Advanced Methods of Structural Analysis

538 14 Dynamics of Elastic Systems

Equations (c) becomes

.1 1 / 2 D0;

 1 C2.1 1 / 2  3 D0;

 2 C2.1 1 / 3 D0:

Solution of these equation are 2 D

p

3=2;  3 D1=2.

The same procedure should be repeated for 2 D1:0and 3 D 1 C

p

3

2

The modal matrixˆis defined as

ˆD

2

4

p11 1

3=2 0 

p

3=2

1=2 11=2

3

5

14.3.4 Comparison of the Force and Displacement Methods

Some fundamental data about application of two fundamental methods for free vi-
bration analysis of the structures with finite number degrees of freedom is presented
in Table14.2.
Generally, for nonsymmetrical beams, the force method is more effective than the
displacement method. However, for frames especially with absolutely rigid cross-
bar, the displacement method is beyond the competition.

14.4 Free Vibrations of One-Span Beams with Uniformly

Distributed Mass

The more precise dynamical analysis of engineering structure is based on the as-
sumption that a structure has distributed masses. In this case, the structure has
infinite number degrees of freedom and mathematical model presents a partial
differential equation. Additional assumptions allow construction of the different
mathematical models of transversal vibration of the beam. The simplest mathemat-
ical models consider a plane vibration of uniform beam with, taking into account
only, bending moments; shear and inertia of rotation of the cross sections are ne-
glected. The beam upon these assumptions is called as Bernoulli-Euler beam.