4.5 Time Domain Analysis: Non-proportional Damping
99
for the homogeneous version of Eq. 4.77, i.e., f(t) = 0, where the 2 n-dimensional
column vector w and the scalar λ are unknown and need to be determined. We show
that the solutions of this type can be used to construct sets of 2 n-dimensional vectors
that span R 2 n so that z(t) can be represented in the coordinate systems defined by
these sets of vectors.
The expressions of z(t) in Eq. 4.79 and Eq. 4.77 with f(t) = 0 give
w λ e
λ t
= a w e
λ t or, equivalently,
a − λ I
w e
λ t
= 0,
which implies
a − λ I
w = 0,
(4.80)
since
a − λ I
w e λ t = 0 must hold at all times and exp(λ t) is not zero for bounded
λ. The condition of Eq. 4.80 defines a linear homogeneous system of equations for
w that admits the trivial solution if det
a − λ I
= 0 and non-trivial solutions, in
addition to the trivial solution, if det
a − λ I
= 0. Since the trivial solution is not
possible for non-zero initial conditions (z(t) = 0 at all times if w = 0), we require
det
a − λ I
= 0.
(4.81)
For simplicity, we assume that the 2 n solutions {λ i }, i = 1, . . . , 2 n, of these
equations, i.e., the eigenvalues of the nonsymmetric matrix a, are distinct.
As for classical modes of vibration, the solutions of the above eigenvalue
problem involve the following two steps.
– Step 1. Eigenvalues: The roots λ 1 , . . . , λ 2 n of the 2 n-degree polynomial det
a −
λ I
in λ, i.e., the solutions of Eq. 4.81, are the eigenvalues of matrix a. Since
det
a − λ I
= 0 and det
a − λ I
T = det
a T − λ I
= 0 have the same roots
(the determinant of a matrix coincides with the determinant of its transposed),
the system has a single set of 2 n eigenvalues. Generally, the eigenvalues are
complex-valued as a is not symmetric.
– Step 2. Eigenvectors: There are two sets of eigenvectors, right eigenvectors and
left eigenvectors, which correspond to the non-trivial solutions of
a−λ I
w = 0,
and
a T − λ I
w = 0, i.e.,
a u i = λ i u i (Right eigenvectors)
a
T v i = λ i v i (Left eigenvectors).
(4.82)
We use the notations u and v for the right and left eigenvectors of matrix a.
99
for the homogeneous version of Eq. 4.77, i.e., f(t) = 0, where the 2 n-dimensional
column vector w and the scalar λ are unknown and need to be determined. We show
that the solutions of this type can be used to construct sets of 2 n-dimensional vectors
that span R 2 n so that z(t) can be represented in the coordinate systems defined by
these sets of vectors.
The expressions of z(t) in Eq. 4.79 and Eq. 4.77 with f(t) = 0 give
w λ e
λ t
= a w e
λ t or, equivalently,
a − λ I
w e
λ t
= 0,
which implies
a − λ I
w = 0,
(4.80)
since
a − λ I
w e λ t = 0 must hold at all times and exp(λ t) is not zero for bounded
λ. The condition of Eq. 4.80 defines a linear homogeneous system of equations for
w that admits the trivial solution if det
a − λ I
= 0 and non-trivial solutions, in
addition to the trivial solution, if det
a − λ I
= 0. Since the trivial solution is not
possible for non-zero initial conditions (z(t) = 0 at all times if w = 0), we require
det
a − λ I
= 0.
(4.81)
For simplicity, we assume that the 2 n solutions {λ i }, i = 1, . . . , 2 n, of these
equations, i.e., the eigenvalues of the nonsymmetric matrix a, are distinct.
As for classical modes of vibration, the solutions of the above eigenvalue
problem involve the following two steps.
– Step 1. Eigenvalues: The roots λ 1 , . . . , λ 2 n of the 2 n-degree polynomial det
a −
λ I
in λ, i.e., the solutions of Eq. 4.81, are the eigenvalues of matrix a. Since
det
a − λ I
= 0 and det
a − λ I
T = det
a T − λ I
= 0 have the same roots
(the determinant of a matrix coincides with the determinant of its transposed),
the system has a single set of 2 n eigenvalues. Generally, the eigenvalues are
complex-valued as a is not symmetric.
– Step 2. Eigenvectors: There are two sets of eigenvectors, right eigenvectors and
left eigenvectors, which correspond to the non-trivial solutions of
a−λ I
w = 0,
and
a T − λ I
w = 0, i.e.,
a u i = λ i u i (Right eigenvectors)
a
T v i = λ i v i (Left eigenvectors).
(4.82)
We use the notations u and v for the right and left eigenvectors of matrix a.