| Due to the features of lightweight,high packaging efficiency,good thermal stability,and high stiffness when deployed,large-scale cable net structures are therefore widely used in space deployable antennas.The vibrations in this type of structure are easy to form and hard to decay due to the lightweight and large size.To quickly and effectively suppress the vibration and wave response of a large cable net structure is of great significance for improving the system’s working accuracy and efficiency and extending the service life of the structure.This thesis is devoted to research on the active control strategy of large-scale cable net structures based on the wave theory,by using methods such as irrational transfer functions,Lax-Friedrichs scheme,spectral analysis on the imaginary axis,etc.The main contributions of the thesis are summarized as follows:(1)Graph theory is used to describe the connection of cable net structures,and onedimensional wave network models for the planar motion of cable net structures are established.The derivation of irrational transfer functions from the external force to the displacement at any point of the structure is provided.Based on the Jury stability criterion,the stability analysis method for irrational transfer functions is proposed.By rewriting the irrational transfer functions into serial forms,treating the exponent terms as delays,and treating the exponential polynomials on the denominators as feedbacks,the Simulink models of the irrational transfer functions of the cable net structures are obtained.Considering interference and control effects and a variety of boundary conditions,the dynamic model of the cable net structures is represented in a vector form.Lax-Friedrichs algorithm is then used to derive the iterative calculation formula of wave propagations in the cable net structures.It is used to study the propagation law of waves in cable net structures and verify the effectiveness of vibration and wave control of these structures.(2)For disturbance coming from the external boundary of the cable net structure,a boundary control strategy based on the wave theory is proposed to control the vibration of the structure.The transfer functions from the disturbance at the exterior boundary to the structure displacement are derived,when the boundary wave controllers are applied to all the boundary vertices of the structure.By pole analysis,it is found that with the boundary wave controllers,all poles corresponding to the flexible modes have negative real parts,so that the vibration caused by the disturbance at the exterior boundary of the cable net structure can be completely suppressed.When the external boundary vertices are partly under the boundary wave control,the transfer functions from the disturbance at the exterior boundary to the structure displacement are derived.By pole analysis,it is found that the structural vibration caused by the disturbance on the strings can still be completely suppressed under the boundary wave controllers,but the vibration caused by the disturbance on strings without applying the control may not be suppressed.(3)For disturbance coming from the internal of the cable net structure,the control effect of the boundary wave absorbing control strategy is studied,based on the irrational transfer function method.When all the external boundary vertices of the structure are under the boundary wave absorbing control,the transfer functions from the disturbance force at the internal vertices and the internal of the strings to the structure displacement are derived.By pole analysis,it is found that when the disturbance is applied on the internal vertices,all the poles,which are corresponding to the flexible modes,of the closed-loop system are located in the left half s plane;while when the disturbance comes from the interior of the internal strings,the closed-loop system may still have a lot of poles,which are corresponding to the flexible modal,located on the imaginary axis.Namely,the boundary wave absorbing control can prevent the formation of the vibration caused by disturbance from the internal vertices,but may not prevent the formation of the vibration caused by the disturbance from the interior of the internal strings.Based on the infinite dimensional transfer functions,modal analyses of the cable net structures are carried out.When the irrational condition of the wave velocity ratios of the adjacent strings at the circuit of the string network structure fails,the structure would have modes in which the internal vertices are the nodes of the corresponding modal shapes.The vertices and the strings whose ends are the nodes of the modal shapes will form closed circuits.If a string network structure contains such a mode,then the boundary wave absorbing control will not be able to prevent the structural vibration caused by the disturbance from the interiors of the strings in the circuits.(4)A controller design method based on spanning tree theory is proposed for disturbance comes from any part of the cable net structure.By choosing a suitable Hilbert space,the closed-loop equation of motion of the cable net structure is written into an abstract state space form.The stability of the closed-loop system is then proved by spectral analysis on the imaginary axis.From the stability proof process,a graphical method to verify whether the controller is effective is extracted.If the graphical method can deduce that the closed-loop system under control has no characteristic roots on the imaginary axis,then the structure under the designed control strategy can achieve asymptotically stability.The graphical method provides a quick and effective analysis method for further reducing the number of the controllers.The controller design method based on the spanning tree theory is then applied to the vibration control of a large cable net structure.Through the graphic method,structural design and vibration control are taken into consideration as a whole to further reduce the number of the controllers.The designed structure with corresponding controllers can completely suppress the disturbance from any locations of the structure,and effectively avoid residual vibrations in the structure. |
