An analytical solution for vibration of a parallel robot where its end-effector is flexible and has a passive prismatic joint(s) has not been presented before. In this research vibration analysis of a PR-PRP parallel robot using an analytical method is investigated. The PR-PRP parallel robot has two planar degrees of freedom and moves by means of two active prismatic joints. The robot moving platform is a flexible link with one passive revolute and one passive prismatic joint. First, the motion equation for a flexible link with the passive prismatic joint is developed. The motion equation is solved by employing an approximate analytical method called “constrained assumed modes method”. A time-variant constraint is written for the passive prismatic joint. The developed model allows for inclusion of the effect of physical length for the passive prismatic joint in contact with the moving platform on the vibration response of the system. For verification of the presented model and solution, three case studies are presented and results of the analytical solution are compared with results of a commercial finite element method software. For each case study, two different lengths for the passive prismatic joint are considered. Finally, active vibration control is performed for an applicable motion of the robot using the proportional-integral-derivative controller.
a b s t r a c tFlexible beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the beam. In the present research, a horizontal flexible link sliding through a passive prismatic joint while attached to a rigid link of a robot moving in a vertical direction is considered. A body coordinate system is used which aids in obtaining a new and rather simple form of the differential equation without the loss of generality. To model the passive prismatic joint, the motion differential equation is written in a form of virtual displacement. Next, a solution method is presented for the lateral vibrations of the beam referred to as ''constrained assumed modes method''. Unlike the traditional assumed modes method, in the proposed constrained assumed modes method, the assumed mode shapes do not each satisfy the geometrical boundary conditions of the point where passive prismatic joint is located. Instead, by writing additional constraint equations the combination of the assumed modes will satisfy the geometrical boundary conditions at location of the passive prismatic joint. Two case studies for the effect of axial motion on lateral vibration of the beam are presented. Approximate analytical results are compared with FEM results.
In the present study, a simple and efficient finite element approach is presented for large deflection analysis of both the straight and curved Euler-Bernoulli beams in the planar static problems. The linear stress-strain relationship is assumed for the Euler-Bernoulli beam. In some of finite element methods, displacements together with rotations, and in some others, positions are considered as the main fields of interpolation. However, in the present study, the main idea for the interpolation is using the dimensions of the deformed element instead of the displacements. Therefore, the slope angle (like the previous works) and the length of beam centroidal axis (unlike the previous works) are used as the main field parameters. This treatment creates simplicity in the constitutive equations. Next, using the weighted residual method, the constitutive equations are applied to the element. Using the equilibrium equations and kinematic equations, the position coordinates of the nodes are related to the internal forces and the main field parameters. In the present study, three-node element and, consequently, Simpson's 1/3 rule is used for integration. For solving nonlinear equations of the beam, the Newton-Raphson method is used. Finally, several numerical examples are presented and compared with the previous works to illustrate the validity and efficiency of the new element.
SUMMARYIn this research, using an approximate analytical method, vibration analysis of a 3-PRP (active prismatic—P, passive revolute—R, passive prismatic—P) planar parallel robot having a flexible moving platform is presented. A specific architecture of the 3-PRP parallel robot, also known as the ST (Star-Triangle) parallel robot, is considered. The moving platform of the robot, called the star, is assumed to be made of three flexible beams shaped like a star. For analytical modeling, each of the three beams of the star is assumed to be a discrete Euler–Bernoulli beam with a passive prismatic joint. Continuity equations at the center of the star are used to relate vibrations of the three beams. The vibration behavior of each beam is modeled using previously developed constrained motion equations for a planar Euler–Bernoulli beam having a prismatic joint. In this paper, previously presented “constrained assumed modes method” is further developed to solve the constrained motion equation for the ST parallel robot. The solution method is used to obtain the vibration of the robot for the inverse dynamics problem and simultaneously provides generalized constraint forces. Furthermore, the solution method can be used for the direct dynamics problem of the ST robot. Several input trajectories are considered to investigate the different behavior for the center of the star. For each of the trajectories, three different groups of mode shapes are considered and their vibrational responses are compared. In this research, for the first time, effects of the passive prismatic joint parameters such as mass, rotational moment of inertia, and its actual length are considered in an analytical model. Finally, the analytical solution and a FEM (Finite Element Method) software solution are compared.
In this research, first an analytical model is presented for dynamic and vibration analysis of a 3-PSP parallel robot with a flexible moving platform. Next, the presented analytical model is solved using an approximate analytical method. The moving platform is assumed to be made of three Euler-Bernoulli beams joined together to form a star. Each of the three beams of the star slides through a passive prismatic joint. Then, three-dimensional vibration analysis of the flexible moving platform, star, with three passive prismatic joints is the main subject of the present research. Only vibration during free motion is considered. Therefore, it is assumed that only inertia forces of the star are the main source of its vibration. First, direct kinematics is used for acceleration analysis of the rigid robot and inertia forces are obtained. For dynamic modeling, the passive prismatic joints and junction point of the three beams are modeled using a new set of geometric constraints. Additionally, a previously developed constrained motion equation for a planar Euler-Bernoulli beam having a prismatic joint is further developed for the three beams of the star. Next, an approximate analytical solution method, called the “constrained assumed modes method”, is used for inverse dynamics and vibration analysis of the robot. Furthermore, the developed model can be used for direct dynamics analysis of the robot. Finally, several input trajectories and two different groups of mode shapes are considered to investigate the model efficiency. The results of the presented model are compared with the results of a commercial finite element method software.
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