Showing papers in "Nonlinear Dynamics in 1997"
TL;DR: In this article, a linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented and proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton-Euler formulation in body co-ordinates.
Abstract: A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.
497 citations
TL;DR: In this paper, an elliptic Lindstedt-Poincare (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form f(x,\dot x) = (c_0 - c_2 x^2 )
Abstract: An elliptic Lindstedt--Poincare (L--P) method is presented for the steady-state analysis of strongly non-linear oscillators of the form
$$\ddot x + c_1 x + c_3 x^3 = \varepsilon f(x,\dot x)$$
, in which the Jacobian elliptic functions are employed instead of the usual circular functions in the classical L--P perturbation procedure. This method can be viewed as a generalization of the L--P method. As an application of this method, three types of the generalized Van der Pol equation with
$$f(x,\dot x) = (c_0 - c_2 x^2 )\dot x$$
are studied in detail.
154 citations
TL;DR: In this paper, the dynamic behavior of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects under the hypothesis that the motion of the rotor mass center is plane and the rotor has five Lagrangian co-ordinates which are represented by the co-ordinate of the mass center and the three angular co-coordinates needed to express the rotor's rotation with respect to its center of mass.
Abstract: The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the π-film, short bearing model After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation
109 citations
TL;DR: In this article, an Euler-Bernoulli beam is analyzed in terms of a nonlinear elastic foundation and the frequency response curves are compared using the Galerkin method.
Abstract: Approximate methods for analyzing the vibrations of an Euler--Bernoulli beam resting on a nonlinear elastic foundation are discussed. The cases of primary resonance (Ω ≈ Ω
n
) and subharmonic resonance of order one-half (Ω ≈ 2 Ω
n
), where Ω is the excitation frequency and Ω
n
is the natural frequency of the nth mode of the beam, are investigated. Approximate solutions based on discretization via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equation and boundary conditions. The amplitude and phase modulation equations show that single-mode discretization leads to erroneous qualitative as well as quantitative predictions. Regions of softening (hardening) behavior of the system, the spatial dependence of the response drift, and frequency-response curves are numerically evaluated and compared using both approaches.
101 citations
TL;DR: In this article, the effect of damping ratio and slope of the torque-speed curve of an elastically mounted unbalanced shaft on a passage/capture threshold is evaluated.
Abstract: Rotation of an elastically mounted unbalanced shaft may be in general affected by its lateral vibration, due to a “vibrational torque”. This interaction is nonlinear, and can be neglected only in case of an unlimited power supply. Whenever the available power of the drive is comparable with power consumption due to vibration, various nonlinear phenomena may be observed, the most well-known of these being the so called Sommerfeld effect – slowing down or complete capture of the shaft at resonance. The corresponding steady-state motions and their stability can be studied by asymptotic methods, as applied to the governing nonlinear set of two second-order equations. However, study of transient motions in general requires numerical solution. This numerical solution is obtained here, and extensive parametric studies are performed of the Sommerfeld effect. In particular, the influence is evaluated of damping ratio and slope of the torque-speed curve of the drive on a passage/capture threshold. The results of numerical simulation, as well as experiments with a physical model, also demonstrate the effect of smooth passage through resonance with a limited power supply, based on using a “switch” of suspension stiffness from a certain artificially increased value to the design one. A brief description is presented also of time-variant components of the resonant amplitude and rotational frequency responses in the case of capture, as observed both in numerical simulation studies and in experiments. Whilst these components are small compared with the corresponding constant ones, i.e. steady-state vibration amplitude and rotational frequency, the above observations indicate the possibility for periodic or chaotic nonstationarity in the system's response.
95 citations
TL;DR: In this paper, an experimental confirmation of the theoretical results is sought, where the rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the λ parameter assigned in the first part of the present investigation, focusing particular attention on its nonlinear aspects.
Abstract: In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the λ parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity es was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C1, C and C2) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.
85 citations
TL;DR: In this article, the bifurcations of limit cycles in a rather general class of nonlinear dynamic systems were studied and the harmonic balance approach was applied in control engineering to derive the frequency conditions.
Abstract: The paper studies the bifurcations of limit cycles in a rather general class of nonlinear dynamic systems. Relying on the classical harmonic balance approach as applied in control engineering neat frequency conditions for such bifurcations are derived. These results, approximate in nature, make clear the structural mechanism of the considered phenomena and can be applied to predict the occurrence of bifurcations as a function of system parameters. The application to several examples of different complexity shows the simplicity and accuracy of the proposed method for solving complicated problems of nonlinear dynamics.
72 citations
TL;DR: In this article, an active nonlinear vibration absorber is proposed to take advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one.
