80 results on '"Valery N. Pilipchuk"'
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2. Professor Leonid I. Manevitch
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Oleg Gendelman, Margarita Kovaleva, Igor V. Andrianov, Yuri V. Mikhlin, and Valery N. Pilipchuk
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Engineering ,Control and Systems Engineering ,business.industry ,Applied Mathematics ,Mechanical Engineering ,Aerospace Engineering ,Mechanical engineering ,Ocean Engineering ,Electrical and Electronic Engineering ,business - Published
- 2020
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3. Analytical criterion of a multimodal snap-through flutter of thin-walled panels with initial imperfections
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Valery N. Pilipchuk
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Physics ,Multiple equilibrium ,Applied Mathematics ,Mechanical Engineering ,Airflow ,Aerospace Engineering ,Perturbation (astronomy) ,Ocean Engineering ,Thin walled ,Mechanics ,Snap through ,Critical speed ,Flexural strength ,Control and Systems Engineering ,Flutter ,Electrical and Electronic Engineering ,Computer Science::Distributed, Parallel, and Cluster Computing - Abstract
This work deals with snap-through flutter dynamics of thin-walled shallow panels accompanied by flexural mode transitions assuming cylindrical bending conditions. The problem is therefore multimodal and, in addition, essentially non-local due to the presence of multiple equilibrium positions. The corresponding analysis is based on the asymptotic of a perfectly flexible panel with a continuous manifold of equilibrium configurations. It is assumed that trajectories of the snap-through dynamics are close to such a manifold, which is interpreted as a family of generating solutions. It is shown that the two-mode approximation depicts major physical specifics of the snap-through process, whereas higher modes can be reasonably treated as a perturbation. As a main result of the analysis, the analytical estimate for the critical speed of airflow leading to the cyclical snap-through flutter is derived.
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- 2020
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4. Control of the nonlinear dynamics of a truck and trailer combination
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AmirReza Latif, Nabil G. Chalhoub, and Valery N. Pilipchuk
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Truck ,Computer science ,Applied Mathematics ,Mechanical Engineering ,Trailer ,Flat tire ,Aerospace Engineering ,Ocean Engineering ,Differential (mechanical device) ,Nonlinear system ,Control and Systems Engineering ,Control theory ,Torque ,Electrical and Electronic Engineering ,Reference model - Abstract
The directional instability problem of a truck and trailer combination in a forward motion has been investigated by developing a dynamic model of the planar rigid-body dynamics of the system. Two control strategies have been devised based on the reference model following controller configuration. The main objective of the controllers is to ensure that the nonlinear model follows the desired response generated by a linearized version of the truck and trailer model. The first RFMC uses an integral-plus-state feedback controller for its compensator, which was designed based on the eigenstructure assignment methodology. The second RFMC relies on a sliding mode compensator. The control inputs were applied through the actuation of differential wheel torques. Different driving scenarios and road conditions, such as braking and accelerating maneuver with a flat tire on a dry road, and a double-lane change maneuver on a slippery road, were considered. Overall, the simulation results demonstrated the superiority of the sliding mode controller over the integral-plus-state feedback controller in ensuring better tracking of the truck and trailer combination to a target path under different driving maneuvers.
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- 2020
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5. Design of Energy Absorbing Metamaterials Using Stochastic Soft-Wall Billiards
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Valery N. Pilipchuk
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Physics ,Work (thermodynamics) ,Photon ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Metamaterial ,energy absorbing materials ,Mechanics ,Kinetic energy ,Standing wave ,soft wall billiards ,Chemistry (miscellaneous) ,Normal mode ,Energy flow ,QA1-939 ,Computer Science (miscellaneous) ,Dynamical billiards ,Mathematics ,wave cancellation - Abstract
Physical principles for designing cellwise artificial materials with energy-absorbing/harvesting and wave guiding properties are discussed in the present work. We analyzed the evolution of waves in a one-dimensional lattice of 3D massive potential wells with light particles inside. The potential wells were coupled with elastic springs and represented soft-wall versions of the so-called stochastic billiards. A billiard could switch from repelling to the stadium type as the parameter of shape changed its sign from positive to negative. We found that certain shapes of the potential wells/containers provided a one-directional trend of the energy flow from the chain of containers into the chaotically moving light inclusions by increasing their total kinetic energy. As a result, propagating waves became trapped by giving rise to standing waves with chaotic mode shapes with decaying amplitudes.
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- 2021
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6. Stochastic energy absorbers based on analogies with soft-wall billiards
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Valery N. Pilipchuk
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Physics ,Similarity (geometry) ,Applied Mathematics ,Mechanical Engineering ,Aerospace Engineering ,Ocean Engineering ,Dissipation ,Container (type theory) ,01 natural sciences ,Stability (probability) ,Nonlinear system ,Classical mechanics ,Control and Systems Engineering ,Normal mode ,Spring (device) ,0103 physical sciences ,Limit (mathematics) ,Electrical and Electronic Engineering ,010301 acoustics - Abstract
A few degrees-of-freedom Hamiltonian model exhibiting one-directional long-term trends in energy exchange flows is introduced. The model includes a massive potential well—a container with one or few light non-interacting particles—attached to a linearly elastic spring. Intentionally, no phenomenological dissipation is imposed. Nonetheless, due to the similarity of container shapes to various types of stochastic soft-wall billiards, the energy is transferred from the container (donor) to the inner particle (acceptor) in almost irreversible way during physically reasonable time intervals. The potential well is introduced in such a way that, in the rigid-body limit, it resembles one of the two most common types of billiards, with either dispersing- or stadium-type boundaries, as the signature of main geometrical parameter of the well changes. In particular, we found conditions of stochasticity of the model’ dynamics based on the nonlinear normal mode stability concept. Such an approach points to the link between stability properties of normal modes and the typical dynamics of billiards. Possible applications to macro-level energy harvesters and the design of artificial energy-absorbing materials are discussed.
