Non-periodic Responses Research Articles

Experimental evidences of nonlinear programmable electroacoustic loudspeaker

A two degrees-of-freedom system coupling an acoustical mode of a closed tube to an electroacoustic loudspeaker is considered. A model-inversion technique is presented to digitally program the impedance of the loudspeaker for reaching a targeted nonlinear behavior. Experimental results show that the programed nonlinearity can guide the system to both periodic and non-periodic responses while acoustical levels for activation of the nonlinearity are less than the ones for systems with passive nonlinear oscillators (94 and 100 dB in this paper compared to 138 dB in previous papers), allowing building applications. Moreover, it is spotted that the programed nonlinearity of the electroacoustic loudspeaker, even in its non-optimized form, performs nonlinear noise control for some frequency ranges. The main objective of this article is to show that it is possible to program the behavior of an electroacoustic loudspeaker.

Different dynamics of a periodic mass-in-mass nonlinear chain during a single mode excitation

The multiple scale dynamics of a periodic chain composed of nonlinear mass-in-mass cells is studied. Based on a continuous approach of the one-dimensional chain, dispersion equation is obtained which provides the general form of solutions of the linearized system. Taking into account a single harmonic of the chain around a 1:1 resonance with a targeted mode, fast dynamics of the system leads to the detection of the slow invariant manifold and its stability borders. Slow dynamics permits to predict singularities and equilibrium points leading to frequency responses. The developments predict periodic and non-periodic responses and permit to tune parameters of the chain for the aim of localization of vibrating energy and design of periodic or non-periodic responses.

Nonlinear dynamic characteristics analysis of a rotor system supported on aerodynamic bearings

Due to the compressibility of gas, the aerodynamic bearings supported rotor system has strong nonlinearity. In this paper, the finite difference method is adopted to solve the Reynolds equation of the compressible fluid and obtain the forces of the gas film. Then the dynamic model of the aerodynamic bearing-rotor system is established, and the Runge-Kutta method is applied to solve the nonlinear equations of motion. With theories of nonlinear dynamics, the bifurcation characteristics of the rotor system supported by aerodynamic bearings are studied. Results show that the rotational speed and the unbalance has great influence on the nonlinear characteristics of the rotor, both periodic and non-periodic responses might emerge.

Prediction and validation of the strongly modulated forced response of two beams undergoing frictional impacts

We consider two cantilevered beams undergoing frictional impacts at the free end. The beams are designed to be of similar geometry so that they have distinct but close natural frequencies. Under harmonic base excitation near the primary resonance with the higher-frequency fundamental bending mode, the system shows a strongly modulated non-periodic response. The purpose of this work is to analyze to what extent the non-periodic vibro-impact dynamics can be predicted. To this end, we use a recently developed modeling and simulation approach. The approach relies on component mode synthesis, the massless boundary concept and an appropriate time stepping scheme. Unilateral contact and dry friction are modeled as set-valued laws and imposed locally within the spatially resolved contact area. A linear model updating is carried out based on the natural frequencies and damping ratios identified in the regime without impacts. The nonlinear simulation of the steady-state response to forward and backward stepped sine excitation is compared against measurements. The results are in very good agreement, especially in the light of the uncertainty associated with the observed material loss in the contact region and the nonlinear behavior of the clamping.

Non-periodic responses of the Izhikevich neuron model to periodic inputs

Non-periodic responses of the Izhikevich neuron model to periodic inputs

Fabrication and Characteristics of SQIF Based on NbN/AlN/NbN Josephson Junctions

Superconducting quantum interference filter (SQIF) is an array of superconducting quantum interference devices (SQUIDs) with different loop sizes. Such a device is capable of providing a much higher transfer factor and much higher magnetic field sensitivity with respect to those of single SQUID. Series niobium nitride (NbN) SQIF containing 400 SQUIDs with randomly varied loop areas and epitaxial NbN/AlN/NbN Josephson junctions have been developed, fabricated, and measured at helium temperature. The device shows a nonperiodic voltage response with a single sharp peak at zero magnetic field. The measured magnetic flux to voltage trans fer factor reaches 37.8 mV/Ф0 at 4.2 K.

Modeling periodic and non-periodic response of dynamical systems using an efficient Chebyshev-based time-spectral approach

A Chebyshev-based time-spectral method (C-TSM) is developed to model periodic and non-periodic nonlinear dynamical systems. It is shown that the proposed technique can accurately model such problems eliminating the need to use expensive classical dual-timestepping time-accurate integration. Furthermore, for autonomous dynamical systems subjected to single or multiple fundamental frequencies, the C-TSM analysis can be used without the prior knowledge of those frequencies. This offers an apparent advantage over the Fourier-based time-spectral methods. In addition, the current approach lends itself as a very useful tool for transient adjoint-based sensitivity analysis since it greatly reduces the memory requirements by solving and storing time-dependent response at a handful of collocation points instead of storing the entire time-history of the primal solution. The efficacy of the present technique is demonstrated by directly comparing the results with Fourier-based time-spectral, as well as time-accurate methods.

