Modelling Stiffness and Damping by Use of Fractional Calculus with Application to Railpads
When studying the dynamic behaviour of a structure, e g a railway track, it is important for the analyst to model both stiffness and damping accurately. Mathematical models including fractional derivatives have been found to work well for many materials. Such models are linear and causal. They result in linear differential equations with noninteger-order derivatives when incorporated into the equations of motion of the structure. Different methods to solve such equations exist.
A modal synthesis solution method is discussed. The order of the fractional derivatives is assumed (approximated) to be a rational number. A state-space vector containing fractional derivatives of the displacements is introduced and the system of equations is expanded. The eigenmodes of the expanded system are used to transform the equations into a set of decoupled modal equations. The solutions of the modal equations, weighted by the eigenmodes, are superposed to give the physical response of the transiently loaded structure. Ways to reduce the computational effort are proposed. It is found that the matrices of the expanded system need not be numerically established.
Equations of motion with fractional derivatives require initial conditions on both fractional and integer-order derivatives of the displacements. A model with fractional integrals instead of fractional derivatives is suggested. When this model is incorporated into the equations of motion, the solution of the resulting equations require initial conditions on displacements and velocities only. Further, the fractional integral model implies a unique relationship between stress and strain, whereas the fractional derivative model requires initial conditions. A time integration method is outlined for the numerical solution of the equations of motion when they are based on the fractional integral model.
A fractional calculus model of the dynamic behaviour of railpads is proposed. In a railway track, railpads are placed between the steel rails and the concrete sleepers. They protect the sleepers from wear and they provide electrical insulation. More important, when studying the dynamic behaviour of the track, is the influence of the railpads on the stiffness and damping of the whole track structure. The stiffness and damping of studded rubber railpads have been measured as functions of frequency, both in a test rig and in a complete track. The stiffness is found to increase weakly with the frequency, but to increase strongly with the static preload. The loss factor is found to be nearly independent of the preload and to increase only slightly with the frequency. The parameters of the fractional derivative railpad model are fitted to experimental data from the laboratory measurements. A different set of parameters is needed for each level of preload since, as mentioned, the fractional derivative model is linear.