A note on the damped vibrating
systems
Ranislav M. Bulatovi´c
∗
Theoret. Appl. Mech., Vol.33, No.3, pp. 213–221, Belgrade 2006
Abstract
The presence of pure imaginary eigenvalues of the partially damped
vibrating systems is treated. The number of such eigenvalues is
determined using the rank of a matrix which is directly related to
the system matrices.
Keywords: linear system, incomplete dissipation, pure imaginary
eigenvalue
1 Introduction and preliminaries
In this note a “damped vibrating system” is understood to be the classical
model of a linear, viscously damped elastic system with n degrees of
freedom. This system has equations of motion
A¨q + B ˙q + Cq = 0, q ∈ <
n
(1)
where A, B and C are n × n constant real symmetric matrices. The
inertia matrix A and stiffness matrix C are positive definite (> 0), and
the damping matrix B may be positive definite or positive semi-definite
(≥ 0). In the case B > 0 dissipation is complete, and the case B ≥ 0
corresponds to incomplete dissipation. In the latter case the system is
called partially dissipative (damped).
∗
Faculty of Mechanical Engineering, University of Montenegro, 81 000 Podgorica,
Montenegro
213
214 Ranislav M. Bulatovi´c
It is convenient, although not necessary, to rewrite equation (1) in the
form
¨x + D ˙x + Kx = 0, (2)
using the congruent transformation x = A
1/2
q, where A
1/2
denotes the
unique positive definite square root of the matrix A, and D = A
−1/2
BA
−1/2
,
and K = A
−1/2
CA
−1/2
.
All solutions x(t) of the equation (2) (or q(t) of (1)) can be character-
ized algebraically using properties of the quadratic matrix polynomial
L(λ) = λ
2
I + λD + K, (3)
where I is the identity matrix. The eigenvalues of the system are zeros
of the characteristic polynomial
∆(λ) = det(L(λ)) (4)
Since (4) is a polynomial of degree 2n with respect to λ, there are
2n eigenvalues, counting multiplicities. If λ is an eigenvalue, the nonzero
vectors X in the nullspace of L(λ) are the eigenvectors associated with
λ, i. e.,
L(λ)X = 0 (5)
In general, eigenvalues and corresponding eigenvectors may be real or
may appear in complex conjugate pairs.
If the dissipation is complete, it is well-known that the system (2) (or
(1)) is asymptotically stable (x(t) → 0 as t → ∞ for all solutions x(t)).
On the other hand, the partially damped system (2) may or may not
be asymptotically stable, although it is obviously stable in the Lyapunov
sense (any solution of equation (2) remains bounded). Consequently,
all eigenvalues of this system lie in the closed left-half of the complex
plane (Reλ ≤ 0). Notice that if the system is asymptotically stable, then
Reλ < 0.
Recently some attention has been paid to the question whether or not
a damped system has pure imaginary eigenvalues, i. e., in the terminology
of the mechanical vibrations, whether or not undamped motions (also
called “residual motions”) are possible in such system (see [1] and quoted
A note on the damped vibrating systems 215
references). From the above discussion it is clear that nonexistence of
undamped motions is equivalent to the asymptotic stability of the system,
and consequently, any test for asymptotic stability gives the answer of the
question. A survey of the stability criteria for linear second order systems
is given in [2]. Also, it should be mentioned that the paper [1] rediscovered
an old criterion for asymptotic stability of the system [3], as was recently
stressed in [4].
In this note we are interested in the determination of the number of
pure imaginary eigenvalues of the system without computing the zeros
of the characteristic polynomial (4). The main result given in the next
section (Theorem 1) is based on the well-known condition of asymptotic
stability [5], which coincides with the rank condition of controllability of
a linear system (see [6]).
2 Results
Introduce the n × n
2
matrix
Φ =
³
D
.
.
. KD
.
.
.
.
.
.
.
.
. K
n−1
D
´
(6)
which plays key role in a test for asymptotic stability of the system [5].
Theorem 1. The system (2) has r = n − rankΦ conjugate pairs of
purely imaginary eigenvalues, including multiplicity.
Corollary. If rankD = m, then 0 ≤ r ≤ n − m.
This follows immediately from rankD ≤ rankΦ ≤ n.
To prove Theorem 1 we need the following lemmas.
Lemma 1. Let (iω, X), ω ∈ <, i =
√
−1, be an eigenpair of L(λ).
Then (ω
2
, X) and (0, X)are eigenpairs of the matrices K and D, respec-
tively.
