Graduate Thesis Or Dissertation
 

Normal modes in nonlinear nonsymmetrical systems with two degrees of freedom, and applications to air springs

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  • 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.
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