M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation

Over the past few years, nonlinear oscillators have been given growing attention due to their ability to enhance the performance of energy harvesting devices by increasing the frequency bandwidth. Duffing oscillators are a type of nonlinear oscillator characterized by a symmetric hardening or softening cubic restoring force. In order to realize the cubic nonlinearity in a cantilever at reasonable excitation levels, often an external magnetic field or mechanical load is imposed, since the inherent geometric nonlinearity would otherwise require impractically high excitation levels to be pronounced. As an alternative to magnetoelastic structures and other complex forms of symmetric Duffing oscillators, an M-shaped nonlinear bent beam with clamped end conditions is presented and investigated for bandwidth enhancement under base excitation. The proposed M-shaped oscillator made of spring steel is very easy to fabricate as it does not require extra discrete components to assemble, and furthermore, its asymmetric nonlinear behavior can be pronounced yielding broadband behavior under low excitation levels. For a prototype configuration, linear and nonlinear system parameters extracted from experiments are used to develop a lumped-parameter mathematical model. Quadratic damping is included in the model to account for nonlinear dissipative effects. A multi-term harmonic balance solution is obtained to study the effects of higher harmonics and a constant term. A single-term closed-form frequency response equation is also extracted and compared with the multi-term harmonic balance solution. It is observed that the single-term solution overestimates the frequency of upper saddle-node bifurcation point and underestimates the response magnitude in the large response branch. Multi-term solutions can be as accurate as time-domain solutions, with the advantage of significantly reduced computation time. Overall, substantial bandwidth enhancement with increasing base excitation is validated experimentally, analytically, and numerically. As compared to the 3 dB bandwidth of the corresponding linear system with the same linear damping ratio, the M-shaped oscillator offers 3200, 5600, and 8900 percent bandwidth enhancement at the root-mean-square base excitation levels of 0.03g, 0.05g, and 0.07g, respectively. The M-shaped configuration can easily be exploited in piezoelectric and electromagnetic energy harvesting as well as their hybrid combinations due to the existence of both large strain and kinetic energy regions. A demonstrative case study is given for electromagnetic energy harvesting, revealing the importance of higher harmonics and the need for multi-term harmonic balance analysis for predicting the electrical power output accurately.     

Vibration serviceability of footbridges under human-induced excitation: a literature review

Increasing strength of new structural materials and longer spans of new footbridges, accompanied with aesthetic requirements for greater slenderness, are resulting in more lively footbridge structures. In the past few years this issue attracted great public attention. The excessive lateral sway motion caused by crowd walking across the infamous Millennium Bridge in London is the prime example of the vibration serviceability problem of footbridges. In principle, consideration of footbridge vibration serviceability requires a characterisation of the vibration source, path and receiver. This paper is the most comprehensive review published to date of about 200 references which deal with these three key issues.The literature survey identified humans as the most important source of vibration for footbridges. However, modelling of the crowd-induced dynamic force is not clearly defined yet, despite some serious attempts to tackle this issue in the last few years.The vibration path is the mass, damping and stiffness of the footbridge. Of these, damping is the most uncertain but extremely important parameter as the resonant behaviour tends to govern vibration serviceability of footbridges.A typical receiver of footbridge vibrations is a pedestrian who is quite often the source of vibrations as well. Many scales for rating the human perception of vibrations have been found in the published literature. However, few are applicable to footbridges because a receiver is not stationary but is actually moving across the vibrating structure.During footbridge vibration, especially under crowd load, it seems that some form of human–structure interaction occurs. The problem of influence of walking people on footbridge vibration properties, such as the natural frequency and damping is not well understood, let alone quantified.Finally, there is not a single national or international design guidance which covers all aspects of the problem comprehensively and some form of their combination with other published information is prudent when designing major footbridge structures. The overdue update of the current codes to reflect the recent research achievements is a great challenge for the next 5–10 years.     

Higher harmonics of vibrating gas-filled microspheres. Part one: simulations

The acoustic behaviour of an ideal gas bubble in water is considered and the equation of motion is extended to model an Albunex microsphere. Calculations reveal large differences in non-linear behaviour between ideal gas bubbles and Albunex microspheres, due to the additional restoring force of, and friction inside, the shell that surrounds the Albunex microsphere. Simulations with the Albunex contrast agent further reveal that the optimal driving frequency is 1 MHz, resulting in a second harmonic that is 20 dB below the first harmonic at an acoustic pressure of 50 kPa. The difference increases to 25 dB for a driving frequency of 2 MHz.     