Abstract: A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.
69 citations
TL;DR: In this paper, the authors studied the nonstationary vibration of a string with time-varying length and a mass-spring system attached at the lower end, where the string is hung vertically and excited sinusoidally by a horizontal displacement at its upper end.
Abstract: The purpose of this paper is to study the nonstationary vibration of a string with time-varying length and a mass-spring system attached at the lower end. The string is hung vertically and excited sinusoidally by a horizontal displacement at its upper end. The mass is supported by a guide spring horizontally and has two-degrees-of-freedom, vertical and horizontal. It is shown analytically that axial velocity of the string influences the peak amplitude of the string vibration at the passage through resonances. Moreover, it is shown numerically that the amplitudes of both the string and the mass vibrations depend on the sign of the axial velocity, when the natural frequency of the mass-spring system is close to the frequency of the excitation. The above two theoretical results are confirmed experimentally with a simple experimental setup.
67 citations
TL;DR: In this paper, an experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered, and the system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes.
Abstract: An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A ‘local’ analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.
60 citations
TL;DR: In this paper, an investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made, where the original non-autonomous system is reduced to an approximate autonomous system of amplitude and phase variables.
Abstract: An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.
TL;DR: In this article, a bifurcation control method was proposed to stabilize the trivial steady state in the frequency response and to eliminate the jump in the force response by employing a combined linear-plus-nonlinear feedback control.
Abstract: For a parametrically excited Duffing system we propose a bifurcation control method in order to stabilize the trivial steady state in the frequency response and in order to eliminate jump in the force response, by employing a combined linear-plus-nonlinear feedback control. Because the bifurcation of the system is characterized by its modulation equations, we first determine the order of the feedback gain so that the feedback modifies the modulation equations. By theoretically analyzing the modified modulation equations, we show that the unstable region of the trivial steady state can be shifted and the nonlinear character can be changed, by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the suppression of the discontinuous bifurcation due to the change of the nonlinear character allows the elimination of the jump in the quasistationary force response. Furthermore, by performing numerical simulations, and by comparing the responses of the uncontrolled system and the controlled one, we clarify that the proposed bifurcation control is available for the stabilization of the trivial steady state in the frequency response and for the reduction of the jump in the nonstationary force response.
TL;DR: In this article, a dynamical formulation of a recently established shell theory capable of catching finite deformations and falling within the class of geometrically exact shell theories is proposed, which preserves specific features of the continuous system such as conservation of momentum, angular momentum, and energy.
Abstract: The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered
TL;DR: In this paper, the nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated, and the analysis accounts for midplane stretching, a static axial load and a restraining spring at one end, and modal damping.
Abstract: The nonlinear planar response of a hinged-clamped beam to a primary excitation of either its first mode or its second mode is investigated. The analysis accounts for mid-plane stretching, a static axial load and a restraining spring at one end, and modal damping. For a range of axial loads, the second natural frequency is approximately three times the first natural frequency and hence the first and second modes may interact due to a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear partial-differential equation and derive two sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the case of primary resonance of either the first or the second mode. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of primary resonance of the first mode, only two-mode solutions are possible, whereas for the case of primary resonance of the second mode, single- and two-mode solutions are possible. The two-mode equilibrium solutions of the modulation equations may undergo a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. A shooting technique is used to calculate limit cycles of the modulation equations and Floquet theory is used to ascertain their stability. The limit cycles correspond to periodically modulated motions of the beam. The limit cycles are found to undergo cyclic-fold bifurcations and period-doubling bifurcations, leading to chaos. The chaotic attractors may undergo boundary crises, resulting in the destruction of the chaotic attractors and their basins of attraction.
TL;DR: In this article, a two-time version of the Lindstedt-Poincare Method and the multiple scale method are employed to analyze a bifurcation problem of codimension two.
Abstract: It is shown that the logical bases of the static perturbation method, which is currently used in static bifurcation analysis, can also be applied to dynamic bifurcations. A two-time version of the Lindstedt–Poincare Method and the Multiple Scale Method are employed to analyze a bifurcation problem of codimension two. It is found that the Multiple Scale Method furnishes, in a straightforward way, amplitude modulation equations equal to normal form equations available in literature. With a remarkable computational improvement, the description of the central manifold is avoided. The Lindstedt–Poincare Method can also be employed if only steady-state solutions have to be determined. An application is illustrated for a mechanical system subjected to aerodynamic excitation.
TL;DR: In this paper, the authors presented numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation, where the shell, with a rectangular boundary, is simply supported for deflection and the shell is constrained elastically in an in-plane direction.
