Nonlinear Frictional Damping Research Articles

A stability result for viscoelastic piezoelectric system with nonlinear feedback

In this work, we consider a nonlinear viscoelastic piezoelectric beam in the presence of an infinite memory and a nonlinear frictional damping of general type. We investigate the interaction between the two dampings and establish, under general assumptions on the relaxation function and the nonlinear feedback, an explicit formula for the energy decay rates of this system.

Dynamics for wave equations connected in parallel with nonlinear localized damping

Abstract This study investigates the properties of solutions about one-dimensional wave equations connected in parallel under the effect of two nonlinear localized frictional damping mechanisms. First, under various growth conditions about the nonlinear dissipative effect, we try to establish the decay rate estimates by imposing minimal amount of support on the damping and provide some examples of exponential decay and polynomial decay. To achieve this, a proper observability inequality has been proposed and constructed based on some refined microlocal analysis. Then, the existence of a global attractor is proved when the damping terms are linearly bounded at infinity, a special weighting function has been used in this part, which eliminates undesirable terms of the higher order while contributing lower-order terms. Finally, we establish that the long-time behavior of solutions of the nonlinear system is completely determined by the dynamics of large finite number of functionals.

New decay results for Timoshenko system in the light of the second spectrum of frequency with infinite memory and nonlinear damping of variable exponent type

In this study, we consider a one-dimensional Timoshenko system with two damping terms in the context of the second frequency spectrum. One damping is viscoelastic with infinite memory, while the other is a non-linear frictional damping of variable exponent type. These damping terms are simultaneously and complementary acting on the shear force in the domain. We establish, for the first time to the best of our knowledge, explicit and general energy decay rates for this system with infinite memory. We use Sobolev embedding and the multiplier approach to get our decay results. These results generalize and improve some earlier related results in the literature.

Stability Results for a System of Nonlinear Viscoelastic Plate Equations with Nonlinear Frictional Damping and Logarithmic Source Terms

Stability Results for a System of Nonlinear Viscoelastic Plate Equations with Nonlinear Frictional Damping and Logarithmic Source Terms

Global existence and asymptotic behaviour for a viscoelastic plate equation with nonlinear damping and logarithmic nonlinearity

In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Here we prove the existence of the solution to the problem using the Faedo–Galerkin method. Also, we prove few general decay rate results.

Fault detection and identification for rolling mill main drive system based on integrated observer under iterative learning strategy

In this article, the problem of multiple fault detection, isolation and reconfiguration of the rolling mill main drive system containing external disturbances is investigated. Considering the nonlinear frictional damping between the rolls and the rolled parts, a nonlinear mathematical model of the main drive system of the mill is established. A comprehensive fault diagnosis scheme based on observer is addressed for this system subjected to unknown external interference. The proposed scheme is divided into two parts. In the first stage, a set of sliding mode observers is designed for system fault detection, and a fault isolation criterion is proposed based on observer redundancy and generalised residual set theory to reveal the fault source. In the second stage, combined with the iterative learning algorithm, an iterative learning-unknown input observer is constructed to realise the accurate estimation of the fault signal. Unlike the existing fault estimation methods, the iterative learning-unknown input observer designed in this article uses the state estimation error of the previous iteration to estimate the fault signal in the current iteration period. Using [Formula: see text] synthesis to design observers for the system will guarantee fault diagnosis robustness. The Lyapunov theory and linear matrix inequality are introduced to prove the convergence of the proposed observer. The simulation study of a 1780-mm hot strip mill evaluates the proposed scheme. Simulation results demonstrate that the sliding mode observer approach can detect faults in the main drive system and isolate faults accurately. In contrast, the iterative learning-unknown input observer method has the lowest fault reconfiguration error (99.87% smaller than the extended state observer, 99.77% smaller than the unknown input observer) and achieves accurate fault signal tracking.

Global attractors for porous elastic system with memory and nonlinear frictional damping

This paper is concerned with the long‐time behavior of a porous‐elastic system with infinite memory and nonlinear frictional damping. We prove that the dynamical system generated by the solutions of the equations is dissipative, only under the basic conditions (for the well‐posedness) on the memory kernel and the frictional damping . Further, we come up with a condition on , being more general than the usual one (with a positive constant ), under which we prove the asymptotic smoothness and quasi‐stability (the latter needs some stronger condition on ) of the dynamical system. Accordingly, we obtain the existence of a global attractor and show the finite dimensionality of the attractor.

On the energy decay of a viscoelastic piezoelectric beam model with nonlinear internal forcing terms and a nonlinear feedback

In this paper, we consider a piezoelectric beam model with nonlinear source terms and investigate the interaction between a viscoelastic damping and a nonlinear frictional damping. Under general assumptions on the relaxation function and the nonlinear feedback, we establish explicit formulae for the energy decay rates of this system and prove that the energy decays at a rate dictated by the weaker damping. Our results substantially improve some earlier related results in the literature.

