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6 Free Vibration of Multiple Degrees of Freedom System
6.2.2 The Stiffness Matrix
For linear elastic systems, the stiffness matrix [K] is a symmetric matrix. The structure
consists of a number of elements. Total stiffness matrix is formed by assembling the
stiffness matrix of the individual elements. The assembly can be done in a systematic
manner in a computer by writing a suitable computer program. The stiffness matrix
of a structure for dynamic analysis is calculated by utilising the methods used in
standard structural analysis procedures.
6.2.3 The Damping Matrix
The damping matrix [C] given by Eq. (6.3) has been formed by assuming the system
to have viscous damping. The damping in a system depends on various factors. These
values are to be obtained experimentally. Unfortunately, not much data are available
on full-scaled structures. Further, prescribing the damping matrix in the form given
by Eq. (6.3) poses considerable difficulties in the analysis. As such, [C] matrix is
reduced to simpler forms for facilitating the analysis.
6.2.4 Loading Matrix
The dynamic loads are assumed to act at nodal points corresponding to the displacement degrees of freedom. Loads when acting in between nodal points, or when they
are distributed, are converted to equivalent values acting at respective nodal points.
6.3 Free Undamped Vibration Analysis of MDF Systems
The equation of motion for MDF system for free vibration can be written from
Eq. (6.4) as
[M] { ¨
x} + [K ] {x} = {O}
(6.5)
Assuming the solution in the following form
{x} = ae
i pt
{φ }
(6.6)
where
a
is a scalar of dimension L,