Advanced Methods of Structural Analysis

520 14 Dynamics of Elastic Systems

14.2 Free Vibrations of Systems with Finite Number Degrees

of Freedom: Force Method

Behavior of such structures may be described by two types of differential equations.
They are equations in displacement (i.e., in the form of the force method) and equa-
tions in reactions (i.e., in the form of thedisplacement method). In both cases, we
will consider only the undamped vibration.

14.2.1 Differential Equations of Free Vibration in Displacements

In essence, the first approach consists of expressing the forces of inertia as function
ofunit displacements. For the derivation of differential equations, let us consider a
structure with concentrated masses (Fig.14.7).

yn dn 1

m 1

d (^11) mn
m 1
mn dnn
d 1 n
dn 2
m 1
mn
d 12
m 1
mn
y 1
F 1 in F 1 in =^1 F 2 in =^1
Fnin Fnin =^1
F 2 in
y 2
m 2 m 2
d 21
m 2
d 22
m 2
d 2 n
Fig. 14.7 Design diagram and unit states
In case of free vibration, each mass is subjected to forces of inertia only. Dis-
placement of each mass may be presented as
y 1 Dı 11 F 1 inCı 12 F 2 inC:::Cı1nFnin;
y 2 Dı 21 F 1 inCı 22 F 2 inC:::Cı2nFnin;
:::::::::::
ynDın1F 1 inCın2F 2 inC:::CınnFnin;
(14.1)
whereıikis displacement ini-th direction caused by unit force acting in thek-th
direction.
Since the force of inertia of massmiisFiinDmi

yi, then the equations (14.1)
become
ı 11 m 1

y 1 Cı 12 m 2

y 2 C:::Cı1nmn

ynCy 1 D0;
:::::::::::::
ın1m 1

y 1 Cın2m 2

y 2 C:::Cınnmn

ynCynD0:
(14.2)