Static and Dynamic Analysis of Absolute Nodal Coordinate Formulation Planar Elements

Abstract

The paper presents the mathematical and numerical formulation of recently articulated finite element method for flexible structures. The Absolute Nodal Coordinate Formulation (ANCF), proposed by A. Shabana helps to numerically articulate the large deformations, reportedly with precision. Renowned finite element abstraction based on Euler Bernoulli's beam element is used to drive the ANCF approach of beam element. Finite coordinates and slopes are used as nodal variable with respect to global frame of reference. Each node is completely defined by four nodal variables in a planar 2-dimensional formulation. The proposed formulation facilitates to extract the rigid body modes with accuracy. Analytical formulation of Absolute Nodal Coordinate Formulation (ANCF) uses constant mass matrix and generalized gravity but the strain matrix is highly nonlinear as it originates from green Langrage strain. Legendre's quadration integration is used to extract the elastic forces. Dynamic system is solved through direct integration technique by simultaneously solving equations of motions. The joints between multibody are modelled by kinematic constraint as a consequence, nonlinear algebraic equations are formed. Ordinary differential equation (ODE) and (Algebraic Equations) AE are simulated by Newmark's implicit integration. The paper presents the results from 2-dimensional ANCF applications like cantilever beam, pendulum and four bar mechanism. The paper also presents planar 2-dimensional case of Euler Bernoulli's finite element formulation for comparison and static analysis of cantilever beam. The dynamic analysis is provided for cantilever beam and pendulum and elaborated with the help of time history diagram.