Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay

The effect of time-delayed feedback and fast harmonic excitation (FHE) on stationary periodic vibration and quasi-periodic responses in a parametric and self-excited weakly nonlinear oscillator is analyzed in this paper. The method of direct partition of motion and two stages of multiple scales analysis are conducted to obtain analytical approximation for quasi-periodic oscillation envelopes and frequency-locking area near primary resonance. A parameter study shows that, in the absence or the presence of high-frequency excitation, time-delayed feedback may reduce significantly the amplitude and the envelopes of quasi-periodic oscillations leading to a quasi synchronization of the response over the whole frequency range around the resonance. The results presented for the parameters tested agree well with results obtained by numerical simulation.     

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part II—resonance secondary

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part I—primary resonance and free response at low temperatures

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.     

The Simplest Resonance Capture Problem,Using Harmonic Balance Based Averaging

We study the dynamics of capture into, or escape from, resonance in a strongly nonlinear oscillator with weak damping and forcing, using harmonic balance based averaging (HBBA). This system provides the simplest example of resonance capture that we know of. The HBBA technique, here adapted to tackle nonlinear resonances, provides a harmonic balance assisted approximation to the underlying, asymptotically correct, averaged dynamics. Allowing the harmonic balance approximation makes a variety of systems analytically tractable which might otherwise be intractable. The evolution equations for amplitude and phase of oscillations are derived first. Restricting attention near the primary resonance, the slow flow equations are approximately averaged. The resulting flow transparently shows the stable and unstable primary resonant solutions, as well as the trajectories that get captured into resonance and the ones that escape. Good agreement with numerics is obtained, showing the utility of HBBA near resonance manifolds.     

2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator

The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance. Numerical simulations are performed to confirm the analytical results.     

Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

Nonlinear Dynamics - We analyze the effect of a fast harmonic excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1...     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

含裂纹转子在变速过程中的瞬态振动

本文分别针对理想和非理想能源，研究了含裂纹的转子加速通过主共振区、次共振区和参数共振区的瞬态振动。结果表明，转子过次共振及参数共振区的瞬态特性可能成为诊断裂纹的重要依据。另外，本文采用主谐波变换，将转子系统的运动分方程变换成了用主谐波系数表示的方程，用该方程求转系统远离次共振和参数共振点的稳态响应时，可用求解代数方程来代替求解原微分方程，从而使求解更简便。     

Non-linear dynamics of a thermomechanical pseudoelastic oscillator excited by non-ideal energy sources

This paper discusses the non-linear dynamical response of a shape-memory non-ideal oscillator. The non-ideal excitation originates from a DC electric motor with limited power supply driving an unbalanced rotating mass. The restoring force provided by the shape-memory device is described by a thermomechanical model capable of accounting for the hysteretic behavior via the evolution of a suitable internal variable. The non-linear dynamic response of the system is investigated with the voltage as control parameter. Numerical simulations show the occurrence of regular and quasi-periodic motions, which are investigated via bifurcation diagrams and phase plane portraits. The 0–1 test is used for quantitative characterization of chaotic responses. The computation of basins of attraction points out the strong dependence of the response on small changes of initial conditions, along with meaningful modifications of competing basins with variations of the control parameter. Finally, variations of the mechanical and thermal parameters of the pseudoelastic oscillator are considered, with the aim to evaluating the effects produced by the non-ideal excitation source on the non-linear dynamics of the shape memory device.     

Nonlinear characterization of concurrent energy harvesting from galloping and base excitations

An energy harvester is proposed to concurrently harness energy from base and galloping excitations. This harvester consists of a triangular cross-sectional tip mass attached to a multilayered piezoelectric cantilever beam and placed in an incompressible flow and subjected to a harmonic base excitation in the cross-flow direction. A coupled nonlinear-distributed-parameter model is developed representing the dynamics of the transverse degree of freedom and the generated voltage. The galloping force and moment are modeled by using a nonlinear quasi-steady approximation. Under combined loadings and when the excitation frequency is away from the global natural frequency of the harvester, the response of the harvester mainly contains these two harmonic frequencies. Thus, the harvester’s response is generally aperiodic and is either periodic with large period (i.e., period- \(n\) ), or quasi-periodic, or chaotic. To characterize the harvester’s response under a combination of vibratory base excitations and aerodynamic loading, we use modern methods of nonlinear dynamics, such as phase portraits, power spectra, and Poincaré sections. A further analysis is then performed to determine the effects of the wind speed, frequency excitation, base acceleration, and electrical load resistance on the performance of the harvester under separate loadings.     

Bifurcation and pull-in voltages of primary resonance of electrostatically actuated SWCNT cantilevers to include van der Waals effect

In this work the voltage response of primary resonance of electrostatically actuated single wall carbon nano tubes (SWCNT) cantilevers over a parallel ground plate is investigated. Three forces act on the SWCNT cantilever, namely electrostatic, van der Waals and damping. While the damping is linear, the electrostatic and van der Waals forces are nonlinear. Moreover, the electrostatic force is also parametric since it is given by AC voltage. Under these forces the dynamics of the SWCNT is nonlinear parametric. The van der Waals force is significant for values less than 50 nm of the gap between the SWCNT and the ground substrate. Reduced order model method (ROM) is used to investigate the system under soft excitation and weak nonlinearities. The voltage-amplitude response and influences of parameters are reported for primary resonance (AC near half natural frequency).     

