Quasi-periodic Responses Research Articles

External noise synchronizes forced oscillators.

Periodic pulsatile perturbation of nonlinear oscillators generates phase-locking, quasiperiodic, and chaotic responses. This work shows that the application of external noise to ensembles of such forced systems can synchronize oscillations, even in regimes where neither the noise nor the periodic forcing, when applied alone, would lead to such a phenomenon.

DYNAMIC ANALYSIS OF A SELF-EXCITED HYSTERETIC SYSTEM

The dynamic behaviour of a self-excited system with hysteretic non-linearity is investigated in this paper. The averaging method is applied to the autonomous system and the resulting bifurcation equation of the self-excited response is analyzed using the singularity theory. The study of the bifurcation diagrams reveals the multivalued and jumping phenomena due to the effect of the hysteretic non-linearity. Secondly, the steady state response of the averaged system of the non-autonomous oscillator in primary resonance is investigated. Due to the effect of the hysteretic non-linearity, the system exhibits softening spring behaviour. A stability analysis shows that the steady state periodic response exists over a limited excitation frequency range. It loses its stability outside the frequency range through Hopf bifurcation and then the system undergoes quasi-periodic motion. Finally, by using circle maps to get winding numbers, various orders of super- and subharmonic resonance and mode-locking are investigated. The mode-locking, alternating with the quasi-periodic responses, takes place according to the Farey number tree as revealed in many other systems. The increase of the hystereticity can improve the stability of subharmonic resonance.

Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash

Dynamics of gear-pair systems driven by motors and presenting speed-dependent moment resistance is investigated. First, a suitable mechanical model is developed, taking into account the gear mesh backlash and static transmission error as well as the essential non-linearities due to the bearing clearance and contact characteristics. In comparison with earlier related studies, the new element of the present work is that the model developed determines the system response by simply specifying the external loads, rather than by assuming an a priori value for the constant mean angular velocity of the gear shafts. In fact, models employed in earlier research studies are shown to be obtained as special cases of the present model. The study is completed by numerical results. First, classical response diagrams are presented, illustrating the effect of the most important parameters on the system response. Finally, direct integration of the equations of motion is also performed, demonstrating the existence of quasi-periodic and chaotic long time response for selected combinations of the system parameters.

ON GEARED ROTORDYNAMIC SYSTEMS WITH OIL JOURNAL BEARINGS

This study investigates response and stability characteristics of rotordynamic systems interconnected with gear pairs and supported on oil journal bearings. The systems examined are driven by a motor providing a known torque, which is transferred to a load element through shafts and a gear box. Initially, the emphasis is placed on the gear-pair action, by assuming that the interconnecting shafts are rigid. This results in low order mechanical models with strongly non-linear characteristics, whose long-time dynamics are then effectively studied by applying appropriate numerical methodologies. In this way, useful information is obtained about the influence of the loading, the gear mesh and the journal parameters on the system dynamic behaviour, including periodic, quasi-periodic and chaotic response. Efforts are also directed towards predicting the onset of oil whirl instability and investigating its effects on the system dynamics. The focus is then shifted to more realistic models, including the flexibility of the interconnecting shafts. In this case, the order of the system is first reduced by applying a component mode synthesis method, which takes advantage of the fact that the non-linearities are located at the journal bearings and the gear mesh only. As a result, it is possible to study the dynamics of the reduced rotor-bearing model by employing the methodology that was developed for the gear-pair models.

CODIMENSION-2 HOPF BIFURCATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM

Codimension-2 Hopf bifurcation problem of a two-degree-of-freedom system vibrating against a rigid surface is investigated in this paper. The four-dimensional Poincaré map of the vibro-impact system is reduced to a two-dimensional normal form by virtue of a center manifold reduction and a normal form technique. Then the theory of Hopf bifurcation of maps in R2is applied to conclude the existence of codimension-2 Hopf bifurcation of the vibro-impact system. The quasi-periodic response of the system by theoretical analysis is well supported by numerical simulations. It is shown that there exists codimension-2 Hopf bifurcation in multi-degree-of-freedom vibro-impact systems. The codimension-2 tori doubling phenomenon and the routes of quasi-periodic impacts to chaos are reported briefly.

