Advanced Methods of Structural Analysis

14.1 Fundamental Concepts 515

Table 14.1 Types of elastic members and their characteristics
Design diagram CharacteristicP-y Design diagram CharacteristicP-y

y

P

y

P P

y 0 y 0 y

P

P

y 0 y 0

y

P P

y

P

Different types of forces acting on a structure lead to different types of vibration.
Among them are two general classes – they are free and forced vibration.
Vibrations of a system in which disturbing forces are absent are called free vi-
brations. At free vibration, the system is subjected to forces inherent to the system
itself, i.e., the restoring and resisting forces.
To impose free vibrations, nonzero initial conditions should be created, which
means some initial displacement and initial velocity. Free vibration may be linear or
nonlinear depending on the characteristics of restoring and resisting forces. Absence
of resisting forces leads to thefree undampedvibrations; in this case, the system is
subjected only to a restoring force.
Vibration of a system caused by any disturbing forces is called a forced vibration.
Absence of resisting forces leads to theforced undampedvibration. Just as the free
vibration, the forced vibration may be linear and nonlinear.

14.1.3 Degrees of Freedom..........................................

The fundamental difference of the concept of “degrees of freedom” in static and
structural dynamics in spite of the same definition (a number of independent param-
eters, which uniquely defines the positions of all points of a structure) is as follows:
In statics, the number of degrees of freedom is related to a structure consisting of
absolutely rigiddiscs. If the degree of freedom is greater than or equal to one, then
the system is geometrically changeable and cannot be assumed as an engineering
structure; when the degree of freedomequals to zero, it means that a system is
geometrically unchangeable and statically determined. In structural dynamics, the
number of degrees of freedom is determined by just taking into account thede-
formationof the members. If the degree of freedom equals to zero, then a system
presents an absolutely rigid body and all displacements in space are absent.
All structures may be divided into two principal classes according to their degrees
of freedom. They are the structure with concentrated and distributed parameters.
Members with concentrated parameters assume that the distributed mass of the
member itself may be neglected in comparison with the lumped mass, which is
located on the member. The structure with distributed parameters is characterized
by uniform or nonuniform distribution of mass within its parts.