Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

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.     

A cascaded second-order approach to computing third-order scattering of noncollinear acoustic beams

A theoretical development of the third-order nonlinear scattering of sound from two noncollinear ultrasonic beams produced by baffled piston sources is presented. The third-order intermodulation (IM3) frequency components are derived by exploiting cascaded second-order nonlinear effects where the quadratic nonlinear interaction of second-order frequency components with first-order (primary) frequency components is considered. It is shown that cascaded second-order interactions generate intermodulation frequency components that are equivalent to those generated by cubic nonlinear effects. Comparison of measured and modeled amplitude sweeps demonstrate the three-to-one gain in decibels of the amplitudes of the third-order intermodulation frequency components to that of the primary frequency components. Measurements are also presented for the farfield interaction of distantly spaced sources, which results in a highly focused ultrasonic parametric array. Also considered are the nearfield interaction of closely spaced sources, which results in scattering, with good agreement to the theory.     

Accelerometer measurements of acoustic-to-seismic coupling above buried objects

The surface velocity of sand inside a large PVC container, induced by the sound pressure from either a large loudspeaker radiating into an inverted cone and pipe or a Bruel and Kjaer point source loudspeaker mounted with its axis vertical, has been measured using accelerometers. Results of white noise and stepped frequency excitation are presented. Without any buried object the mass loading of an accelerometer creates resonances in the spectral ratio of sand surface velocity to incident acoustic pressure, i.e., the acoustic-to-seismic (A/S) admittance spectra. The A/S responses above a buried compliant object are larger and distinctive. The linear A/S admittance spectra in the presence of a buried electronic components box have been studied as a function of burial depth and sand state. The nonlinear responses above the buried box have been studied as a function of depth, sand state, and amplitude. Predictions of a modified one-dimensional lumped parameter model have been found to be consistent with the observed nonlinear responses. Also the modified model has been used to explain features of the A/S responses observed when using an accelerometer without any buried object.     

Studying the mechanical behavior of a plastic, shock-resisting, antitank landmine

Modal behavior in landmines has recently become a topic of interest for acoustic landmine detection. It is well known that landmines exhibit mechanical resonance behavior that enhances the soil velocity over a buried landmine. Recent experimental work by Zagrai et al. [A. Zagrai, D. Donskoy, and A. Ekimov, J. Acoust. Soc. Am. 118 (6), 3619-3628 (2005)] demonstrates the existence of structural modes in several landmines. The work reported herein parallels the work of Zagrai et al. in studying the structural modes of the pressure plate of a plastic, cylindrically symmetric, antitank landmine. The pressure plate is considered to act as an elastically supported thin elastic plate. An observed perturbation of the first symmetric mode of the pressure plate is caused by the landmine's shock-resisting mechanism. This is validated by a lumped element model for the first symmetric mode coupled to the shock-resisting mechanism.     

Nonlinear acoustic techniques for landmine detection

Measurements of the top surface vibration of a buried (inert) VS 2.2 anti-tank plastic landmine reveal significant resonances in the frequency range between 80 and 650 Hz. Resonances from measurements of the normal component of the acoustically induced soil surface particle velocity (due to sufficient acoustic-to-seismic coupling) have been used in detection schemes. Since the interface between the top plate and the soil responds nonlinearly to pressure fluctuations, characteristics of landmines, the soil, and the interface are rich in nonlinear physics and allow for a method of buried landmine detection not previously exploited. Tuning curve experiments (revealing "softening" and a back-bone curve linear in particle velocity amplitude versus frequency) help characterize the nonlinear resonant behavior of the soil-landmine oscillator. The results appear to exhibit the characteristics of nonlinear mesoscopic elastic behavior, which is explored. When two primary waves f1 and f2 drive the soil over the mine near resonance, a rich spectrum of nonlinearly generated tones is measured with a geophone on the surface over the buried landmine in agreement with Donskoy [SPIE Proc. 3392, 221-217 (1998); 3710, 239-246 (1999)]. In profiling, particular nonlinear tonals can improve the contrast ratio compared to using either primary tone in the spectrum.     

水下近水面锥柱组合壳声固耦合效应的半解析求解方法

针对水下近水面锥柱组合壳声固耦合多借助于数值方法求解的现状,本文提出一种半解析方法从机理上分析此类问题。首先基于能量泛函和Sanders壳体理论、虚拟弹簧法以及力与力矩平衡条件建立锥柱组合壳的结构模型;然后采用Legendre谱元法和二维傅里叶变换得到含自由液面的水下声场模型;最后由非线性迭代法和高斯积分求解耦合系统声振控制方程。通过与参考文献和数值方法结果的对比,验证了本文方法的收敛性、正确性和可靠性。研究结果表明,结构参数、浸没深度和激励频率与远场辐射声压密切相关。本文工作可推广到水下含内部结构的复杂旋转组合壳在不同结构边界及声边界下的声固耦合问题。      

