(2.9) the norm
(2.23)
h, kl.
of e is related
to the system
parameters
and the
llellsmax{h, h, k)Ilqll. By the definition of operator norm, we note that expression bound for e among all upper bounds of the type
(2.24) (2.24) gives the tightest
llells mllqll~
(2.25)
because max {h,, h,, h,} = 11 H/l is the smallest possible value for m. To arrive at a more easily manageable bound, we utilize the inequalities which are established in Appendix B, namely, 0.3 183 =
T--’s
= 0.3619,
e -’G 5, exp(x(6))l These inequalities
upper
(Bl) and (B2)
OS&Sl, 5, 3 1.
Y(5;) GO.5,
(2.26) (2.27)
imply that 4K’
(+I max-Gh,a4eP’ max I,, 25iwt
u,
U,
-<2maxI,,
25Pt
I,,
25Pi
I,,
25iw,
(2.28) '
and ui
457 -’ max --44-l I,, 2&W! Using expressions
u,
(2.28), (2.20) and (2.29), in equation 4~~’ max ITi< I--ICfl 2[,Wi
g,
max-
(2.29) ’
(2.23), we obtain
lIfill~2 ,my+
(2.30) ,w,
Now, the relation between IIHII and the sums of the absolute values of the off-diagonal elements u, and the diagonal elements 25,~; of 6 is transparent. Clearly, IIiill and consequently the error upper bound are small if and only if u,/2&wi << 1, for all i = I, . . , n, i.e., if and only if the off-diagonal elements of 6 on each row are much smaller than the corresponding diagonal elements. Intuitively, this conclusion is expected, for which we have furnished a rigorous proof. We continue with further analysis of the approximation error. Suppose that for all i=l,..., n, o,/24w, << I, and that the decoupled system (2.2) has been readily solved to obtain qa. It would be of interest to derive a neighborhood of qa in which the exact solution q lies. Since q = e + qa, by the triangle inequality llqll s Ijell + IIqa11; using this inequality in expression (2.9), we obtain
llell~{llfill/(l -llW)Illq~ll. That is, q lies in the {/Ifi II/( I- 1)fi 11)}11 qUII neighborhood of qa. It has been assumed 1 - [[fill > 0, which is always the case if ui/25,wi is small for all i = 1, . . _, n.
(2.31) that
56
S. M. SHAHRUZ
3. REPLACEMENT
In this section
we consider
BY ARBITRARY
the system
q+Q+,fPq=g(t),
DIAGONAL
MATRICES
(2.1),
4(O) = UTM&,, t 3 0, (3.1) I and evaluate the error introduced by replacing C with an arbitrary fixed diagonal matrix C, =diag(2r],w,, . . . , 277,,0~), where for all i = 1,. . . , n, 7, > 0. By i,, i,. and i,, respectively, we denote the set of indices i, for which 0 < vi < 1, n, = 1, 2nd ni > 1. Let p denote the solution of the decoupled equations resulted from replacing C with C,. Thus we have q(0) = 0,
jj+c,li+n’p=g(t),
f 2 0,
with p(O) = q(0) = 0 and d(O) = 4(O). Subtracting denoting the error q-p by e, we obtain
equation
(3.2)
(3.2) from equation
;;+C,f?+fl’e+(C-C,)q=O. By an approach
similar
to that in the previous
(3.3)
section,
we obtain
Z(s) = &)q*(s), where
e^ is the Laplace
transform
(3.1) and
(3.4)
of e, and
d(s)=-S(znS2+CdS+n2)-‘(~-cc,).
(3.5)
The ith row of C?(s) is
(3.6) Furthermore,
we obtain (3.7)
where
JIGI/ is the L,-induced
norm of the linear
operator
Gq=G”q=e, A and given by (Gq)(t)=(G*q)(t)=L-‘(G(s)q*(s)),
G, defined
by (3.8)
ta0.
In this case
IIGil= maxk, g,,go>,
(3.9)
where g, = max 6,
~i+245,-77,1 471,exp(V(77,)) 2r)i0i
g,. = max
vi +245, - ?il 277iwi
4
g, = max ill
O< TJi< 1,
(3.10)
l-exp(Wni))
ui+2wi15z-t)il
2WI
477i
(4 e-‘I,
rli=
exp (X(771)) Y(%)
’
1,
(3.11)
rl,’ 1,
(3.12)
with (TV,V, W, X and Y given by equations (2.11), (2.19a), (2.19b), (2.22a) and (2.22b), respectively. Equation (3.8) is an exact expression for the approximation error. Applying inequalities (Bl) and (B5) of Appendix B to equations (3.10) and (3.12).respectively, we obtain 4??_‘pG
IJCII<2p,
(3.13)
APPROXIMATE
DECOUPLING
OF
LINEAR
57
SYSTEMS
where p=
max I s Is ,I
ai+wi15i-17il
21714
(3.14) .
Obviously, expression (3.13) is an extension of expression (2.30). t-11p(( o f a neighborhood, centered Likewise, we evaluate the half-width the exact solution q lies. We have that
at p, in which
r= lIGll/(l -II~ll). 4. ON MINIMIZATION
OF ERROR
(3.15)
BOUND
of development is followed. We consider C, = In this section, a different direction n,,) as an unknown matrix, and assume that the normalized damping matrix djag(nl,..., C is diagonally dominant, that is, us G
(4.1)
25iwr,
for all i = 1,. . . , n. We determine C,, namely n,, in order to have minimum upper bound for the norm of error between the solution of the approximately decoupled system (3.1) and the exact solution of system (3.1). It will turn out that the error upper bound is minimum when 7, = &, for all i = 1, . . . , n; for this reason neglecting the off-diagonal elements of e is probably the most appropriate approach to decouple system (3.1), when C? is diagonally dominant. The error upper bound in expression (3.7) is the smallest when IJGll is minimum. In order to obtain the minimum of )IG (1, we first establish the following facts: (i) The function r.(~)=~(~i+20,1&-~)
exp(V(n)) l-exp(W(t7))’
O
(4.2)
is minimum at 77= 5,. (ii) The function (4.3) is minimum at 77= 6. (iii) The function (4.4) is minimum at 77= 6. Clearly, the function yc is minimum at n = 5, = 1. In the following, we prove (i) and (iii) separately. Case (i). The function y,, is differentiable over (0, I), except at n = 6. Let 77> 5, ; then I&,-nl=n-6, and
=1
Yi4(17)
wi
exp(V(?7))
{2wi[(l+nV’(n)(l-exp(W(n)))
[l-exp(W(?)))]*
+17W’(77)ev(W77))1 +((+,-25;w,)[Vf(rl)(1-exp(W(77)))+
W77)exp(W17))11.
(4.5)
S.
58
M. SHAHRUZ
Using expressions (4.1) and (B2), (B3) and (B4) of Appendix B, we conclude that y:( 7) > 0, for .$,< r) < 1. Hence, y,, is monotonically increasing over (tl, l), and achieves its infimum at 5,. Let n < 5, ; then 15, - n\= & - 7, and
Y:(7))
exp (V(77)) =2[l -exp i-24(1 (Wrl))l’
+ vV’(77)(1 -exp (W(77)))
wi
+77W’(77)exp ( Wrl))l +(~,+22S,w,)[V’(77)(1_exp(W(77)))+W77)exp(W(?7))11.
(4.6)
Using expressions (B2), (B3) and (B4) of Appendix B, we conclude that yI,( r]) < 0, for O< n < &. Hence, y,, is monotonically decreasing over (0, &), and achieves its infimum at [,. Therefore, yu is minimum at n = 5,. Case (iii). The function y(, is differentiable over (1, cc), except at 7 = 6,. Let r] > 5, ; then \ti-n(=n-[t, and
+tg, Using
-2!%4)[X’(77)Y(v)- Y’(rl)ll.