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- 2019
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7. Oscillators and Oscillatory Signals From Smooth to Discontinuous : Geometrical, Algebraic, and Physical Nature
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Valery N. Pilipchuk and Valery N. Pilipchuk
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- Mechanics, Multibody systems, Vibration, Mechanics, Applied, Engineering, Dynamics, Nonlinear theories
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This updated and enriched new edition maintains its complementarity principle in which the subgroup of rotations, harmonic oscillators, and the conventional complex analysis generate linear and weakly nonlinear approaches, whereas translations and reflections, impact oscillators, and hyperbolic Clifford's algebras, give rise to the essentially nonlinear “quasi-impact” methodology based on the idea of non-smooth temporal substitutions. In the years since “Nonlinear Dynamics: Between Linear and Impact Limits,” the previous edition of this book, was published, due to a widening area of applications, a deeper insight into the matter has emerged leading to the rudimentary algebraic view on the very existence of the complementary smooth and non-smooth base systems as those associated with two different signs of the algebraic equation j2 =± 1. This edition further includes an overview of applications found in the literature after the publication of first edition, and new physical examples illustrating both theoretical statements and constructive analytical tools.
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- 2023
8. Obituary
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Igor V. Andrianov, Oleg V. Gendelman, Yuri V. Mikhlin, and Valery N. Pilipchuk
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Mechanics of Materials ,Applied Mathematics ,Mechanical Engineering - Published
- 2021
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9. Closed Form Solutions for Nonlinear Oscillators Under Discontinuous and Impulsive Periodic Excitations
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Valery N. Pilipchuk
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Physics ,Physics and Astronomy (miscellaneous) ,010308 nuclear & particles physics ,General Mathematics ,lcsh:Mathematics ,Mathematical analysis ,02 engineering and technology ,exact solutions ,lcsh:QA1-939 ,01 natural sciences ,discontinuous loading ,Vibration ,Nonlinear oscillators ,symbols.namesake ,Nonlinear system ,strongly nonlinear oscillators ,Chemistry (miscellaneous) ,0103 physical sciences ,Poincaré conjecture ,Homogeneous space ,0202 electrical engineering, electronic engineering, information engineering ,Computer Science (miscellaneous) ,symbols ,Elementary function ,020201 artificial intelligence & image processing - Abstract
Periodic responses of linear and nonlinear systems under discontinuous and impulsive excitations are analyzed with non-smooth temporal transformations incorporating temporal symmetries of periodic processes. The related analytical manipulations are illustrated on a strongly nonlinear oscillator whose free vibrations admit an exact description in terms of elementary functions. As a result, closed form analytical solutions for the non-autonomous strongly nonlinear case are obtained. Conditions of existence for such solutions are represented as a family of period-amplitude curves. The family is represented by different couples of solutions associated with different numbers of vibration half cycles between any two consecutive pulses. Poincaré, sections showed that the oscillator can respond quite chaotically when shifting from the period-amplitude curves.
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- 2019
10. Friction induced pattern formations and modal transitions in a mass-spring chain model of sliding interface
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Valery N. Pilipchuk
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Physics ,0209 industrial biotechnology ,Mesoscopic physics ,Mechanical Engineering ,Phase (waves) ,Mode (statistics) ,Aerospace Engineering ,02 engineering and technology ,Mechanics ,01 natural sciences ,Computer Science Applications ,symbols.namesake ,Nonlinear system ,020901 industrial engineering & automation ,Amplitude ,Control and Systems Engineering ,0103 physical sciences ,Signal Processing ,symbols ,Dissipative system ,Rayleigh scattering ,010301 acoustics ,Excitation ,Civil and Structural Engineering - Abstract
Friction induced modal transitions and pattern formations are investigated both analytically and numerically based on a mesoscopic mass-spring chain model of sliding interfaces. The case of smooth self-sustained oscillations is considered first based on a two-degrees-of-freedom model with a generalized Rayleigh’s dissipative term by means of a new type of descriptive variables characterizing total excitation level, its distribution between the oscillators, and coherency of oscillations. In particular, it is shown that there exists a threshold of excitation above which self-sustained nonlinear normal and local oscillations are possible. The existence of threshold for generation of standing and propagating waves in the corresponding mass-spring chain is confirmed by both analytical estimates and numerical simulations. Also, it is shown that friction induced oscillations with a creeping phase develop according to a qualitatively different scenario leading to amplitude modulated waves with the highest possible carrying mode.
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- 2021
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11. Analysis of micro-structural effects on phononic waves in layered elastic media with periodic nonsmooth coordinates
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Valery N. Pilipchuk, Bernd Markert, and Igor V. Andrianov
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Physics ,Subharmonic ,Wave propagation ,Applied Mathematics ,Acoustics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,01 natural sciences ,Homogenization (chemistry) ,Computational Mathematics ,Acoustic wave propagation ,Normal mode ,Modeling and Simulation ,0103 physical sciences ,Perpendicular ,010307 mathematical physics ,0101 mathematics ,Anisotropy - Abstract
Propagation of elastic phononic waves in layered composite materials is analyzed by introducing nonsmooth periodic coordinates associated with structural specifics of the materials. Spatial scales of the original (smooth) coordinates are estimated by the wave lengths. In terms of the new coordinates, the homogenization procedure occurs naturally from the continuity conditions imposed on elastic displacements and forces at layer interfaces. As a result, higher-order asymptotic approximations describing spatiotemporal ‘macro’- and ‘micro’-effects of wave propagation are obtained in closed form. Such solutions provide visualizations for the wave shapes illustrating their structure induced local details. In particular, beat-wise mode shapes and effective anisotropy of acoustic wave propagation are revealed. The subharmonic beating in wave modes occur when wave lengths orthogonal to layers is about to ‘resonate’ with layer’ thickness. If the wave speed has a non-zero projection along the layers, then phase shifts between the beats are observed in different cross sections perpendicular to the layers.
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- 2016
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12. Effective Hamiltonians for Resonance Interaction Dynamics and Interdisciplinary Analogies
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Valery N. Pilipchuk
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Physics ,Work (thermodynamics) ,Polynomial ,Slosh dynamics ,Quantum dynamics ,02 engineering and technology ,General Medicine ,macroscopic quantum dynamics ,01 natural sciences ,Resonance (particle physics) ,010305 fluids & plasmas ,Resonance ineractions ,liquid sloshing ,Nonlinear system ,020303 mechanical engineering & transports ,Classical mechanics ,0203 mechanical engineering ,Quantum mechanics ,Nonlinear resonance ,0103 physical sciences ,Restoring force ,nonlinear beating - Abstract
Resonance interactions of oscillators are responsible for fundamental effects in different areas of physics and classical mechanics. The resonance between any two oscillators/modes destroys their individuality by generating a new effective oscillator of energy flow between the two parent oscillators, which is known as beating. In particular, the fundamental character of such energy exchange oscillators is revealed by the fact of their exact integrability in many physically reasonable cases. The present work illustrates such a standpoint on elastic oscillators, discrete liquid sloshing models, and ‘macroscopic’ quantum dynamics related to Josephson's effects. In the case of elastic oscillations, a strongly nonlinear conservative oscillator describing the dynamics of energy partition between two identical linearly coupled oscillators with polynomial restoring force characteristics is analyzed.