Abstract: This paper presents numerical results on chaotic vibrations of a shallow cylindrical shell-panel under harmonic lateral excitation. The shell, with a rectangular boundary, is simply supported for deflection and the shell is constrained elastically in an in-plane direction. Using the Donnell--Mushtari--Vlasov equation, modified with an inertia force, the basic equation is reduced to a nonlinear differential equation of a multiple-degree-of-freedom system by the Galerkin procedure. To estimate regions of the chaos, first, nonlinear responses of steady state vibration are calculated by the harmonic balance method. Next, time progresses of the chaotic response are obtained numerically by the Runge--Kutta--Gill method. The chaos accompanied with a dynamic snap-through of the shell is identified both by the Lyapunov exponent and the Poincare projection onto the phase space. The Lyapunov dimension is carefully examined by increasing the assumed modes of vibration. The effects of the in-plane elastic constraint on the chaos of the shell are discussed.
TL;DR: In this article, the authors study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems and show that the hovering motions are asymptotically stable.
Abstract: In this paper, we study emergent behaviors elicited by applying open-loop, high-frequency oscillatory forcing to nonlinear control systems. First, we study hovering motions, which are periodic orbits associated with stable fixed points of the averaged system which are not fixed points of the forced system. We use the method of successive approximations to establish the existence of hovering motions, as well as compute analytical approximations of their locations, for the cart and pendulum on an inclined plane. Moreover, when small-amplitude dissipation is added, we show that the hovering motions are asymptotically stable. We compare the results for all of the local analysis with results of simulating Poincare maps. Second, we perform a complete global analysis on this cart and pendulum system. Toward this end, the same iteration scheme we use to establish the existence of the hovering periodic orbits also yields the existence of periodic orbits near saddle equilibria of the averaged system. These latter periodic orbits are shown to be saddle periodic orbits, and in turn they have stable and unstable manifolds that form homoclinic tangles. A quantitative global analysis of these tangles is carried out. Three distinguished limiting cases are analyzed. Melnikov theory is applied in one case, and an extension of a recent result about exponentially small splitting of separatrices is developed and applied in another case. Finally, the influence of small damping is studied. This global analysis is useful in the design of open-loop control laws.
TL;DR: In this article, the authors present an extensive investigation of several discretization techniques of flexible beams, in a pure multibody context, and show that shape functions based on power series monomials are very suitable and versatile to model beams being part of a multi-body system and thus constitutes an interesting alternative to finite element analysis.
Abstract: The modelling of flexible elements in mechanical systems has been widely investigated through several methods issuing from both the area of structural mechanics and the field of multibody dynamics. As regards the latter discipline, beside the problem of the generation of the multibody equations of motion, the choice of a spatial discretization method for modelling flexible elements has always been considered as a critical phase of the modelling. Although this subject is abundantly tackled in the open-literature, the latter probably lacks an objective comparison between the most commonly used approaches. This contribution presents an extensive investigation of several discretization techniques of flexible beams, in a pure multibody context. In particular, it is shown that shape functions based on power series monomials are very suitable and versatile to model beams being part of a multibody system and thus constitutes an interesting alternative to finite element analysis. For this purpose, a symbolic multibody program, in which various discretization techniques were implemented, was generalized to compute the equations of motion of a general multibody system containing flexible beams.
TL;DR: In this article, the authors describe a one-dimensional map generated by a two-degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction.
Abstract: This paper describes a one-dimensional map generated by a two degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction. The iterated map allows a much simpler representation and a better understanding of some dynamic features of the system. Some applications of the map are illustrated and its behaviour is simulated by means of an analytically defined one-dimensional map. A method of reconstructing one-dimensional maps from experimental data from the system is introduced. The method uses cubic splines to approximate the iterated mappings. From a sequence of such time series the parameter dependent bifurcation behaviour is analysed by interpolating between the defined mappings. Similarities and differences between the bifurcation behaviour of the exact iterated mapping and the reconstructed mapping are discussed.
TL;DR: In this paper, an active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations of a controlled suspended elastic cable under inplane excitation.
Abstract: The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.
TL;DR: In this article, the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable are studied and the governing equations of motion for two-mode interaction are derived on the basis of a general continuum model.
Abstract: The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.
TL;DR: In this article, the authors studied the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane.
Abstract: We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor These equations, valid in the neighborhood of the resonance, possess a small parameter e which is related to the imbalance In the limit e → 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for e ≠ 0 It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape
TL;DR: In this article, the nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed using the Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms.
Abstract: The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.
TL;DR: In this article, the authors proposed a method for simultaneous, automatic identification of the ten inertia parameters of a rigid body using the complete information hidden in the nonlinear model equations of the test body.
