Dynamic modelling of mechanical systems with opposing restrained preloaded stiffnesses

Preloaded springs are used in many mechanical systems such as suspension systems and linkages, as well as household items such as clothing clasps. Usually, the preload has no effect on small amplitude dynamic responses which occur about the mean preloaded position. However if the system has opposing preloaded elastic elements, where unloading is limited by restraints such as central rods which preserve the preload, then the preload has a dramatic effect on a systemʹs dynamic response. When such a system is set into free oscillation, the period of motion steadily decreases as the amplitude diminishes. Until now the authors have not seen this behavior described elsewhere. esent work is motivated by the modelling of joysticks which are used to actuate hydraulic systems in a wide variety of mobile construction and forestry equipment. Their use for long periods of time may lead to repetitive strain injuries in an operatorʹs upper limbs, neck and back. In order to assess the total force required to move a joystick, the stiffness, damping and inertia characteristics must be determined. A mathematical model for joystick dynamics is presented and the effect of restrained spring preloads on the changing period of free vibration is explained. In addition, procedures to estimate the non-symmetric non-linear damping from experimental data are described. By simulating the dynamic response in MATLABTM, the damping is fine-tuned by comparing the joystickʹs simulated free oscillation response with its experimental response. Finally, the torque waveform required to perform a simple joystick motion is estimated. The methods developed in this paper could be applied to any lumped-parameter mechanical system where there are opposing restrained preloaded elastic elements.