Research On Optimal Control Of Nonlinear Vibration Of Beams

An optimal control method is presented for the vibration control of the primary and super-harmonic resonances of nonlinear vibration system. The nonlinear vibration of beams is controlled bythe control methods. Some new control methods, such as optimal piezoelectric self-feedback, optimalpiezoelectric delayed self-feedback, optimal magnetic delayed feedback and non-linear grade optimalcontrol scheme, are proposed in the control of the nonlinear vibration. The nonlinear vibrations offlexible structures and nano structures are mitigated using the control methods.The main works and novel researched performed in this dissertation include:1. The optimal control method is utilized to control the vibration of the nonlinear system withlinear and nonlinear feedback controllers. The stable conditions of the nonlinear vibration systems aregiven from Jacobi eigenvalue equations. The equations of parameters function are constructed. Therange of feedback gains for a stable vibration is gotten according to the conditions of solutions. Theattenuation of vibration reduction is defined by the ratio of peak value for the primary resonance andharmonic resonance with control and without control. Taking the attenuation of vibration reductionsystem and the energy function as objective functions and taking the stable feedback controlparameters as constraint conditions, the optimal control parameters of the velocity and displacementare calculated by the optimization method. The linear and nonlinear optimal controllers are designedto control the dynamic behavior of the nonlinear vibration system. An optimal self-feedback controlmethodology is provided to mitigate the primary and superharmonic resonances of a cantilever beam.The controller is designed to control the vibration of primary and superharmonic resonances ofintelligent structures. An experimental control system is designed and control experiment is carriedout.2. The primary and superharmonic resonances of flexible beams are mitigated by a piezoelectricoptimal time delay self-feedack control methodology. The piezoelectric optimal controllers aredesigned to control the dynamic behaviors of the nonlinear dynamical systems. It is found that boththe optimal feedback gains and time delay can change the control performance.3. The optimal delayed feedback control of the nonlinear vibration is studied with parametricexcitation. The equations of the amplitude and phase are obtained by the average method. A vibrationdecay rate is defined by the energy ratio of the nonlinear vibration with control and without control.Taking the decay rate as the objective function and the stable conditions and the optimal delayconditions as constraints, the linear and non-linear optimal feedback control parameters arecalculated by optimization method. 4. A non-linear grade optimal control scheme is proposed and used in the piezoelectric vibrationreduction control of non-linear cantilever beam. The differential equation is linearized into a set oflinear equations using perturbation method. The state space equations are obtained by decoupling inthe space coordinates. The non-linear grade controllers are designed to control the nonlinear vibration.5. The natural frequency is gotten and the principal resonance is studied considering non-localeffect and axial nonlinear elongation. The numerical results show that the nonlocal effect has aneffect on the natural frequency and the relationship between frequency and amplitude. Using thepiezoelectric elements to sign and control driving actuators, the influence of the nonlocal effects ofparameters and time delay on the control parameters is studied. An optimal maganetic delayed self-feedback control method is provided to mitigate the primary resonance of a single-walled carbonnanotube. The nonlinear governing equations of motion for the SWCNT under longitudinal magneticfield are derived. The optimal controllers are designed to control the dynamic behaviors of thenonlinear vibration systems.