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- 2016
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13. Non-smooth Spatial and Temporal Substitutions in Impact Dynamics
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Valery N. Pilipchuk
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State variable ,Dynamical systems theory ,Computer science ,Differential equation ,Numerical analysis ,Coordinate system ,Classification of discontinuities ,01 natural sciences ,010305 fluids & plasmas ,Transformation (function) ,0103 physical sciences ,Applied mathematics ,010301 acoustics ,Smoothing - Abstract
This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. A general purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different approaches are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Also a new type of substitutions eliminating both infinite and step-wise discontinuities and thus completely smoothing the system is suggested. Different illustrating examples are introduced.
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- 2018
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14. Closed-form solutions for oscillators with inelastic impacts
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Valery N. Pilipchuk
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Acoustics and Ultrasonics ,Scale (ratio) ,Mechanical Engineering ,Mechanics ,Condensed Matter Physics ,Collision ,Vibration ,Classical mechanics ,Amplitude ,Mechanics of Materials ,Limiter ,Triangle wave ,Energy (signal processing) ,Variable (mathematics) ,Mathematics - Abstract
A class of vibrating systems with perfectly stiff amplitude limiters is considered by means of non-smooth time substitutions. The motion is represented as a combination of oscillating component, which is due to cyclic collisions with the limiters, and a slow decay caused by the gradual energy loss at collision times. A specific modification of the two variable expansions is used, where the non-smooth (triangle wave) temporal argument is viewed as a fast time while the energy decay is described in a slow time scale. As a result, closed-form analytical solutions are obtained that automatically satisfy collision conditions with the energy loss. Three qualitatively different basic types of vibrations are considered to cover periodic, frequency modulated, and amplitude–frequency modulated motions.
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- 2015
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15. Transient friction-induced vibrations in a 2-DOF model of brakes
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Jan Awrejcewicz, Valery N. Pilipchuk, and Paweł Olejnik
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Engineering ,Acoustics and Ultrasonics ,business.industry ,Mechanical Engineering ,Dynamics (mechanics) ,Process (computing) ,Phase (waves) ,Mechanics ,Condensed Matter Physics ,Vibration ,Normal load ,Mechanics of Materials ,Brake ,Transient (oscillation) ,business ,Simulation - Abstract
Non-stationary effects in the friction-induced dynamics of a two-degree-of-freedom brake model are examined in this paper. The belt–spring–block model is designed to take into account variations of the normal load during the braking process. It is shown that due to the adiabatically slowing down velocity of the belt, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. In particular, the creep-slip leading to a significant widening of the spectrum of the dynamics is observed at the final phase of the process.
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- 2015
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16. Non-conventional phase attractors and repellers in weakly coupled autogenerators with hard excitation
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Margarita Kovaleva, Valery N. Pilipchuk, and Leonid I. Manevitch
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Physics ,Phase portrait ,Phase (waves) ,General Physics and Astronomy ,FOS: Physical sciences ,Pattern Formation and Solitons (nlin.PS) ,Phase plane ,Nonlinear Sciences - Chaotic Dynamics ,Nonlinear Sciences - Pattern Formation and Solitons ,01 natural sciences ,Stationary point ,010305 fluids & plasmas ,Nonlinear system ,Classical mechanics ,Phase space ,Limit cycle ,34D06 ,0103 physical sciences ,Attractor ,Chaotic Dynamics (nlin.CD) ,010306 general physics - Abstract
In our earlier studies, we found the effect of non-conventional synchronization, which is a specific type of nonlinear stable beating in the system of two weakly coupled autogenerators with hard excitation given by generalized van der Pol-Duffing characteristics. The corresponding synchronized dynamics are due to a new type of attractor in a reduced phase space of the system. In the present work, we show that, as the strength of nonlinear stiffness and dissipation are changing, the phase portrait undergoes a complicated evolution leading to a quite unexpected appearance of difficult to detect "repellers" separating a stable limit cycle and equilibrium points in the phase plane. In terms of the original coordinates, the limit cycle associates with nonlinear beatings while the stationary points correspond to the stationary synchronous dynamics similar to the so-called nonlinear local modes.
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- 2017
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17. Nonlinear interactions and energy exchange between liquid sloshing modes
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Valery N. Pilipchuk
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Physics::Fluid Dynamics ,Physics ,Nonlinear system ,Classical mechanics ,Slosh dynamics ,Diagonal ,Phase (waves) ,Elliptic function ,Elementary function ,Statistical and Nonlinear Physics ,Phase plane ,Condensed Matter Physics ,Hamiltonian system - Abstract
Non-stationary effects of the resonance energy exchange between liquid sloshing modes for square-base tanks are analyzed in terms of new descriptive variables namely energy partition and coherency (phase shift) indexes. In particular, it is shown that such a couple represents conjugate variables of an effective Hamiltonian system whose phase plane captures all possible effects of the modal interaction. Furthermore, the presence of damping affects only the temporal scale of the dynamics but still preserves the Hamiltonian structure of equations that provides the existence of first integral. As a result, analytical solution for the nonlinear interaction of predominant modes is obtained in quadratures. The dynamic properties of the modal interaction are controlled by a single parameter which depends upon the fluid depth. Transitions of the phase plane diagram reveal that, above some critical depth, the diagonal in-phase and out-of-phase sloshing modes disappear while only clockwise and counterclockwise swirling, and running phase modes are possible. Whereas the quadratures are invertible within the class of elliptic functions, an explicit solution for the important critical case is obtained in terms of elementary functions. The approach is illustrated on the sloshing model for a square-base tank suggested by Ikeda et al. (T. Ikeda, R.A. Ibrahim, Y. Harata, T. Kuriyama, Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation, J. Fluid Mech., 700 (2012) 304–328.) However, formulated in general terms of coupled oscillators, the approach seems to have a wider area of applicability dealing with non-stationary effects of resonance interactions with nonlinear beating effects in physics and classical mechanics.
