Research On Continuum Modeling And Control Of Lrge Periodic Structures Considering The Impact Of Cables

Large periodic structures constructed via many repeatable fundamental elements having the advantages of easy deployment,large stiffness-to-mass ratio,simplicity of construction and on-orbit assembly,are being widely used in space missions as a kind of special space structure.However,due to having the characteristic of light-weight,large size and flexibility,once been excited by external disturbance or internal excitation,large periodic structures may produce large-scale vibrations that is difficult to damp by itself.In addition,the presence of commonly used signal and power cables may result in a big drift in system natural frequency,which will affect the control accuracy and stability of the system.So,it is necessary to fully understand the dynamic characteristics of the Large periodic structures before launch.Limited by the ground gravity and structural size,obtaining an accurate system-level dynamics model completely through experiments is difficult.Therefore,this paper will analyze the dynamic characteristics of Large periodic structures through mathematical methods,and then will conduct research on the control methods on this basis.The cable wrapping geometry is optimized to minimize the impact of cables on the system dynamics so that the cables can be ignored in the modeling process.In order to simplify the problem,two kinds of periodic cable wrapping pattern,which are zigzag and diagonal patterns,are considered in this paper.Based on the energy-equivalence principle,a continuum model of the cable-harnessed beam structure described by partial differential equations is developed.Compared to finite element model,the continuum model can more intuitively reflect the effects of cable parameters on the system dynamics.It is found from the continuum model that the natural frequency of the cable-harnessed beam structure is related to the cable wrapping angle,which is determined by the number of fundamental elements.Using the sum of the squared natural frequency differences between the beam with and without cables as the objective function and the number of fundamental elements as optimization variable,an optimization model is established based on the continuum model.According to the necessary and sufficient conditions for the minimum value of the objective function,the analytical expressions of the optimal solution for the two periodic wrapping patterns are found,and the conditions for the existence of the optimal solution are given.The effects of cable and beam parameters on the optimal solution are investigated and discussed using numerical simulations.Finally,an experimental platform is built,and the effectiveness of the continuum model under two periodic wrapping patterns is verified by comparing the theoretical and experimental frequency response functions.This indicates that it is feasible to study the optimization of cable wrapping geometry via this model.The experimental results show that the frequency response functions of the optimized cable-harnessed beams match well with those of the bare beams,which proves that the cable wrapping geometry given in this paper is correct and effective.For the large deformation and small strain problem of the truss-type periodic structure,a geometric nonlinear dynamic analysis method based on the continuum modeling method and co-rotational formulation is developed.This method decomposes the motion of the periodic structure into rigid body motion and pure deformations by introducing a co-rotational coordinate system.Using the energy-equivalence principle,a linear continuum model of periodic structure is established in the local coordinate system.Based on the equivalent stiffness and mass matrix of the linear continuum model,the nonlinear equivalent stiffness and mass matrix of the periodic structure in the global coordinate system are derived,and the relationship between them is given.Finally,the virtual work principle and Lagrange’s equations are used to establish the nonlinear dynamic equilibrium equations of the system,and the influence of the boundary mass is introduced by modifying the boundary conditions.Compared with the non-linear finite element model based on the truss element,the proposed model has fewer degrees of freedom,lower model dimensions,and higher computational efficiency,while obtaining the desired accuracy.For the attitude tracking and vibration suppression of Large periodic structures,the control method is studied based on the continuum model.Consider the situation that the attitude manuver is slow and the structure deformation is small,the attitude-vibration coupled dynamic model of the periodic structure that described by ordinary differential equations and partial differential equations is derived based on the linear continuum model.The control scheme is designed without discretizing the partial differential equations,which can effectively avoid the problems induced by model discretization and truncation.Meanwhile,the system stability conditions are analyzed by the Lyapunov direct method and the control parameter determination method is given.Due to the large size of the Large periodic structure,the input saturation of the actuator is easy to occur during the attitude maneuver.In this paper,the auxiliary system commonly used to anti input saturation is improved,and the boundary controller is redesigned to ensure that the system is uniformly ultimately bounded.For the control problem of large deformation of the structure induced by the rapid attitude maneuver,Variable Step Size Firefly Algorithm is used to plan the maneuver path of large periodic structures so that the residual vibration is minimized,which is performed based on the nonlinear continuum model.Then using the obtained optimal path,a controller for attitude control and vibration suppression is designed by combining the feedforward control and PD feedback control.Finally,the effectiveness of the proposed control method is verified by simulation.