Sliding-mode observer-based fault diagnosis and fault-tolerant control of the main drive system of rolling mill

In order to address the problem that the main drive system of rolling mill is easily affected by the impact of biting steel, and considering the nonlinear friction damping and the external perturbations of the main drive system of rolling mill during the rolling process, a fault model of the main drive system of rolling mill is established, and a fault diagnosis and fault tolerance control method of the main drive system of rolling mill based on the nonlinear sliding-mode observer is proposed. In order to suppress the influence of external perturbations on fault diagnosis, a nonlinear sliding-mode observer is constructed for fault diagnosis and fault reconfiguration of the system, and the robustness of the observer to fault reconfiguration is improved by using the sliding-mode control rate [Formula: see text], and the stability of the designed nonlinear sliding-mode observer is proved using Lyapunov’s stability theorem. In order to ensure that the system can operate normally even after a fault occurs, a reference model is designed, and a new controller is redesigned for fault-tolerant control of the system by adding a fault compensation term to the original control scheme using fault estimation information. Through the simulation study of the main drive system of stand F4 of 2030 mm cold rolling mill, it is verified that the observer can accurately track the system state with an angular velocity error of 2.45% and detect and estimate the main drive system failure of rolling mill with an estimation error of no more than 0.04% after a fault occurs; the fault-tolerant control of the main drive system of rolling mill is carried out by using the fault information to restore the system to its normal state, and the angular velocity error is 1.89%.

Optimal energy decay for a viscoelastic Kirchhoff equation with distributed delay acting on nonlinear frictional damping

In this paper, we have analysed the influence of viscoelastic and frictional damping on the decay rate of solutions for a Kirchhoff-type viscoelastic wave equation with a distributed delay acting on nonlinear internal damping. Taking the relaxation function of a fairly large class and using the method of energy in which we introduce an adapted Lyapunov functional and by exploiting certain properties of convex functions, under certain assumptions on the constants of system, we obtain the optimal decay rate of energy in the sense that it is compatible with the decay rate of the relaxation function.

Theoretical and computational decay results for a memory type wave equation with variable-exponent nonlinearity

<p style='text-indent:20px;'>In this paper we are concerned with a viscoelastic wave equation with infinite memory and nonlinear frictional damping of variable-exponent type. First, we establish explicit and general decay results with a very general assumption on the relaxation function. Then, we remove the constraint imposed on the boundedness condition on the initial data used in the earlier results in the literature. Finally, we perform several numerical tests to illustrate our theoretical findings. This study generalizes and improves previous literature outcomes.</p>

Multi-fault diagnosis scheme based on robust nonlinear observer with application to rolling mill main drive system

A fault diagnosis solution is proposed in this article for the uncertainty and external disturbance problems in the main drive system of a rolling mill based on analytical redundancy. Considering the nonlinear friction damping, a system is mathematically modeled, and a bank of nonlinear Luenberger observers is designed. The generalized residual set theory is used to isolate two typical fault types of the system. To optimize the anti-disturbance ability of the observer, the disturbance attenuation factor is introduced. The influence of external disturbance on fault estimation is reduced by adjusting the disturbance attenuation factor. Using Lyapunov stability theory, the convergence analysis of the observer dynamic error equation is provided. The observer gain is obtained through resolving the optimality issue under the linear matrix inequality constraint. Numerical simulations of a 2030-mm tandem cold rolling mill were conducted to demonstrate the practicability of the approach. This fault diagnosis scheme is important to improve the stability of system operation.

Numerical and Theoretical Stability Study of a Viscoelastic Plate Equation with Nonlinear Frictional Damping Term and a Logarithmic Source Term

This paper is designed to explore the asymptotic behaviour of a two dimensional visco-elastic plate equation with a logarithmic nonlinearity under the influence of nonlinear frictional damping. Assuming that relaxation function g satisfies g′(t)≤−ξ(t)G(g(t)), we establish an explicit general decay rates without imposing a restrictive growth assumption on the damping term. This general condition allows us to recover the exponential and polynomial rates. Our results improve and extend some existing results in the literature. We preform some numerical experiments to illustrate our theoretical results.

