Real-Time Simulation of Fluid Power Systems Containing Small Oil Volumes, Using the Method of Multiple Scales

Machinery devices often consist of mechanical mechanisms that are actuated by fluid power systems. In many applications, the mechanical system can be modelled and analysed in terms of the multi-body system dynamics. Fluid power systems, in turn, can be analysed via the lumped-fluid theory, with which simulation of fluid power systems requires smaller integration time steps than needed by multi-body solvers. This leaves simulation of the entire machinery device beyond reach for a real-time framework, with the main reason for the very small time steps in modelling of fluid power systems being the presence of a small hydraulic volume, which creates a numerical stiffness problem. The stiffness issue may arise from numerical singularity emerging in the fluid power system, which implies that solving the governing equations involves different time scales - small and large. To resolve the numerical singularity in hydraulic circuits, the authors developed a perturbed model to alleviate the stiffness problem demonstrated that it can increase the integration time step by an order of magnitude. Since the perturbed model does necessitate a correction factor for the volumetric flow rate, the method of multiple scales is applied to compute the pressure within the small volume to second-order accuracy, O(ε
²
), in comparison with the perturbed model's O(ε). The results reveal that if the correction parameter is not set, the perturbed model's cumulative error leads to considerable deviation in piston position with respect to the reference model, whereas the multiple-scale model eliminates the issue of cumulative error without demanding any flow-rate correction factor.