(4.7)
expressions
YL(V)>O,for&
its infimum
(4.1) and (B6), (B7) and (B8) of Appendix B, we conclude that < co. Hence, y(, is monotonically increasing over (&, CO),and achieves at 5. Let 7~6,; then I&i-n[=~,-n~ and
YWW,
2 ew (X(77)) {-2wL(l+ 77X’(?))Y(T) - rlY’(v)l
w?)
+(a,+25iw,)[X'(77)Y(rl)-
y’h)lj.
(4.8)
Using expressions (B6), (B7) and (B8) of Appendix B, we conclude that y:,(n) < 0, for 1~ n < 5,. Hence, y,, is monotonically decreasing over (1, tZ), and achieves its infimum at 5,. Therefore, y,, is minimum at 77= 5,. Recall that 11G II IS ’ g’iven by equation (3.9), in which g,,, g,. and g,, are given by equations (3.10), (3.11) and (3.12), respectively. In order to achieve the minimum of IIc?II, we have to minimize g,, g,. and g,, by choosing 7,. By (i), (ii) and (iii), we conclude that if ni = [,, and i,, = I,,, i, = I,. and for all i = 1, . . , n, then the functions g,,, g,. and g,, are minimum, i,,==I,,.Thus, if q,=,$,, forall i=l,..., n, then IIGll is minimum, and is equal to IIfill given in equation (2.3). This shows that the diagonal matrix which minimizes the error upper bound is obtained by simply neglecting the off-diagonal elements of the diagonally dominant normalized damping matrix. 5. EXAMPLES
In this section, we give two examples to illustrate possible application of the theory developed in the previous sections; for convenience, low order systems are considered. Example 1. We consider a system represented in the normalized form by [:,
:][3+[_gZi -;::][;;]+[; :.41][;;]=[:]sin2r, (5.1)
t 3 0. We have w, = 2, w2 = 2.1, 5, = & = 0.5, I,, = { 1,2}, I,, = 4, and I,, = 4 (4 denotes empty set). An approximate solution of equations (5.1) is obtained by solving decoupled equations [b
:][::I+[:
:.J[;::]+[:
04.411[~.11=[:lsin2r~
the the
(5.2)
APPROXIMATE
t 5 0. The steady
state solution
DECOUPLING
of equations
OF LINEAR
(5.2) is
0.25 sin (2t - 1.571) ss(t) =
0.237 sin (2t - 1.473)
1’
59
SYSTEMS
t 20.
(5.3)
Forthe system (5.2), llqa I( = llqsjl = 0.25. By equations (2.18), (2.20) and (2.21), respectively, (2.23) and (2.31), we obtain we obtain h, = 0.196, h, = 0 and h,, = 0. By expressions IIe I/ s 0.243 11qa 1)= 0.061. The exact solution q, and the corresponding approximate soluof qa, , in which q, lies are plotted in Figure 1. Since tion qa, , as well as a neighborhood the exact solution has been calculated, we have j/q11= 0.29, and by expression (2.24), 11 ell G 0.196llqll = 0.057. This error upper bound is the tightest possible in the functional form specified by expression (2.25), where the coefficient m is independent of the driving force. Example 2. We consider a system represented in the normalized form by 1
0
0
0
1
0
[0
0
1
I[][ 4,
2
&
+
ij3
-0.15
-I:”
-:::“I[
-0.15
-0.2
6.6
“:I+[: C&
0
Y.41
:][
:I]=[
:.2]l(t),
0
9
q3
2.5 tzo,
(5.4)
where l(t), t 2 0 denotes the unit-step function. We have w, = 2, w2 = 2.1, C+ = 3, 5, = 0.5, & = 1, & = 1.1, I, = {l}, I, = (2) and I,, = (3). An approximate solution of system (5.4) is obtained by solving the decoupled equations
The solution
of system 0.25(1-l.55eP’sin(l*732t+1.047)) &Z(f) =
0.375
0.272(1-e-2”‘(1+2+lt)) 0.278(1+0.7 ep4.675’_ 1.7 ,-1+25’)
Naghborhood
1 ,
tZ0.
(5.6)
of q,,
t
0,125
0.0
-0,125
-0.25 0.0
2.5
5.0
7.5
IO.0
f Figure 1. The exact solution neighborhood of q‘,, , in which
q, of the system q, lies.
(5.1), the corresponding
approximate
solution
q,,
, and the
60
S. M. SHAHRUZ
For the system (5.5),I(qaII = //qo,I/ =O-29. By equations (2.18),(2.20)and (2.21), respectively, we obtain h, = O-196, h,. = O-123 and h,, = 0.08. By expressions (2.23) and (2.31), we obtain IIell G 0.243 II qa II= 0.07. The exact solution q1 and the corresponding approxiof qa, , in which q, lies are plotted in Figure mate solution qa, , as well as a neighborhood 2. Since the exact solution has been calculated, we have llql/ = 0.3, and by expression (2.24), /IelI G 0.196llqll = 0.059.
f
NeIghborhood of q.,
Figure 2. The exact solution neighborhood of qu,, in which
q, of the system
(5.4), the corresponding
approximate
solution
qul, and the
q, lies.
6.
CONCLUSIONS
One common procedure in solving the normalized equations of motion of an n-degreeof-freedom linear system with non-zero off-diagonal damping elements is to replace the normalized damping matrix e by a selected diagonal matrix C,, and hence decouple the system equations. In this paper, we considered an n-degree-of-freedom damped linear system and studied the errors between the exact solution of the system and that of the corresponding decoupled system. We obtained tight upper bounds on the norm of errors. We derived the error between the exact solution ?f the system and that of the approximately deco_upled system when Cd is obtained from C by neglecting the off-diagonal elements of C (see equation (2.7)); we gave an upper bound on the norm of this error (see expression (2.24)). We derived an upper bound on the norm of error in the solution of the approximately decoupled equations when 6 is replaced by an arbitrary fixed diagonal matrix C, (see expression (3.7)). The upper bounds on the norm of errors are the tightest possible in the functional form specified-by expression (2.25). We gave a proof of the following fact: Let C, be obtained from C by neglecting the off-diagonal elements of c; if the off-diagonal elements of d are small, then the error in the solution of the approximately decoupled system is small. Furthermore, we showed that, if 6 is diagonally dominant, then among diagonal matrices C, the one which results in the minimum error upper bound is obtained by simply neglecting the off-diagonal elements of c.
