Advanced Methods of Structural Analysis

522 14 Dynamics of Elastic Systems

The equations (14.4) are homogeneous algebraic equations with respect to un-
known amplitudesA. Trivial solutionAi D 0 corresponds to the system at rest.
Nontrivial solution (nonzero amplitudesAi) is possible, if the determinant of the
coefficients of amplitude is zero.

DD

2

6

6

4

m 1 ı 11!^2 1m 2 ı 12!^2 ::: mnı1n!^2

m 1 ı 21!^2 m 2 ı 22!^2 1 ::: mnı2n!^2

::: ::: ::: :::

m 1 ın1!^2 m 2 ın2!^2 ::: mnınn!^2  1

3

7

7

5 D0: (14.5)

This equation is called the frequency equation in terms of displacements. So-
lution of this equation! 1 ;! 2 ;:::;!npresents the eigenfrequencies of a structure.
The number of the frequencies of free vibration equals to the number of degrees of
freedom.

14.2.3 Mode Shapes Vibration and Modal Matrix..................

The set of equations (14.4) are homogeneous algebraic equations with respect to
unknown amplitudesA. This system does not allow us to find these amplitudes.
However, we can find the ratios between different amplitudes. If a structure has two
degrees of freedom, then the system (14.4) becomes



m 1 ı 11!^2  1



A 1 Cm 2 ı 12!^2 A 2 D0;

m 1 ı 21!^2 A 1 C



m 2 ı 22!^2  1



A 2 D0:

(14.4a)

From these equations, we can find the following ratios

A 2

A 1

D

m 1 ı 11!^2  1

m 2 ı 12!^2

or

A 2

A 1

D

m 1 ı 21!^2

m 2 ı 22!^2  1

: (14.6)

If we substitute the first frequency of vibration! 1 into any of the two equations
(14.6), then we can find.A 2 =A 1 /! 1. Then we can assume thatA 1 D 1 and calculate
the correspondingA 2 (or vice versa). The numbersA 1 D 1 andA 2 defines the
distribution of amplitudes at the first frequency of vibration! 1 ; such distribution
is referred as the first mode shape of vibration. This distribution is presented in the
form of vector-column' 1 , whose elements areA 1 D 1 and the calculatedA 2 ;this
column vector is called a first eigenvector' 1. Thus the set of equations (14.4a)for
! 1 define the first eigenvector to within an arbitrary constant.
Second mode shape of vibration or second eigenvector, which corresponds to the
second frequency vibration! 2 , can be found in a similar manner. After that we can
construct a modal matrixˆD
' 1 ' 2

̆

.