Advanced Methods of Structural Analysis

14.2 Free Vibrations of Systems with Finite Number Degrees of Freedom: Force Method 521

Each equation of (14.2) presents the compatibility condition. The differential
equations of motion are coupled dynamicallybecause the second derivative of all
coordinates appears in each equation.
We can see that the idea of force method has not been used above. Alternatively,
these same equations may be obtained by force method. In this case, unknown
inertial forces should be considered as primary unknowns of the force method.
Therefore, hereafter (14.2) will be called the differential equations of free undamped
vibration indisplacementsor canonical equations in form of theforce method.
In matrix form this system may be presented as

FM



YCYD^0 ; (14.2a)

whereFis the flexibility matrix (or matrix of unit displacements),Mis the diagonal
mass matrix andYrepresents the vector displacements

FD

2

6

6

4

ı 11 ı 12 ::: ı1n

ı 21 ı 22 ::: ı2n

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

ın1 ın2 ::: ınn

3

7

7

5 ;MD

2

6

6

4

m 1 0 ::: 0

0m 2 ::: 0

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

0 0 ::: mn

3

7

7

5 ;YD

2 6 6 6 6 6

y 1

y 2

:::

yn

3 7 7 7 7 7

:

(14.2b)

14.2.2 Frequency Equation..........................................

Solution of system of differential equations (14.2)is

y 1 DA 1 sin.! tC' 0 /; y 2 DA 2 sin.! tC' 0 /; y 3 DA 3 sin.! tC' 0 /;

(14.3)

whereAiare the amplitudes of the corresponding massesmiand' 0 is the initial
phase of vibration
The second derivatives of these displacements over time are



y 1 DA 1!^2 sin.! tC' 0 /;



y 2 DA 2!^2 sin.!tC' 0 /;



ynDAn!^2 sin.!tC' 0 /: (14.3a)

By substituting (14.3)and(14.3a)into(14.2) and reducing by!^2 sin.!tC' 0 /
we get

m 1 ı 11!^2  1



A 1 Cm 2 ı 12!^2 A 2 C:::Cmnı1n!^2 AnD0;

m 1 ı 21!^2 A 1 C



m 2 ı 22!^2  1



A 2 C:::Cmnı2n!^2 AnD0;

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

m 1 ı 31!^2 A 1 Cm 2 ı 32!^2 A 2 C:::C



mnınn!^2  1



AnD0:

(14.4)