Research On Nonlinear Vibration And Active Control Of T-shaped Beam Structures

A specific structure composed of multiple beams is usually used as component in large-scale space structures within the fields of mechanical,aeronautical and civil engineering.In the dynamics of a flexible multi-beam system,the deformation of the flexible beam will have a great impact on the dynamic behavior of the system,and the elastic deformation movement between the flexible beams and the rigid body movement will be coupled.In order to acquire higher precision,reliability and stability,a nonlinear dynamical analysis of such systems is of important practical significance for predicting and understanding their behavior under the effect of applied loadings.Dynamic analysis and nonlinear vibration control research for the T-beam structure are investigated in this paper.Firstly,establish the dynamic model of the T-beam structure.For a T-beam structure composed of three lightweight slender beams fixed on a horizontally movable support,assuming that the shear deformation and warpage of all beams can be ignored,and the beams are not stretchable,the in-plane motion is considered.Considering the in-plane transverse movement of the structure,the nonlinear partial differential equations are derived for each beam by using the generalized Hamiltonian principle,along with their nonlinear matching and boundary conditions.The global mode method is used to determine the natural frequencies and global mode shapes of the linearized system,and the orthogonality relations of the global mode functions are proved.Using the global mode shapes and their orthogonality relations,an explicit set of reduced-order nonlinear ordinary differential equations(ODEs)of motion for the structure are obtained by the Galerkin truncation of the original dynamic model.Then,the dynamic analysis of the T-beam structure is carried out by numerical simulation calculation method.The validity and accuracy of the derived model are verified by a comparison with the results from FEM.Based on the low-dimensional high-precision discrete dynamics model,a study on the variation of dynamic responses for the system with different number of global modes is performed,and the results of which is used to determine the number of modes taken in nonlinear vibration analysis.The results show that the dynamical responses are dominated by the low-order modes of the system,and the responses of the systems under different mode numbers have a strong dependence on the excitation amplitude.A comparison between the responses of the system with linear and nonlinear matching and boundary conditions is given to evaluate the importance of neglecting and reserving the nonlinear terms in matching and boundary conditions.It is found that ignoring the nonlinear terms in matching and boundary conditions may significantly alter the responses while developing the discretized governing equations of the global modal.Finally,the active control of nonlinear vibration of T-beam structure is studied.Finally,using the forward and inverse piezoelectric effects of piezoelectric materials,based on the global modal nonlinear discrete control equation of the T-beam structure,the optimal control law obtained from the linearized system is adopted to realize the active control of the nonlinear vibration of the structure.And explore the influence of different piezoelectric film positions on the control effect.