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4 Multi-Degree of Freedom (MDOF) Systems
The presentation is organized as follows. Methods for calculating the forced
vibrations of damped MDOF systems are discussed in Sect. 4.4.1. These methods
are applied in Sect. 4.4.2 to find the response of MDOF to seismic events. The
remaining subsections are special cases of that in Sect. 4.4.1. They first suggest
how to specialize the results of Sect. 4.4.1 and then, for completeness, construct
the solutions for the special cases under consideration by direct arguments. The
free vibration of undamped MDOF systems is examined in Sect. 4.4.3. Torsional
vibration in Sect. 4.4.4 provides an illustration for these systems. The forced
vibration of undamped MDOF systems is considered in Sect. 4.4.5. Experimental
estimation of modal frequencies in Sect. 4.4.6 provides an application for this case.
The free vibration of damped systems is examined in Sect. 4.4.7.
4.4.1 Damped Systems: Forced Vibration
We have seen that the displacement vector x(t) defined by Eq. 4.2 with initial
displacement x 0 and velocity ˙
x 0 admits the representation in Eq. 4.19 since it is
the element of R n at any time t ≥ 0 and the eigenvectors { i } span this space.
We have seen that the eigenvectors { i } of this representation can be obtained from
the mass and stiffness matrices of the dynamical system under consideration. To
find the projections {q i (t)} of x(t) on the eigenvectors { i }, referred to as modal
coordinates, we require that x(t) given by Eq. 4.19 satisfies the equation of motion
Eq. 4.2, i.e.,
m ¨
q(t) + c ˙
q(t) + k q(t) = f(t).
(4.21)
To break this system of coupled equations into a set of independent equations for
the modal coordinates {q i (t)}, we use the orthogonality of modal shapes. The left
multiplication of the above equation by T gives
T m ¨
q(t) +
T c ˙
q(t) +
T k q(t) =
T f(t),
(4.22)
or
diag{ ˜
m i } ¨
q(t) +
T c ˙
q(t) + diag{ ˜
k i } q(t) =
T f(t)
(4.23)
by the orthogonality of the modal shapes (see Eq. 4.10), where diag{ ˜
m i } and
diag{ ˜
k i } are (n, n)-diagonal matrices whose non-zero entries are the modal masses
{m i } and modal stiffnesses {k i }.
Generally, the matrix T c is not diagonal so that it is not possible to decouple
the system of equations of motion. However, it is diagonal under the assumption of
proportional damping in Eq. 4.20. Under this assumption, Eq. 4.23 becomes
diag{ ˜
m i } ¨
q(t) + diag{ ˜
c i } ˙
q(t) + diag{ ˜
k i } q(t) =
T f(t),
(4.24)
4 Multi-Degree of Freedom (MDOF) Systems
The presentation is organized as follows. Methods for calculating the forced
vibrations of damped MDOF systems are discussed in Sect. 4.4.1. These methods
are applied in Sect. 4.4.2 to find the response of MDOF to seismic events. The
remaining subsections are special cases of that in Sect. 4.4.1. They first suggest
how to specialize the results of Sect. 4.4.1 and then, for completeness, construct
the solutions for the special cases under consideration by direct arguments. The
free vibration of undamped MDOF systems is examined in Sect. 4.4.3. Torsional
vibration in Sect. 4.4.4 provides an illustration for these systems. The forced
vibration of undamped MDOF systems is considered in Sect. 4.4.5. Experimental
estimation of modal frequencies in Sect. 4.4.6 provides an application for this case.
The free vibration of damped systems is examined in Sect. 4.4.7.
4.4.1 Damped Systems: Forced Vibration
We have seen that the displacement vector x(t) defined by Eq. 4.2 with initial
displacement x 0 and velocity ˙
x 0 admits the representation in Eq. 4.19 since it is
the element of R n at any time t ≥ 0 and the eigenvectors { i } span this space.
We have seen that the eigenvectors { i } of this representation can be obtained from
the mass and stiffness matrices of the dynamical system under consideration. To
find the projections {q i (t)} of x(t) on the eigenvectors { i }, referred to as modal
coordinates, we require that x(t) given by Eq. 4.19 satisfies the equation of motion
Eq. 4.2, i.e.,
m ¨
q(t) + c ˙
q(t) + k q(t) = f(t).
(4.21)
To break this system of coupled equations into a set of independent equations for
the modal coordinates {q i (t)}, we use the orthogonality of modal shapes. The left
multiplication of the above equation by T gives
T m ¨
q(t) +
T c ˙
q(t) +
T k q(t) =
T f(t),
(4.22)
or
diag{ ˜
m i } ¨
q(t) +
T c ˙
q(t) + diag{ ˜
k i } q(t) =
T f(t)
(4.23)
by the orthogonality of the modal shapes (see Eq. 4.10), where diag{ ˜
m i } and
diag{ ˜
k i } are (n, n)-diagonal matrices whose non-zero entries are the modal masses
{m i } and modal stiffnesses {k i }.
Generally, the matrix T c is not diagonal so that it is not possible to decouple
the system of equations of motion. However, it is diagonal under the assumption of
proportional damping in Eq. 4.20. Under this assumption, Eq. 4.23 becomes
diag{ ˜
m i } ¨
q(t) + diag{ ˜
c i } ˙
q(t) + diag{ ˜
k i } q(t) =
T f(t),
(4.24)