High-Frequency Nonlinear Response of Superconducting Cavity-Grade Nb Surfaces

Nb Superconducting Radio-Frequency (SRF) cavities are observed to break down and lose their high-Q superconducting properties at accelerating gradients below the limits imposed by theory. The microscopic origins of SRF cavity breakdown are still a matter of some debate. To investigate these microscopic issues temperature and power dependent local third harmonic response was measured on bulk Nb and Nb thin film samples using a novel near-field magnetic microwave microscope between 2.9K-10K and 2GHz-6GHz. Both periodic and non-periodic response as a function of applied RF field amplitude are observed. We attribute these features to extrinsic and intrinsic nonlinear responses of the sample. The RF-current-biased Resistively Shunted Junction (RSJ) model can account for the periodic response and fits very well to the data using reasonable parameters. The non-periodic response is consistent with vortex semi-loops penetrating into the bulk of the sample once sufficiently high RF magnetic field is applied, and the data can be fit to a Time-Dependent Ginzburg-Landau (TDGL) model of this process. The fact that these responses are measured on a wide variety of Nb samples suggests that we have captured the generic nonlinear response of air-exposed Nb surfaces.

On vibration transmission between interactive oscillators with nonlinear coupling interface

This paper investigates the dynamic characteristics and vibration transmission behaviour of interactive oscillators with nonlinearities at their coupling interface. Three different types of stiffness nonlinearities, i.e., hardening stiffness, softening stiffness and double-well potential type stiffness and cubic damping nonlinearity are considered. Both analytical approximations based on the method of averaging and also numerical integrations are employed to obtain the steady-state response and to determine the vibration transmission level. The time-averaged vibration power variables and kinetic energies of the system and the force transmissibility are formulated and obtained analytically and numerically. Time-averaged transmitted power is used as an index to quantify vibration transmission associated with both periodic responses and non-periodic responses such as chaos. It is found that hardening stiffness nonlinearity at the interface can lead to higher vibration power transmission at high excitation frequencies. In comparison, softening stiffness nonlinearity at the coupling interface can result in higher vibration transmission at lower excitation frequencies. It is shown that the interface with double-well potential stiffness nonlinearity may yield chaotic responses that can significantly affect vibration transmission as indicated by time-averaged transmitted power. It is also found that cubic damping nonlinearity may cause lower time-averaged transmitted and dissipated powers at the interface in the vicinity of resonant frequency. These findings provide better understanding of the effects of nonlinearity at the interface on vibration transmission, and facilitate better designs of coupling interface for control of vibration transmission.

Dynamic analysis of a 3-PRR parallel mechanism by considering joint clearances

The joint clearance causes poor dynamic mechanism performance, and the effect of multiple clearance joints on many complex mechanisms is unclear. This paper therefore performs dynamic analysis of a 3-PRR parallel mechanism by considering joint clearances. A concise method based on Newton–Euler equations is presented for the planar multibody system, and the root mean square error (RMSE) of the angular acceleration is proposed as an index to quantify the effect of the joint clearance on the dynamic behaviour. To improve the performance of the mechanism, this paper discusses the dynamic response of three parameters: the clearance size, restitution coefficient of the material and movement trajectory. Moreover, the simulation results indicate that: (i) the maximum absolute value of the displacement deviation is approximately triple the clearance size, and the nonlinearity of the mechanism rises obviously with increasing clearance size; (ii) different movement trajectories have various dynamic responses, and the nonperiodic response decreases due to the coupling effect of certain special movement trajectories; and (iii) a material with a moderate value of coefficient restitution is beneficial to the dynamic response, because the trend curve of the RMSE of the angular acceleration appears similar to a bathtub curve with increasing restitution coefficient of the material. This study provides support for the design and control of the parallel mechanism in the field of precision machinery.

High-Tc superconducting quantum interference filters (SQIFs) made by ion irradiation

Superconducting quantum interference filters (SQIFs) are arrays of superconducting loops of different sizes including Josephson junctions (JJ). For a random distribution of sizes, they present a non-periodic response to an applied magnetic field, with a large transfer function and a magnetic field sensitivity potentially improved with respect to that of a single SQUID. Such properties make SQIFs interesting devices to detect the magnetic component of electromagnetic waves at microwave frequencies. We have used the highly scalable technique of ion irradiation to make SQUIDs and SQIFs based on commercial YBa2Cu3O7 films, and studied their properties. Both display optimal performance as a function of temperature and bias current, that can be understood in the frame of numerical simulations that we developed. The role of asymmetries and dispersion in JJ characteristics (routinely found in high Tc superconductors technologies) is also studied. We have found that none of them impede the existence of a SQIF effect but both play a role on the emergence of the optimal point. We finally present results on SQIF made with 2000 SQUIDs in series, showing a transfer function .