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- 2013
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18. New type of synchronization of oscillators with hard excitation
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L. I. Manevich, Valery N. Pilipchuk, and Margarita Kovaleva
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Physics ,Nonlinear system ,Classical mechanics ,Synchronization networks ,Normal mode ,Synchronization (computer science) ,Attractor ,Dissipative system ,Phase (waves) ,General Physics and Astronomy ,Phase synchronization - Abstract
It is shown that stable limiting cycles corresponding to nonlinear beats with complete energy exchange between oscillators can exist in a system of two weakly coupled active oscillators (generators). The oscillatory regime of this type, which implements a new type of synchronization in an active system, is an alternative to the well-studied synchronization in a regime close to a nonlinear normal mode. In this case, the ranges of dissipative parameters corresponding to different types of synchronization do not intersect. The analytic description of attractors revealed in analysis is based on the concept of limiting phase trajectories, which was developed earlier by one of the authors for conservative systems. A transition (in the parametric space) from the complete energy exchange between oscillators to predominant localization of energy in one of the oscillators can be naturally described using this concept. The localized normal mode is an attractor in the range of parameters in which neither the limiting phase trajectory nor any of the collective normal modes is an attractor.
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- 2013
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19. Low temperature brittle debond damage under normal compression of sandwich plates: Analytical modeling and experimental validation
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Raouf A. Ibrahim, Ihab M. Grace, and Valery N. Pilipchuk
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Cracking ,Brittleness ,Materials science ,Strain (chemistry) ,Ceramics and Composites ,Jump ,Experimental validation ,Composite material ,Compression (physics) ,Displacement (fluid) ,Quasistatic process ,Civil and Structural Engineering - Abstract
Experimental study and analytical modeling of brittle debond damage formation in polymethacrylimide (PMI) foam sandwich plates under slow normal compression are reported in this paper. The major cause of damage formation under quasi static compression is found to be low temperature brittle cracking effects taking place below some critical temperature levels within the range from −20 °C to −30 °C. The magnitude of critical temperature is increasing as the displacement controlled loading speed increases. The maximal strain achieved during the compression phase is found to be less influential, however it still significantly affects the shape and size of damage. The phenomenon is described in terms of the theory of thin elastic plates (beams) on a piece-wise linear foundation of Kelvin–Voight type. In particular, exact analytical solutions are derived to describe the damage formation. The dependence of the corresponding load on effective strain reveals a jump of slope at yielding point, which is in perfect match with experimental observations.
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- 2013
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20. Nonconventional synchronization and energy localization in weakly coupled autogenerators
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Margarita Kovaleva, Leonid I. Manevitch, and Valery N. Pilipchuk
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Physics ,Work (thermodynamics) ,Phase (waves) ,Phase plane ,Dissipation ,01 natural sciences ,010305 fluids & plasmas ,Synchronization (alternating current) ,Nonlinear system ,Normal mode ,0103 physical sciences ,Statistical physics ,010306 general physics ,Stationary state - Abstract
The present work follows our previous study dealing with a new type of synchronization in a system of two weakly coupled generalized van der Pol-Duffing autogenerators. The essence of the effect revealed is that the synchronized oscillations are not stationary but accompanied by the most intensive energy exchange between the oscillators. The phase shift between the generators remains constant most of the time, except for vanishingly small transitional intervals. The current analysis deals with a generalized model in order to clarify the frequency detuning effect. We found that varying the frequency detuning, nonlinearity, and dissipation parameters can lead to structural changes in phase diagrams of the energy exchange dynamics, with important transitions from the intensive energy exchange to its localization on one of the two oscillators. The main conclusion is that stationary and nonstationary synchronizations associate with nonlinear normal and local modes, respectively. The analysis uses phase plane diagrams, including the concept of limiting phase trajectories, whose role in nonstationary synchronization appears to be similar to the role of nonlinear normal modes in conventional stationary states.
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- 2016
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21. Temperature effect on non-stationary compressive loading response of polymethacrylimide solid foam
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Valery N. Pilipchuk, Emmanuel Ayorinde, Raouf A. Ibrahim, and Ihab M. Grace
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Materials science ,Brittleness ,Acoustic emission ,Phase (matter) ,Phenomenological model ,Stress–strain curve ,Composite number ,Ceramics and Composites ,Deformation (engineering) ,Composite material ,Plateau (mathematics) ,Civil and Structural Engineering - Abstract
This paper presents the results of experimental investigation and modeling the polymethacrylimide (PMI) solid foam response on slightly non-stationary compressive loads at different loading speed and temperature. The PMI represents a typical material used as a core component of composite sandwich structures. For all conditions, the strain–stress curves exhibit three definite regions, such as linearly elastic, plateau, and densification as it is known from the literature. However, it is shown in the present study that even minor fluctuations of the loading speed may lead to significant qualitative effects within the plateau segment. In particular, spike-wise load drops occur in a regular way during the loading phase when the specimen temperature is below −20 °C. Acoustic emission tests lead to the conclusion that the load drops are associated with generation of brittle fractures in the clusters of broken foam cells. Also, cyclic loading tests were conducted to evaluate the energy loss in cells permanent deformation during the loading cycle. The amount of energy dissipated for each cycle is reduced at higher temperatures. Finally, based on the experimental results, a phenomenological model of the foam load–displacement response at different loading speeds and temperatures is presented in the form of a single analytical expression.
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- 2012
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22. Inelastic impact dynamics of ships with one-sided barriers. Part II: experimental validation
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Ihab M. Grace, Raouf A. Ibrahim, and Valery N. Pilipchuk
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Engineering ,business.industry ,Applied Mathematics ,Mechanical Engineering ,Chaotic ,Aerospace Engineering ,Ocean Engineering ,Mechanics ,Response amplitude operator ,Classical mechanics ,Control and Systems Engineering ,Position (vector) ,Coefficient of restitution ,Orbit (dynamics) ,Reflection (physics) ,Electrical and Electronic Engineering ,Material properties ,business ,Roll moment - Abstract
This paper is the second part of a two-part study of impact interaction of a ship roll motion with one-sided ice barrier. The first part was devoted to analytical and numerical simulations for the case of inelastic impact. The analytical approach was based on Zhuravlev and Ivanov non-smooth coordinate transformations. Extensive numerical simulations were carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave-induced roll moment. The basins of attraction of safe operation revealed the coexistence of different response regimes such as non-impact periodic oscillations, modulated impact motion, period added impact oscillations, chaotic impact motion and roll-over dynamics. This part presents an experimental investigation conducted on a small ship model in a tow tank. In particular, the experimental tests reveal complex dynamic response characteristics such as multi-frequency wave motion caused by the wave reflection from the tank end wall. Measured results show a good agreement with the predicted results by for small angles of the barrier relative to the ship unbiased position. However, deviation becomes significant as the angle increases. This deviation is mainly attributed to the uncertainty of the coefficient of restitution, which is found to depend on the velocity of impact in addition to the geometry and material properties of the model and barrier.
