Advanced Methods of Structural Analysis

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 533

can find the ratios between different amplitudes. If a structure has two degrees of
freedom, then system (14.9) becomes



r 11 m 1!^2



A 1 Cr 12 A 2 D0;

r 21 A 1 C



r 22 m 2!^2



A 2 D0:

From these equations, we can find following ratios

A 2

A 1

D

r 11 m 1!^2

r 12

or

A 2

A 1

D

r 21

r 22 m 2!^2

: (14.11)

If we assumeA 1 D 1 , then entries

1A 2

̆T
defines for each eigenfrequency, the
corresponding column®of the modal matrixˆ. The formulas (14.11)and(14.6)
lead to the same result.
Let us show application of the displacement method for free vibration analysis of
a beam with three equal lumped masses (Fig.14.12a); previously this structure had
been analyzed by force method (Example14.3). It is necessary to find eigenvalues
and modal matrix.

m 1 m 2 m 3

m 1 m 2 m 3

EI

123

aaaa

EI

y 1 y 2 y 3

a

1

1 2 3

3.6429

2.5714

0.6429

r 11

r 21 r 31

2

1 1 3

M 2

M 1

2.5714 2.5714

4.2857

r 12

r 22

r 32

r 11 =9.8572 EI/a^3

3.6429 6.2143

1

r 12 = –9.4285 EI/a^3

6.8571

1

2.5714

b

Fig. 14.12 (a) Design diagram of the beam and primary system; (b) Unit displacements and cor-
responding bending moment diagrams (factorEI=a^2 ); calculation ofr 11 andr 12