Experiments on nonlinear vibrations of a combined structure with segments of beams and a disc subjected to axial tensile force

This paper presents experimental results on nonlinear vibrations of a combined structure with segments of beams and a disc, which is a fundamental model of a micro scanner. Both ends of the structure are clamped for deflection and one of the ends of the structure is constrained with an axial spring. Under an axial tensile force, the structure is excited with lateral periodic acceleration. Sweeping the excitation frequency, nonlinear frequency response curves are obtained. In the typical frequency region, non-periodic responses are generated. The responses are inspected by the Fourier spectrum, the Poincare projection, the maximum Lyapunov exponents and the principal component analysis. The non-periodic responses are confirmed as chaotic responses coupled with torsional and the lowest flexural modes of the combined structure.

Identification of regular and chaotic isothermal trajectories of a shape memory oscillator using the 0–1 test

Shape memory oscillators are thermomechanical hysteretic systems that, in a wide range of model parameters, can exhibit complex non-periodic non-linear dynamic responses under the excitation of a periodic force. In this study, the statistical 0–1 test based on the asymptotic properties of a Brownian motion chain is applied to periodic and non-periodic isothermal trajectories to examine the type of motion. The analysis is based on the computation of the control parameter K that approaches asymptotically 0 or 1 for regular and chaotic motions, respectively. The presented approach is independent of the integration procedure, being based on the characteristic sampling distance between the points of the analysed time series. The numerical results show that the test is able to unambiguously distinguish periodic from chaotic trajectories.

A novel polymer-on-silicon Mach-Zehnder interferometer optical filter and the performance analysis based on spectrum-periodized theory

The analysis on the traditional asymmetric Mach-Zehnder interferometer (AMZI) optical filter based on two 3 dB directional couplers (DCs) shows that by adding an additional nonlinear phase generated by phase-generating coupler (PGC) to the original phase difference of the AMZI, its non-periodic frequency response can be modified, and a strictly periodic spectrum can be obtained. A novel structure of the AMZI filter using two PGCs before and after the AMZI region is proposed. With the needed free spectrum range (FSR) of 20 nm, the design and optimization of the device are performed using polymer SU-8 as the core and PMMA-GMA as the buffer. Though the insertion loss (IL) gets larger than that of the traditional AMZI filter, the FSR is nearly uniform as the expected period of 20 nm.

808 面内非対称弾性拘束を受ける長方形板の非線形振動実験

Experimental results are presented on vibrations of a rectangular plate with an in-plane elastic constraint. One of the plate edges is clamped by rigid blocks. The other three edges are simply supported. The simply support edges are connected with elastic springs and are movable in in-plane direction. Effects of in-plane asymmetric force on nonlinear vibrations are investigated. Under the in-plane asymmetric force, the deflection of the plate shows the type of a softening-and-hardening spring. Under periodic lateral excitation, non-periodic responses are obtained in the typical frequency regions. The non-periodic responses are examined with the Fourier spectra and the maximum Lyapunov exponents. It is found that the chaotic responses are generated from internal resonance coupled with the lowest and second modes. In-plane asymmetric force increases the maximum Lyapunov exponents.

327 両端を弦拘束された弓状はりの形状関数近似による曲げ振動解析

Analytical results are presented on nonlinear vibrations of a bowed-type beam deformed by stretched strings at both ends with a mode shape function approach. This procedure combines the procedures of modal expansion with finite element analysis. The mode shape function is introduced with the product of truncated power series and trigonometric function. The unknown coefficients of the mode shape function satisfy both of the geometric and dynamical boundary condition at the nodes of segments. First, the governing equation of the beam is obtained including the geometrical nonlinearity due to the stretch of the beam and of the string. Then, linear vibration modes of the beam are obtained under the condition of pre-buckling. Furthermore, applying the modified Galerkin procedure based on the linear mode to the nonlinear governing equation. Nonlinear periodic and non-periodic responses are calculated. The chaotic response of the beam is generated from the internal response of the sub-harmonic response of 1/4 order with the asymmetric lowest mode of vibration and of the principal resonance with the symmetric second mode accompanied by the dynamic snap-thorough.

