An analytical model for vibration and control of a P R- P RP parallel robot with flexible platform and prismatic joint

J Braz. Soc. Mech. Sci. Eng.

2018

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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.

Section: And Mahdi Sharifniamentioning

confidence: 99%

A new beam element for analysis of planar large deflection

J Braz. Soc. Mech. Sci. Eng.

2018

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“…Authors of the present paper in ref. [14] have presented a differential equation for planar vibrating motion of an Euler-Bernoulli beam having a prismatic joint. The authors assumed both vibration amplitude and slope angle of deformation to be small in their analytical model.…”

Section: Motion Equation and Applying The Constraintsmentioning

confidence: 99%

“…To the best of the author's knowledge, aside from a few FEM studies, there exists only one approximate analytical study on vibration analysis of flexible parallel robots with passive prismatic joints. 14 Authors of the present paper have presented a motion equation in variational form for vibration of an Euler-Bernoulli beam having a passive prismatic joint. 14 The flexible beam is used as a moving platform for a PR-PRP parallel robot.…”

Section: Introductionmentioning

confidence: 99%

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Approximate analytical solution for vibration of a 3-PRP planar parallel robot with flexible moving platform

Akbarzadeh

2014

Robotica

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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.

“…Finally, it seems that the prismatic joint constraints are not used in mesh-free methods such as assumed modes method. Authors of the present paper previously performed dynamic and vibration of PR-PRP, 3-PRP and 3-PSP parallel robots in which flexible moving platforms have prismatic joints [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

A constrained assumed modes method for solution of a new dynamic equation for an axially moving beam

Computers & Mathematics with Applications

Akbarzadeh

2016

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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.