锥形渐变截面驻波管及其极高纯净驻波场的实验研究

锥形渐变截面驻波管是用锥形管代替突变截面驻波管突变截面部分的驻波管。为对比研究锥形渐变截面驻波管与突变截面驻波管的声学及其极高纯净驻波场性质,首先借助传递矩阵,对锥形渐变截面驻波管的声学性质进行了实验研究。研究表明,与突变截面驻波管一样,锥形渐变截面驻波管也属于失谐驻波管。利用其失谐性,在一阶共振频率激励下,锥形渐变截面驻波管获得了181dB的极高纯净驻波场。在对锥形渐变截面驻波管和等长的突变截面驻波管的驻波饱和性质进行对比实验研究后发现,在一阶共振频率下,锥形渐变截面驻波管不仅能很好地抑制管内驻波场高次谐波的增长,而且能有效地降低管内驻波场的能量损耗,在相同扬声器激励电压下获得声压级更高的大振幅纯净驻波场。实验研究还发现,在三阶共振频率激励下,锥形渐变截面驻波管的大振幅驻波场三次谐波频率接近声压级传递函数谷值对应的声源端七阶阻抗共振频率,三次谐波随基波快速增长并表现出趋于二次谐波的饱和性质。     

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.     

Three-dimensional nonlinear theory of travelling wave tubes and simulation

A three-dimensional (3D) nonlinear theory of travelling wave tubes (TWTs) is developed, which includes a fundamental radio frequency (RF) and harmonics. When the instantaneous bandwidth exceeds an octave, the harmonic is generated and the mutual coupling between the harmonic and the fundamental RF can be observed in TWTs due to nonlinear interaction between the electron beam and the RF. At low frequencies the harmonic has an obvious effect.Based upon Tien‘‘s disc model, a plastic 3D super-particle model is proposed to improve the nonlinear analysis of TWTs.Numerical results employing a periodic magnetic focusing field are presented.     

Force–frequency effect of thickness mode langasite resonators

Langasite resonators are of recent interest for a variety of applications because of their good temperature behavior, good piezoelectric coupling, low acoustic loss and high Q factor. The force–frequency effect describes the shift in resonant frequency a resonator experiences due to the application of a mechanical load. A clear understanding of this effect is essential for many design applications such as pressure sensors. In this article, the frequency shift is analyzed theoretically and numerically for thin, circular langasite plates subjected to a pair of diametrical forces. In addition, the sensitivity of the force–frequency effect is analyzed with respect to the nonlinear material constants. The results are anticipated to be valuable for experimental measurements of nonlinear material constants as well as for device design.     

Photon-bunching, photon-antibunching and the nonclassical photon statistics of coherent light coupled to a driven harmonic oscillator of time dependent mass and frequency

The analytical solution of the quantum driven harmonic oscillator of time dependent mass and frequency (DHTDMF) is exploited to obtain the photon-bunching, photon-antibunching and the nonclassical photon statistics of the input coherent light coupled to the oscillator. We observe that the model Hamiltonian of a DHTDMF coincides with those of the model Hamiltonian of a two-photon coherent state due to H.P. Yuen [Phys. Rev. A 13 (1976) 226]. To substantiate our analytical results, we give a special case corresponding to a damped harmonic oscillator subject to an external periodic force. The quantum statistical properties of the radiation field are discussed for under-, over-, and for critically-damped situations of the oscillator coupled to an initial coherent field. The off-resonance and the near- or on-resonance of the oscillator with respect to the frequency of the external force are discussed. In the determination of quantum statistical properties of the radiation field, we observe that the effects of the driving force are important only when the frequency of the oscillator is in near- or in on-resonance condition with that of the frequency of the external periodic force.     

On the third harmonic generation in a medium with negative pump wave refraction

The propagation of the pump and its third harmonic pulses in a cubically nonlinear medium is considered theoretically, provided that the linear properties of the medium are characterized by a negative refractive index at the pump frequency and a positive refractive index at the harmonic frequency. For low-intensity interacting waves, the pump and third harmonic pulses propagate in opposite directions, but sufficiently intense pulses can produce a simulton—a solitary two-frequency wave that propagates in a certain direction as a single whole. The system of equations is investigated numerically for a model that, apart from the harmonic generation, includes the second-order group velocity dispersion and the nonlinear self- and cross-phase modulations of the interacting waves. The separation of the pump and harmonic pulses due to the difference in the directions of their group velocities and peculiarities of the Manley-Rowe relation for parametric processes in metamedia are discussed.     

