190
6 Free Vibration of Multiple Degrees of Freedom System
m 2 ¨
x
(1)
2
inertia force of mass m 2 for the first mode and
x
(2)
2
displacement of mass m 2 for the second mode.
Equation (6.25) can thus be interpreted as the work done by the inertia forces
occurring in the first mode, in going through the displacements of the second mode,
is equal to zero. We know from elementary mechanics that if the line of action of the
force and displacement is orthogonal, the work done by the force is equal to zero.
Equation (6.25) reveals that normal modes are orthogonal. Thus, this is known as
orthogonality relationship.
Another physical interpretation of the orthogonality is described in the following:
The equivalent static force acting at the rth mode
{F
(r )
} = [K ] {φ
(r )
} a e
i p r t
(6.26)
and the corresponding displacement in the sth mode is
{x
(s)
} = {φ
(s)
} a e
i p s t
(6.27)
The work done by the equivalent static force in rth mode in undergoing
displacement in the sth mode is
W =
({φ
(r )
})
T
[K ] {φ
(s)
}
a
2 e
i p r t e
i p s t
(6.28)
which is zero due to the orthogonality relation given by Eq. (6.20).
Therefore, another interpretation of orthogonality properties is that the work done
by the equivalent static forces in the rth mode in undergoing displacement in the sth
mode is zero [2].
6.5 Eigenvalue Problem
For the solution of the eigenvalue problem, the most direct approach is to compute the
zeroes of the characteristic polynomial and then obtain eigenvectors for each root of
λ. The value of the determinant | [D] − λ [I ] | in Eq. (6.14) involves a computation,
which, if the size of the determinant is large, may lead to round-off errors. Hence,
when the size of [D] is large, the determination of the roots of the determinant is not
attempted.
For the general case, no explicit formulas are available for computation of roots
of f (λ) given by Eq. (6.15), if the degree of the polynomial is larger than four. As
such, various methods have been developed for the solution of the problem. As the
problem demands the determination of the roots of the polynomial, the proposed
methods are basically iterative in nature. However, for the economic evaluation of
the eigensystem, [M] and [K] matrices may be suitably modified. Various methods
6 Free Vibration of Multiple Degrees of Freedom System
m 2 ¨
x
(1)
2
inertia force of mass m 2 for the first mode and
x
(2)
2
displacement of mass m 2 for the second mode.
Equation (6.25) can thus be interpreted as the work done by the inertia forces
occurring in the first mode, in going through the displacements of the second mode,
is equal to zero. We know from elementary mechanics that if the line of action of the
force and displacement is orthogonal, the work done by the force is equal to zero.
Equation (6.25) reveals that normal modes are orthogonal. Thus, this is known as
orthogonality relationship.
Another physical interpretation of the orthogonality is described in the following:
The equivalent static force acting at the rth mode
{F
(r )
} = [K ] {φ
(r )
} a e
i p r t
(6.26)
and the corresponding displacement in the sth mode is
{x
(s)
} = {φ
(s)
} a e
i p s t
(6.27)
The work done by the equivalent static force in rth mode in undergoing
displacement in the sth mode is
W =
({φ
(r )
})
T
[K ] {φ
(s)
}
a
2 e
i p r t e
i p s t
(6.28)
which is zero due to the orthogonality relation given by Eq. (6.20).
Therefore, another interpretation of orthogonality properties is that the work done
by the equivalent static forces in the rth mode in undergoing displacement in the sth
mode is zero [2].
6.5 Eigenvalue Problem
For the solution of the eigenvalue problem, the most direct approach is to compute the
zeroes of the characteristic polynomial and then obtain eigenvectors for each root of
λ. The value of the determinant | [D] − λ [I ] | in Eq. (6.14) involves a computation,
which, if the size of the determinant is large, may lead to round-off errors. Hence,
when the size of [D] is large, the determination of the roots of the determinant is not
attempted.
For the general case, no explicit formulas are available for computation of roots
of f (λ) given by Eq. (6.15), if the degree of the polynomial is larger than four. As
such, various methods have been developed for the solution of the problem. As the
problem demands the determination of the roots of the polynomial, the proposed
methods are basically iterative in nature. However, for the economic evaluation of
the eigensystem, [M] and [K] matrices may be suitably modified. Various methods