A nonlinear spring mechanism incorporating a bistable composite plate for vibration isolation

The High Static Low Dynamic Stiffness (HSLDS) concept is a design strategy for a nonlinear anti-vibration mount that seeks to increase isolation by lowering the natural frequency of the mount whilst maintaining the same static load bearing capacity. It has previously been proposed that an HSLDS mount could be implemented by connecting linear springs in parallel with the transverse flexure of a composite bistable plate — a plate that has two stable shapes between which it may snap. Using a bistable plate in this way will lead to lightweight and efficient designs of HSLDS mounts. This paper experimentally demonstrates the feasibility of this idea. Firstly, the quasi-static force–displacement curve of a mounted bistable plate is determined experimentally. Then the dynamic response of a nonlinear mass–spring system incorporating this plate is measured. Excellent agreement is obtained when compared to theoretical predictions based on the measured force–displacement curve, and the system shows a greater isolation region and a lower peak response to base excitation than the equivalent linear system.     

On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets

The frequency range over which a linear passive vibration isolator is effective is often limited by the mount stiffness required to support a static load. This can be improved upon by incorporating a negative stiffness element in the mount such that the dynamic stiffness is much less than the static stiffness. In this case, it can be referred to as a high-static-low-dynamic stiffness (HSLDS) mount. This paper is concerned with a theoretical and experimental study of one such mount. It comprises two vertical mechanical springs between which an isolated mass is mounted. At the outer edge of each spring, there is a permanent magnet. In the experimental work reported here, the isolated mass is also a magnet arranged so that it is attracted by the other magnets. Thus, the combination of magnets acts as a negative stiffness counteracting the positive stiffness provided by the mechanical springs. Although the HSLDS suspension system will inevitably be nonlinear, it is shown that for small oscillations the mount considered here is linear. The measured transmissibility is compared with a comparable linear mass-spring-damper system to show the advantages offered by the HSLDS mount.     

Simulation of the nonlinear vibration of a string using the cellular automation method

In this study, the nonlinear dynamic responses of a string were simulated using the cellular automaton (CA) method. The local rules were set for the amplitude of vibration and the decay rate of amplitude. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. Thus, a new treatment of the time step was proposed, i.e., the time step in the CA method is adjusted to real time by considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic were simulated and a typical resonance curve could be obtained. Secondly the dynamic responses of a string with nonlinear characteristic were simulated. Some characteristic types of vibration could be obtained. It was concluded that the linear and nonlinear dynamic responses of a string could be simulated using the CA method.     

Indirect measurement of dynamic force transmitted by a nonlinear hydraulic mount under sinusoidal excitation with focus on super-harmonics

Direct measurement of forces is not practical in many real-life applications since the interfacial conditions may change. Thus indirect force estimation methods must be developed though they pose special difficulty for nonlinear mounts or isolators. The hydraulic engine mount is examined as an illustrative example in this article since it exhibits spectrally varying and amplitude-sensitive parameters. First, we propose linear time-invariant, nonlinear and quasi-linear fluid and mechanical system models. Second, models are utilized to predict the transmitted force time history under sinusoidal excitation conditions given measured (or calculated) motion and/or internal pressure time histories. Experimental data from the non-resonant dynamic stiffness test is investigated in both time and frequency domains. In particular, the super-harmonic contents in fluid chamber pressure and force time histories are investigated using both measurements and mathematical models. This paper examines several alternate indirect schemes for estimating dynamic forces and highlights their strengths. The quasi-linear model with effective system parameters, say in terms of force to pressure or force to motion transfer functions, is found to correlate well with measured dynamic forces though linear and nonlinear models could be employed as well.     

Transient response of a hydraulic engine mount

Linear and non-linear transient responses of a typical hydraulic engine mount are analytically and experimentally studied in this paper. First, a lumped parameter linear model is used to approximate the typical step response and to suggest parameters that must be experimentally determined. Various configurations as related to inertia track and decoupler are analyzed. Two bench experiments are constructed for the identification of non-linear compliances and resistances. One of the main non-linear characteristics, however, comes from the decoupler mechanism. To accurately predict the time events of the decoupler opening and closing, an equivalent viscous damper model is employed along with a multi-staged switching mechanism. Additionally, non-linear behavior arising due to the vacuum formation in the top chamber is studied by defining a bi-linear asymmetric stiffness curve. New transient experiments are conducted on an elastomer test system, and measured transmitted force and top chamber pressure signals are analyzed. Results of the proposed simulation model match well with measured responses when step up, step down and triangular waveforms are applied. Areas for future research are identified.     