Non-linear dynamics of a slender beam carrying a lumped mass under principal parametric resonance with three-mode interactions

The non-linear response of a base-excited slender beam carrying a lumped mass subjected to principal parametric resonance is investigated. The attached mass and its location are so adjusted that the system exhibits 1 : 3 : 5 internal resonances. Method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and stability of the system. The steady-state response thus obtained is compared with those found by single- and two-mode analyses and very significant differences are observed in the bifurcation and stability of the response curves. The effects of external and internal detuning, amplitude of excitation and damping on the non-linear steady state, periodic, quasi-periodic and chaotic responses of the system are investigated. Funnel-shaped chaotic orbits, fractal orbits, cascade of period-doubling, torus doubling and intermittency routes to chaos are observed in this system. A simple illustration is given to control chaos by changing the system parameters.

SIMULTANEOUS COMBINATION, PRINCIPAL PARAMETRIC AND INTERNAL RESONANCES IN A SLENDER BEAM WITH A LUMPED MASS: THREE-MODE INTERACTIONS

The non-linear dynamics of a parametrically base-excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined. The mass and the position of the attached element are so adjusted that the system exhibits 1:3:5 internal resonance by virtue of which a combination resonance of typeω1 +ω3occurs simultaneously in the system. The method of multiple scales (mms) is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state, periodic, quasi-periodic and chaotic responses of the system for different control parameters, namely, frequency and amplitude of base excitation, damping, internal detuning, etc. The results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion. Poincaré section and Lyapunov exponents are used to characterize the chaotic responses.

On the parametric excitation of electrothermal instability in a dielectric liquid layer using an alternating electric field

The onset of electrothermal convective instability of a liquid dielectric subjected to an unsteady electric field is studied in the EHD approximation, when charge formation is produced only due to dielectrophoresis. Convective thresholds are found in two different cases: (i) instability of the liquid equilibrium in a horizontal layer, and (ii) instability of the liquid flow in a vertical layer. The stability boundaries are obtained when there is interaction of dielectrophoretic and gravitational forces. Stability plots of electrical Rayleigh number versus thermal Rayleigh number are given. We show that only synchronous response to variations of the external electric field of finite frequency exists when heating a horizontal layer from above. Quasiperiodic response to the external alternating action is possible in the case of a vertical layer. The influence of the Prandtl number on the stability thresholds is also examined. The asymptotic behavior of the critical parameters in the limiting case of low-frequency modulation is studied using the Wentzel–Kramers–Brillouin method.

Coherence resonance in chaotic electronic circuits

It is experimentally demonstrated that a chaotic electronic circuit exhibits the main features of coherence resonance. This phenomenon shows up as a quasi-periodic response when the circuit is subjected to an external white-noise source of the appropriate level.

Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

INTERNAL RESONANCES IN WHIRLING STRINGS INVOLVING LONGITUDINAL DYNAMICS AND MATERIAL NON-LINEARITIES

Internal resonance mechanisms between near-commensurate longitudinal and transverse modes of a taut spatial string are identified and studied using an asymptotic method, and the influence of material non-linearities on the resulting solutions is considered. Geometrical non-linearities couple longitudinal motions to in-plane and out-of-plane transverse motions, resulting in resonant and non-resonant interactions between linearly orthogonal string modes. Past studies have included only transverse modes in the description of string motions and have predicted periodic, quasi-periodic, and chaotic whirling motions arising from the geometrical non-linearities. This study considers further the inclusion of longitudinal motions and a non-linear material law, which are both appropriate for the study of rubber-like strings. An asymptotic analysis captures the aforementioned whirling motions, as well as a new class of whirling motions with significant longitudinal content. Periodic, quasi-periodic, and aperiodic (likely chaotic) responses are included among these motions. Their existence, hardening–softening characterization, and stability are found to be highly dependent on the magnitude of the material non-linearities.

FORCING INDUCED ASYMMETRY ON DYNAMICAL SYSTEMS WITH CUBIC NON-LINEARITIES

The present work investigates the dynamics of a class of two-degree-of-freedom oscillators with cubic non-linearity in the restoring forces. These oscillators are under the action of an external load including constant and harmonic components. Initially, a perturbation analysis is applied to the equations of motion, demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1:1 and 1:3 internal resonances, as well as on the possibility of the appearance of a first order 1:2 internal resonance. Next, sets of slow-flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1:1 internal resonance. The analytical findings are then complemented by numerical results, obtained by examining the dynamics of a two-degree-of-freedom mechanical system. First, the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow-flow equations is illustrated by presenting a sequence of response diagrams. Finally, the dynamics of the system used as an example is investigated further by direct integration of the slow-flow equations. This shows the existence of a period-doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response. It also demonstrates a new transition scenario from phase-locked to phase-entrained and drift response.

