Dynamic Analysis Of Axially Moving Beam Structure With Double-Layer Time-Varying Section

In the design of solid rocket,it is an important problem that the unstable combustion of solid rocket with large aspect ratio at the last phase of the flight.As shown in published literatures,the main cause of unstable combustion is related to the unstable internal flow field,but the real mechanism is not clearly confirmed.In recent years,during the research process of solid rocket motor(SRM)with large aspect ratio,the problem of pressure oscillation occurs frequently.At the same time,it can be observed that the whole missile body is vibrating with the fundamental frequency or multiple frequency of the engine combustion sound field.Therefore,the change of the engine structure in the flight process will also play an important role in the unstable combustion phenomenon.Based on the observed phenomena,in this paper the inner diameter of the inner structure is regarded as a function of time and combustion rate,and then a double-layer time-varying cross-section axially moving beam model with two free ends is proposed according to the characteristics of the steady-state flying solid rocket in cruise state.The transverse vibration of the double-layer time-varying cross-section beam model in high-speed motion is analyzed by theoretical method,and the transient dynamics of the finite element model of solid rocket motor under the pressure oscillation of static test on the ground is studied by means of experiment and simulation.All of this is hoped to provide a new idea to study the unsteady combustion of SRM and enrich the research on the transverse vibration of the axial moving continuum.First of all,according to the established mechanical model with radial time-varying section,the dynamic equation of its transverse vibration is derived by using Hamilton principle,and the dynamic equation of free-free boundary condition are dimensionless.The Galerkin method is used to solve the linear dynamic equation of the double-layer time-varying cross-section axially moving beam.The time-varying modal function coefficient is obtained by the specific normalization method.The above results are verified by the complex modal analysis method and the finite difference numerical algorithm.Based on the solution result of linear dynamic equation,the nonlinear dynamic equation is solved by multi-scale method.At the same time,according to the characteristics of the engine cavity,the theoretical calculation method of the natural frequency of the cylindrical cavity in the axially moving beam is given,and the transient dynamic analysis and Fourier transform method are described.The above results provide the theoretical support for the later calculation.Secondly,the wide applicability of the dynamic equation and the accuracy of the solution method both of the double-layer time-varying cross-section axially moving beam are verified,by means of degenerating dynamic equation,comparing the results of literature and the results of different algorithms.Based on the verified theoretical method,the effects of axial velocity,aspect ratio,combustion time(mass reduction),combustion rate,nonlinear term and other parameters on the transverse free vibration of the double-layer time-varying cross-section axially moving beam are analyzed.At the same time,the acoustic modes of the inner cavity of the moving beam with different parameters are studied,and the acoustic field vibration is determined there is the possibility of coupling amplification between dynamic and structural oscillations.Thirdly,the multi-scale method is used to calculate the steady-state response under forced vibration and the instability boundary under combined resonance of sum type both of a doublelayer axially moving beam with time-varying cross-section.Based on the study of free vibration,the influence of axial velocity,ratio of length to diameter and combustion time on the amplitude frequency response curve of the structure is discussed in detail by numerical examples under the condition of main resonance and the effect of these factors on the instability region under subharmonic resonance.The response law under different working conditions is obtained,and the dangerous working conditions are determined also.Finally,for the cylinder like structure with large aspect ratio,a sound instability test platform with multiple measurement functions is built.The cavity modes and structural modes under different working conditions are verified and calculated,the results are complementary to the results of vibration analysis of axially moving beams with time-varying cross-section under free constraints at both ends.Based on the wide applicability of finite element software modeling analysis,the influence of parameters such as the thickness of the combustion surface,the thickness of the shell,the ratio of length to diameter,the amplitude of the pressure oscillation on the transient dynamic response of the simplified engine model is discussed in detail through the simulation calculation.Then the structural response law under different working conditions and the resonance amplification phenomenon under specific working conditions are obtained respectively.Combined with the vibration characteristics of the time-varying cross-section axial moving beam previously mentioned,the simulation results are also of reference significance.