Nonlinear model and system identification of a capacitive dual-backplate MEMS microphone

This paper presents the nonlinear identification of a capacitive dual-backplate microelectromechanical systems (MEMS) microphone. First, a nonlinear lumped element model of the coupled electromechanical microphone dynamics is developed. Nonlinear finite element analyses are performed to verify the accuracy of the lumped linear and cubic stiffnesses of the diaphragm. In order to experimentally extract the system parameters, an approximate solution using the second-order multiple scales method is synthesized for a nonlinear microphone model, subject to an electrical step input. A nonlinear least-squares technique is then implemented to extract system parameters from laser vibrometry data of the diaphragm motion. The results indicate that the theoretical fundamental resonant frequency, damping ratio and nonlinear stiffness parameter agree with the corresponding extracted experimental parameters with 95% confidence interval estimates.     

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.     

Transient analysis of dielectric step discontinuity of microstrip lines containing a nonlinear layer

Tansient signal propagation characteristics through step discontinuities on microstrip lines with nonlinear substrate are presented. In the proposed model structure, the transient electromagnetic wave propagates from microstrip line with linear substrate to one with nonlinear substrate or vice versa, where two structures are connected to create abrupt discontinuities. The distortions of transient signal through this two abrupt discontinuity structures are numerically simulated by three dimensional FD-TD method. Results show that the distortions of transient signal are evidently influenced by the density of electric field propagating along the microstrip line with nonlinear substrate.     

Linear and nonlinear Biot waves in a noncohesive granular medium slab: transfer function, self-action, second harmonic generation

Experimental results are reported on second harmonic generation and self-action in a noncohesive granular medium supporting wave energy propagation both in the solid frame and in the saturating fluid. The acoustic transfer function of the probed granular slab can be separated into two main frequency regions: a low frequency region where the wave propagation is controlled by the solid skeleton elastic properties, and a higher frequency region where the behavior is dominantly due to the air saturating the beads. Experimental results agree well with a recently developed nonlinear Biot wave model applied to granular media. The linear transfer function, second harmonic generation, and self-action effect are studied as a function of bead diameter, compaction step, excitation amplitude, and frequency. This parametric study allows one to isolate different propagation regimes involving a range of described and interpreted linear and nonlinear processes that are encountered in granular media experiments. In particular, a theoretical interpretation is proposed for the observed strong self-action effect.     

A numerical algorithm for the solution of a phase-field model of polycrystalline materials

We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton–Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate-projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton–Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

Nonlinear focusing of acoustic shock waves at a caustic cusp

The present study investigates the focusing of acoustical weak shock waves incoming on a cusped caustic. The theoretical model is based on the Khokhlov-Zabolotskaya equation and its specific boundary conditions. Based on the so-called Guiraud's similitude law for a step shock, a new explanation about the wavefront unfolding due to nonlinear self-refraction is proposed. This effect is shown to be associated not only to nonlinearities, as expected by previous authors, but also to the nonlocal geometry of the wavefront. Numerical simulations confirm the sensitivity of the process to wavefront geometry. Theoretical modeling and numerical simulations are substantiated by an original experiment. This one is carried out in two steps. First, the canonical Pearcey function is synthesized in linear regime by the inverse filter technique. In the second step, the same wavefront is emitted but with a high amplitude to generate shock waves during the propagation. The experimental results are compared with remarkable agreement to the numerical ones. Finally, applications to sonic boom are briefly discussed.     

The effects of nonlinearity on the output frequency response of a passive engine mount

In this paper, the new concept of output frequency-response function (OFRF) that was derived recently by the authors from the Volterra-series theory of nonlinear systems is briefly introduced. An effective algorithm is proposed to determine the monomials in the OFRF-based representation of the output frequency response of nonlinear systems. The results are then used to analyze the output frequency response of a passive engine mount. Important conclusions regarding the effects of system nonlinearity on the output frequency-response behaviors of the engine mount are reached via theoretical analysis and verified by simulation studies. These conclusions are of significant importance for the analysis and design of vibration isolators such as engine mounts in practice.     

