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

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.