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Derivation Matrices in Mechanics – Data Approach

Great part of mechanics is involved with derivation of some kind, e.g., differential equations. In the long history of mechanics, a lot of mathematical and numerical methods have been developed for their solution. However, today we have an additional component that strongly influences approach to the solution of problems in mechanics. Namely, existence of data allows us to use solution methods that would otherwise be inapplicable. A significant tool towards formalization of the solution process is use of derivation matrices that reduce the derivation operation and solution of differential equations to linear algebra operations [1]. Derivation matrices are formulated by applying numerical methods in matrix notation, like finite difference schemes. Here we are developing a novel formulation based on Lagrange polynomials. 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. Prerequisite for application of derivation matrices is availability of data, recorded displacements or velocities or accelerations. Let us assume quadratic function interpolation and interpolation of the function value in some point 'j' is represented using the equation p_j (x)=u_l L_l (x)+u_j L_j (x)+u_d L_d (x) where 'u's are unknown function values in discretization points, 'L's are Lagrange interpolation polynomials, and indices 'l' and 'd' represent the left and the right point in the adopted discretization scheme. Value of the derivative in point 'j' is w_j (x)=u_l L_l (x) that gives the equation 〖p'〗_j (x). It is clear that we only have to differentiate the Lagrange polynomials, which is straightforward. Special care should be taken at boundary points. In order to preserve a uniform precision, usually an extra point is added at the boundary. 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 us to successfully differentiate our 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. Additional adjustments are needed when one records a reduced set of parameters (less then one set of data for each degree of freedom). It is also possible to determine the initial conditions and damping at the same time. Acknowledgment: This work has been supported through project HRZZ 7926 ”Separation of parameter influence in engineering modelling and parameter identification” and project KK.01.1.1.04.0056 ”Structure integrity in energy and transportation”, which is gratefully acknowledged.

derivation matrices ; partial differential equations ; finite difference method ; linear algebra

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