Harmonic distortion in intracochlear pressure and its analysis to explore the cochlear amplifier

Intracochlear pressure was measured close to the basal basilar membrane in gerbil with pure-tone stimulation. This report describes harmonic distortion in the pressure. The harmonic components were tuned in frequency and physiologically vulnerable, implying that they were related to the cell-derived force that sharpens tuning at low levels in healthy cochleae. For stimulus frequencies in the vicinity of the best frequency the harmonic distortion appeared to be produced locally, at the place of measurement. Therefore, it could be explored with a local nonlinear model. The combined model and observations demonstrate two specific points: First, the harmonics in the cell-based force were likely similar in size to the harmonics in pressure (multiplied by area) close to the basilar membrane. This is distinctly different than the situation for the fundamental component, where the cell-based force is apparently much smaller than the pressure (times area). Second, although the fundamental component of the measured pressure was much larger than its harmonic components, the harmonic and fundamental components of the active force were likely much more similar in size. This allows the harmonic components in the pressure to be used as an indirect measure of the active force.     

Forced vibrations of a torsional oscillator with Coulomb friction under a periodically varying normal load

The forced vibration response of a single degree of freedom torsional system with Coulomb friction, under a periodically varying normal load, is studied. First, an enhanced multi-term harmonic balance method is developed to calculate nonlinear responses directly in frequency domain; this should be in general applicable to periodically varying nonlinear systems. Second, a pulse width modulated normal load is approximated by a truncated Fourier series with a reasonable number of harmonic components and utilized for case studies. Finally, the effects of duty ratio on nonlinear frequency responses are examined.     

一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

Series resonance circuit with ferroelectric capacitor

A series resonance circuit under sinuousoidal driving is investigated experimentally. The inductance consists of an air coil. The capacitance is made up of a ferroelectric material that introduces its nonlinear dielectric properties into the circuit. The dynamical system linear coil-nonlinear capacaitor shows an interesting behaviour. The phase portrait differs in general from the ellipse of the harmonic oscillator. For appropriate external conditions period doubling sequences, chaos and therein enclosed periodic windows might occur. Starting from a cubic nonlinearity of the dielectric properties a Duffing equation is proposed as a model for periodic behaviour of the series resonance circuit. Simulations of experimentally recorded phase portraits yield good agreement between experiment and model.     

Crack as modulator,detector and amplifier in structural health monitoring

A model of a one-dimensional cracked cantilever bar subjected to longitudinal harmonic excitation is used to analyse a nonlinear response as a way to monitor structural health. The effect of the bilinear (nonlinear) character of the crack on the dynamics of the structure is studied. Simulation and experiments were performed to analyse the nonlinear behaviour of the cracked bar. In simulation the nonlinear information is obtained based on a combination of the analytical technique and the Matlab–Simulink computation. From analysis and experiment, it is found that the crack-induced nonlinearity leads to the generation of higher harmonics, whose intensity is a function of a distance from the crack. Side band frequencies were clearly revealed as well. The latter indicate modulation of exciting frequency due to systematic interaction of crack faces. The nonlinear transformation of modulated vibration by crack leads to generation of a low frequency periodic component. Its intensity is proportional to the forced response of the cracked bar at the exciting frequency. The phenomenology revealed can be effective for Structural Health Monitoring.     

Numerical study of friction-induced instability and acoustic radiation – Effect of ramp loading on the squeal propensity for a simplified brake model

This paper presents a numerical study of the influence of loading conditions on the vibrational and acoustic responses of a disc brake system subjected to squeal. A simplified model composed of a circular disc and a pad is proposed. Nonlinear effects of contact and friction over the frictional interface are modelled with a cubic law and a classical Coulomb?s law with a constant friction coefficient. The stability analysis of this system shows the presence of two instabilities with one and two unstable modes that lead to friction-induced nonlinear vibrations and squeal noise. Nonlinear time analysis by temporal integration is conducted for two cases of loadings and initial conditions: a static load near the associated sliding equilibrium and a slow and a fast ramp loading. The analysis of the time responses shows that a sufficiently fast ramp loading can destabilize a stable configuration and generate nonlinear vibrations. Moreover, the fast ramp loading applied for the two unstable cases generates higher amplitudes of velocity than for the static load cases. The frequency analysis shows that the fast ramp loading generates a more complex spectrum than for the static load with the appearance of new resonance peaks. The acoustic responses for these cases are estimated by applying the multi-frequency acoustic calculation method based on the Fourier series decomposition of the velocity and the Boundary Element Method. Squeal noise emissions for the fast ramp loading present lower or higher levels than for the static load due to the different amplitudes of velocities. Moreover, the directivity is more complex for the fast ramp loading due to the appearance of new harmonic components in the velocity spectrum. Finally, the sound pressure convergence study shows that only the first harmonic components are sufficient to well describe the acoustic response.     

