Derivation matrix in mechanics – data approach

Abstract

Existence of data allows use of solution methods for differential equations that would otherwise be inapplicable. The solution process for this is formalized by using derivation matrices that reduce the time necessary for derivation and solving of differential equations. Derivation matrices are formulated by applying numerical methods in matrix notation, like finite difference schemes. In this work, a novel formulation is developed based on Lagrange polynomials with special care taken at boundary points in order to persevere a uniform precision. The main advantage of the approach is straightforward formulation, clear engineering insight into the process and (almost) arbitrary precision through choice of the interpolation order. The result of this procedure is the derivation matrix of the dimension [n×n], where 'n' is the number of data points. The resulting matrix is singular (of rank 'n-1') until boundary/initial conditions are introduced. However, that does not prevent the user to successfully differentiate its unknown function represented with the recorded data points. Derivation matrix approach is easily applicable to a wide range of engineering problems. This methodology could be extended to dynamic systems with multiple degrees of freedom and adapted when velocities or accelerations are recorded instead of displacements.