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- 2011
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23. Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations
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Raouf A. Ibrahim, Ihab M. Grace, and Valery N. Pilipchuk
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Physics ,Work (thermodynamics) ,Applied Mathematics ,Mechanical Engineering ,Dynamics (mechanics) ,Rotation around a fixed axis ,Chaotic ,Aerospace Engineering ,Equations of motion ,Ocean Engineering ,Mechanics ,Dissipation ,Classical mechanics ,Control and Systems Engineering ,Orbit (dynamics) ,Electrical and Electronic Engineering ,Roll moment - Abstract
This two-part paper deals with impact interaction of ships with one-sided ice barrier during roll dynamics. The first part presents analytical and numerical studies for the case of inelastic impact. An analytical model of a ship roll motion interacting with ice is developed based on Zhuravlev and Ivanov non-smooth coordinate transformations. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. Such approaches, however, account for the energy loss at impact times in different ways. The present work, in particular, demonstrates that the impact dynamics may have qualitatively different response characteristics to different dissipation models. The difference between localized and distributed equivalent damping approaches is discussed. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequency of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period added impact oscillations, chaotic impact motion and rotational motion. The second part will consider experimental validations of predicted results.
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- 2011
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24. Analysis of crack formation in T-joint structures under dynamic loading
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Raouf A. Ibrahim and Valery N. Pilipchuk
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Engineering ,Mathematical model ,business.industry ,Mechanical Engineering ,Mode (statistics) ,Aerospace Engineering ,Poison control ,Structural engineering ,Modal ,Mechanics of Materials ,Normal mode ,Dynamic loading ,Hull ,Automotive Engineering ,Fracture (geology) ,General Materials Science ,business - Abstract
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.
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- 2010
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25. Closed-form periodic solutions for piecewise-linear vibrating systems
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Valery N. Pilipchuk
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Piecewise linear function ,Control and Systems Engineering ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Aerospace Engineering ,Perturbation (astronomy) ,Ocean Engineering ,Restoring force ,Electrical and Electronic Engineering ,Mathematics - Abstract
In this paper, closed-form asymptotic solutions are derived for piecewise-linear one- and two-degrees-of-freedom systems. Deviations from perfectly linear restoring force characteristics play the role of small but nonsmooth perturbations. It is shown that the nonsmooth transformation of temporal argument enables one of justifying at least first several steps of the classic perturbation procedure which eventually gives the unit-form solutions. The form of the solutions is suitable for further manipulations including possible generalization on non-periodic cases.
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- 2009
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26. Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena
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Valery N. Pilipchuk
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Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Aerospace Engineering ,Perturbation (astronomy) ,Beat (acoustics) ,Ocean Engineering ,Phase variable ,Nonlinear oscillators ,Nonlinear system ,Classical mechanics ,Control and Systems Engineering ,Special functions ,Electrical and Electronic Engineering ,Trigonometry ,Mathematics - Abstract
In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.
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- 2007
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27. Strongly nonlinear vibrations of damped oscillators with two nonsmooth limits
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Valery N. Pilipchuk
- Subjects
Acoustics and Ultrasonics ,Viscous damping ,Mechanical Engineering ,media_common.quotation_subject ,Zero (complex analysis) ,Condensed Matter Physics ,Infinity ,Power (physics) ,Vibration ,Nonlinear system ,Nonlinear oscillators ,Classical mechanics ,Mechanics of Materials ,Exponent ,Mathematics ,media_common - Abstract
A family of strongly nonlinear oscillators with a generalized power form elastic force and viscous damping is considered. An explicit analytical solution is obtained as a combination of smooth and nonsmooth functions. Two different nonsmooth functions involved into the solution are associated with two different nonsmooth limits of the oscillator as the exponent becomes either zero or infinity. As a result, the solution is drastically simplified to give the best match with numerical tests if approaching any of the two limits.
- Published
- 2007
- Full Text
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28. Transient mode localization in coupled strongly nonlinear exactly solvable oscillators
- Author
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Valery N. Pilipchuk
- Subjects
Physics ,Applied Mathematics ,Mechanical Engineering ,Mode (statistics) ,Aerospace Engineering ,Ocean Engineering ,Resonance (particle physics) ,Nonlinear system ,Classical mechanics ,Control and Systems Engineering ,Normal mode ,Bounded function ,Electrical and Electronic Engineering ,Bifurcation ,Equipartition theorem ,Energy (signal processing) - Abstract
Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.
- Published
- 2007
- Full Text
- View/download PDF
29. Non-linear system identification based on Lie series solutions
- Author
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Valery N. Pilipchuk and C.A Tan
- Subjects
Power series ,State variable ,Series (mathematics) ,Mechanical Engineering ,Direct method ,Mathematical analysis ,System identification ,Degrees of freedom (physics and chemistry) ,Aerospace Engineering ,Computer Science Applications ,Control and Systems Engineering ,Phase space ,Signal Processing ,Relaxation (approximation) ,Algorithm ,Civil and Structural Engineering ,Mathematics - Abstract
A direct method of system identification and parameters monitoring is introduced for a general class of non-linear systems. The only requirement is that the system characteristics must be modelled by analytic or sufficiently smooth functions of the state variables, including the time parameter. The approach is based on the operator Lie representations and the corresponding Lie series solutions. This kind of solutions can be obtained for a general class of non-linear systems in the form of analytical power series including the system parameters. Despite the local characterisation of such solutions, the corresponding information carried by the solutions appears to be sufficiently complete and provides precise estimates of the system parameters. A non-linear vertical cantilever beam, excited at its support, is considered as a sample problem. Both constant and relaxation stiffness parameters are determined at every sample point, and thus the stiffness relaxation curves are recovered from the phase space dynamics. The method is found to be stable with respect to artificially introduced noisy components of the numerically simulated data and an increase in the number of degrees of freedom.