APPROXIMATE
DECOUPLING
61
OF LINEAR SYSTEMS
REFERENCES 1. L. MEIROVITCH 1967 Analytical Merhods in Vibrations. New York: Macmillan. The Netherland: 2. P. C. MIIILLER and W. 0. SCHIEHLEN 1985 Linear Vibrations. Dordrecht, Martinus Nijhoff. LORD RAYLEIGH 1945 The Theory ofSound, Volume 1. New York: Dover. T. K. CAUGHEY 1960 American Society of Mechanical Engineers Journal of Applied Mechanics 27, 269-271. Classical normal modes in damped linear dynamic systems. and M. E. J.O'KELLY 1965 American Society of Mechanical Engineers Journal T. K. CAUGHEY ofApplied Mechanics 32, 583-588. Classical normal modes in damped linear dynamic systems. K. A. FOSS 1958 American Society of Mechanical Engineers Journal of Applied Mechanics 25, which uncouple the equations of motion of damped linear dynamic 361-364. Co-ordinates systems. 7. D. L. CRONI N 1976 American Society of Mechanical Engineers Journal of Engineeringfor Industry 98, 43-47. Approximation for determining harmonically excited response of nonclassically damped systems. and C. W. LEE 1986 Journal of Sound and Vibration 111, 37-50. Dynamic 8. K. R. CHUNG reanalysis of weakly non-proportionally damped systems. 9. G. PRATER, JR.and R. SINGH 1986 Journal of Sound and Vibration 104,109-125.Quantification of the extent of non-proportional viscous damping in discrete vibratory systems. of 10. S. S. NAIR and R. SINGH 1986 Journal of Sound and Vibration 104,348-350. Examination the validity of proportional damping approximations with two further numerical indices. 11. K. H. YAE and D. J. INMAN 1987 American Society of Mechanical Engineers Journal of Applied Mechanics 54, 419-423. Response bounds for linear underdamped systems. 12. D. W. NICHOLSON 1987 American Society of Mechanical Engineers Journal of Applied Mechanics 54, 430-433. Response bounds for nonclassically damped mechanical systems under transient loads. and P.CARAVANI 1974 Earthquake EngineeringandStructural 13. W.T. THOMSON,T.CALKIN Dynamics 3, 97-103. A numerical study of damping. 14. S. M. SHAHRUZ and F. MA 1988 American Society of Mechanical Engineers Journal of Applied Mechanics 55, 716-720. Approximate decoupling of the equations of motion of linear underdamped systems. 15. L. S. SHIEH, M. M. MEHIO and H. M. DIB 1987 Institute of Electrical and Electronics Engineers Transactions on Automatic Control AC-32, 231-233. Stability of the second-order matrix polynomial. 16. C. T. CHEN 1984 Linear System Theory and Design. New York: Holt, Rinehart and Winston. 1975 Feedback Systems: Input-Output Properties. New 17. C. A. DESOER and M. VIDYASAGAR York: Academic Press.
APPENDIX Derivation
of equation
(2.18).
For each
at t = 0, and is zero at tk = {l/wi( by equation (2.14). We have
I^ Ihi, 0
dT= I”“,,~~) 0 1 =-exp wi +lexp w,
A
i in I,,, hii is given by equation
(2.13);
h,,(t)
= 1
kn + 4;), for k = 0, 1, 2, . . . , where +I is given
1 - tf)“‘}(
dT+
(
h -cl_t:)l12tan
-
(
-, (1 - sfY 5,
)
_, (l-[j)""
5t (l-$)riztan
&
-I
('
-f"")
)[
ki,,
1+2,$,exp((~~~J
exp( cl12;,2).
(Al)
62
5..M.SHAHRUZ
Since jJ exp(-~~,k/(1--~~)‘~‘)=1/{1-exp(-~~,/(1-~~)”~)}, I,-0 equation
(2.18) follows. Derivation of equation (2.20). For at t = 0, and is zero at t = l/w,. The Hence, 52 lhii(r)I dr = 2 Jbl”’ I?,,(T) Derivation of equation (2.21). For write equation (2.16) as
each i in I,, h,, is given by equation (2.15); h,,(t) = 1 function h,, in equation (2.15) satisfies I,” A;,( T) d7 = 0. dT= 2 e-‘/w,, and equation (2.20) follows. each i in I,,, h,, is given by equation (2.16); we can
h,,(t)={l/(b-~)}(f7e~~~‘-~
t 30,
em”‘),
(AZ)
where a=w;(&-([f-l)‘:‘),
b=W;(&+([f-1)“‘).
(A3)
The function hii( t) = 1 at t = 0, and is zero at t* = { l/( b - a)} In (b/u). Furthermore, h,, satisfies j; hi;( 7) dr = 0. Hence Jz lhii(T)ldT=2 Ji hii(~) dr=2exp (X(&))/w,Y([,), where X and Y are given by equations (2.22a) and (2.22b), respectively. Thus equation (2.21) holds.
APPENDIX
The following
inequalities ?T
h(rl)=
have been drawn
-‘~Y,Lt)=
B
upon
in sections
5 exp (V(5))
(Bl)
OGlSl;
l-exp(W(&))~e~‘7
V’(rl)(l-exp(W(q)))+
2, 3 and 4:
OSnGl;
W’(n)exp(W(71))<0,
(B2) (B3)
and y3(v)+~, as v+Oo; y4(77)=(l+nV’(77))(1--xp(W(n)))+nW’(~)exp(W(n))~O, and the equality
Oansl,
holds only for n = 0; e ‘,z,([)=5exp(x(S))<0 Y(5)
and z,(t)+O.5
Y’(‘I)
rlZ 1,
(X(n))/
Y’(n)>O,
as ~+cc; (B8)
921,
as n+m.
Proof of expression
(Bl ). We have
exp ( V(5)) “(‘)=[l
(B7)
7721,
z4(77)=(1+77X’(n))Y(n)-rlY’(n)>O, and z,(n)+0
(B5)
52 1,
as n-+cc; z3(rl) =exp
and z3(n)+0
.5 5
as (+a; zJn)=X’(n)Y(n)-
and zz(~)-+O
(B4)
-exp
( W([))12
[(I +tY’(5))(1
-ew
(W(5)))+5W’(5)
exp
(W(O)l.
(B9)
APPROXIMATE
DECOUPLING
OF
LINEAR
SYSTEMS
63
By expressions (B3) and (B4), to be proved in the following, y{(t) > 0, for 0 < 5~ 1, and y{(O) = 0. Therefore, y, is monotonically increasing over [0, 11, and achieves its minimum and maximum at 5 = 0 and 5 = 1, respectively. Using L’HBpital’s rule we obtain y,(O) := 7~~’ and y,(l) = ee’. Proof qf expression (B2). We have “,(77)
T(l -12Y2 -tan’
((1-n’)“2/n) (1 _ 71?)3/2
=
We let n = cos 13in equation
(BlO)
(BlO); then
V’(B)=
sin 28 -26’
0s
9
2sin”o
0 S 7rJ2.
(Bll)
By L’H6pital’s rule, V’( 0 = 0) = -2/3. For 0 s 0 d ~12, we have sin 20 < 20, and sin 8 > 0; hence, V’(0) ~0, for 0~ 8 G 7r/2. Thus, for 0~ n S 1, V’(n)
having
W defined
by equation
(B12)
(2.19b),
for OG n G 1, we have (B13)
V’(n)(l-exp(W(n)))cO, where -7rJ2.
the equality We have
holds
only for r] = 0. By L’HBpital’s
W’(77)exp(W(77))= (1 _ -77 $)3/2 for 0~ ‘15 1. Clearly, W’(q) exp (W(T))
V’(1) (1 -exp
(W(1)))
for
Ocn<
=
(B14)
exp(ll_~),/?).
W’(rl)exp(W17))~0, where equality holds Proofofexpression Furthermore, ~~(7) + (B3) holds. Proof of expression and hence is omitted. Proof of expression
rule,
1. By
L’H6pital’s
OGnGl,
rule,
(B15)
only for n = 1. Hence, (B2) follows. (B3). By expression (2.19a), it is clear that y,(n) > 0, for O< 77< 1. cc as n + 0, and by L’Hopital’s rule, y?( 1) = e-‘. Hence, expression (B4).
Proof of expression
(BS).
We have
z,(5)
=
1
(B4) is straightforward,
but laborious,
ev W(5)) [Cl +5x’(5)) Y(5) - tx5)l. Y’(5)
(B16)
By expressions (B7) and (B8), to be proved in the following, z{( 5) > 0, for 1 < 5 < ~0, and z{( 1) = 0. Thus z, is monotonically increasing over [ 1, OO), and is minimum at .$ = 1, and achieves its supremum as t-*00. We let 4=ln (4+(&l)“‘); then [*(&1)“2=e’ti, and expression (B5) reads z,($)=$(l+e-‘“)
exp
By L’HGpital’s rule, we obtain (equivalently as 5 + a).
