Nonlinear dynamics of composite plates and other physical systems
The computer algebra system MACSYMA is used to derive the nonlinear expression for the Lagrangian and the nonlinear equations of motion of composite plates undergoing large deformations by using a higher-order shear-deformation theory. When computer algebra is not used, the derivation of these equations is very involved and time consuming.
A time-averaged-Lagrangian technique is developed for the nonlinear analysis of the response of a wide variety of physical systems. It is a perturbation method that produces accurate second-order approximate solutions in the neighborhoods of different resonances. As an application of the technique, the nonlinear response of a fluid-relief valve is discussed in detail. The different resonances are studied, and in each case the responses are compared to those obtained by using the Galerkin procedure. The shortcomings of the latter procedure are pointed out.
The time-averaged-Lagrangian technique is implemented in a MACSYMA code that produces second-order perturbation solutions. The effects of the quadratic nonlinearities are incorporated into the solution and different cases of resonances are fully investigated. First-order differential equations are derived for the evolution of the amplitudes and phases for the following resonances: primary resonance, subharmonic resonance of order one-half, and superharmonic resonance of order two. The evolution equations are used to determine the fixed point or constant solutions and the results are then used to obtain representative frequency-response and force-response curves for each case. The stability of the fixed points is investigated. The results show that stable and unstable solutions may coexist when multi-valued solutions are possible, the initial conditions determine which describes the response. The multi-valuedness of the solutions lead to the jump phenomenon. The results show that subharmonic resonances of order one-half cannot be activated unless the excitation amplitude exceeds a threshold value.
Lastly, a numerical-perturbation approach is used to study modal interactions in the response of the surface of a liquid in a cylindrical container to a principal parametric resonant excitation in the presence of a two-to-one internal (autoparametric) resonance. The force-response curves exhibit saturation, jumps, and Hopf bifurcations. They also show that the response does not start until a certain threshold level of excitation is exceeded. The frequency-response curves exhibit jumps, pitchfork bifurcations, and Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf bifurcation values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, the limit-cycles deform and lose their stability through either pitchfork or cyclic-fold (saddie-node) bifurcations. Some of these saddle-node bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles. Period-three motions are observed over a narrow range of excitation frequencies.