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Abstract Recent studies in passively-isolated systems have shown that mode coupling is desirable for best vibration suppression, thus refuting the long-standing rule of mode decoupling. However, these studies have ignored the non-linearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear free undamped vibrations of a passively-isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present closed-form expressions using the method of multiple scales for two cases viz. the non-resonant case and the bounded internal resonance case. We validate our theoretical findings through direct numerical simulations. For the non-resonant case, we show that the system behaves similar to a linear system. However, for the nearly internal resonance case, we demonstrate strong energy exchange between the modes stemming from nonlinear mode coupling. We further study the effect of nonlinear mode coupling on the vibration isolation performance and demonstrate that mode coupling is not always desirable. 

Abstract The application of servocontrolled mechanical-bearing-based precision motion stages (MBMS) is well-established in advanced manufacturing, semiconductor industries, and metrological applications. Nevertheless, the performance of the motion stage is plagued by self-excited friction-induced vibrations. Recently, a passive mechanical friction isolator (FI) has been introduced to reduce the adverse impact of friction in MBMS, and accordingly, the dynamics of MBMS with FI were analyzed in the previous works. However, in the previous works, the nonlinear dynamics components of FI were not considered for the dynamical analysis of MBMS. This work presents a comprehensive, thorough analysis of an MBMS with a nonlinear FI. A servocontrolled MBMS with a nonlinear FI is modeled as a two DOF spring-mass-damper lumped parameter system. The linear stability analysis in the parametric space of reference velocity signal and differential gain reveals that including nonlinearity in FI significantly increases the local stability of the system's fixed-points. This further allows the implementation of larger differential gains in the servocontrolled motion stage. Furthermore, we perform a nonlinear analysis of the system and observe the existence of sub and supercritical Hopf bifurcation with or without any nonlinearity in the friction isolator. However, the region of sub and supercritical Hopf bifurcation on stability curves depends on the nonlinearity in FI. These observations are further verified by a detailed numerical bifurcation, which reveals the existence of nonlinear attractors in the system. 

The utilization of mechanical-bearing-based precision motion stages (MBMS) is prevalent in the advanced manufacturing industries. However, the productivity of the MBMS is plagued by friction-induced vibrations, which can be controlled to a certain extent using a friction isolator. Earlier works investigating the dynamics of MBMS with a friction isolator considered a linear friction isolator, and the source of nonlinearity in the system was realized through the friction model only. In this work, we present the nonlinear analysis of the MBMS with a nonlinear friction isolator for the first time. We consider a two-degree-of-freedom spring-mass-damper system to model the servo-controlled motion stage with a nonlinear friction isolator. The characteristic of the dynamical friction in the system is captured using the LuGre friction model. The system’s stability and nonlinear analysis are carried out using analytical methods. More specifically, the method of multiple scales is used to determine the nature of Hopf bifurcation on the stability lobe. The analytical results indicate the existence of subcritical and supercritical Hopf bifurcations in the system, which are later validated through numerical bifurcation. This observation implies that the nonlinearity in the system can be stabilizing or destabilizing in nature, depending on the choice of operating parameters.