- Published
- 2005
- Full Text
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30. Creep–Slip Capture as a Possible Source of Squeal During Decelerated Sliding
- Author
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Chin An Tan and Valery N. Pilipchuk
- Subjects
Engineering ,business.industry ,Applied Mathematics ,Mechanical Engineering ,Aerospace Engineering ,Equations of motion ,Ocean Engineering ,Natural frequency ,Mechanics ,Structural engineering ,Dissipation ,law.invention ,Brake pad ,Vibration ,Control and Systems Engineering ,law ,Brake ,Bending moment ,Disc brake ,Electrical and Electronic Engineering ,business - Abstract
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).
- Published
- 2004
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31. Disc Brake Ring-Element Modeling Involving Friction-Induced Vibration
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P. G. Blaschke, Raouf A. Ibrahim, and Valery N. Pilipchuk
- Subjects
0209 industrial biotechnology ,Engineering ,Friction force ,business.industry ,Mechanical Engineering ,Ring element ,Aerospace Engineering ,020302 automobile design & engineering ,02 engineering and technology ,Structural engineering ,law.invention ,Vibration ,020901 industrial engineering & automation ,Classical mechanics ,0203 mechanical engineering ,Mechanics of Materials ,law ,Automotive Engineering ,General Materials Science ,Disc brake ,Astrophysics::Earth and Planetary Astrophysics ,business - Abstract
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.
- Published
- 2002
- Full Text
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32. Non-smooth time decomposition for nonlinear models driven by random pulses
- Author
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Valery N. Pilipchuk
- Subjects
Sequence ,Van der Pol oscillator ,Series (mathematics) ,Dynamical systems theory ,General Mathematics ,Applied Mathematics ,Monte Carlo method ,Mathematical analysis ,Phase (waves) ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Pulse (physics) ,Nonlinear Sciences::Chaotic Dynamics ,Nonlinear system ,Classical mechanics ,Mathematics - Abstract
In this work, the time argument is decomposed according to impact interactions in a chain of absolutely rigid and perfectly elastic particles. Such a decomposition leads to explicit equations of the stroboscopic mapping for a general class of dynamical systems. Between the times of observation, the system motion is approximated by the Lie series. For illustration, Duffing's oscillator with no linear stiffness under the sine-modulated sequence of Dirac's pulses, that is a modified Ueda's model, was considered. In some cases, a slight randomization of the pulse times could significantly suppress the mapping chaos which is caused by the system nonlinearity. Monte Carlo simulation showed also that such a small random irregularity of the input can bring out the system orbits more clearly in stroboscopic phase plots of the dynamics. An asymmetric Van der Pol equation under the regular and random pulse sequences was considered as another example which is related to the nerve pulse propagation modeling adopted in Biology. In some cases, irregularity of the pulse times resulted in more organized structures of the stroboscopic diagrams.
- Published
- 2002
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33. NON-INVERTIBLE TEMPORAL TRANSFORMATIONS AND POWER SERIES PERIODIC SOLUTIONS
- Author
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Valery N. Pilipchuk
- Subjects
Power series ,Pure mathematics ,Invertible matrix ,Acoustics and Ultrasonics ,Mechanics of Materials ,law ,Mechanical Engineering ,Condensed Matter Physics ,law.invention ,Mathematics - Published
- 2002
- Full Text
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34. Asymptotic of 'Rigid-Body' Motions for Nonlinear Dynamics: Physical Insight and Methodologies
- Author
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Valery N. Pilipchuk
- Subjects
Vibration ,Nonlinear system ,Transformation (function) ,Classical mechanics ,Basis (linear algebra) ,Computer science ,Simple (abstract algebra) ,Euclidean geometry ,Rigid body ,Rigid transformation - Abstract
The purpose of the present work is to show that an adequate basis for understanding the essentially nonlinear phenomena must also be essentially nonlinear but still simple enough to play the role of a basis. It is shown that such types of “elementary” nonlinear models can be revealed by tracking the hidden links between analytical tools of analyses and subgroups of the rigid-body motions or, in other terms, rigid Euclidean transformation. While the subgroup of rotations is linked with linear and weakly nonlinear vibrations, the translations with reflections can be viewed as a geometrical core of the strongly nonlinear dynamics associated with the so-called vibro-impact behaviors. It is shown that the corresponding analytical approach develops through non-smooth temporal substitutions generated by the impact models.
- Published
- 2014
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35. Recent Advances in Liquid Sloshing Dynamics
- Author
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Raouf A. Ibrahim, Valery N. Pilipchuk, and T. Ikeda
- Subjects
Engineering ,business.industry ,Slosh dynamics ,Mechanical Engineering ,Aerospace engineering ,business ,Passive control - Abstract
A liquid free surface in partially filled containers can experience a wide spectrum of motions such as planar, non-planar, rotational, quasi-periodic, chaotic, and disintegration. Civil engineers and seismologists have been studying liquid sloshing effects on large dams, oil tanks and elevated water towers under ground motion. Since the early 1960’s, the problem of liquid sloshing dynamics has been of major concern to aerospace engineers studying the influence of liquid propellant sloshing on the flight performance of jet vehicles. Since then, new areas of research activities have emerged. The modern theory of nonlinear dynamics has indeed promoted further studies and uncovered complex nonlinear phenomena. These include rotary sloshing, Faraday waves, nonlinear liquid sloshing interaction with elastic structures, internal resonance effects, stochastic sloshing dynamics, hydrodynamic sloshing impact dynamics, g-jitter under microgravity field, cross-waves, and spatial resonance. The dynamic stability of liquid gas tankers and ship cargo tankers, and liquid hydrodynamic impact loading are problems of current interest to the designers of such systems. This article will address the means of passive control of liquid sloshing and the use of liquid sloshing forces to control vibratory structures. Other important contributions include the development of digital computer codes to solve complex problems that were difficult to handle in the past. The purpose of this article is to review the research work developed in different applications. It will highlight the major achievements and results reported in the literature. Some early work will be cited very briefly in order to provide an updated bibliography of liquid sloshing dynamics. This review article contains 1,319 references.
- Published
- 2001
- Full Text
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36. Modeling and Simulation of Elastic Structures with Parameter Uncertainties and Relaxation of Joints
- Author
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S. L. Qiao, Raouf A. Ibrahim, and Valery N. Pilipchuk
- Subjects
Monte Carlo method ,General Engineering ,Geometry ,Mechanics ,Modeling and simulation ,Vibration ,Nonlinear system ,Joint stiffness ,medicine ,Relaxation (physics) ,Boundary value problem ,medicine.symptom ,Random variable ,Mathematics - Abstract
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.