Proof. From
L(iω)X = (−ω
2
I + iωD + K)X = 0 , (7)
we obtain
< X, (K − ω
2
I)X > +iω < X, DX >= 0, (8)
216 Ranislav M. Bulatovi´c
where < ., . > denotes the inner product, and < X, ( K − ω
2
I)X >,
and < X, DX > are real quantities, since K and D are real symmetric
matrices. Then < X, DX >= 0, which implies DX = 0, since D ≥ 0.
This together with L(iω)X = 0 gives KX = ω
2
X. 2
It is clear that the eigenvector X in Lemma 1 can be taken to be unit
(< X, X >= 1) and real.
Lemma 2. a) If (iω
1
, X
(1)
) and (iω
2
, X
(2)
) are eigenpairs of L(λ) with
ω
2
1
6= ω
2
2
, then < X
(1)
, X
(2)
>= 0.
b) If the eigenvalue iω of L(λ) has multiplicity k, it possesses k eigenvec-
tors which are mutually orthogonal.
Proof. a) The result follows from Lemma 1 and the additional fact that
eigenvectors associated with distinct eigenvalues of a symmetric matrix
are orthogonal.
b) Since the system (2) is stable, the multiple eigenvalue iω must be
semi-simple, which means that the eigenvalue has k linearly independent
eigenvectors. Since a linear combination of these k vectors is also an
eigenvector of L(λ) associated with iω, the Gram-Schmidt process (see
[7]) can be used to obtain k mutually orthogonal eigenvectors. 2
Lemma 3. Let ±iω
1
, ..., ±iω
r
be eigenvalues of L(λ). Then there
exists an orthogonal matrix Q such that
Q
T
DQ =
ˆ
D =
µ
0
r
0
0
ˆ
D
n−r
¶
, (9)
and
Q
T
KQ =
ˆ
K =
µ
Ω
r
0
0
ˆ
K
n−r
¶
, (10)
where 0
r
is the zero square matrix of order r, and Ω
r
= diag(ω
2
1
, ..., ω
2
r
).
Proof. By lemmas 1 and 2, there exists an orthonormal set of r vectors
X
(1)
, ..., X
(r)
, such that
DX
(j)
= 0, KX
(j)
= ω
2
j
X
(j)
, j = 1 , . . . , r (11)
Now, consider an orthogonal matrix Q having the vectors X
(1)
, ..., X
(r)
as its first r columns,
Q = (X
(1)
, ..., X
(n)
) (12)
A note on the damped vibrating systems 217
The matrices D and K are then orthogonally congruent to matrices
ˆ
D and
ˆ
K, respectively, described by
ˆ
D = Q
T
DQ = (< X
(i)
, DX
(j)
>) (13)
and
ˆ
K = Q
T
KQ = (< X
(i)
, KX
(j)
>), (14)
where i, j = 1, . . . , n. Using (11) and < X
(i)
, X
(j)
>= δ
ij
, where δ
ij
is the
Kronecker delta and i, j = 1, . . . , n, we compute
< X
(i)
, DX
(j)
>= 0 (15)
and
< X
(i)
, KX
(j)
>= ω
2
j
δ
ij
, (16)
where i = 1, . . . , nandj = 1, . . . , r. The relations (15) and (16) show that
ˆ
D and
ˆ
K have the partitioned forms (9) and (10). 2
Proof of Theorem 1. Suppose that ∆(±iω
j
) = 0,ω
j
∈ <, j = 1, . . . , r
and that remaining zeros of ∆(λ) take places on the open left-half of
the complex plane. Then from Lemma 3 it follows that there exists an
orthogonal coordinate transformation
x = Q
µ
y
z
¶
, y ∈ <
r
, z ∈ <
n−r
, (17)
which transforms equation (2) to the form
µ
¨y
¨z
¶
+
ˆ
D
µ
˙y
˙z
¶
+
ˆ
K
µ
y
z
¶
=
µ
0
0
¶
(18)
where
ˆ
D and
ˆ
K have the partitioned forms (9) and (10). Under the above
assumptions it is clear that the (n − r) dimensional subsystem of (18)
¨z +
ˆ
D
n−r
˙z +
ˆ
K
n−r
z = 0, z ∈ <
n−r
(19)
is asymptotically stable and, according to well-known result [5], we have
rank
³
ˆ
D
n−r
.
.
.
ˆ
K
n−r
ˆ
D
n−r
.
.
.
.
.
.
.
.
.
ˆ
K
n−r−1
n−r
ˆ
D
n−r
´
= n − r (20)