Harmonic pulsed excitation and motion detection of a vibrating reflective target

Elasticity imaging is an emerging medical imaging modality. Methods involving acoustic radiation force excitation and pulse-echo ultrasound motion detection have been investigated to assess the mechanical response of tissue. In this work new methods for dynamic radiation force excitation and motion detection are presented. The theory and model for harmonic motion detection of a vibrating reflective target are presented. The model incorporates processing of radio frequency data acquired using pulse-echo ultrasound to measure harmonic motion with amplitudes ranging from 100 to 10,000 nm. A numerical study was performed to assess the effects of different parameters on the accuracy and precision of displacement amplitude and phase estimation and showed how estimation errors could be minimized. Harmonic pulsed excitation is introduced as a multifrequency radiation force excitation method that utilizes ultrasound tonebursts repeated at a rate f(r). The radiation force, consisting of frequency components at multiples of f(r), is generated using 3.0 MHz ultrasound, and motion detection is performed simultaneously with 9.0 MHz pulse-echo ultrasound. A parameterized experimental analysis showed that displacement can be measured with small errors for motion with amplitudes as low as 100 nm. The parameterized numerical and experimental analyses provide insight into how to optimize acquisition parameters to minimize measurement errors.     

Spatial distribution of the nonlinear acoustic parameters in a thin polycrystalline plate of alloy with defects

By the results of experimental measurements of the Lamb wave’s amplitude at the fundamental frequency and its second and third harmonics, the spatial distributions of the quadratic and cubic nonlinear acoustic parameters in a thin circular plate of D16 polycrystalline aluminum alloy with defects were calculated. The correlation between these distributions and defect positions in the plate was determined.     

Monte Carlo simulations of a D-T neutron generator shielding for landmine detection

Shielding for a D–T sealed neutron generator has been designed using the MCNP5 Monte Carlo radiation transport code. The neutron generator will be used in field for the detection of explosives, landmines, drugs and other ‘threat’ materials. The optimization of the detection of buried objects was started by studying the signal-to-noise ratio for different geometric conditions.     

Nonlinear seismo-acoustic land mine detection and discrimination

A novel technique for detection and discrimination of artificial objects, such as land mines, pipes, containers, etc., buried in the ground, has been developed and tested. The developed approach utilizes vibration (using seismic or airborne acoustic waves) of buried objects, remote measurements of soil surface vibration (using laser or microwave vibrometers), and processing of the measured vibration to extract mine's "vibration signatures." The technique does not depend upon the material from which the mine is fabricated whether it be metal, plastic, wood, or any other material. It depends upon the fact that a mine is a "container" whose purpose is to contain explosive materials and associated detonation apparatus. The mine container is in contact with the soil in which it is buried. The container is an acoustically compliant article, whose compliance is notably different from the compliance of the surrounding soil. Dynamic interaction of the compliant container and soil on top of it leads to specific linear and nonlinear effects used for mine detection and discrimination. The mass of the soil on top of a compliant container creates a classical mass-spring system with a well-defined resonance response. Besides, the connection between mass (soil) and spring (mine) is not elastic (linear) but rather nonlinear, due to the separation of the soil/mine interface in the tensile phase of applied dynamic stress. These two effects, constituting the mine's vibration signature have been measured in numerous laboratory and field tests, which proved that the resonance and nonlinear responses of a mine/soil system can be used for detection and discrimination of buried mines. Thus, the fact that the mine is buried is turned into a detection advantage. Because the seismo-acoustic technique intrinsically detects buried containers, it can discriminate mines from noncompliant false targets such as rocks, tree roots, chunks of metal, bricks, etc. This was also confirmed experimentally in laboratory and field tests.     

Baseline-free estimation of residual fatigue life using a third order acoustic nonlinear parameter

Prediction of crack growth and fatigue life estimation of metals using linear/nonlinear acousto-ultrasound methods is an ongoing issue. It is known that by measuring nonlinear parameters, the relative accumulated fatigue damage can be evaluated. However, there is still a need to measure two crack propagation states to assess the absolute residual fatigue life. A procedure based on the measurement of a third-order acoustic nonlinear parameter is presented to assess the residual fatigue life of a metallic component without the need of a baseline. The analytical evaluation of how the cubic nonlinear-parameter evolves during crack propagation is presented by combining the Paris law to the Nazarov-Sutin crack equation. Unlike other developed models, the proposed model assumes a crack surface topology with variable geometrical parameters. Measurements of the cubic nonlinearity parameter on AA2024-T351 specimens demonstrated high sensitivity to crack propagation and excellent agreement with the predicted theoretical behavior. The advantages of using the cubic nonlinearity parameter for fatigue cracks on metals are discussed by comparing the relevant results of a quadratic nonlinear parameter. Then the methodology to estimate crack size and residual fatigue life without the need of a baseline is presented, and advantages and limitations are discussed.     

Interaction of two sea surface waves over a sloping bottom

A nonlinear model is constructed for the motion of long surface waves over a sloping bottom in the shallow-water approximation. It is shown that if a linear solution is known, the quadratic and cubic nonlinear corrections can be expressed in terms of the derivatives of the functions determining the linear solution. The developed method is applied to the interaction of two periodic waves in the near-shore area.     

