Advanced Methods of Structural Analysis

532 14 Dynamics of Elastic Systems

where the mass and stiffness matricesas well as displacement vector are

MD

2

6

6

4

m 1 0 ::: 0

0m 2 ::: 0

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

0 0 ::: mn

3

7

7

5 ; SD

2

6

6

4

r 11 r 12 ::: r1n

r 21 r 22 ::: r2n

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

rn1 rn2 ::: rnn

3

7

7

5 ; YD

2 6 6 6 6 6

y 1

y 2

:::

yn

3 7 7 7 7 7

;

Solution of system (14.7)

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

(14.8)

whereAiare amplitudes of the displacement of massmiand' 0 is the initial phase
of vibration.
Substituting (14.8)into(14.7a) leads to algebraic homogeneous equations with
respect to unknown amplitudes of lumped masses



r 11 m 1!^2



A 1 Cr 12 A 2 C:::Cr1nAnD0;

r 21 A 1 C



r 22 m 2!^2



A 2 C:::Cr2nAnD0;

::::::::::

rn1A 1 Crn2A 2 C:::C



rnnmn!^2



AnD0:

(14.9)

14.3.2 Frequency Equation..........................................

Nontrivial solution (nonzero amplitudesAi) is possible, if the determinant of the
coefficients to amplitude is zero.

DD

2

6

6

4

r 11 m 1!^2 r 12 ::: r1n

r 21 r 22 m 2!^2 ::: r2n

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

rn1 rn2 ::: rnnmn!^2

3

7

7

5 D0: (14.10)

This equation is called the frequency equation in form of the displacement
method. Solution of this equation presents the eigenfrequencies of a structure. The
number of the frequencies of free vibration is equal to the number of degrees of
freedom.

14.3.3 Mode Shape Vibrations and Modal Matrix..................

Equations (14.9) are homogeneous algebraic equations with respect to unknown
amplitudesA. This system does not allow us to find these amplitudes. However, we