An official website of the United States government
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.
more »
« less
BIFURCATION ANALYSIS OF A PD CONTROLLED MOTION STAGE WITH A NONLINEAR FRICTION ISOLATOR
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.
more »
« less
- Award ID(s):
- 2000984
- PAR ID:
- 10412489
- Date Published:
- Journal Name:
- ASME: IDETC-CIE 2022
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Motion stages are widely used for precision positioning in manufacturing and metrology applications. However, they suffer from nonlinear premotion (i.e. “static”) friction, which adversely affects their speed and motion precision. In this article, a friction isolator is used as a simple and robust solution to mitigate the undesirable effects of premotion friction in precision motion stages. For the first time, a theoretical study is carried out to understand the dynamic phenomena associated with using a friction isolator on a motion stage. Theoretical analysis and numerical simulation are conducted to examine the dynamical effects of friction isolator on a proportional–integral–derivative-controlled motion stage under LuGre friction dynamics. The influence of friction isolator on the response and stability of the system is examined through theoretical and numerical analyses. Parametric analysis is also carried out to study the effects of friction isolator and friction parameters on the eigenvalue and stability characteristics. The numerical results validate the theoretical findings and demonstrate several other interesting nonlinear phenomena associated with the introduction of friction isolator. This motivates deeper nonlinear dynamical analyses of friction isolator for precision motion control.more » « less
-
Abstract Motion stages are widely used for precision positioning in manufacturing and metrology applications. However, they suffer from nonlinear pre-motion (i.e., “static”) friction which adversely affects their precision and motion speed. Existing friction compensation methods are not robust enough to handle the highly nonlinear and variable dynamic behavior of pre-motion friction. Therefore, the first two authors have proposed the concept of a friction isolator as a simple and robust solution to mitigate the undesirable effects of pre-motion friction in precision motion stages. They experimentally demonstrated that a motion stage with friction isolator can achieve significantly improved precision, speed and robustness to variations in pre-motion friction. However, a theoretical study was not carried out to fundamentally understand the dynamic phenomena associated with using a friction isolator on a motion stage. This introductory paper investigates the dynamics of a PD-controlled motion stage with friction isolator. The influence of the friction isolator on the response and stability of the system is examined through theoretical and numerical analysis. It is shown, using a case study, that the addition of a friction isolator shrinks the range of P and D gains that can stabilize the motion stage. Several other case studies that include the effects of external excitation and integral controller are carried out to motivate deeper dynamic analyses of the friction isolator for precision motion control.more » « less
-
Abstract Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.more » « less
-
We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov–Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
