Theoretical Studies On The Kinematics Of Loaded Pin-jointed Bar Mechanisms For Pantadomes

In the erection of long-span space frames by "mechanism method",typical of Pantadome,the structure is efficiently jacked or hoisted up with rigid-body displacements from a lower working surface to its design configuration by removing some members to transform the structure into a mechanism.Based on the Pantadome system,the following four aspects on the kinematics of loaded pin-jointed bar mechanisms are investigated theoretically in this paper.(1)A geometric strategy for the partition of pin-jointed bar assembly is proposed in order to realize a specified rigid-body movement.The square sum of the elongations of all structural members is adopted as the index to describe the structural deformation and expressed as the quadratic form of nodal displacements.The eigenvectors of the quadratic matrix corresponding to the small eigenvalues could be used to construct the rigid body displacement approximately.If the displacements which are constantly construct using those eigenvectors with smaller angle to the moving trend,the structure will theoretically move towards the working surface with small deformations.From the design configuration,the space frame is forced to move towards and finally approached the working surface with the estimated displacement.Based on the structural deformation,a method is proposed for the partition of space frames.(2)The study on the layout of jacking struts of the pin-jointed bar assembly is carried out at the view of geometric stability of the system along the special rigid-body movement.A method is developed to determine the direction of rigid-body motion along the shortest path between the assembly surface and the design configuration of the pin-jointed bar assembly.The obtained direction of rigid-body motion is converted into the main vector in the matrix of mechanism displacement modes.Mathematically,the introduction of jacking strut is described as the positive perturbation to the zero eigenvalues for all modes of mechanism displacement of the pin-jointed bar assembly.By comparing the magnitudes of perturbation along the kinematic path,the quality of setting jacking strut under different joints can be evaluated quantitatively.A method to determine the optimal joint areas for the layout of jacking struts is established on those analysis above.(3)The investigation is implemented on the path planning of loaded pin-jointed bar mechanisms.The basic kinematic equation of loaded pin-jointed bar mechanisms is hereby established based on FEA,and the driving condition of internal rigid-body(mechanism)displacements is presented for the length actuation of active members.A numerical strategy is proposed to trace the shortest path of pure mechanism displacement from the initial configuration to the given target configuration.With the emphasis on the constraints of structural stability and range of motion,the Rapidly exploring Random Tree(RRT)method is adopted for the path planning of the loaded pin-jointed bar mechanisms.The RRT method is further modified to be applicable for the path planning when the target configuration is implicitly defined by its distance to be as close as possible to a geometrical boundary,at which the conventional RRT algorithm fails to track multiple feasible paths.An adaptive strategy is also suggested to improve the sampling efficiency of RRT.The modified RRT method is established to solve the path planning of loaded pin-jointed bar mechanisms.(4)For assemblies with large number of freedoms and bars,two methods are proposed to improve the numerical efficiency to calculate the modes mechanism displacement.The basic idea in the first method is to convert the equilibrium matrix of the assembly into an upper triangle matrix by the economical QR decomposition,which is further adopted to construct a sparse symmetric matrix.The null-space of this matrix is calculated by the Lancsoz method,and the independent mechanism modes are carried out by the orthogonal transformation of the null-space.In the second method,the equilibrium matrix of the assembly is converted into an ill-conditioned symmetrical matrix by transpose multiplication,the null-space of which is just the independent mechanism modes.Considering the ill-condition and symmetry of this matrix,the LDL decomposition can be adopted to calculate the independent mechanism modes approximatively.Efficiency promotion in the first method is brought out by the avoidance of the major calculation for the non-zero eigenvalues as well as their corresponding eigenvectors.In the second method,the core of the calculation is replaced by the non-iterative LDL decomposition,which improves the efficiency further.