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

Abstract

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.

Nonlinear dynamics of flexible links in planar parallel robots using a new beam element

“…Parallel robots can be used in many fields of applications, such as astronomy, flight simulators, and surgery robots, and have been recently used for acquiring more accurate machine tools. Using some parallel mechanisms, such as 3-RRR (Zhang and Zhang, 2014), 3-PRR (Staicu, 2009), 3-PRP (Sharifnia and Akbarzadeh, 2016b), and 3-PSP (Sharifnia and Akbarzadeh, 2016c), in recent machine tools, required properties such as accurate and rapid motions can be achieved. To decrease energy consumption and increase motion speeds, using low-inertia linkages and, consequently, flexible linkages is unavoidable.…”

“…Two main computational methods are used in flexible multibody dynamics: assumed mode method (AMM) and finite element method (FEM). The AMM can still be extended for use in more complicated systems (Sharifnia and Akbarzadeh, 2014, 2016a, 2016b, 2016c, 2017). Some researchers compared the results of the AMM with experimental results (Ettefagh et al, 2018; Korayem et al, 2014, 2016; Luo, 1993; Nagaraj et al, 1997; Yim et al, 1993).…”

A new beam element for analysis of planar large deflection

“…Beside the two most popular methods of GEF and ANCF, new researches with new ideas are also performed for different problems of beam large deflection; see [33][34][35][36][37][38][39]. Analytical methods such as assumed modes method [40][41][42][43][44] can also be developed or combined with the finite element method for use in the large deflection of the beam structures.…”

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

“…Moreover, other modes are as follows ϕ n (x) = A n [(cos (β n L) − cosh (β n L)) (sin (β n x) + sinh (β n x)) − (sin (β n L) − sinh (β n L)) (cos (β n x) + cosh (β n x))] , A n =  L cosh 2 (β n L) − L cos 2 (β n L) − 2L sin (β n L) sinh (β n L)  −0.5 (24) in which β n = k n π /L and A n is obtained using mode normalization. As can be observed, A n is a function of L and β n .…”

“…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].…”