A Dual Number Approach for Numerical Calculation of Velocity and Acceleration in the Spherical 4R Mechanism

TL;DR

This paper introduces a dual number-based method to efficiently compute both velocity and acceleration in spherical 4R mechanisms, implemented in Fortran, enhancing kinematic analysis accuracy and efficiency.

Contribution

It presents a novel nested dual number approach for simultaneous calculation of first and second derivatives in rotational kinematics, specifically applied to spherical 4R mechanisms.

Findings

Effective computation of velocity and acceleration using dual numbers.

Implementation in Fortran improves computational efficiency.

Validated on spherical four-bar mechanism example.

Abstract

This paper proposes a methodology to calculate both the first and second derivatives of a vector function of one variable in a single computation step. The method is based on the nested application of the dual number approach for first order derivatives. It has been implemented in Fortran language, a module which contains the dual version of elementary functions as well as more complex functions, which are common in the field of rotational kinematics. Since we have three quantities of interest, namely the function itself and its first and second derivative, our basic numerical entity has three elements. Then, for a given vector function $f:\mathbb{R}\to \mathbb{R}^m$, its dual version will have the form $\tilde{f}:\mathbb{R}^3\to \mathbb{R}^{3m}$. As a study case, the proposed methodology is used to calculate the velocity and acceleration of a point moving on the coupler-point curve generated by a spherical four-bar mechanism.