Output Regulation in Bimodal Systems With Application to Surface Tracking in the Presence of Contact Vibrations

Motivated by a class of surface tracking problems in mechanical systems subject to contact vibrations, this paper considers a regulation problem for discrete-time switched bimodal linear systems where it is desired to achieve output regulation against exogenous input signals featuring known deterministic and unknown random components. A first step in the regulator design involves constructing a set of observer-based parameterized stabilizing controllers that satisfy a sufficient regulation condition for the switched system against the known deterministic disturbance or reference signals. In the second step, an additional performance constraint is added to identify, from among the already constructed regulators, those that provide the best regulation performance against the unknown random disturbances. A corresponding regulator synthesis algorithm is developed based on iteratively solving properly formulated bilinear matrix inequalities. The proposed regulator is successfully evaluated on an experimental setup involving a switched bimodal mechanical system subject to contact vibrations, hence demonstrating the effectiveness of the proposed regulation approach.

Nonlinear Vibration of Gear Pairs With Tooth Surface Modifications at Primary Resonance Using a Perturbation Method

An analytical solution for the nonlinear vibration of gear pairs that exhibit partial and total contact loss is found. The gear teeth can have arbitrary tooth surface modifications. Such modifications and dynamic displacements separate parts of gear tooth surface otherwise designed to be in contact. This is partial contact loss. The excitation and the nonlinearity are not specified but are found from the force-deflection function of the gear pair, which comes from independent analysis, such as a finite element model. Fourier and Taylor series expansions of the force-deflection function capture the flexibility, nonlinearity, and the excitation in a few coefficients. The gear elastic behavior includes Hertz contact, bending, and shear. The nonlinearity arises chiefly from tooth surface modifications due to the dependence of contact upon the instantaneous dynamic mesh force. Although this work focuses on gear pairs with tooth surface modifications, the physical system from which the force-deflection function comes is not limited to gear pairs. Sphere/half-space contact vibrations are also analyzed. The dynamic frequency-amplitude relation at the steady-state is found using the method of multiple scales. Comparisons with experiments from the literature on gear vibrations and sphere/half-space contact vibrations verify the method.

Normal and Angular Motions at Rough Planar Contacts During Sliding With Friction

The planar dynamics of a rough block in nominally stationary or sliding contact with a counter-surface is studied in this work. Starting with the Greenwood-Williamson model of a rough surface, the analysis of elastic contact deflections is extended to accommodate angular as well as normal motions. The real area of contact and the normal contact force are obtained in terms of the relative approach and orientation of the surfaces. It is shown that angular and normal motions at frictional contacts are generally coupled. The contact area and normal contact force are shown to be nonlinearly related to the normal and angular motions. However, the contact area remains proportional to the normal load, even in the presence of angular motions. When the friction force is assumed to be proportional to the real area of contact, the coefficient of sliding friction will be unchanged by small relative rotations between the sliding bodies. Based on this contact and friction model, the nonlinear equations of motion that describe the planar contact vibrations of a sliding block can be written directly. Although a detailed analysis of the stability and response characteristics of these nonlinear equations is beyond the scope of the present work, a limited comparison of calculations and measurements taken on both stationary and sliding blocks indicate that the small amplitude contact vibrations are reasonably well captured by the model developed in this work.

Simulation of Random Surface Roughness-Induced Contact Vibrations at Hertzian Contacts During Steady Sliding

Random normal contact vibrations, excited by surface irregularities swept through the contact region of Hertzian contacts during sliding, are studied using digital simulation techniques. The input disturbances are modeled as random time processes with specified spectral content in the spatial wavenumber and frequency domains. The Hertzian contact stiffness is modeled directly or through a bilinear approximation. The contact vibration spectra and resulting mean square contact loading are obtained from the simulations. A comparison with previous measurements shows good agreement between the simulation and experimental results.