mechanical vibration

1.  INTRODUCTION:-
 There are systems such as beams, cables, rods etc,
which have their mass and elasticity distributed
continuously throughout the length. Such systems are
known as continuous systems. Since such systems
are supposed to be made of infinite numbers of
particles, so they have infinite number of degrees of
freedom and hence infinite natural frequencies of the
systems. The vibratory motions of such systems are
described by space and time and partial differential
equations are formulated for analysis of the systems.
Partial differential equations consist of many
constants which can be determined from boundary
conditions and initial conditions as well.

2.  The value of unknown constants in the partial
differential equations can be determined by
applying either geometric or natural or both
boundary conditions.
 Geometric boundary conditions are caused
because of geometric compatibility. For example,
if the bar is fixed at both ends, the displacement
and slope will be zero.
 Natural boundary conditions are caused due to
force and moments. For example, if the bar is
hinged at one end, the bending moment at the
hinged end will be zero and so on so forth. Initial
conditions are related to time.

3.  Consider a vibrating string of mass ρ per unit
length having transverse vibrations under
tension T as shown.
 It is assumed that for a very small amplitude
of string vibration the tension T remains
constant.