THE NONLINEAR DYNAMICS OF SPINNING PARABOLOIDAL ANTENNAS (SHELL, STRUCTURES, VIBRATIONS)

A method of systematically generating the discrete nonlinear equations of dynamics up to and including third degree nonlinearities for spinning paraboloidal antennas is presented. The antennas considered are thin shells such that Kirchoff assumptions are valid and are of linearly elastic material with properties which are symmetric with respect to the tangent plane of the middle surface. The spinning motion, which may be a function of time and about an axis which may or may not coincide with the axis of revolution of the antenna, alters the stiffness and damping characteristics through centrifugal and Coriolis effects. The middle surface may have imperfections and the antenna may be subjected to any conservative loading. The deformations may be non-axisymmetric and they may be large enough to produce geometric nonlinearities.;A special purpose computer program is developed and provided for the application of the method outlined to the linear axisymmetric free vibration problem of a spinning paraboloid. Natural frequencies and mode shapes are provided for a spinning paraboloid fixed at its apex. Numerical results showing the effect of spin rate and bending rigidity on the natural frequencies is provided. Results for spinning disks were also obtained using the formulation as a means of verification.;The Rayleigh-Ritz procedure is used in conjunction with Hamilton's Principle of Dynamics whereby the elastic deflections are approximated as a family of admissible trial solutions that are expressed in terms of undetermined functions of time and known yet unspecified functions of spatial variables. The choice of coordinate functions is optional and may range from smooth continuous functions to piecewise continuous functions. The list of strains, position vector of a material particle on the imperfect stressed antenna, and the elastic deflections needed for the strain energy, kinetic energy, and loss of potential energy of prescribed forces, respectively, are expressed in terms of admissible trial solutions as a summation of constant, linear, quadratic, and cubic components with respect to the undetermined functions of time. These sums are then substituted into the appropriate terms of the equation of motion and expressions for the coefficient matrices of the discretized equations of dynamics are obtained.