DYNAMICS OF A SLENDER BEAM WITH AN ATTACHED MASS UNDER COMBINATION PARAMETRIC AND INTERNAL RESONANCES, PART II: PERIODIC AND CHAOTIC RESPONSES

The governing second order temporal differential equation of a slender beam with an attached mass at an arbitrary position under vertical base excitation which retains the cubic non-linearities of geometric and inertial type is reduced to a set of first order differential equations by the method ofnormal formsfor combination parametric and internal resonances of 3:1. These equations are used to find the periodic, quasi-periodic and chaotic responses of the system for various bifurcating parameters, namely, damping, amplitude and frequency of base motion, attached mass and its location. Bifurcation set, mixed-mode oscillation, period-doubling, quasi-periodic orbits and different routes to chaos, namely, alternate periodic-chaotic transition, torus breakdown and intermittency have been studied for the above mentioned bifurcating parameters using phase portrait, Poincaré section, time and power spectra.

Chaotic Response of Panel Vibrations Forced by Turbulent Boundary Layer and Sound

Experimental data are presented to show evidence of chaotic response of two adjacent aircraft panels forced by a turbulent boundary layer and pure tone sound. The experiments are a simulation of boundary-layer and fan noise loads on a fuselage sidewall with Reynolds number per meter of 2 :85 £ 10 5 . The response of the panels is purely random and assumed linear when forced by the turbulent boundary-layer e ow and clearly becomes nonlinear with the appearance of the interspersed periodic to chaotic motion when forced by the boundary layer with superimposed pure tone sound. The initial periodic response of two tori of two commensurate frequencies changes with an increase in pure tone sound level. The response of period-doubling bifurcations then makes a transition to chaos, which alternates with quasiperiodic response as the wave loses the spatial homogeneity. The objective is to demonstrate the existence of strong nonlinear effects on the structure response, which is not yet well understood. I. Background M OST studies of nonlinear deterministic and stochastic dynamic problems examine externally excited systems. A typical example of an externally excited system is an aircraft fuselage structure interacting with a turbulent boundary layer and jet engine noise. Periodic, aperiodic, and chaotic responses can occur along the sidewall of the fuselage structure during the acceleration from takeoff, as well as at cruise altitude. One type of load is the socalled buzz-saw noise in high-bypass-ratio turbofan engines. The present experiment is designed to simulate such loads, as well as structural nonlinear responses that result from turbulent boundarylayer e ow and high-intensity sound interaction. Such experiments must be conducted in a wind tunnel with an anechoic test section to prevent standing wave formation between the test panel surface and the opposite sidewall of the tunnel. As a rule, the panel tension and curvaturedependon theloading. Thistension, therefore, constitutes a coupling between the loading and the response. One manifestation of this coupling is the spontaneous surface deformation of the panel, giving rise e rst to the regular and then to the irregular spatial patterns as the load increases. In the previous experiments, panels with periodic nonlinear responses to sound and e ows of constant or accelerated speeds and their active control were considered. 1;2 The present study simulates the abnormal processes of e ow and sound loads, the unsteady loads of the boundary-layer pressure e uctuations coupled with the panel responses,andthesoundradiationbythepanel.Nonlinearbehaviors result fromincreasing levels of pure tone sound as evidenced by the response changes in the panel from periodic motions to broadband chaos. In the past, chaotic signals were not recognized as a physical behavior but were hidden in the broad view given by stochastic processes. At present, the interest is to distinguish between periodic, quasiperiodic,and nonperiodic responses. 3‐6 In the experiments reported herein, the input of the acoustic load superimposed on the turbulent boundary-layer e ow was gradually increased. First, the boundary-layer instability and then responses changing from periodic to quasiperiodic and e nally to chaotic were observed, analo

Arnold tongues and mode–jumping in the supercritical post–buckling of an archetypal elastic structure

The buckling and post–buckling of a classical simply supported elastic strut on a stiffening elastic foundation is treated as a nonlinear dynamical system. For supercritical loads this fourth–order system has a pair of imaginary eigenvalues, indicating the possibility of either a double or a quasi–periodic response. Finite lengths impose constraints that restrict such responses to the rational positions on the real line of the ratio of competing wavelengths: for irrational positions the response locks onto a single periodic wave. The extent of the mode–locking is described in parameter space by the geometry of Arnold tongues. Critical positions on the tongues are identified which mark conditions for mode–jumping from one wavelength to another.

