Controlling the Nature of Bifurcation in Friction-Induced Vibrations with a Dynamic (Lugre) Friction Model by Delayed Position Feedback (original) (raw)
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Delayed feedback for controlling the nature of bifurcations in friction-induced vibrations
Journal of Sound and Vibration, 2011
We analyse the control of friction-induced vibrations using time-delayed displacement feedback. We have used the exponential model for the drooping characteristics of the friction force for which the bifurcation is subcritical in nature. With an appropriate choice of the control parameters we have managed to change the nature of the bifurcation to supercritical along with increasing the stability boundaries. A nonlinear controller is required when the control force is applied in a direction parallel to the friction force. In contrast, a linear time-delayed displacement feedback applied in a direction normal to the friction force achieves our dual objective of controlling the nature of the bifurcation as well as quenching the vibrations. We also consider a dynamic friction model (the LuGre model) and observe that the qualitative change in the nature of the bifurcation is independent of the complexity considered in modeling the friction force.► We control the nature of bifurcation associated with friction-induced vibrations. ► Time-delayed feedback control has been used. ► Linear controller normal to the contact surface is found to be sufficient. ► Results are robust with respect to the friction model used for the study.
International Journal of Non-Linear Mechanics, 2014
We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.
A comparative study on the control of friction-driven oscillations by time-delayed feedback
Nonlinear Dynamics, 2010
We perform a detailed study of two linear time-delayed feedback laws for control of frictiondriven oscillations. Our comparative study also includes two different mathematical models for the nonlinear dependence of frictional forces on sliding speed. Linear analysis gives stability boundaries in the plane of control parameters. The equilibrium loses stability via a Hopf bifurcation. Dynamics near the bifurcation is studied using the method of multiple scales (MMS). The bifurcation is supercritical for one frictional force model and subcritical for the other, pointing to complications in the true nature of the bifurcation for frictiondriven oscillations. The MMS results match very well with numerical solutions. Our analysis suggests that one form of the control force outperforms the other by many reasonable measures of control effectiveness.
Analysis and control of friction-induced oscillations in a continuous system
Journal of Vibration and Control, 2011
We analyse and control friction-induced oscillations in a continuous system due to the drooping characteristics of the friction force. The model consists of a cantilever beam with an end mass that is in frictional contact with a rigid rotating disc. Time-delayed displacement feedback applied normal to the disc surface is used to control the vibrations. Linear stability analysis yields the stability boundary corresponding to the Hopf bifurcation point. Nonlinear analysis is performed to obtain conditions on the control parameters for which the nature of the bifurcation is subcritical such that these values can be avoided. The control parameters for effective quenching of the vibrations are obtained. An interesting regime of control parameters for which the system is stable for low and high velocities but unstable for intermediate velocities is also observed.
Experimental investigation is performed on a test setup representing a single-degree-of-freedom friction-induced system. The experimental setup consists of a rigid mass (oscillator) connected to a fixed support through a spring and the mass in frictional contact with a moving belt. The major objectives of the experiments are to characterize (i) the nature of friction-induced oscillations, (ii) the nature of bifurcation associated with frictional instability in the system, and (iii) the nature of friction force that is responsible for the oscillations observed from the experiment. The phase portrait of the system shows significant overshoot of the oscillator velocity above the belt velocity indicating the existence of hys-teretic loop around zero relative velocity (pre-sliding regime). The bifurcation diagram clearly demonstrates the subcriticality of the Hopf bifurcation associated with the system negating all empirical friction models which yield supercritical Hopf bifurcation. The friction force-relative velocity curve shows significant hysteretic behavior, both in the pre-sliding as well as in the pure sliding domains. This observation hints towards a dynamic or an acceleration-dependent friction model as an appropriate choice for representing the friction force obtained from our experimental setup .
5-Time-delayed absorber for controlling friction-driven vibration
The efficacy of an active absorber based on the time-delayed displacement difference feedback in controlling friction-driven vibrations is discussed. Mainly two types of absorbers are considered: the tuned absorber having the natural frequency same as that of the primary system and the high-frequency absorber with the natural frequency higher than that of the primary system. The local stability analysis clearly demonstrates that the static equilibrium can be locally stabilized by appropriately selecting the control gain and the time-delay. The regions of stability are delineated in the plane of the control parameters. The robustness analysis is performed to help select the control parameters for the best performance. A method of optimizing the robustness of the system is presented. The influences of the absorber parameters on the degree of stability and the robustness are discussed. Numerical simulations of the system demonstrate that proper choices of the control parameters can also attain the global stability of the system. Numerical simulations reveal that apart from the globally stable static equilibrium or the coexisting locally stable static equilibrium with the stationary limit cycle vibrations, unbounded motions are also possible for some parameter values. Thus, care should be exercised in selecting the absorber parameters. r .in (S. Chatterjee).