On the detection of objects buried at a shallow depth using seismic wave reflections

This paper concerns the detection of shallow (of the order 1 m) buried objects using seismic excitation. Time-extended signals are used to generate a compressional wave using a shaker attached to the ground. The wave propagates through the ground, reflects off a buried object and is captured by an array of geophones on the surface. The envelopes of the cross-correlation functions between the measured ground velocities and the excitation signal are calculated and summed to generate a cross-sectional image of the ground. The wide cross-correlation peaks caused by high ground attenuation are partially compensated for by using the generalized cross-correlation function called the phase transform. Simple simulations are conducted to demonstrate the method, and some field experiments have been carried out aimed at the detection of a buried concrete pipe. In the experiments the pipe could be detected using the method proposed, with experimental and simulated data producing good agreement.     

Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

This work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity, through the use of the harmonic balance method (HBM). In order to study the influence of nonlinearity due to the large deflection of blades, a simplified model has been developed. This approach leads to a system of linearly coupled, second-order nonlinear differential equations, in which nonlinearity appears via cubic terms. Periodic solutions, in both the free and forced cases, are sought by applying HBM coupled with an arc-length continuation method. Solution stability has been investigated using Floquet's theorem. In addition to featuring similar and nonsimilar nonlinear modes, the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes. In the forced case, these nonlinear modes give rise to a complex dynamic behavior. Many bifurcations can take place, thus leading to strong or weak localization that may or may not be stable. In this study, special attention has been paid to the influence of excitation on dynamic responses. Several cases of excitation have been analyzed herein: localized excitation, and low-engine-order excitation. In the case of low-engine-order excitation, sensitivity of the response to a perturbation of this excitation type has been investigated, and it has been shown that for a localized, or sufficiently detuned excitation, several solutions can coexist, some of which are represented by closed curves in the Frequency-Amplitude domain. These various solutions overlap when increasing the force amplitude, leading to forced nonlinear localization. Because closed curves are not tied up with the basic nonlinear solution, they can easily be overlooked. In this study, they have been calculated using a sequential continuation with the force amplitude as a parameter.     

带有次近邻相互作用的非谐性线性链中亚声速和超声速孤子

基于相干态方法，在次近邻相互作用近似下研究了非谐性线性链中电子与声学晶格振动之间相互作用所引起的非线性效应，采用自治地处理运动方程的连续极限，立方非谐性形变势导致新的电子概率幅的非线性方程，并由此给出新的超椭圆积分形式的孤波解。证明除了通常的钟形孤子外，还存在另一种带有扭结状的孤波激发。 关键词：     

The equation of state of a microinhomogeneous medium and the frequency dependence of its elastic nonlinearity

In the framework of a rheological model, a nonlinear dynamic equation of state of a microinhomogeneous medium containing nonlinear viscoelastic inclusions is derived. The frequency dependences of the effective nonlinear parameters are determined for the difference frequency and second harmonic generation processes in the case of a quadratic elastic nonlinearity. It is shown that the frequency dependence of the nonlinear elasticity of the medium is governed by the linear relaxation response of the inclusions at the primary excitation frequency, as well as by the relaxation of the inclusions at the nonlinear generation frequencies.     

The design and instrumentation of an experimental rig to investigate acoustic methods for the detection and location of underground piping systems

A major UK initiative, entitled ‘Mapping the Underworld’, is seeking to improve our capability of locating buried utility service infrastructure without resorting to extensive excavations. One of the four projects aims to develop and prove the efficacy of a multi-sensor device for remote buried utility service detection, location and, where possible, identification. An essential technology to be combined into the device is low-frequency acoustics, and suitable techniques for detecting buried infrastructure, in particular buried plastic water pipes, have been proposed. In order to develop and test these techniques, an experimental rig has been built. It is the design and instrumentation of this rig along with the rationale for the chosen design which is the main focus of this paper. Preliminary measurements have been made on the rig, to determine the most appropriate acoustic excitation method and to confirm that the rig is behaving as anticipated. The results of these investigations are also reported.     

Delay dependence for the origin of the nonlinear derived transient evoked otoacoustic emission

In the guinea pig it has been shown that the nonlinear derived transient evoked otoacoustic emission (TEOAEnl) is comprised of significant amounts of intermodulation distortion energy. It is expected that intermodulation distortion arising from a nonlinear distortion mechanism will contribute to the overall TEOAE in a stimulus-level-dependent manner, being greatest when basilar-membrane vibration in response to a click stimulus is greatest; with decay of vibration of the basilar membrane subsequent to stimulation by a click, nonlinear interaction along the cochlear partition should reduce and so provide for a linear mechanism to dominate TEOAEnl generation, i.e., the contributions of each of these mechanisms should be delay dependent. To examine this delay dependence, TEOAEnl evoked by acoustic clicks of varying bandwidth were time-domain windowed using a recursive exponential filter in an attempt to separate two components with amplitude and phase properties consistent with different mechanisms of OAE generation. It was found that the part of the TEOAEnl occurring first in time can have a relatively constant amplitude and shallow phase slope, consistent with a nonlinear distortion mechanism. The latter part of the TEOAEnl has an amplitude microstructure and a phase response more consistent with a place-fixed mechanism.