### Abstract:

The study of nonlinear vibrations of systems having two
degrees of freedom has met considerable attention during the past
few years. Nonlinear symmetrical systems have received most of it.
Normal mode motion of a system is defined as a periodic motion
such that the masses of the system assume repeated displacements
after some interval of time, called the period of oscillation.
In this kind of motion there is a definite relation between the displacements
of the masses which is called the modal relation. The
modal line is defined as the locus of all points, in the plane representing
the displacements, which set the system in normal mode
motion when started from rest. This line passes through the origin of the plane. Systems vibrating in normal modes have also what is
called the orthogonality property. That is, the modal relation
curves intersect the total energy line orthogonally.
There exists a type of normal mode which has a straight modal
relation. The linear systems belong to it. In this type of mode the
orthogonality property is used to determine the modal relations and
this in turn enables the equations of motion to be decoupled to form
two separate systems, each with a single degree of freedom. This
of course simplifies analysis of the system. Generally speaking the
modal relations are not straight and they could be determined easily
by numerical means. For this purpose an algorithm for determining
the normal mode motions was developed. The application of the
orthogonality property is also useful when applied to small displacements.
One striking phenomenon
have more normal modes than
The existence of this excess of
of nonlinear systems is that they may
the number of degrees of freedom.
modes can be easily detected by
using the orthogonality property.
The stability of systems oscillating in normal modes could be
studied by using Liapounovrs theorem of stability. The total energy
equation of a dynamical system is a Liapounov function. The idea of
the modal phase plane, which is a plot of the total energy line and the modal line, is introduced. This helps in deducing the stability of
normal modes. The concepts of singularities in this plane are
defined.
The mathematical procedure for determining the normal modes
was applied to an air spring system with two degrees of freedom. An
experimental apparatus was constructed to compare the theoretical
and the experimental results. This comparison showed fair agreement.
The forced motion of the air spring system was studied experimentally.
Over some range of the exciting frequency two resonances
appeared. Each one corresponds to one normal mode. The results
showed that the system eventually oscillates essentially in normal
modes when the exciting frequency is equal to a natural frequency of
the system.