Analysis of friction fluctuations mechanism of a preloaded roller linear motion guide based on a new 5-DOF dynamic stiffness model

Accurate modeling of friction in a roller linear motion guide (RLMG) bears significance in the positional deviation compensation of heavy-load mechanical systems. However, few models can predict the friction fluctuations performance of the RLMG. To address this problem, we propose a calculation method for calculating the time-varying friction based on a new five degree-of-freedom (5-DOF) dynamic stiffness model. In this approach, the rolling contact mechanism between the roller and raceway is analyzed. A new 5-DOF dynamic stiffness model is derived by considering the roller pitch shift, working phase and nonlinear friction damping, which is iteratively solved by four-order Runge-Kutta method under computing the initial value of each iteration. Based on this, the time-varying contact load of each roller under the periodic variations of roller-contact positions is calculated. Then, a time-varying friction calculation model is formulated considering the combined effect of the time-varying contact load, contact area and lubrication viscosity variations between the roller and raceway. The effects of the load magnitude, load orientation, roller pitch shift, preload and velocity on the 5-DOF real-time displacement, time-varying contact load and time-varying friction are discussed. The experimental results demonstrate the superiority of the proposed approach compared to the traditional model.

Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term

<p style='text-indent:20px;'>In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.</p>

Simplification of the forced response of mistuned bladed disks using multiple scales techniques

Mistuning can produce a considerable increase of the vibratory forced response of the blades compared with that of the tuned bladed disk. This situation can lead to high cycle fatigue failure, and, therefore, it is of crucial importance for the prediction of the safe operation limit of the bladed disk. Nonlinear friction forces determine the final vibration amplitude of the blades, and, because of mistuning, the complete bladed disk has to be considered, increasing considerably the difficulty of the problem. For this reason, we consider the application of asymptotic multiple scale techniques to derive simplified models that give the final vibration states at a very low CPU cost. This asymptotic method is based on the idea that all significant effects (forcing, nonlinear friction damping, and mistuning) produce, in most practical situations, just small corrections of the tuned blade elastic oscillation. These small effects develop in a much longer time scale than that associated with the natural elastic frequency of the tuned system. The reduced models produced by this asymptotic methodology retain only the slow time dynamics, filtering out the fast scale oscillation. In this paper, the bladed disk is described using a mass-spring system with nonlinear friction and with an external forcing acting on a blade-dominated modal family where all modes have very similar vibration frequencies. The derivation of the asymptotic model from the mass-spring system is explained in detail, and validated against the results from the original mass-spring model.

Experimental and Numerical Assessment of Mistuning Effects on Vibratory Response of a Bladed Disk With Underplatform Dampers

Abstract This paper presents experimental and numerical investigation of mistuned forced responses of an integrally bladed disk with full set of underplatform dampers (UPDs). This research aims at providing: 1. An experimental benchmark for nonlinear dynamics of a mistuned bladed disks with UPDs. 2. A numerical model that can account for features of a mistuned forced response level. Accordingly, a detailed experimental campaign is conducted on a static test rig called Octopus. This rig is specifically designed to investigate the dynamics of a full-scale integrally bladed disk (blisk) with UPDs in a noncontact manner so that the dynamic response of the system is not modified. The effect of mistuning on experimental forced response levels is assessed and a linearized model is proposed to predict the modulation of frequency response functions (FRFs) due to the frequency splitting. In the development of the model, the mistuning pattern identified from the linear blisk without UPDs is used and it is assumed that adding the dampers does not change the structural mistuning of the blisk. In this study, the fundamental mistuning model identification (FMM ID) was employed to identify the mistuning pattern of the blisk. It is shown that the proposed model successfully predicts the modulation of linear mistuned FRFs. The linearized model is also able to predict the modulation of nonlinear mistuned FRFs in stick condition (when nonlinear friction damping is negligible) with a good accuracy validating this assumption that adding the dampers does not change the mistuning pattern.

General decay results for a viscoelastic wave equation with a variable exponent nonlinearity

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

Effect of Modal Interactions on Friction-Damped Self-Excited Vibrations

Abstract It is well-known that nonlinear dry friction damping has the potential to bound the otherwise unboundedly growing vibrations of self-excited structures. An important technical example are the flutter-induced friction-damped limit cycle oscillations of turbomachinery blade rows. Due to symmetries, natural frequencies are inevitably closely spaced, and they can generally be multiples of each other. Not much is known on the nonlinear dynamics of self-excited friction-damped systems in the presence of such internal resonances. In this work, we analyze this situation numerically by regarding a two degrees-of-freedom system. We demonstrate that in the case of closely-spaced natural frequencies, the self-excitation of the lower-frequency mode gives rise to non-periodic oscillations, and the occurrence of unbounded behavior well before reaching the maximum friction damping value. If the system is close to a 1:3 internal resonance, limit cycles associated with much higher frictional damping appear, however, most of these are unstable. If more than one mode is subjected to self-excitation, the maximum resistance against self-excitation is at least given by the damping capacity of the most weakly friction-damped mode. These results are of high technical relevance, as the prevailing practice is to analyze only periodic limit states and argue the stability solely by the slope of the damping-amplitude curve. Our results demonstrate that this practice leads to considerable mis- and overestimation of the resistance against self-excitation, and a more rigorous stability analysis is required.

Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation: \begin{document}$ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $\end{document} together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup \begin{document}$ S(t) $\end{document} .