Lie reductions for the Euler-Bernoulli model of flexible beam robots

Mehdi Nadjafikhah,Leila Hamedi Mobarra

doi:10.9790/5728-10633846

Abstract

The Euler-Bernoulli model of flexible beam robots is a four-dimensional under-determined partial differential equation, related to define the unique link's deflection of this type of robots. In this paper, Lie point symmetries of the equation have been investigated by a Lie group method. Then, reductions of the Euler-Bernoulli equation to ordinary equations have been obtained by repeat the method . Two of the reduced equations are of the first order.

Highlights

1.1 Classical Lie point symmetries of PDEsSophus Lie and et al innovated the methods for finding Lie point symmetries and reductions of differential equations in the 19th century
The linearized condition for a classical infinitesimal one-dimensional Lie point symmetry X corresponded to a k-dimensional PDE, = 0, is given by: pr(k) X ( ) | =0 = 0. (1)
According to [2], Yu acquired a mathematical model for this class of flexible robots in 1994. He with citing to [3, 4, 5, 6, 7, 8, 9], at first, produced a system which is composed of two elastic partial differential equations and three nonlinear integrodifferential equations for a general robot of that class

Summary

Classical Lie point symmetries of PDEs

Sophus Lie and et al innovated the methods for finding Lie point symmetries and reductions of differential equations in the 19th century. According to [2], Yu acquired a mathematical model for this class of flexible robots in 1994 He with citing to [3, 4, 5, 6, 7, 8, 9], at first, produced a system which is composed of two elastic partial differential equations and three nonlinear integrodifferential equations for a general robot of that class. 1.3 Euler-Bernoulli model Consider a motor driven beam robot (refer to [10]) with a joint and an elastic link that can deform along a direction perpendicular to its longitudinal axis. The reductions of the equation to ODEs by optimal sub-algebras, are found

The symmetry group

Classification of sub-algebras

Conclusion