Design and stiffness analysis of three-dimensional（3-PRRP4R）parallel vibration reduction mechanism

Abstract: In order to solve the problems of multi-dimensional vibration in industry and agriculture effectively, the optimization design of the 3-PRC parallel mechanism is carried out for improving its kinematic performances and mechanical properties. A classic 3-PRC parallel mechanism with 3 translational degrees of freedom (DOF) has some defects such as asymmetry, and low mobility. Therefore, for eliminating these defects, the 3-PRC parallel mechanism is upgraded into a novel 3-PRRP4R parallel mechanism by adopting some optimal measures. Firstly, change the cylindrical pair into a revolute one and apply a 4R structure which includes 4 revolute pairs in the mechanism so as to make the parallel mechanism move more smoothly. Additionally, change the arrangement form of sub-chains from square to equilateral triangle, which makes the force distributed more homogeneously on platforms. Finally, change the position relationship between the sliding pair and the revolute pair, and eliminate the unwanted additional torque. After the improvement of the mechanism, based on the screw theory, the degree of freedom and the motion form of the 3-PRRP4R parallel mechanism are analyzed. The result shows that the 3-PRRP4R parallel mechanism is still a 3-DOF translation mechanism which can be used as three-translation vibration reduction platform. Using the homogeneous coordinate transformation method, the kinematics of the 3-PRRP4R parallel mechanism is formulated. The inverse kinematics solution describes the displacement relationship between upper platform and active joints which are on the lower fixed platform. Jacobian matrix, which describes the velocity relationship between upper platform and active joints which are on lower platform, is obtained by the analysis of velocity. Through the coordinate searching method, the workspace of the 3-PRRP4R parallel mechanism is obtained, which has better symmetricity than the classic 3-PRC parallel mechanism. The comparison of the workspaces shows the 3-PRRP4R parallel mechanism is more suitable as a multi-dimensional vibration reduction mechanism. With the help of Adams software, the mechanical properties of cylindrical pair and 4R structure are analyzed, and the result proves 4R structure has better mechanical properties than cylindrical pair. The multi-dimensional vibration reduction system is established by using the 3-PRRP4R parallel mechanism as the main body and installing the spring damping between active joints and lower platform. Based on the principle of virtual work, the statics of the 3-PRRP4R parallel mechanism is analyzed. Static equilibrium equation, which describes the force relationship between upper platform and active joints, is established. Static equilibrium equation can be used to set the structure parameters of mechanism and the preload of spring at the initial time. The stiffness of parallel mechanism describes the relationship between the force on upper platform and the corresponding displacement of upper platform, and it is very complex. In order to describe the spatial distribution law of the stiffness visually and briefly, the equivalent stiffness is introduced, and its formula is derived as well by means of dividing the magnitude of the force on upper platform by the magnitude of the displacement of the upper platform. With this formula, the equivalent stiffness in any arbitrary position can be obtained. With the help of MATLAB, the obtained figures, which describe the distribution law of equivalent stiffness in some special positions, show the general distribution law of equivalent stiffness in workspace. The distribution law of equivalent stiffness can help us to make sure whether the special position is useful or not, so it is very important for the application of parallel mechanism. Therefore, the paper provides some valuable guidance for the optimization design, the dynamic analysis, the equivalent stiffness inspection of parallel mechanisms, and the application of multi-dimensional vibration reduction system.