On a Nonlinear Model in Adiabatic Evolutions

In this paper, we study a kind of nonlinear model of adiabatic evolution in quantum search problem. As will be seen here, for this problem, there always exists a possibility that this nonlinear model can successfully solve the problem, while the linear model can not. Also in the same setting, when the overlap between the initial state and the final stare is sufficiently large, a simple linear adiabatic evolution can achieve O(1) time efficiency, but infinite time complexity for the nonlinear model of adiabatic evolution is needed. This tells us, it is not always a wise choice to use nonlinear interpolations in adiabatic algorithms. Sometimes, simple linear adiabatic evolutions may be sufficient for using.     

Critical examination of isolation system design paradigms for a coupled powertrain and frame: Partial torque roll axis decoupling methods given practical constraints

The torque roll axis motion decoupling concept is analytically and computationally studied in a realistic coupled powertrain and frame system using discrete, proportionally damped linear models. Recently, Hu and Singh (2012 [1]) (Journal of Sound and Vibration 331 (2012) 1498–1518) proposed new paradigms to fully decouple such a system. However, critical examination shows that the derivation does not always lead to a physically realizable system, as each powertrain mount is not referenced to a single location. This deficiency is overcome by deriving mount compatibility conditions to ensure realistic mount positions which are incorporated into proposed decoupling conditions. It is mathematically shown that full decoupling is not possible for a practical system, and therefore partial decoupling paradigms are pursued. Powertrain mount design using only the decoupled powertrain achieves better decoupling than minimizing conditions for the coupled system using a total least squares method. Further decoupling is obtained through frame isolation design using a decoupled frame model such that the torque roll mode is dominant over the frequency range considered. Other methods for limiting frame coupling are also briefly discussed.     

Field Penetration into a Plasma with Current Saturating Conductivity

The problem of electric field penetration into a plasma is investigated employing a nonlinear current saturating conductivity. The case of step electric field applied at the boundary of semi-infinite plasma is studied. It is observed that the skin penetration depth is much larger than the one predicted by the linear theory and is comparable with the rapid field penetration reported in some turbulent heating experiments.     

Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

Nonlinear structures in electroencephalogram signals

We apply a nonlinear prediction algorithm to investigate the presence of nonlinear structure in electroencephalogram (EEG) recordings. The EEG signal could be modeled as a realization of a nonlinear model plus a residual noise (uncorrelated Gaussian noise). Using linear and nonlinear models we analyze the statistical nature of these residual noises in the case of epileptic patients and normal subjects. We found that the residual noise presents Gaussian distribution for epileptic patients if a nonlinear model is used whereas in the case of normal subjects the residual noise will exhibit a Gaussian distribution only if a linear model (autoregressive) is used. These results provide another evidence of the nonlinear character of the epileptic seizure recordings, while the normal EEG seems to be better described as linearly correlated noise.     

基于快速全线性预测控制的混沌系统控制与同步

针对连续时间混沌(超混沌)系统的控制问题, 提出了一种基于扩张状态观测器的快速全线性广义预测控制算法. 利用线性扩张状态观测器估计和补偿混沌(超混沌)系统的非线性动力学和存在的不确定性, 将原始对象近似转化为积分器形式, 随后针对单积分器设计广义预测控制, 解决了预测控制计算量大的问题. 阶跃系数矩阵可以直接得到解析解, 而对于未来输出的预测则可以根据最近两个时刻的输出采样值直接计算得到, 避免了使用自校正算法和在线求解丢番图方程. 该线性算法可以直接应用于非线性对象的控制系统设计. 将该算法应用于典型Lorenz混沌系统的控制中, 数学仿真结果验证了有效性.     

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

We investigate the acoustic wave propagation in bubbly liquid inside a pilot sonochemical reactor which aims to produce antibacterial medical textile fabrics by coating the textile with ZnO or CuO nanoparticles. Computational models on acoustic propagation are developed in order to aid the design procedures. The acoustic pressure wave propagation in the sonoreactor is simulated by solving the Helmholtz equation using a meshless numerical method. The paper implements both the state-of-the-art linear model and a nonlinear wave propagation model recently introduced by Louisnard (2012), and presents a novel iterative solution procedure for the nonlinear propagation model which can be implemented using any numerical method and/or programming tool. Comparative results regarding both the linear and the nonlinear wave propagation are shown. Effects of bubble size distribution and bubble volume fraction on the acoustic wave propagation are discussed in detail. The simulations demonstrate that the nonlinear model successfully captures the realistic spatial distribution of the cavitation zones and the associated acoustic pressure amplitudes.