Experiments on chaotic vibrations of a post-buckled cantilevered beam connected by a string to an axial spring

Experimental results are presented on chaotic vibrations of a post-buckled cantilevered beam constrained by a string. The string is stretched between the top end of the beam and an axial spring. The axial spring consists of a leaf spring with an attached mass and is fixed on a base frame close to the clamped end of the beam. The length of the string is less than that of the beam. The beam is excited by lateral periodic acceleration. By increasing the attached mass on the axial spring, nonlinear responses of the beam are examined. Nonperiodic response is observed in a typical frequency region. The response is examined by the Fourier spectrum, maximum Lyapunov exponents, Poincaré projection, and principal component analysis. Predominant chaotic response is generated by the internal resonance with a frequency ratio of one-to-three. The fundamental mode and the second mode of vibration are strongly coupled in the chaotic response. When the attached mass of the axial spring is increased, the natural frequency of the axial spring approaches the region of chaotic response. Therefore, the number of vibration modes that contribute to the chaos increases. Furthermore, the Poincaré projection of the chaotic response shows more scattered figures.

Modeling Non-Locally Coupled DC SQUID Arrays

In this paper we provide numerical simulations for modeling an array of non-locally coupled DC SQUIDs that are coupled through the magnetic field created by the circulating current. The motivation is based on work using an array of non-identical SQUID loops or SQIFs to produce a non-periodic voltage response with a unique anti-peak centered at the zero applied flux. Our approach differs by using an array of identical SQUID loops and varying the spacing between each of these loops. Certain distributions of spacing between SQUIDs demonstrate the anti-peak response as seen in the SQIF.

Experiments on Chaotic Vibrations of an Outer-Free and Inner-Clamped Annular Plate with Initial Deflection

This paper presents experimental results on chaotic vibrations of an annular plate with initial deflection. The effect of modal interaction on chaotic vibrations is main scope in this paper. The plate is free on the outer edge and clamped on the inner edge. The initial deflection of the plate is nine thousandth times to the difference between the outer radius and the inner radius. The lowest and the third modes with one nodal-diameter have different natural frequencies due to the initial deflection. The second mode corresponds to the mode without nodal-diameter. The fourth and fifth modes have two cross nodal-diameters. The linear natural frequencies of the second and the third modes coincide with a slight deviation. The plate is subjected to lateral periodic acceleration. Sweeping the excitation frequency, nonlinear response curves of the plate are recorded. In the wide frequency ranges, non-periodic responses are generated. The responses are inspected by the Fourier spectrum, the Poincaré projection, the maximum Lyapunov exponent and the principal component analysis. The predominant non-periodic response is confirmed as the chaotic response generated from one-to-one and two-to-three internal resonances. The lowest, the second and the third modes are coupled in the internal resonances. The principal component analysis is applied to the chaotic response selected with short-time intervals sequentially collected from the long-time response. It is found that exchanges of the contribution ratio are observed among the lowest, the second and the third modes. Furthermore, the nodal diameters of the fourth and the fifth modes are fluctuated along the circumferential direction in the chaotic response.

356 面内非対称弾性拘束を受ける長方形平板の非線形振動解析

This paper presents theoretical results on nonlinear vibrations of a rectangular plate subjected to gravitational and periodic acceleration laterally. The rectangular plate is simply supported at all edges, which are surrounded with an elastic material. Both uniform in-plane displacements and asymmetric in-plane displacements act as in-plane constraints on the opposite edges via the elastic material. Other opposite edges are compressed with uniform in-plane displacements. Restricting nonlinear vibrations of a thin plate in lower frequency range where bending vibrations are dominant, the effects of in-plane inertia forces can be neglected in the analysis. The Karman type equation modified with lateral inertia is applied as the governing equations of the plate. The response of lateral deflection is assumed with multiple natural modes of linear vibration including unknown time functions. Stress function including nonlinear coupling of the deflection is derived from the compatibility equation. The stress function satisfies both equilibrium conditions of in-plane forces and in-plane moments of forces at the corresponding boundaries. Substituting the deflection and the stress function to the governing equation, a set of nonlinear ordinary differential equations is derived with the Galerkin procedure. Characteristics of restoring force of the post-buckled plate, due to gravity and the in-plane constraints, show the type of a softening-and-hardening spring. Periodic responses are calculated by the harmonic balance method. Non-periodic responses are integrated numerically by the Runge-Kutta-Gill method. Frequency response curves including responses of multiple modes are obtained. Increasing the in-plane asymmetric constraints at the boundaries, responses of the plate are inspected. The amplitude of the principal response corresponding to the lowest mode of vibration is affected by the in-plane constraint. The in-plane asymmetric constraint generates a stretched area and a compressed area in the plate. Within the relatively small amplitude of response, which corresponds to the characteristic of a softening spring, the response shifts to the higher frequency region, as the asymmetric constraint is increased. On the other hand, within the relatively large amplitude of response, which corresponds to the characteristic of a hardening spring, the response becomes smaller at the stretched area, and becomes larger at the compressed area. Moreover principal resonance of the 3rd mode of vibration appears more remarkably with asymmetric constraints.