The Transient Dynamical Analysis Of Time-Varying Parameter Systems Based On The Deploying/Retracting Cantilever Beam

Time-varying parameter structure system is a classical class of systems in aerospace engineering, such as the deploying/retracting wing as well as large developable antenna, etc. Compared with time-independent parameter continuous system, the differential equations of lateral motions and the corresponding boundary conditions of time-varying parameter structure system vary at any instant, which belongs to the non-conservative system. Along with energy transfer and the mutation of the stability, its dynamic behaviors appear more complex than the vibration of the time-independent parameter system. The classical theory of ordinary differential equation is difficult to be carried out for the time-varying system, qualitatively or quantitatively. Therefore, this research is of importance in the study of the structural dynamics modeling, dynamic characteristic, kinetic stability and experimental research of the time-varying parameter structures.This paper proposes a set of modified analytical methods for the length varying deploying beam. In addition, for the time-varying structure, the invariant is proposed and the dynamic behavior and stability are discussed by both the numerical method and the experimental method. The result of this research provides a rich theoretical foundation for studies of the transient dynamics of time-varying complex system. The research contents mainly include the following parts:(1) Mathematical modelling. The deploying/retracting structure is simplified as the axial moving beam whose length and mass vary with time. The moment of momentum theory is used to obtain the partial differential vibration equations, and these equations are converted to ordinary partial equations based on the assumed-mode method.(2) Energy analysis. Based on the ordinary differential equations, the analytical expressions of lateral energy which vary with length are obtained, as well as the situation of the energy that transfers among each modal. For the transverse vibrations of the axially retracting beam, the energy-like invariant is derived by both the averaging method and the Bessel function method. Afterwards, the results of the two methods are compared.(3) Stability analysis. The stability of the axially deploying beam is studied by analyzing the pseudo-eigenvalues of the ordinary differential equations and the determinant of stiffness matrix, the results of the two methods are also compared. The effect of axial acceleration on stability is also discussed at last.(4) Modified perturbation methods analysis. At the stable state of the deploying beam, the perturbation methods are modified and applied to solve the ordinary differential equations based on slow time-varying parameter system, then the analytical solutions are obtained. Finally, the numerical solution and the analytical solution of the modified perturbation methods and Bessel function are compared.(5) Experiment research. The experimental equipment of the deploying beam is designed. Moreover, the damping of the deploying cantilever beam is studied by experimental method and its analytical expressions are substituted into the vibration equation of first order truncation. The comparison of the displacement response curve which are acquired by numerical method and the retracting experiment has been implemented to show the effectiveness of both methods.