,
3 (
~~([=l)=z,(rl,=O)=e~‘,
)
$30. and
(B17) zl($)-+0.5
as 4-a
S. M. SHAHRUZ
64 Proof
of expression
zz(77) =X’(n) =
(B6). We have Y(n) - Y’(n)
77+(+1)“Z,n 2($_1)‘/2
n+(+l)“Z n_($-l)‘/~
1 n?_1
n+($-1)“’ ($I)‘/2
(Bl8)
*
We let $=ln(n+(n2~1)“‘). Then,
~f(n1-1)“2=e*Q,and(n1-l)“2=
(B19)
sinh 7; thus equation
(Bl8)
can be simplified
to e”‘(24 - sinh 2+) zz($) =
2 sinh3 IJ
(B-20)
* 2 0.
’
By L’HGpital’s rule, z2 at I,G= 0 (equivalently at 77= 1) is -2/3. For $ > 0, we have sinh 2$>2+, and sinh I+!J>O; hence, z2($)0 (equivalently n > 1). We write
e*+
z*(‘b)=------
e"
(B21)
sinh 4 tanh $ ’
sinh3 Q
as r] + cc). Then it is easy to conclude that z2( 4) -+ 0 as Cc,+ 00 (equivalently Proof of expression (B7). Applying the change of variable (B19) to expression we obtain z3( I/I) = em’* exp
2*
( 1 l-ezG
’
(B7),
(B22)
* 30.
Clearly, z3( 4) > 0 for rF,> 0. By L’Hepital’s rule, z3( 7) = 1) = z3( $ = 0) = e-‘, and z3( 4) + 0 as + + cc (equivalently n + co). Proof of expression (B8). Applying the change of variable (B19) to expression (B8), we obtain e”( $ - tanh +!I)cash + zq($) =
sinh3 $
By L’Hhpital’s rule, Z,(T) = 1) = zq(+ = 0) = l/3. sinh $ > 0; hence, z~(I,!J)> 0 for Cc,> 0 (equivalently
z,(‘b)= .
e”*
smh’ J,!Jtanh $
Then it is trivial
to conclude
’
(B23)
$20.
For $> 0, we have r] > 1). We write
$> tanh $, and
____e*
(~24)
sinh’ $ ’
that zq( $) + 0 as $ + a~ (equivalently
n + CO).
APPROXIMATE DECOUPLING OF THE EQUATIONS OF MOTION OF DAMPED LINEAR SYSTEMS S. M. SHAHRUZ Department of Electrical Engineering and Computer Sciences, and the Electronics Research Laboratory, University of California, Berkeley, California 94720, U.S.A. (Received 27 September 1988, and in revised form 18 April 1989)
One common procedure in solving a normalized damped linear system with non-zero off-diagonal damping elements is to replace the normalized damping matrix by a selected diagonal matrix. The extent of approximation introduced by this method of decoupling the system is evaluated, and tight upper bounds on the norm of errors are derived. Moreover, when the normalized damping matrix is diagonally dominant, it is shown that decoupling the system by neglecting the off-diagonal elements indeed minimizes the error upper bound. The results expounded in this paper are applicable to any damped linear dynamical system the coefficient matrices of which are symmetric and positive definite. 1. INTRODUCTION The equations of motion can be written as
of an n-degree-of-freedom
Mjl+Ci+Kx=f(t),
x(O) = x0,
linear system under external
i(0) = i”,
tso,
excitation
(1.1)
where the mass matrix M, the damping matrix C and the stiffness matrix K are n x n real matrices; the displacement vector x(t) and the external excitation f(t) are ndimensional vectors for all t 2 0. For strictly passive systems, the matrices M, C and K are symmetric and positive definite. Let U denote the n x n modal matrix (see, e.g., references [ 1,2]) corresponding to the matrix the columns of which are system (1.1). The modal matrix is an n x n non-singular the eigenvectors of the symmetric generalized eigenvalue problem Ku’” = &Q’),
(1.2)
where W: > 0, and uci), i = 1, . . . , n, are the eigenvalues (undamped natural frequencies of the system) and the corresponding eigenvectors, respectively. The modal matrix is commonly orthonormalized according to U*MU = Z, (UT denotes the transpose of U, and Z,, denotes the n x n identity matrix), and hence satisfies UTKU = diag (a:, . . . , co’,) = 0’. It is well known that by the linear change of co-ordinates x(t) = Uq( t), for all t z 0, the system (1.1) can be written in the normalized form G+&j+fFq=g(t),
q(0) = UTMx,,
$0)
= UTM&,
t 3 0,
(1.3)
where c = UTCU, and for all t 30,-g(t) = U’f(t), and q is the vector of normalized co-ordinates. The symmetric matrix C is called the normalized dampins matrix. The normalized damping matrix 6 is not diagonal in general; when C is not diagonal, the system (1.1) is said to be non-classically damped. If the damping matrix C is a linear combination of the mass and stiffness matrices, then e is diagonal; this is a sufficient condition under which c is diagonal, and was originally given by Lord Rayleigh 51 0022-460X/90/010051 +14%03.00/O @ 1990 Academic Press Limited
52
S. M. SHAHRUZ
[3, p. 1301. The necessary and sufficient condition under given by Caughey [4] and Caughey and O’Kelly [5].
which
C is diagonal
has been
When e is diagonal, system (1.3) is a set of n decoupled second order differential equations, which can be solved for q conveniently. Then, the solution of system (1.1) is obtained by x(t) = Uq(t), for all f 2 0. When e is not diagonal, system (1.1) may be solved for x either by direct numerical integration or by the method proposed by Foss [6]; these methods, although quite accurate, are involved. Several authors have studied non-classically damped linear systems, as well as proposed solutions for these systems by approximate techniques. For instance, Cronin [7] obtained an approximate solution for a non-classically damped system under harmonic excitation by perturbation technique. Chung and Lee [8] applied a perturbation technique to obtain the eigensolutions of damped systems with weakly non-classical damping. Prater and Singh [9], and Nair and Singh [lo] developed several indices to determine quantitatively the extent of nonproportional damping in discrete vibratory systems. Yae and Inman [ 111 obtained bounds on the response of non-classically underdamped systems. Nicholson [12] gave upper bounds for the response of a non-classically damped system under impulsive loads and step loads. An approximate technique for solving system (1.3) is to replace C by a diagonal matrix C,, and hence decouple equations (1.3). The solution of the decoupled equations leads to an approximate solution for system (1.1); this solution would be close to the exact solution of system (1.1) if a reasonable C, were to be chosen. Several approaches to et al. [13], and Cronin [7]; choosing C, have been given by Meirovitch [I], Thomson most of these approaches are quite involved. The simplest approach to choosing C, is to neglect the off-diagonal elements of (?’and take the remaining diagonal matrix for C,, whereby the system is decoupled. Intuitively, the solution of such an approximately dec_oupled system is close to the exact solution of system (1.3) if the off-diagonal elements of C are small compared to its diagonal elements. The extent of approximation introduced by replacing c with a selected C, for the linear systems, the approximately decoupled system of which has only underdamped modes, has been reported in reference [14]. In this paper, we examine the errors introduced by approximate decoupling when C is replaced by a selected diagonal matrix C,, and tight upper bounds on the norm of errors are obtained by analytical methods. The outline of the paper is as follows. In section 2, C, is taken to be the diagonal matrix obtained by omitting the off-diagonal elements of 6. An expression for the error in the solution of the approximately decoupled system is derived, and a relation between the norm of error and the elements of e is obtained; by this relation a rigorous proof is furnished for the fact that if the off-diagonal elements of E are small compared to its diagonal elements, then the approximately decoupled system obtained by neglecting the off-diagonal elements of d in system (1.3) has a solution close to the exact solution of system (1.3). In section 3, C, is assumed to be any diagonal matrix, and an upper bound on the norm of error is again derived. In section 4, it is shown that if e is diagonally dominant, then among all diagonal matrices C,, the one that minimizes the error upper bound is simply the diagonal matrix obtained by neglecting the off-diagonal elements of C. Examples in section 5 illustrate the application of the theory developed herein. There are two appendices in which the proofs of several expressions are given. The results in this paper are an extension of those in reference [14] for the systems the approximately decoupled system of which can have underdamped, critically damped, or overdamped modes.
