The Analysis Of Nonlinear Vibration Of An Annular Thin Plate

The essay analyses the nonlinear vibration of an annular thin plate.The fourth-order power function as trial function of the deflection of anannular thin plate, which is substituted into the consistent equation, gives theAiry Stress Function. Together with the deflection function, the Stress Functionis substituted into the Dynamic von Karman Partial Differential Equation, andby the Galerkin Method the residuals are eliminated, thereby gives thegoverning equation for forced vibration under the four types of boundaryconditions, that is, rigidly clamped edge, clamped but free to slip edge, simplyhinged edge, simply supported edge. This equation is also called the hard-springtype of Duffing Equation. Using the gradualness method (also is called KBMmethod) to solve the governing equation to obtain the first-order approximatesolutions of both non-resonance and primary resonance, then to compare anddiscuss the impact of four types of parameter--boundary conditions, damp ratio,external excitation, ratio of inner and outer radius-to the vibration amplitude. Ina word, due to considering the nonlinear term, there appear both jumpphenomenon and lag phenomenon in both amplitude-frequency and phasedifference-frequency maps. Finally the dynamic stability of hard-spring type ofDuffing equation is discussed. By stability theory and numerical simulation, wetake the increment of the external excitation as the controlling parameter, and itwill make dynamic system from periodic motion around one focus to periodicmotion around two focuses.