Visualization of crowd synchronization on footbridges

Abstract  

This paper proposes a framework for the visualization of crowd walking synchronization on footbridges. The bridge is modeled as a mass-spring system, which is a weakly damped and driven harmonic oscillator. Both the bridge and the pedestrians walking on the bridge are affected by the movement of each other. The crowd acts according to local behavioral rules. Each pedestrian is provided with a kinematic walking system. We extend a current mathematical model of crowd synchronization on footbridges to include human walking model and crowd simulation techniques. We run experiments to evaluate the influence of these extensions on synchronization.     

Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

This paper deals with geometrically nonlinear vibrations of sandwich beams with viscoelastic materials. For this purpose, a new finite element formulation has been developed, in which a zig-zag model is used to describe the displacement field. The viscoelastic behaviour is handled by using hereditary integrals and their relationships with complex moduli. An efficient solution procedure based on the harmonic balance method is also developed. To demonstrate its abilities, various problems of nonlinear vibrations of sandwich beams are considered. First, the results derived from the proposed approach are compared with those of nonlinear dynamic analyses using direct time integration and to experimental data. Then, the influence of the vibration amplitude on the damping properties of sandwich beams is investigated. The effect of an initial axial strain is also examined.     

An investigation into the vibration analysis of a plate with a surface crack of variable angular orientation

An analytical approach is presented for the forced vibration analysis of a plate containing a surface crack of variable angular orientation, based on three different boundary conditions. The method is based on classical plate theory. Firstly, the equation of motion is derived for the plate containing the angled surface crack with respect to one side of the plate and subjected to transverse harmonic excitation. The crack formulation representing the angled surface crack is based on a simplified line-spring model. Then, by employing the Berger formulation, the derived governing equation of motion of the cracked plate model is transformed into a cubic nonlinear system. The nonlinear behaviour of the cracked plate model is thus investigated from the amplitude–frequency equation by use of the multiple scales perturbation method. For both cracked square and rectangular plate models, the influence of the boundary conditions, the crack orientation angle, crack length, and location of the point load is demonstrated. It is found that the vibration characteristics and nonlinear characteristics of the plate structure can be greatly affected by the orientation of the crack in the plate. Finally the validity of the developed model is shown through comparison of the results with experimental work.     

Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation

The magneto-rheological damper(MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom(2-DOF) MR suspension system was established first, by employing the modified Bouc–Wen force–velocity(F –v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface.The largest Lyapunov exponent(LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density(PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy(K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system.     

Numerical assessment of fore-and-aft suspension performance to reduce whole-body vibration of wheel loader drivers

While driving off-road vehicles, operators are exposed to whole-body vibration acting in the fore-and-aft direction. Seat manufacturers supply products equipped with fore-and-aft suspension but only a few studies report on their performance. This work proposes a computational approach to design fore-and-aft suspensions for wheel loader seats. Field tests were conducted in a quarry to analyse the nature of vibration to which the driver was exposed. Typical input signals were recorded to be reproduced in the laboratory. Technical specifications are defined for the suspension. In order to evaluate the suspension vibration attenuation performance, a model of a sitting human body was developed and coupled to a seat model. The seat model combines the models of each suspension component. A linear two-degree-of-freedom model is used to describe the dynamic behaviour of the sitting driver. Model parameters are identified by fitting the computed apparent mass frequency response functions to the measured values. Model extensions are proposed to investigate postural effects involving variations in hands and feet positions and interaction of the driver's back with the backrest. Suspension design parameters are firstly optimized by computing the seat/man model response to sinusoidal acceleration. Four criteria including transmissibility, interaction force between the driver's back and the backrest and relative maximal displacement of the suspension are computed. A new suspension design with optimized features is proposed. Its performance is checked from calculations of the response of the seat/man model subjected to acceleration measured on the wheel loader during real work conditions. On the basis of the computed values of the SEAT factors, it is found possible to design a suspension that would increase the attenuation provided by the seat by a factor of two.