2. SMALL
OFF-DIAGONAL
ELEMENTS
We consider the system ( 1.1) under bounded external excitation f(t), x0 = 0. Since the coefficient matrices M, C’ and K are positive definite,
and assume that the system (1.1)
APPROXIMATE
DECOUPLING
OF
LINEAR
53
SYSTEMS
with zero input is asymptotically stable (see, e.g., reference [15]). Moreover, it can be shown that the system (1.1) is bounded-input bounded-output stable (see, e.g., reference [16]). In the following, we use L,-norm of vectors defined by IIv]]= ,m,“lf,suf? ]Vi(t)l, . . ZS for ~(f)=[~,(f)... ~,(t)]~, t>O. We rewrite system (1.3) as ij+(ed+cJ4+.n’s=g(t),
q(0) = 0,
4(O) = iYkfx(),
t 2 0,
(2.1)
where ed + e, = e, cd = diag (25,~~). . . ,2&w,) is a diagonal matrix, the elements of which are the diagonal elements of 6;, and er = [&,I is an n x n symmetric matrix with zero diagonal elements. Note that by the positive definiteness of C, the damping ratios &>O, for all i=l,..., n. We decouple system (2.1) by neglecting & and denote the solution of the resulting equation by qa. Thus we have &7+Z;r4,+fi2%=g(t),
tso,
(2.2)
with qa(0) = q(0) = 0, and &(O) = Q(0). The ith approximate mode, denoted by qoi, is underdamped if 0 < ti < 1, critically damped if 5; = 1, and overdamped if & > 1. We denote the set of indices i, for which O< 4 < 1, & = 1, and b > 1, by I,, I, and I,, respectively. An analysis of the error introduced by approximate decoupling is now pursued. Subtracting equation (2.2) from equation (2.1), and denoting the n-dimensional vector of error q - qa by e, we obtain ?+Ej+f12e+er;,g=0.
(2.3)
Applying the Laplace transform to equation (2.3), and noting that e(0) = P(0) = 0, and q(0) = 0, we arrive at e^(s)=-s(I,s2+~~;ds+~2)-‘~~~(s),
(2.4)
where e^= Le is the Laplace transform of e. Let G(S) denote the n x n matrix &(S, = -s(Z,?+
d;ds+P)‘&
(2.5)
The ith row of I?(s) is
(2.6) Rewriting equation (2.4) as e^(S) = k(.s)g*( s), we can write the error in the time domain as I e(t)=(H*q)(t)=
50
H(r-T)q(T)dT,
rz0,
(2.7)
* where H(t) = L-‘(H(s)), t 20, and H*q denotes the convolution of the time functions H and q, The above expression for the approximation error e is exact. We define the linear operator fi by Rq = fJ*q,
(2.8)
where (fiq)(t) = (H*q)( t) for all t ~0, and is given by equation (2.7). From equations (2.7) and (2.8), we obtain llell = where ]jfi]] is the &-induced we obtain
Ilfiqll G11~11 11~11~
(2.9)
norm of n. By an identity given on p. 26 of reference [ 171,
54
S. M. SHAHRUZ
where for all i=l,..., off-diagonal elements
n, the quantity (+i is the sum on the ith row of e, i.e.,
of the absolute
values
u, = i 1q,
of the
(2.11)
IPI IfI
hi, is given by
and the function
hi,(t) = The function h,, has different the following three cases: (i) For 0~ & < 1,
s= + 25,w,s + w f
forms depending
-1 hii( t, = (* _
s
L-’
[f)1/2
’
t
20.
on the values
(2.12)
that 6 assumes.
e~‘*“~‘sin(Wi(l-,$)“‘t-~i)r
t30,
We have
(2.13)
where 4; = tan’
((1 - &)“2/&}.
(2.14)
(ii) For & = 1, h,,(t) = e-“1’ (1 - w,t), (iii)
t 3 0.
(2.15)
For 5, > 1, hii( t) =
-1 e-“t”‘l’sin h(w;(&f - 1)“‘t - +i), (8; - 1)“2
tzo,
(2.16)
where *,=In(&+([f-l)“2). We carry out the integration A), and denote
I,” Jh,,( T)/ d7, for the functions
max (T;
,
evaluated
(2.17)
on I,, I,. and I,,, respectively,
hji, given above (see Appendix
x lkb)l d7, J0
by h,, h,. and h,,. We have
u, 45, ew h,, = max I,, 25,~~ 1 -exp
(V(L))
O<&
(W(L))
(2.18)
’
where V(&)=-
5s (*_g)ldan
~,(1-g)“’ 5,
’
W(&) = -
GZi (1 -[f)“’
(2.19a, b)
Also h, = max -!?- (4ee’), I, 24% u, 45, exp (X(5,)) h,, = max I,. 2&W! Y(5,)
5,= 1,
’
5z> 1,
(2.20)
(2.21)
APPROXIMATE
DECOUPLING
OF
LINEAR
SYSTEMS
55
where X([.)=_&-(5i-I)“‘,n5’+(li-I)“2 _I 2(5f- I)“2 Therefore,
the norm in equation
(2.10) is IIfill = max {h,
Thus, by expression norm of q by
(2.22a, b)
y(5,)=&+(5f-l)“‘.
t;-(,$f_l)‘l”
(2.9) the norm
(2.23)
h, kl.
of e is related
to the system
parameters
and the
llellsmax{h, h, k)Ilqll. By the definition of operator norm, we note that expression bound for e among all upper bounds of the type
(2.24) (2.24) gives the tightest
llells mllqll~
(2.25)
because max {h,, h,, h,} = 11 H/l is the smallest possible value for m. To arrive at a more easily manageable bound, we utilize the inequalities which are established in Appendix B, namely, 0.3 183 =
T--’s
= 0.3619,
e -’G 5, exp(x(6))l These inequalities
upper
(Bl) and (B2)
OS&Sl, 5, 3 1.
Y(5;) GO.5,
(2.26) (2.27)
imply that 4K’
(+I max-Gh,a4eP’ max I,, 25iwt
u,
U,
-<2maxI,,
25Pt
I,,
25Pi
I,,
25iw,
(2.28) '
and ui
457 -’ max --44-l I,, 2&W! Using expressions
u,
(2.28), (2.20) and (2.29), in equation 4~~’ max ITi< I--ICfl 2[,Wi
g,
max-
(2.29) ’
(2.23), we obtain
lIfill~2 ,my+
(2.30) ,w,
Now, the relation between IIHII and the sums of the absolute values of the off-diagonal elements u, and the diagonal elements 25,~; of 6 is transparent. Clearly, IIiill and consequently the error upper bound are small if and only if u,/2&wi << 1, for all i = I, . . , n, i.e., if and only if the off-diagonal elements of 6 on each row are much smaller than the corresponding diagonal elements. Intuitively, this conclusion is expected, for which we have furnished a rigorous proof. We continue with further analysis of the approximation error. Suppose that for all i=l,..., n, o,/24w, << I, and that the decoupled system (2.2) has been readily solved to obtain qa. It would be of interest to derive a neighborhood of qa in which the exact solution q lies. Since q = e + qa, by the triangle inequality llqll s Ijell + IIqa11; using this inequality in expression (2.9), we obtain
llell~{llfill/(l -llW)Illq~ll. That is, q lies in the {/Ifi II/( I- 1)fi 11)}11 qUII neighborhood of qa. It has been assumed 1 - [[fill > 0, which is always the case if ui/25,wi is small for all i = 1, . . _, n.