Dynamic instability of a circular cylindrical shell carrying a top mass under base excitation: Experiments and theory

The present paper is focused on the experimental and theoretical analysis of circular cylindrical shells under base excitation. The shell axis is vertical, it is clamped at the base and connected to a rigid body on the top; the base provides a vertical seismic-like excitation. The goal is to investigate the shell response when a resonant harmonic forcing is applied: the first axisymmetric mode is excited around the resonance at relatively low frequency and low amplitude of excitation. A violent resonant phenomenon is experimentally observed as well as an interesting saturation phenomenon close to the previously mentioned resonance. A theoretical model is developed to reproduce the experimental evidence and provide an explanation of the complex dynamics observed experimentally; the model takes into account geometric shell nonlinearities, electrodynamic shaker equations and the shell shaker interaction.     

Non-isothermal oscillations of pseudoelastic devices

The non-linear dynamic response of a pseudoelastic oscillator embedded in a convective environment is studied taking into account the temperature variations induced, during oscillations, by the latent heat of transformation and by the heat exchange with the surroundings. The asymptotic periodic response under harmonic excitation is characterized by frequency-response curves in terms of maximum displacement, maximum and mean temperature. The periodic thermomechanical response is computed by a multi-component harmonic balance method implemented within a continuation algorithm that enables to trace out multivalued frequency-response curves. The accuracy of the results is checked by comparison with the results of the numerical integration of the basic equations governing the dynamics of the system. The response is investigated for various excitation amplitude levels and in various material parameters ranges. The resulting picture of the mechanical response shows, in some cases, features similar to other hysteretic oscillators, while, in other cases, points out peculiar behaviors. It turns out that the temperature variations induced by the phase transformations influence the mechanical response and that the results obtained under the simplifying assumption of isothermal behavior can be rather different from those obtained in a fully thermomechanical setting.     

谐和与随机噪声联合作用下非线性系统的响应

研究了Duffing振子在谐和与随噪声联合激励下的响应和稳应性问题。用谐波平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统晌应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象。数值模拟表明本文提出的方法是有效的。     

Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation

In this paper, the nonlinear vibration of a single-walled carbon nanotube conveying fluid is investigated utilizing a multidimensional Lindstedt–Poincaré method. Considering the geometric large deformation of the single-walled carbon nanotube and external harmonic excitation force, based on nonlocal elastic theory and Euler–Bernoulli beam theory, the nonlinear vibration equation of a fluid-conveying single-walled carbon nanotube is established. Analyzing the equation through the multidimensional Lindstedt–Poincaré method, and from the solvability condition of the nonlinear vibration equation, the cubic algebraic equation which indicates the amplitude–frequency relation is obtained. Based on the root discriminant of the cubic equation, the first order primary response of the pinned–pinned carbon nanotube is discussed. The relations among internal resonance, the amplitude and frequency of the external excitation force are analyzed in detail. When the external excite force frequency is around the first mode natural frequency, the first mode primary resonance occurs. If simultaneously the first two modes natural frequency ratio is around 3, internal resonance occurs and the internal resonance region depends on the amplitude of external excitation force.     

A Theoretical and Experimental Implementation of a Control Method Based on Saturation

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.A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.     

Nonlinear dynamic characteristic of a face gear drive with effect of modification

Face gear drive is one of the main directions of research for aeronautical transmission for its advantages, but the vibration induced gear noise and dynamic load are rarely involved by researchers. The present work examines the complex, nonlinear dynamic behavior of a 6DOF face gear drive system combining with time varying stiffness, backlash, time varying arm of meshing force and supporting stiffness. The mesh pattern of the face gear drive system is analyzed when the modification strategy is applied and the effect of modification on the dynamics response, the time varying arm of meshing force based on the TCA is deduced. The dynamic responses of the face gear drive system show rich nonlinear phenomena. Nonlinear jumps, chaotic motions, period doubling bifurcation and multiple coexisting stable solutions are detected but different from the spur and bevel gear dynamics, which don’t occur near primary and higher harmonic resonance.     

Combination Internal Resonances in Heated Annular Plates

Nonlinear modal interactions in the forced vibrations of a thermally loaded pre-buckled annular plate with clamped–clamped immovable boundary conditions are investigated. The mechanism responsible for the interaction is a combination internal resonance involving the natural frequencies of the three lowest axisymmetric modes. The in-plane thermal load acting on the plate is assumed to be axisymmetric and the plate is externally excited by a harmonic force. The nonlinear von Kármán plate equations along with the heat conduction equation are combined to model the behavior of the system. An analytical/numerical approach is used to examine the plate vibrations to a harmonic excitation near primary resonance of one of the modes.     

Subharmonic response of single-degree-of-freedom linear vibroimpact system to narrow-band random excitation

The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated. The analysis is based on a special Zhuravlev transformation, which reduces the system to the one without impacts or velocity jumps, and thereby permits the applications of asymptotic averaging over the period for slowly varying the inphase and quadrature responses. The averaged stochastic equations are exactly solved by the method of moments for the mean square response amplitude for the case of zero offset. A perturbation-based moment closure scheme is proposed for the case of nonzero offset. The effects of damping, detuning, and bandwidth and magnitudes of the random excitations are analyzed. The theoretical analyses are verified by the numerical results. The theoretical analyses and numerical simulations show that the peak amplitudes can be strongly reduced at the large detunings.