Abstract: Standard experiments for identifying inertia parameters of a rigid body only provide a subset of the inertia parameters of the body [1–10]. In addition, they do not use in the estimation process the complete information included in the equations of motion of the rigid test body. The objective of the work described in this paper is the simultaneous, automatic experimental identification of the ten inertia parameters of a rigid body using the complete information hidden in the nonlinear model equations of the test body. This task has been solved in several steps:
TL;DR: In this article, the Fokker-Planck-Kolmogorov (FPK) equation was used to estimate the response statistics of a suspended cable with simultaneous internal resonances, where the internal resonance can take place among the first in-plane and the first two out-of-plane modes.
Abstract: The random excitation of a suspended cable with simultaneous internal resonances is considered. The internal resonances can take place among the first in-plane and the first two out-of-plane modes. The external loading is represented by a wide-band random process. The response statistics are estimated using the Fokker-Planck-Kolmogorov (FPK) equation, together with Gaussian and non-Gaussian closures. Monte Carlo simulation is also used for numerical verification. The unimodal in-plane motion exists in regions away from the internal resonance condition. The mixed mode interaction is manifested within a limited range of internal detuning parameters, depending on the excitation power spectrum density and damping ratios. The Gaussian closure scheme failed to predict bounded solutions of mixed mode interaction. The non-Gaussian closure results are in good agreement with the Monte Carlo simulation. The on-off intermittency of the autoparametrically excited modes is observed in the Monte Carlo simulation over a small range of excitation levels. The influence of the cable parameters, such as damping ratios, sag-to-span ratio, internal detuning parameters, and excitation level on the autoparametric interaction, is studied. It is found that the internal detuning and excitation level are the two main parameters which affect the autoparametric interaction among the three modes. Due to the system's nonlinearity, the response of the three modes is strongly non-Gaussian and the coupled modes experience irregular modulation.
TL;DR: In this paper, a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable, one of them is a chaotic attractor.
Abstract: Cooperrider's mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting nonlinear dynamics (see True [1] for a survey). In this article a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable. One of them is a chaotic attractor.
TL;DR: In this paper, the authors examine chaotic single-degree-of-freedom stick-slip friction models and use the method of delays to reconstruct the phase space, and illustrate that this reconstruction process can cause pseudo trajectories to collapse in a way that is unlike, yet related to, the dimensional collapse in the original phase-space.
Abstract: Nonsmooth processes such as stick-slip may introduce problems with phase-space reconstructions. We examine chaotic single-degree-of-freedom stick-slip friction models and use the method of delays to reconstruct the phase space. We illustrate that this reconstruction process can cause pseudo trajectories to collapse in a way that is unlike, yet related to, the dimensional collapse in the original phase-space. As a result, the reconstructed attractor is not topologically similar to the real attractor. Standard dimensioning tools are applied in effort to recognize this situation. The use of additional observables is examined as a possible remedy for the problem.
TL;DR: In this article, a non-linear parametrically self-excited system with two degrees of freedom under harmonic external excitation was analyzed and the amplitude and the width of synchronization areas were determined.
Abstract: Vibration analysis of a non-linear parametrically self-excited system with two degrees of freedom under harmonic external excitation was carried out in the present paper. External excitation in the main parametric resonance area was assumed in the form of standard force excitation or inertial excitation. Close to the first and second free vibrations frequency, the amplitudes of the system vibrations and the width of synchronization areas were determined. Stability of obtained periodic solutions was investigated. The analytical results were verified and supplemented with the effects of digital and analog simulations.
TL;DR: In this article, the dynamics of a rigid body simply supported on a moving rigid ground in the presence of dry friction is investigated, and a variational formulation is adopted to describe the dynamics and to analyze, in the case of contact, the connection between dynamically possible motions and actual evolution of the system.
Abstract: The dynamics of a rigid body simply supported on a moving rigid ground in the presence of dry friction is investigated. Rigid body kinematics, in terms of generalized coordinates, and features of contact are discussed. According to the contact laws, a variational formulation is adopted to describe the dynamics and to analyze, in the case of contact, the connection between dynamically possible motions and actual evolution of the system. A geometric method is then proposed which allows the dynamic evolution to be determined without any direct evaluation of the contact forces. Though different situations are possible, depending on the instantaneous values of relative position, velocity and active forces, a unique solution is identified. Examples illustrating applications of the theory are presented.
TL;DR: In this paper, an axisymmetric, spatially localized standing wave solution with periodic time dependence (breathers) of a nonlinear partial differential equation is derived in the "continuum approximation" of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators.
Abstract: We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrodinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Pade approximants, expressing the Pade coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Pade approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.