Valery N. Pilipchuk

Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT)

Abstract A nonlinear boundary value problem (NBVP) formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented. The formulation is based on a non-smooth temporal transformation (NSTT), to replace the independent temporal variable of the problem with two new piecewise-linear periodic temporal variables. As a result, the problem of computing the subharmonic motions is reduced to a set of NBVPs with homogeneous boundary conditions. Some numerical and analytical solutions of the reduced NBVPs are given, and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed. In particular, the formulation shows a definite connection between vibro-impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit. An interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system.

Creep-slip capture as a possible source of squeal during decelerated sliding

Friction-induced vibration of a two-degree-of-freedom mass-damper-spring system interacting with a decelerating rigid strip is investigated. The friction law is approximated by an analytical function to facilitate the analyses and numerical integrations. It is shown that, after a quasi-harmonic transient period, accompanied by viscous energy dissipation, a short period of intensive ‘creep-slip’ vibration occurs, which generates a series of ‘micro-impacts’ on the strip. Because of the impulsive character of such kind of loading, its Fourier spectrum is rich and quite broadband. Using an averaging technique, the ‘normal form’ equations of motion show that the out-of-phase vibration mode absorbs more energy from the decelerating strip when its natural frequency satisfies certain resonance conditions. The study is then applied to an automotive disc brake model to gain useful insight into the generation of squeal. It is shown that the out-of-phase creep-slip vibration (in the longitudinal direction) of the brake pads generates an impulsive bending moment on the decelerating strip (disc rotor). This impulsive load may be considered as a possible source for brake squeal. The technique developed in this paper may be extended to other ‘squealing systems’ including models for geophysical faults (earthquakes).

Non-conventional Synchronization of Weakly Coupled Active Oscillators

We present a new type of self-sustained vibrations in the fundamental physical model covering a broad area of applications from wave generation in radiophysics and nonlinear optics to the heart muscle contraction and eyesight disorder in biophysics. Such a diversity of applications is due to the universal physical phenomenon of synchronization. Previous studies of this phenomenon, originating from Huygens famous observation, are based mainly on the model of two weakly coupled Van der Pol oscillators and usually deal with their synchronization in the regimes close to nonlinear normal modes (NNMs). In this work, we show for the first time that, in the important case of threshold excitation, an alternative synchronization mechanism can develop when the conventional synchronization becomes impossible. We identify this mechanism as an appearance of dynamic attractor with the complete periodic energy exchange between the oscillators, which is the dissipative analogue of highly intensive beats in a conservative system. This type of motion is therefore opposite to the NNM-type synchronization with no energy exchange by definition. The analytical description of these vibrations employs the concept of Limiting Phase Trajectories (LPTs) introduced by one of the authors earlier for conservative systems. Finally, within the LPT approach, we describe the transition from the complete energy exchange between the oscillators to the energy localization mostly on one of the two oscillators. The localized mode is an attractor in the range of model parameters wherein the LPT as well as the in-phase and out-of-phase NNMs become unstable.

Application of Special Nonsmooth Temporal Transformations to Linear and Nonlinear Systems under Discontinuous and Impulsive Excitation

Linear and nonlinear mechanical systems under periodic impulsive excitation are considered. Solutions of the differential equations of motion are represented in a special form which contains a standard pair of nonsmooth periodic functions and possesses a convenient structure. This form is also suitable in the case of excitation with a periodic series of discontinuities of the first kind (a stepwise excitation). The transformations are illustrated in a series of examples. An explicit form of analytical solutions has been obtained for periodic regimes. In the case of parametric impulsive excitation, it is shown that a nonequidistant distribution of the impulses with dipole-like temporal shifts may significantly effect the qualitative characteristics of the response. For example, the sequence of instability zones loses its different subsequences depending on the parameter of the shifts. It is shown that the methods applicability can be extended for nonperiodic regimes by involving the idea of averaging.

Study of the Oscillations of a Nonlinearly Supported String Using Nonsmooth Transformations

An analytical method for analyzing the oscillations ofa linear infinite string supported by a periodic array of nonlinear stiffnesses is developed. The analysis is based on nonsmooth transformations of spatial variable, which leads to the elimination of singular terms (generalized functions) from the governing partial differential equation of motion. The transformed set of equations of motion are solved by regular perturbation expansions, and the resulting set of modulation equations governing the amplitude of the motion is studied using techniques from the theory of smooth nonlinear dynamical systems. As an application of the general methodology, localized time-periodic oscillations ofa string with supporting stiffnesses with cubic nonlinearities are computed, and leading-order discreteness effects in the spatial distribution of the slope of the motion are detected.

