Tangential dynamic behaviors of the machine hydrostatic slide with a magnetorheological (MR) fluid damper are studied, and the effect of the MR damper to control the vibration of the hydrostatic slide is discussed. The dynamic model of the hydrostatic slide with the MR damper is established, and the tangential vibration equation of linear and nonlinear is derived. The multidimensional incremental harmonic balance method (MIHBM) with discrete Fourier transform (DFT) is derived by which the nonlinear response and stability of the system are studied. The resonance response of the Duffing equation under the combined action of harmonic excitation and constant excitation is obtained. In order to investigate the vibration response of the hydrostatic slide with the MR damper in detail, the bifurcation diagram, phase diagram, and Poincaré map are given. Finally, the dynamic response of the machine hydrostatic slide with the MR damper is discussed, and it is verified that the MR damper can suppress the tangential vibration of the hydrostatic slide effectively and the constant controller can control the chaotic behavior of the system well.

In this paper, the nonlinear dynamic responses of the hydrostatic slide were investigated and the effects of damping and external force to control the vibration system were discussed. The dynamic model of the system was established, and the tangential vibration equation taking into account nonlinear factors was derived. The heteroclinic orbit parameter equations of the vibration system were solved, and the Melnikov function of vibration system is derived. And the chaos condition and judging criterion of the vibration system were obtained by Melnikov’s method. The vibration equation of the hydrostatic slide was solved using the numerical method. The bifurcation diagram, phase diagram, wave diagram of displacement, and Poincaré map were obtained, and the nonlinear dynamic responses were analyzed. Finally validation experiments were conducted, and the results agree well with the results obtained by the Melnikov method and numerical method.

Safety is the most fundamental problem of safety‐critical systems. Safety control addresses the problem whether a given unsafe region of the state space can be avoided by a specific control‐input. Moreover, linearly parameterized dynamical system is a general assumption in most safety‐critical adaptive control literature; however, unknown parameters in real systems are usually nonlinear. The control problem of nonlinearly parameterized systems is really difficult without linear‐in‐the‐parameters (LIP) assumptions, which tends to be complicated and computationally intensive. In this paper, a novel model reference safety‐critical adaptive control (MRAC) approach is proposed for a class of nonlinearly parameterized systems. The proposed approach involves a novel controller architecture with a modified update law, which specifically filters out the unsafe behavior (the system is in a state which the system cannot operate normally, i.e., the given unsafe region), while preserving favorable tracking capability and robustness. The novelty of this paper is that the nonlinearly parameterized systems can be enforced safety, without LIP assumptions and complex calculations. Most importantly, this approach is effective for nonlinearly parameterized systems as well as linearly parameterized systems. Finally, three illustrative numerical examples are presented to demonstrate the effectiveness of the proposed design approach.