Journal of Sound and Vibration, 2021
The use of active vibration control may induce a delay leading to detrimental degradation of the performance of active vibration control. This is particularly true in the case of mechanical systems subjected to friction-induced vibration and noise for which such time-delays can lead to the appearance of undesirable instability. Furthermore, conducting a stability analysis of time-delay systems and estimation of the critical time delay are challenging, due to the infinite nature of the characteristic (quasi) polynomial of the associated closed-loop system, having an infinite number of roots. The objective of this paper is to discuss a strategy for the estimation of the critical time delay for the problem of Friction-Induced Vibration and noisE (FIVE). To achieve such an objective, the prediction of the stability analysis of time delay systems and the estimation of the associated critical time delay are first performed by applying the frequency sweep test and the eigenvalue problem approximation using the Taylor series expansion of the delayed term. In a second time, a mixed approach is proposed to predict effectively the real critical time delay of autonomous controlled systems subjected to friction-induced vibration. The efficiency of the proposed approach is illustrated by numerical examples for the prediction of self-sustaining vibrations of a phenomenological model with two degrees of freedom for which it is possible to provide a clear understanding and illustration of the phenomena involved and observed.
Friction-induced vibration considering multiple types of nonlinearities
Nonlinear Dynamics, 2020
The friction-induced vibration of a novel 5-DoF (degree-of-freedom) mass-on-oscillating-belt model considering multiple types of nonlinearities is studied. The first type of nonlinearity in the system is the nonlinear contact stiffness, the second is the non-smooth behaviour including stick, slip and separation, and the third is the geometrical nonlinearity brought about by the moving-load feature of the mass slider on the rigid belt. Both the linear stability of the system and the nonlinear steady-state responses are investigated, and rich dynamic behaviours of the system are revealed. The results of numerical study indicate the necessity of the transient dynamic analysis in the study of friction-induced-vibration problems as the linear stability analysis fails to detect the occurrence of self-excited vibration when two stable solutions coexist in the system. The bifurcation behaviour of the steady-state responses of the system versus some parameters is determined. Additionally, th...
Friction-Induced Vibration Suppression via the Tuned Mass Damper: Optimal Tuning Strategy
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Friction-induced vibrations are a significant problem in various engineering applications, while dynamic vibration absorbers are an economical and effective tool for suppressing various kinds of vibrations. In this study, the archetypal mass-on-moving-belt model with an attached dynamic vibration absorber was considered. By adopting an analytical procedure, the optimal tuning of the absorber’s parameters was defined. Furthermore, the bifurcations occurring at the loss of stability were analytically investigated; this analysis illustrated that a properly chosen nonlinearity in the absorber’s stiffness permits controlling the supercritical or subcritical character of the bifurcation. However, a numerical analysis of the system’s dynamics, despite confirming the analytical results, also illustrated that the system’s global behavior is only slightly affected by the bifurcation character. Indeed, a dynamic vibration absorber possessing a perfectly linear restoring force function seems to...
Utilizing time-delays to quench the nonlinear vibrations of a two-degree-of-freedom system
Meccanica, 2017
Although, time-delays are considered an undesirable phenomenon in the active control system, this article shows how to utilize time-delays to mitigate the oscillations of a two-degree-of-freedom nonlinear model simulating a horizontally supported Jeffcott-rotor system. The multiple scales method is conducted to obtain a second-order asymptotic solution to the system governing equations. The slow-flow modulating equations of both the amplitudes and phases are extracted. The conditions that make the time-delayed controller works as a damper or exciter are clarified. The effects of the control gains and timedelays on the system stability are investigated. The analyses illustrated that both negative and positive displacement feedback control can mitigate the system vibrations to an excellent level if the time-delays are chosen within their optimal values. A method for selecting the optimal values of the time-delays is included. Then, the acquired analytical results are approved by solving the system equations numerically. The analytical and numerical results confirmed that the vibration peak has been reduced by about 80% without time-delays, and by 95% with the optimal values of the time-delays. Finally, a comparison with recently published articles concerning time-delayed feedback control is included.