Disc Brake Ring-Element Modeling Involving Friction-Induced Vibration

This paper presents the analytical modeling and dynamic characteristics of disc brake systems under equal contact loads on both sides of the disc. The friction force acting on the pad is assumed to be concentrated along its trailing edge due to the moment arising from the friction force, and thus results in a redistribution of normal forces. In view of equal contact forces, the disc will not experience transverse motion but only tangential and radial vibrations. The only nonlinearity involved in the model arises mainly from contact forces. The dependence of the friction coefficient between the pad and disc is smoothed at zero relative velocity to avoid the problem of differential inclusion. Some preliminary numerical results of the disc and pad are obtained. The results exhibit the occurrence of stick-slip with a relatively small high frequency component during the sliding regime. The later component is mainly due to higher elastic in-plane modes of the disc, whereas the stick-slip component is a global disc-pad motion involving the lowest pad mode.

Modeling and Simulation of Elastic Structures with Parameter Uncertainties and Relaxation of Joints

Joint preload uncertainties and associated geometrical nonlinearities have a direct impact on the design process and decision making of structural systems. Thus, it is important to develop analytical models of elastic structures with bolted joint stiffness uncertainties. The conventional boundary value problem of these systems usually involves time-dependent boundary conditions that will be converted into autonomous ones using a special coordinate transformation. The resulting boundary conditions will be combined with the governing nonhomogeneous, nonlinear partial differential equation that will include the influence of the boundary conditions uncertainty. Two models of the joint stiffness uncertainty are considered. The first represents the uncertainty by a random variable, while the second considers the relaxation process of the joint under dynamic loading. For a single mode random excitation the response statistics will be estimated using Monte Carlo simulation. The influence of joint uncertainty on the response center frequency, mean square, and power spectral density will be determined for the case of clamped-clamped beam. For the case of joints with time relaxation the response process is found to be nonstationary and its spectral density varies with time. Under random excitation, the response bandwidth is found to increase as the excitation level increases and becomes more stationary. Under sinusoidal excitation, it is shown that the relaxation process of the joints may result in bifurcation of the response amplitude, when even all excitation parameters are fixed.

Dynamics of a two-pendulum model with impact interaction and an elastic support

This paper examines the dynamic behavior of a double pendulummodel with impact interaction. One of the masses of the two pendulumsmay experience impacts against absolutely rigid container wallssupported by an elastic system forming an inverted pendulum restrainedby a torsional elastic spring. The system equations of motion arewritten in terms of a non-smooth set of coordinates proposed originallyby Zhuravlev. The advantage of non-smooth coordinates is that theyeliminate impact constraints. In terms of the new coordinates, thepotential energy field takes a cell-wise non-local structure, and theimpact events are treated geometrically as a crossing of boundariesbetween the cells. Based on a geometrical treatment of the problem,essential physical system parameters are established. It is found thatunder resonance parametric conditions of the linear normal modes thesystems response can be either bounded or unbounded, depending on thesystems parameters. The ability of the system to absorb energy from anexternal source essentially depends on the modal inclination angle,which is related to the principal coordinates.

Analysis of Crack Formation in T-Joint Structures under Dynamic Loading

T-joints play a vital role in the hulls of ships and bulkheads made of sandwich materials. Their design aspects and crack detection are important in preserving the safety of ships navigating through violent sea waves. In this work, an overview of the state-of-the-art of design aspects of T-joints used mainly in ship structures is first provided. The design aspects are focused on estimating static and quasi-static failure loads, fracture and fatigue characteristics. This study also develops a reduced order dynamic model for identification of cracks in T-joints. The reduced model constitutes three modal equations with piecewise-linear asymmetric characteristics in which the influence of the crack appears in terms of its length parameter. In particular, it is shown that the presence of a crack essentially affects both the amplitude and frequency content of the dynamic response due to nonlinear coupling between normal modes. Under external dynamic loading with a frequency close to the first mode frequency, the development of a crack is identified by the evolution of attractors on the configuration planes created by different combinations of modal coordinates.

Sensitive Dependence on Initial Conditions of Strongly Nonlinear Periodic Orbits of the Forced Pendulum

We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can ‘jump’ from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.