(2.31) that
56
S. M. SHAHRUZ
3. REPLACEMENT
In this section
we consider
BY ARBITRARY
the system
q+Q+,fPq=g(t),
DIAGONAL
MATRICES
(2.1),
4(O) = UTM&,, t 3 0, (3.1) I and evaluate the error introduced by replacing C with an arbitrary fixed diagonal matrix C, =diag(2r],w,, . . . , 277,,0~), where for all i = 1,. . . , n, 7, > 0. By i,, i,. and i,, respectively, we denote the set of indices i, for which 0 < vi < 1, n, = 1, 2nd ni > 1. Let p denote the solution of the decoupled equations resulted from replacing C with C,. Thus we have q(0) = 0,
jj+c,li+n’p=g(t),
f 2 0,
with p(O) = q(0) = 0 and d(O) = 4(O). Subtracting denoting the error q-p by e, we obtain
equation
(3.2)
(3.2) from equation
;;+C,f?+fl’e+(C-C,)q=O. By an approach
similar
to that in the previous
(3.3)
section,
we obtain
Z(s) = &)q*(s), where
e^ is the Laplace
transform
(3.1) and
(3.4)
of e, and
d(s)=-S(znS2+CdS+n2)-‘(~-cc,).
(3.5)
The ith row of C?(s) is
(3.6) Furthermore,
we obtain (3.7)
where
JIGI/ is the L,-induced
norm of the linear
operator
Gq=G”q=e, A and given by (Gq)(t)=(G*q)(t)=L-‘(G(s)q*(s)),
G, defined
by (3.8)
ta0.
In this case
IIGil= maxk, g,,go>,
(3.9)
where g, = max 6,
~i+245,-77,1 471,exp(V(77,)) 2r)i0i
g,. = max
vi +245, - ?il 277iwi
4
g, = max ill
O< TJi< 1,
(3.10)
l-exp(Wni))
ui+2wi15z-t)il
2WI
477i
(4 e-‘I,
rli=
exp (X(771)) Y(%)
’
1,
(3.11)
rl,’ 1,
(3.12)
with (TV,V, W, X and Y given by equations (2.11), (2.19a), (2.19b), (2.22a) and (2.22b), respectively. Equation (3.8) is an exact expression for the approximation error. Applying inequalities (Bl) and (B5) of Appendix B to equations (3.10) and (3.12).respectively, we obtain 4??_‘pG
IJCII<2p,
(3.13)
APPROXIMATE
DECOUPLING
OF
LINEAR
57
SYSTEMS
where p=
max I s Is ,I
ai+wi15i-17il
21714
(3.14) .
Obviously, expression (3.13) is an extension of expression (2.30). t-11p(( o f a neighborhood, centered Likewise, we evaluate the half-width the exact solution q lies. We have that
at p, in which
r= lIGll/(l -II~ll). 4. ON MINIMIZATION
OF ERROR
(3.15)
BOUND
of development is followed. We consider C, = In this section, a different direction n,,) as an unknown matrix, and assume that the normalized damping matrix djag(nl,..., C is diagonally dominant, that is, us G
(4.1)
25iwr,
for all i = 1,. . . , n. We determine C,, namely n,, in order to have minimum upper bound for the norm of error between the solution of the approximately decoupled system (3.1) and the exact solution of system (3.1). It will turn out that the error upper bound is minimum when 7, = &, for all i = 1, . . . , n; for this reason neglecting the off-diagonal elements of e is probably the most appropriate approach to decouple system (3.1), when C? is diagonally dominant. The error upper bound in expression (3.7) is the smallest when IJGll is minimum. In order to obtain the minimum of )IG (1, we first establish the following facts: (i) The function r.(~)=~(~i+20,1&-~)
exp(V(n)) l-exp(W(t7))’
O
(4.2)
is minimum at 77= 5,. (ii) The function (4.3) is minimum at 77= 6. (iii) The function (4.4) is minimum at 77= 6. Clearly, the function yc is minimum at n = 5, = 1. In the following, we prove (i) and (iii) separately. Case (i). The function y,, is differentiable over (0, I), except at n = 6. Let 77> 5, ; then I&,-nl=n-6, and
=1
Yi4(17)
wi
exp(V(?7))
{2wi[(l+nV’(n)(l-exp(W(n)))
[l-exp(W(?)))]*
+17W’(77)ev(W77))1 +((+,-25;w,)[Vf(rl)(1-exp(W(77)))+
W77)exp(W17))11.
(4.5)
S.
58
M. SHAHRUZ
Using expressions (4.1) and (B2), (B3) and (B4) of Appendix B, we conclude that y:( 7) > 0, for .$,< r) < 1. Hence, y,, is monotonically increasing over (tl, l), and achieves its infimum at 5,. Let n < 5, ; then 15, - n\= & - 7, and
Y:(7))
exp (V(77)) =2[l -exp i-24(1 (Wrl))l’
+ vV’(77)(1 -exp (W(77)))
wi
+77W’(77)exp ( Wrl))l +(~,+22S,w,)[V’(77)(1_exp(W(77)))+W77)exp(W(?7))11.
(4.6)
Using expressions (B2), (B3) and (B4) of Appendix B, we conclude that yI,( r]) < 0, for O< n < &. Hence, y,, is monotonically decreasing over (0, &), and achieves its infimum at [,. Therefore, yu is minimum at n = 5,. Case (iii). The function y(, is differentiable over (1, cc), except at 7 = 6,. Let r] > 5, ; then \ti-n(=n-[t, and
+tg, Using
-2!%4)[X’(77)Y(v)- Y’(rl)ll.
(4.7)
expressions
YL(V)>O,for&
its infimum
(4.1) and (B6), (B7) and (B8) of Appendix B, we conclude that < co. Hence, y(, is monotonically increasing over (&, CO),and achieves at 5. Let 7~6,; then I&i-n[=~,-n~ and
YWW,
2 ew (X(77)) {-2wL(l+ 77X’(?))Y(T) - rlY’(v)l
w?)
+(a,+25iw,)[X'(77)Y(rl)-
y’h)lj.
(4.8)
Using expressions (B6), (B7) and (B8) of Appendix B, we conclude that y:,(n) < 0, for 1~ n < 5,. Hence, y,, is monotonically decreasing over (1, tZ), and achieves its infimum at 5,. Therefore, y,, is minimum at 77= 5,. Recall that 11G II IS ’ g’iven by equation (3.9), in which g,,, g,. and g,, are given by equations (3.10), (3.11) and (3.12), respectively. In order to achieve the minimum of IIc?II, we have to minimize g,, g,. and g,, by choosing 7,. By (i), (ii) and (iii), we conclude that if ni = [,, and i,, = I,,, i, = I,. and for all i = 1, . . , n, then the functions g,,, g,. and g,, are minimum, i,,==I,,.Thus, if q,=,$,, forall i=l,..., n, then IIGll is minimum, and is equal to IIfill given in equation (2.3). This shows that the diagonal matrix which minimizes the error upper bound is obtained by simply neglecting the off-diagonal elements of the diagonally dominant normalized damping matrix. 5. EXAMPLES
In this section, we give two examples to illustrate possible application of the theory developed in the previous sections; for convenience, low order systems are considered. Example 1. We consider a system represented in the normalized form by [:,
:][3+[_gZi -;::][;;]+[; :.41][;;]=[:]sin2r, (5.1)
t 3 0. We have w, = 2, w2 = 2.1, 5, = & = 0.5, I,, = { 1,2}, I,, = 4, and I,, = 4 (4 denotes empty set). An approximate solution of equations (5.1) is obtained by solving decoupled equations [b
:][::I+[:
:.J[;::]+[:
04.411[~.11=[:lsin2r~
the the
(5.2)
APPROXIMATE
t 5 0. The steady
state solution
DECOUPLING
of equations
OF LINEAR
(5.2) is
0.25 sin (2t - 1.571) ss(t) =
0.237 sin (2t - 1.473)
1’
59
SYSTEMS
t 20.