- Published
- 2000
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37. [Untitled]
- Author
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Raouf A. Ibrahim and Valery N. Pilipchuk
- Subjects
Applied Mathematics ,Mechanical Engineering ,Log-polar coordinates ,Pendulum ,Aerospace Engineering ,Equations of motion ,Ocean Engineering ,Action-angle coordinates ,Inverted pendulum ,Generalized coordinates ,Classical mechanics ,Orthogonal coordinates ,Control and Systems Engineering ,Normal mode ,Electrical and Electronic Engineering ,Mathematics - Abstract
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 thesystem's response can be either bounded or unbounded, depending on thesystem's 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.
- Published
- 2000
- Full Text
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38. NON-LINEAR MODAL INTERACTIONS IN SHALLOW SUSPENDED CABLES
- Author
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Valery N. Pilipchuk and Raouf A. Ibrahim
- Subjects
Engineering ,Acoustics and Ultrasonics ,business.industry ,Differential equation ,Mechanical Engineering ,Equations of motion ,Bending ,Mechanics ,Condensed Matter Physics ,Vibration ,Transverse plane ,Nonlinear system ,Classical mechanics ,Modal ,Mechanics of Materials ,Mode coupling ,business - Abstract
This paper examines different regimes of non-linear modal interactions of shallow suspended cables. In a high-energy level, the equations of motion in terms of in-plane and out-of-plane co-ordinates are strongly coupled and cannot be linearized. For this type of problem, a special co-ordinate transformation is introduced to reduce the number of strongly non-linear differential equations by one. The resulting equations of motion are written in terms of stretching, transverse (geometrical bending), and swinging co-ordinates, and are suitable for analysis using standard quantitative and qualitative techniques. Both free and forced vibrations of the cable are considered for in-plane and out-of-plane motions. The cable stretching free vibrations results in parametric excitation to the cable transverse motion. Under in-plane forced excitation the stretching motion is directly excited while the transverse motion is parametrically excited.
- Published
- 1999
- Full Text
- View/download PDF
39. Application of the Lie Group Transformations to Nonlinear Dynamical Systems
- Author
-
Raouf A. Ibrahim and Valery N. Pilipchuk
- Subjects
Double pendulum ,Mechanical Engineering ,Mathematical analysis ,Equations of motion ,Lie group ,Geometry ,Condensed Matter Physics ,Resonance (particle physics) ,Nonlinear system ,Mechanics of Materials ,Nonlinear resonance ,Mode coupling ,Group theory ,Mathematics - Abstract
This paper describes the theory of Lie group operators in a form suitable for the applied dynamics community. In particular, it is adapted to analyzing the dynamic behavior of nonlinear systems in the presence of different resonance conditions. A key ingredient of the theory is the Hausdorff formula, which is found to be implicitly reproduced in most averaging techniques during the transformation process of the equations of motion. The method is applied to examine the nonlinear modal interaction in a coupled oscillator representing a double pendulum. The system equations of motion are reduced to their simplest (normal) form using operations with the linear differential operators according to Hausdorff's formula. Based on the normal form equations, different types of resonance regimes are considered. It is shown that the energy of the parametrically excited first mode can be regularly (or nonregularly) shared with the other mode due to the internal resonance condition. If the second mode is parametrically excited, its energy is localized and is not transferred to the first mode, even in the presence of internal resonance.
- Published
- 1999
- Full Text
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40. [Untitled]
- Author
-
Valery N. Pilipchuk
- Subjects
Sequence ,Series (mathematics) ,Differential equation ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Aerospace Engineering ,Ocean Engineering ,Classification of discontinuities ,Periodic function ,Nonlinear system ,Distribution (mathematics) ,Control and Systems Engineering ,Electrical and Electronic Engineering ,Parametric statistics ,Mathematics - Abstract
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 method's applicability can be extended for nonperiodic regimes by involving the idea of averaging.
- Published
- 1999
- Full Text
- View/download PDF
41. Study of the Oscillations of a Nonlinearly Supported String Using Nonsmooth Transformations
- Author
-
Alexander F. Vakakis and Valery N. Pilipchuk
- Subjects
Vibration ,Nonlinear system ,Amplitude ,Partial differential equation ,Generalized function ,Singularity ,Numerical analysis ,Mathematical analysis ,General Engineering ,Equations of motion ,Mathematics - Abstract
An analytical method for analyzing the oscillations of a linear infinite string supported by a periodic array of nonlinear stiffnesses is developed. The analysis is based on nonsmooth transformations of a 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 of a 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.
- Published
- 1998
- Full Text
- View/download PDF
42. [Untitled]
- Author
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Valery N. Pilipchuk, M. A.F. Azeez, and Alexander F. Vakakis
- Subjects
Butterfly effect ,Oscillation ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Chaotic ,Aerospace Engineering ,Ocean Engineering ,Nonlinear system ,Classical mechanics ,Control and Systems Engineering ,Pendulum (mathematics) ,Piecewise ,Electrical and Electronic Engineering ,Invariant (mathematics) ,Saddle ,Mathematics - Abstract
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.
- Published
- 1998
- Full Text
- View/download PDF
43. THE DYNAMICS OF A NON-LINEAR SYSTEM SIMULATING LIQUID SLOSHING IMPACT IN MOVING STRUCTURES
- Author
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Valery N. Pilipchuk and Raouf A. Ibrahim
- Subjects
Physics ,Hydroelasticity ,Acoustics and Ultrasonics ,Slosh dynamics ,Oscillation ,Mechanical Engineering ,Mechanics ,Condensed Matter Physics ,Symmetry (physics) ,Amplitude ,Classical mechanics ,Mechanics of Materials ,Phenomenological model ,Parametric oscillator ,Parametric statistics - Abstract
In this paper the non-linear modal interaction is examined between liquid hydrodynamic impact and an elastic support structure. The liquid impact is modelled based on a phenomenological concept by introducing a power non-linearity with a higher exponent. A special saw-tooth time transformation (STTT) technique is used analytically to describe the in-phase and out-of-phase strongly non-linear periodic regimes. Based on explicit forms of analytical solutions, all basic characteristics of the non-linear free and forced response regimes, such as the time history, the amplitude–frequency dependence and the non-linear parametric resonance curves, are estimated. The response behaviour reveals that a high frequency out-of-phase non-linear mode takes place with a relatively small tank amplitude, and is more stable than the in-phase oscillation mode under small perturbations. The in-phase mode has relatively large tank amplitudes and does not preserve its symmetry under periodic parametric excitation.
