Advanced Methods of Structural Analysis

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 531

constraints which prevent each displacement of the mass. Thus, the total number of
constraints equal to the number of degrees of freedom. Let us consider a structure
with concentrated massesmi;iD 1;:::;n(Fig.14.11a). This structure hasn
degrees of freedom. They are lateral displacements of the frame points at the each
mass. Primary system of the displacement method is shown in Fig.14.11b.

m 1

mn

m 2

n

m 1

mn

1

m 2

2

ab

r 11

m 1

mn rn 1

Z 1 =1

m 2

r 21

c First unit state

r 12 mn

rn 2

1

r 22

m 1 m 2

Second unit state

m 1

mn

r 1 n

rnn

m 2

r 2 n

n-th unit state

Zn=1

Fig. 14.11 (a) Design diagram; (b) Primary system; (c) Unit states

Displacement of each lumped mass (or displacement of each introduced con-
straints) areyi. Inertial forces of each mass may be presented in terms of unit
reactionsrikas follows

m 1



y 1 Dr 11 y 1 Cr 12 y 2 C:::Cr1nyn;

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

mn



ynDrn1y 1 Crn2y 2 C:::Crnnyn:

(14.7)

The coefficientrikpresents the reaction ini-th introduced constraint caused by
unit displacement ofk-th introduced constraint. The termrikykmeans reaction in
i-th introduced constraint caused by real displacement ofk-th introduced constraint.
Each equation of (14.7) describes the equilibrium condition.
Equations (14.7) lead to the following differential equations of undamped free
vibration of the multi-degree of freedom system

m 1



y 1 Cr 11 y 1 Cr 12 y 2 C:::Cr1nynD0;

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

mn



ynCrn1y 1 Crn2y 2 C:::CrnnynD0:

(14.7a)

These equations are coupled statically, because the generalized coordinates ap-
pears in each equation.
In matrix form, the system (14.7a) may be written as

M



YCSYD 0 ; (14.7b)