(5.3)
Forthe system (5.2), llqa I( = llqsjl = 0.25. By equations (2.18), (2.20) and (2.21), respectively, (2.23) and (2.31), we obtain we obtain h, = 0.196, h, = 0 and h,, = 0. By expressions IIe I/ s 0.243 11qa 1)= 0.061. The exact solution q, and the corresponding approximate soluof qa, , in which q, lies are plotted in Figure 1. Since tion qa, , as well as a neighborhood the exact solution has been calculated, we have j/q11= 0.29, and by expression (2.24), 11 ell G 0.196llqll = 0.057. This error upper bound is the tightest possible in the functional form specified by expression (2.25), where the coefficient m is independent of the driving force. Example 2. We consider a system represented in the normalized form by 1
0
0
0
1
0
[0
0
1
I[][ 4,
2
&
+
ij3
-0.15
-I:”
-:::“I[
-0.15
-0.2
6.6
“:I+[: C&
0
Y.41
:][
:I]=[
:.2]l(t),
0
9
q3
2.5 tzo,
(5.4)
where l(t), t 2 0 denotes the unit-step function. We have w, = 2, w2 = 2.1, C+ = 3, 5, = 0.5, & = 1, & = 1.1, I, = {l}, I, = (2) and I,, = (3). An approximate solution of system (5.4) is obtained by solving the decoupled equations
The solution
of system 0.25(1-l.55eP’sin(l*732t+1.047)) &Z(f) =
0.375
0.272(1-e-2”‘(1+2+lt)) 0.278(1+0.7 ep4.675’_ 1.7 ,-1+25’)
Naghborhood
1 ,
tZ0.
(5.6)
of q,,
t
0,125
0.0
-0,125
-0.25 0.0
2.5
5.0
7.5
IO.0
f Figure 1. The exact solution neighborhood of q‘,, , in which
q, of the system q, lies.
(5.1), the corresponding
approximate
solution
q,,
, and the
60
S. M. SHAHRUZ
For the system (5.5),I(qaII = //qo,I/ =O-29. By equations (2.18),(2.20)and (2.21), respectively, we obtain h, = O-196, h,. = O-123 and h,, = 0.08. By expressions (2.23) and (2.31), we obtain IIell G 0.243 II qa II= 0.07. The exact solution q1 and the corresponding approxiof qa, , in which q, lies are plotted in Figure mate solution qa, , as well as a neighborhood 2. Since the exact solution has been calculated, we have llql/ = 0.3, and by expression (2.24), /IelI G 0.196llqll = 0.059.
f
NeIghborhood of q.,
Figure 2. The exact solution neighborhood of qu,, in which
q, of the system
(5.4), the corresponding
approximate
solution
qul, and the
q, lies.
6.
CONCLUSIONS
One common procedure in solving the normalized equations of motion of an n-degreeof-freedom linear system with non-zero off-diagonal damping elements is to replace the normalized damping matrix e by a selected diagonal matrix C,, and hence decouple the system equations. In this paper, we considered an n-degree-of-freedom damped linear system and studied the errors between the exact solution of the system and that of the corresponding decoupled system. We obtained tight upper bounds on the norm of errors. We derived the error between the exact solution ?f the system and that of the approximately deco_upled system when Cd is obtained from C by neglecting the off-diagonal elements of C (see equation (2.7)); we gave an upper bound on the norm of this error (see expression (2.24)). We derived an upper bound on the norm of error in the solution of the approximately decoupled equations when 6 is replaced by an arbitrary fixed diagonal matrix C, (see expression (3.7)). The upper bounds on the norm of errors are the tightest possible in the functional form specified-by expression (2.25). We gave a proof of the following fact: Let C, be obtained from C by neglecting the off-diagonal elements of c; if the off-diagonal elements of d are small, then the error in the solution of the approximately decoupled system is small. Furthermore, we showed that, if 6 is diagonally dominant, then among diagonal matrices C, the one which results in the minimum error upper bound is obtained by simply neglecting the off-diagonal elements of c.
APPROXIMATE
DECOUPLING
61
OF LINEAR SYSTEMS
REFERENCES 1. L. MEIROVITCH 1967 Analytical Merhods in Vibrations. New York: Macmillan. The Netherland: 2. P. C. MIIILLER and W. 0. SCHIEHLEN 1985 Linear Vibrations. Dordrecht, Martinus Nijhoff. LORD RAYLEIGH 1945 The Theory ofSound, Volume 1. New York: Dover. T. K. CAUGHEY 1960 American Society of Mechanical Engineers Journal of Applied Mechanics 27, 269-271. Classical normal modes in damped linear dynamic systems. and M. E. J.O'KELLY 1965 American Society of Mechanical Engineers Journal T. K. CAUGHEY ofApplied Mechanics 32, 583-588. Classical normal modes in damped linear dynamic systems. K. A. FOSS 1958 American Society of Mechanical Engineers Journal of Applied Mechanics 25, which uncouple the equations of motion of damped linear dynamic 361-364. Co-ordinates systems. 7. D. L. CRONI N 1976 American Society of Mechanical Engineers Journal of Engineeringfor Industry 98, 43-47. Approximation for determining harmonically excited response of nonclassically damped systems. and C. W. LEE 1986 Journal of Sound and Vibration 111, 37-50. Dynamic 8. K. R. CHUNG reanalysis of weakly non-proportionally damped systems. 9. G. PRATER, JR.and R. SINGH 1986 Journal of Sound and Vibration 104,109-125.Quantification of the extent of non-proportional viscous damping in discrete vibratory systems. of 10. S. S. NAIR and R. SINGH 1986 Journal of Sound and Vibration 104,348-350. Examination the validity of proportional damping approximations with two further numerical indices. 11. K. H. YAE and D. J. INMAN 1987 American Society of Mechanical Engineers Journal of Applied Mechanics 54, 419-423. Response bounds for linear underdamped systems. 12. D. W. NICHOLSON 1987 American Society of Mechanical Engineers Journal of Applied Mechanics 54, 430-433. Response bounds for nonclassically damped mechanical systems under transient loads. and P.CARAVANI 1974 Earthquake EngineeringandStructural 13. W.T. THOMSON,T.CALKIN Dynamics 3, 97-103. A numerical study of damping. 14. S. M. SHAHRUZ and F. MA 1988 American Society of Mechanical Engineers Journal of Applied Mechanics 55, 716-720. Approximate decoupling of the equations of motion of linear underdamped systems. 15. L. S. SHIEH, M. M. MEHIO and H. M. DIB 1987 Institute of Electrical and Electronics Engineers Transactions on Automatic Control AC-32, 231-233. Stability of the second-order matrix polynomial. 16. C. T. CHEN 1984 Linear System Theory and Design. New York: Holt, Rinehart and Winston. 1975 Feedback Systems: Input-Output Properties. New 17. C. A. DESOER and M. VIDYASAGAR York: Academic Press.
APPENDIX Derivation
of equation
(2.18).
For each
at t = 0, and is zero at tk = {l/wi( by equation (2.14). We have
I^ Ihi, 0
dT= I”“,,~~) 0 1 =-exp wi +lexp w,
A
i in I,,, hii is given by equation
(2.13);
h,,(t)
= 1
kn + 4;), for k = 0, 1, 2, . . . , where +I is given
1 - tf)“‘}(
dT+
(
h -cl_t:)l12tan
-
(
-, (1 - sfY 5,
)
_, (l-[j)""
5t (l-$)riztan
&
-I
('
-f"")
)[
ki,,
1+2,$,exp((~~~J
exp( cl12;,2).