- Published
- 1997
- Full Text
- View/download PDF
44. Strong nonlinear modal interaction in shallow suspended cables with oscillating ends
- Author
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Raouf A. Ibrahim and Valery N. Pilipchuk
- Subjects
Quantitative Biology::Biomolecules ,General Mathematics ,Applied Mathematics ,General Physics and Astronomy ,Resonance ,Motion (geometry) ,Statistical and Nonlinear Physics ,Natural frequency ,Bending ,Mechanics ,Manifold ,Nonlinear system ,Classical mechanics ,Position (vector) ,Configuration space ,Mathematics - Abstract
This paper examines essentially nonlinear regimes of the stretching-bending interactions in suspended cables with horizontally vibrating ends. Using a special nonlinear transformation of coordinates in the configuration space, the Lagrange function of the cable is written in terms of physical coordinates such as stretching and bending-swinging which possess different time scales. The transformed system is considered as a quasi-linear one with respect to the fast varying stretching generalized coordinate. The relatively slow bending-swinging component of the motion is described by strongly nonlinear equations on an ‘unstretched manifold’. It has been shown that the stretching natural frequency strongly depends on the bending position of the cable. As a result, the cable performs a chaos-like stretching-bending interaction around the resonance region, i.e. when the frequency of shaking is equal to the natural frequency of stretching computed for an undeformed position of the cable. Outside the resonance region the motion has a regular character.
- Published
- 1997
- Full Text
- View/download PDF
45. Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT)
- Author
-
Alexander F. Vakakis, Valery N. Pilipchuk, and M. A.F. Azeez
- Subjects
Nonlinear system ,Transformation (function) ,Connection (vector bundle) ,Mathematical analysis ,Statistical and Nonlinear Physics ,Heteroclinic orbit ,Limit (mathematics) ,Boundary value problem ,Nonlinear Oscillations ,Condensed Matter Physics ,Variable (mathematics) ,Mathematics - 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.
- Published
- 1997
- Full Text
- View/download PDF
46. ANALYTICAL STUDY OF VIBRATING SYSTEMS WITH STRONG NON-LINEARITIES BY EMPLOYING SAW-TOOTH TIME TRANSFORMATIONS
- Author
-
Valery N. Pilipchuk
- Subjects
Transformation (function) ,Classical mechanics ,Acoustics and Ultrasonics ,Mechanics of Materials ,Simple (abstract algebra) ,Mechanical Engineering ,Computation ,Numerical analysis ,Mathematical analysis ,Condensed Matter Physics ,Dynamical system ,Mathematics - Abstract
A special saw-tooth temporal transformation technique is formulated. It is simple enough to allow analytical computation of strongly non-linear free and forced dynamic responses, but, at the same time, can be applied to the analysis of general classes of non-linear problems.
- Published
- 1996
- Full Text
- View/download PDF
47. Oscillators with a generalized power-form elastic term
- Author
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Valery N. Pilipchuk
- Subjects
Physics ,Classical mechanics ,Acoustics and Ultrasonics ,Mechanics of Materials ,Mechanical Engineering ,Condensed Matter Physics ,Term (time) ,Power (physics) - Published
- 2004
- Full Text
- View/download PDF
48. Impact Modes and Parameter Variations
- Author
-
Valery N. Pilipchuk
- Subjects
Computer science ,Substitution (logic) ,Mode (statistics) ,Harmonic (mathematics) ,Sine ,Statistical physics - Abstract
In this chapter, a new parameter variation and averaging tools are introduced for impact modes. It is also shown that a specific combination of two impact modes gives another impact mode. The corresponding manipulations with impact modes become possible due to the availability of closed form exact solutions obtained by means of the triangular sine temporal substitution for impulsively loaded and vibroimpact systems. In particular, the idea of Van-der-Pol and averaging tool are adapted for the case of impact oscillator. For illustrating purposes, a model of coupled harmonic and impact oscillators is considered. Further, mass-spring systems with multiple impacting particles are considered in order to illustrate impact localization phenomena on high-energy levels.
- Published
- 2010
- Full Text
- View/download PDF
49. Principal Trajectories of Forced Vibrations
- Author
-
Valery N. Pilipchuk
- Subjects
Vibration ,Nonlinear system ,Basis (linear algebra) ,Normal mode ,Mathematical analysis ,Principal (computer security) ,Harmonic (mathematics) ,Boundary value problem ,Eigenvalues and eigenvectors ,Mathematics - Abstract
As shown earlier by Zhuravlev (1992) that harmonically loaded linear conservative systems possess an alternative physically reasonable basis, which is generally different from that associated with conventional principal coordinates. Briefly, such a basis determines directions of harmonic loads along which the system response is equivalent to a single oscillator. The corresponding definition (principal directions of forced vibrations) is loosing sense in nonlinear case, when the linear tool of eigen vectors becomes inapplicable. However, it will be shown in this chapter that nonlinear formulation is still possible in terms of eigen vector-functions of time given by NSTT boundary value problems. Physical meaning of the corresponding nonlinear definitions for both discrete and continual models is discussed.
- Published
- 2010
- Full Text
- View/download PDF
50. Sawtooth Power Series
- Author
-
Valery N. Pilipchuk
- Subjects
Power series ,Periodic function ,Polynomial ,Dynamical systems theory ,Differential equation ,Applied mathematics ,Algebraic number ,Polynomial expansion ,Mathematics ,Trigonometric series - Abstract
In this chapter, we introduce polynomials and power series expansions with respect to the triangular sine-wave. These can be used for approximations of periodic signals and unknown periodic solutions of dynamical systems. Such approximations may appear to be effective in those cases when trigonometric series converge slowly due to step-wise discontinuities or spikes. Another reason for using polynomial expansions is that they are usually more convenient for algebraic manipulations. If the process under consideration is smooth then sufficient class of smoothness of approximations is achieved by imposing specific constraints on the coefficients. Other equations for the coefficients may appear either as a result of optimization procedures, that minimize the error of approximation, or as an outcome of iterative procedures dictated by the differential equations of motion. It is also shown in this chapter that using operators Lie associated with dynamical systems essentially facilitates construction of the periodic power series.
- Published
- 2010
- Full Text
- View/download PDF
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