Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element

Abstract

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.

New analytical formalisms used in finite element analysis of robots with elastic elements

“…In the rigid kinematics, the orthonormal operator [T] is used to change the components of a vector from the mobile reference frame to the fixed one (global), {t} G = [T]{t} L . Differentiating the matrix [T], it obtains the angular velocity and operator/vector andthe angular acceleration operator/vector ( [20]).…”

“…We note that we have to make three differentiations {∂L/∂ḋ}, (d/dt){∂L/∂ḋ}, {∂L/∂d} whereas in the case of applying the Gibbs-Appell equations it is only necessary to make a single differentiation {∂E a /∂d} [20]. This can reduce the computation time and thus an advantage when applying FEA to MBS with elastic elements.…”

Maggi’s Equations Used in the Finite Element Analysis of the Multibody Systems with Elastic Elements

“…where q is the vector of independent coordinates of the whole system. Introducing (28) in (27), it obtains:…”

“…The analytical methods of studying MBS and the possibilities of obtaining these equations are presented in [22][23][24][25]. The Gibbs-Appell method is a method rarely found in studies, but it may be of interest to researchers to use this method in analyzing MBS systems with elastic elements [26,27].…”

New analytical method based on dynamic response of planar mechanical elastic systems

“…For nonholonomic systems, Kane's equations can also be used, and it seems that in the last decade, their use has begun to be applied to problems that require the dynamic analysis of complex systems. The latest research on applying the method to MBS systems with elastic elements can be found in [13][14][15][16][17].…”