(Al)
62
5..M.SHAHRUZ
Since jJ exp(-~~,k/(1--~~)‘~‘)=1/{1-exp(-~~,/(1-~~)”~)}, I,-0 equation
(2.18) follows. Derivation of equation (2.20). For at t = 0, and is zero at t = l/w,. The Hence, 52 lhii(r)I dr = 2 Jbl”’ I?,,(T) Derivation of equation (2.21). For write equation (2.16) as
each i in I,, h,, is given by equation (2.15); h,,(t) = 1 function h,, in equation (2.15) satisfies I,” A;,( T) d7 = 0. dT= 2 e-‘/w,, and equation (2.20) follows. each i in I,,, h,, is given by equation (2.16); we can
h,,(t)={l/(b-~)}(f7e~~~‘-~
t 30,
em”‘),
(AZ)
where a=w;(&-([f-l)‘:‘),
b=W;(&+([f-1)“‘).
(A3)
The function hii( t) = 1 at t = 0, and is zero at t* = { l/( b - a)} In (b/u). Furthermore, h,, satisfies j; hi;( 7) dr = 0. Hence Jz lhii(T)ldT=2 Ji hii(~) dr=2exp (X(&))/w,Y([,), where X and Y are given by equations (2.22a) and (2.22b), respectively. Thus equation (2.21) holds.
APPENDIX
The following
inequalities ?T
h(rl)=
have been drawn
-‘~Y,Lt)=
B
upon
in sections
5 exp (V(5))
(Bl)
OGlSl;
l-exp(W(&))~e~‘7
V’(rl)(l-exp(W(q)))+
2, 3 and 4:
OSnGl;
W’(n)exp(W(71))<0,
(B2) (B3)
and y3(v)+~, as v+Oo; y4(77)=(l+nV’(77))(1--xp(W(n)))+nW’(~)exp(W(n))~O, and the equality
Oansl,
holds only for n = 0; e ‘,z,([)=5exp(x(S))<0 Y(5)
and z,(t)+O.5
Y’(‘I)
rlZ 1,
(X(n))/
Y’(n)>O,
as ~+cc; (B8)
921,
as n+m.
Proof of expression
(Bl ). We have
exp ( V(5)) “(‘)=[l
(B7)
7721,
z4(77)=(1+77X’(n))Y(n)-rlY’(n)>O, and z,(n)+0
(B5)
52 1,
as n-+cc; z3(rl) =exp
and z3(n)+0
.5 5
as (+a; zJn)=X’(n)Y(n)-
and zz(~)-+O
(B4)
-exp
( W([))12
[(I +tY’(5))(1
-ew
(W(5)))+5W’(5)
exp
(W(O)l.
(B9)
APPROXIMATE
DECOUPLING
OF
LINEAR
SYSTEMS
63
By expressions (B3) and (B4), to be proved in the following, y{(t) > 0, for 0 < 5~ 1, and y{(O) = 0. Therefore, y, is monotonically increasing over [0, 11, and achieves its minimum and maximum at 5 = 0 and 5 = 1, respectively. Using L’HBpital’s rule we obtain y,(O) := 7~~’ and y,(l) = ee’. Proof qf expression (B2). We have “,(77)
T(l -12Y2 -tan’
((1-n’)“2/n) (1 _ 71?)3/2
=
We let n = cos 13in equation
(BlO)
(BlO); then
V’(B)=
sin 28 -26’
0s
9
2sin”o
0 S 7rJ2.
(Bll)
By L’H6pital’s rule, V’( 0 = 0) = -2/3. For 0 s 0 d ~12, we have sin 20 < 20, and sin 8 > 0; hence, V’(0) ~0, for 0~ 8 G 7r/2. Thus, for 0~ n S 1, V’(n)
having
W defined
by equation
(B12)
(2.19b),
for OG n G 1, we have (B13)
V’(n)(l-exp(W(n)))cO, where -7rJ2.
the equality We have
holds
only for r] = 0. By L’HBpital’s
W’(77)exp(W(77))= (1 _ -77 $)3/2 for 0~ ‘15 1. Clearly, W’(q) exp (W(T))
V’(1) (1 -exp
(W(1)))
for
Ocn<
=
(B14)
exp(ll_~),/?).
W’(rl)exp(W17))~0, where equality holds Proofofexpression Furthermore, ~~(7) + (B3) holds. Proof of expression and hence is omitted. Proof of expression
rule,
1. By
L’H6pital’s
OGnGl,
rule,
(B15)
only for n = 1. Hence, (B2) follows. (B3). By expression (2.19a), it is clear that y,(n) > 0, for O< 77< 1. cc as n + 0, and by L’Hopital’s rule, y?( 1) = e-‘. Hence, expression (B4).
Proof of expression
(BS).
We have
z,(5)
=
1
(B4) is straightforward,
but laborious,
ev W(5)) [Cl +5x’(5)) Y(5) - tx5)l. Y’(5)
(B16)
By expressions (B7) and (B8), to be proved in the following, z{( 5) > 0, for 1 < 5 < ~0, and z{( 1) = 0. Thus z, is monotonically increasing over [ 1, OO), and is minimum at .$ = 1, and achieves its supremum as t-*00. We let 4=ln (4+(&l)“‘); then [*(&1)“2=e’ti, and expression (B5) reads z,($)=$(l+e-‘“)
exp
By L’HGpital’s rule, we obtain (equivalently as 5 + a).
,
3 (
~~([=l)=z,(rl,=O)=e~‘,
)
$30. and
(B17) zl($)-+0.5
as 4-a
S. M. SHAHRUZ
64 Proof
of expression
zz(77) =X’(n) =
(B6). We have Y(n) - Y’(n)
77+(+1)“Z,n 2($_1)‘/2
n+(+l)“Z n_($-l)‘/~
1 n?_1
n+($-1)“’ ($I)‘/2
(Bl8)
*
We let $=ln(n+(n2~1)“‘). Then,
~f(n1-1)“2=e*Q,and(n1-l)“2=
(B19)
sinh 7; thus equation
(Bl8)
can be simplified
to e”‘(24 - sinh 2+) zz($) =
2 sinh3 IJ
(B-20)
* 2 0.
’
By L’HGpital’s rule, z2 at I,G= 0 (equivalently at 77= 1) is -2/3. For $ > 0, we have sinh 2$>2+, and sinh I+!J>O; hence, z2($)
e*+
z*(‘b)=------
e"
(B21)
sinh 4 tanh $ ’
sinh3 Q
as r] + cc). Then it is easy to conclude that z2( 4) -+ 0 as Cc,+ 00 (equivalently Proof of expression (B7). Applying the change of variable (B19) to expression we obtain z3( I/I) = em’* exp
2*
( 1 l-ezG
’
(B7),
(B22)
* 30.
Clearly, z3( 4) > 0 for rF,> 0. By L’Hepital’s rule, z3( 7) = 1) = z3( $ = 0) = e-‘, and z3( 4) + 0 as + + cc (equivalently n + co). Proof of expression (B8). Applying the change of variable (B19) to expression (B8), we obtain e”( $ - tanh +!I)cash + zq($) =
sinh3 $
By L’Hhpital’s rule, Z,(T) = 1) = zq(+ = 0) = l/3. sinh $ > 0; hence, z~(I,!J)> 0 for Cc,> 0 (equivalently
z,(‘b)= .
e”*
smh’ J,!Jtanh $
Then it is trivial
to conclude
’
(B23)
$20.
For $> 0, we have r] > 1). We write
$> tanh $, and
____e*
(~24)
sinh’ $ ’
that zq( $) + 0 as $ + a~ (equivalently
n + CO).