Two-mode response of simply supported, rectangular laminated plates

Two-mode responses of thin rectangular laminated plates subjected to a harmonic excitation are studied by using the method of multiple scales (MMS). The plates are assumed to be simply supported along all edges, and internal resonances between two symmetric modes of vibration are analysed. In order to use MMS properly, we introduce new detuning parameters which indicate the relationship of natural frequencies of each mode. It is found analytically that two-mode responses can occur at each primary resonance. The accuracy of analytical results is confirmed in comparison with results of numerical integration of the equation of motion. Furthermore, it is detected that the two-mode response near the primary resonance of the second mode loses its stability via a Hopf bifurcation, giving rise to quasi-periodic response.

HOPF BIFURCATION OF A TWO-DEGREE-OF-FREEDOM VIBRO-IMPACT SYSTEM

The bifurcation problem of a two-degree-of-freedom system vibrating against a rigid surface is studied in this paper. It is shown that there exist Hopf bifurcations in the vibro-impact systems with two or more degrees of freedom under suitable system parameters. In the paper, a centre manifold theorem technique is applied to reduce the Poincaré map of the vibro-impact system to a two-dimensional one, and then the theory of Hopf bifurcation of maps inR2is applied to conclude the existence of Hopf bifurcation of the vibro-impact system. The theoretical solutions are verified by numerical computations. The quasi-periodic response of the system, represented by invariant circles in the projected Poincaré sections, is obtained by numerical simulations, and routes of quasi-periodic impacts to chaos are stated briefly.

Stabilization of Periodic Solutions of Nonlinear Mechanical Systems: Controllability and Stability

In nonlinear mechanical systems, which show stable (sub)harmonic, quasi-periodic and/or chaotic responses, together with coexisting (unstable) harmonic solutions, a large reduction of maximum subharmonic, quasi-periodic, or chaotic displacement might be established if the coexisting unstable harmonic solution could be made stable. The control effort to obtain this goal can be very small. In this article, a method is presented for determining the controllability of periodically excited nonlinear mechanical systems. This method determines if an unstable periodic solution can be stabilized using control. Furthermore, a method for calculating the local stability of an earlier developed control method (i.e., sliding computed torque control) for the above-mentioned goal is presented. Simulation results are presented for a periodically excited beam system supported by a one-sided spring to demonstrate the capabilities of the above-presented methods.

Nonlinear Dynamics of a Magnetically Supported Rotor on Safety Auxiliary Bearings

This study concerns the dynamic response of a rotor landed on auxiliary (catcher) bearings in an Active Magnetic Bearing (AMB) supported rotor, following postulated loss of power or overload of the AMB. An analytical model involving a disk, a shaft and auxiliary bearings on damped flexible supports is constructed and appropriate equations of the nonlinear dynamic system are developed. The equations include a switch function to indicate contact/non-contact events and determine the existence of contact normal forces and tangential friction forces between the shaft and the bearings. Steady state solutions are obtained. An analytical method was formulated and used to yield solutions for cases with well balanced rotors, in absence of any side forces. The Fixed Point Algorithm (FPA) is used to obtain steady state periodic solutions of the unbalanced rotor for various parameters. The FPA is used to determine the stability of the periodic solutions and the type of bifurcation involved. Multiple periodic solutions, quasi-periodic and chaotic responses are detected and discussed. A set of preliminary guidelines for selection of the parameters of the catcher bearings is given.

Bifurcation and Chaos Response of a Nonlinear Cracked Rotor

The dynamic responses of a cracked rotor affected by nonlinear whirl speed are investigated, with particular focus on the behaviors of bifurcation and chaos. A great deal of numerical simulations show that the system has many nonlinear dynamic behaviors, such as period doubling bifurcation, quasiperiodic responses and chaotic motions at the speed ratio near 1/2Ωc and 2/3Ωc when the stiffness change ratio ΔK is large. Periodic vibrations turn to quasiperiodic motions by two different ways at speed ratio 1/2Ωc and 2/3Ωc, one bifurcates a low frequency precession component, and the other bifurcates from fraction harmonic component. With further increasing of ΔK, chaotic vibration occurs first at the speed ratio of 2/3Ωc, the response turns to chaos along various routes, it may be from period doubling to chaos, from quasiperiod to chaos, or relating to period-3 response which may turn to other kinds of periodic motions as initial values changed. When ΔK is very large, chaos from quasiperiod is found near speed ratio 1/2Ωc. The results of this paper show that some signals which were always considered as random noise and filtered off may represent the chaotic characteristics of system, and can be used in future fault diagnosing of rotating machinery.