Generation of elastic stress waves at a corner junction of square rods

Fourier techniques are used to predict the transmitted and reflected waves at an L-joint in rods of square cross-section. The expressions for both longitudinal and flexural wave components are derived for a variable angle of connection for the rods. These components are evaluated for a 90° angle of connection and an arbitrary longitudinal input pulse. The predicted waves are compared with experimental results at a number of locations away from the joint for an input pulse with wavelengths which are large compared with the cross-sectional dimensions of the rods. Good agreement is obtained for all waves. For the flexural wave this agreement is shown to improve with distance from the joint. This confirms the adequacy of elementary and Timoshenko beam theory to describe the longitudinal and flexural wave motions respectively. The results demonstrate the applicability of Fourier techniques to the solution of stress wave propagation in rods.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

Active control of the plate energy transmission in a semi-infinite ribbed plate

Active control of the plate flexural wave transmission through the beam in a semi-infinite beam-reinforced plate is analytically investigated. The ribbed plate is modeled as a continuous system, using equations of motion to describe the plate in flexure and the beam in both flexure and torsion. The maximum transmission of the plate flexural waves through the reinforcing beam is found to occur at resonance frequencies corresponding to the optimal coupling between the plate flexural waves and the flexural and torsional waves in the beam. A single control force is applied to the beam, and a cost function is developed to attenuate the far-field flexural energy transmission. It can be observed that the transmission peaks corresponding to the flexural resonances in the beam are reduced. Similarly, the transmission peaks corresponding to the torsional resonance conditions in the beam can be attenuated using a single control moment applied to the beam. Significant attenuation of all the resonance peaks in the flexural wave transmission can also be achieved with the application of a single force and a single moment collocated on the beam. In this paper, the feasibility of attenuating the flexural wave transmission due to both the flexural and torsional resonance conditions by using a single point force and point moment collocated on the beam is demonstrated.     

Coupled flexural-longitudinal wave motion in a periodic beam

Periodic structure theory is used to study the interactions between flexural and longitudinal wave motion in a beam (representing a plate) to which offset spring-mounted masses (representing stiffeners) are attached at regular intervals. An equation for the propagation constants of the coupled waves is derived. The response of a semi-infinite periodic beam to a harmonic force or moment at the finite end is analyzed in terms of the characteristic free waves corresponding to these propagation constants. Computer results are presented which show how the propagation constants are affected by the coupling, and how the forced response varies with distance from the excitation point. The spring-mounted masses can provide very high attenuation of both longitudinal and flexural waves when no coupling is present, but when coupling is introduced the two waves combine to give very low (or zero) attenuation of the longitudinal wave. The influence of different damping levels on spatial attenuation is also studied.     

Decay of solitons in an isotropic collisionless quasineutral plasma with isothermal pressure

Soliton-type solutions of the complete unreduced system of transport equations describing the plane-parallel motions of an isotropic collisionless quasineutral plasma in a magnetic field with constant ion and electron temperatures are studied. The regions of the physical parameters for fast and slow magnetosonic branches, where solitons and generalized solitary waves—nonlocal soliton structures in the form of a soliton “core” with asymptotic behavior at infinity in the form of a periodic low-amplitude wave—exist, are determined. In the range of parameters where solitons are replaced by generalized solitary waves, soliton-like disturbances are subjected to decay whose mechanisms are qualitatively different for slow and fast magnetosonic waves. A specific feature of the decay of such disturbances for fast magnetosonic waves is that the energy of the disturbance decreases primarily as a result of the quasistationary emission of a resonant periodic wave of the same nature. Similar disturbances in the form of a soliton core of a slow magnetosonic generalized solitary wave essentially do not emit resonant modes on the Alfvén branch but they lose energy quite rapidly because of continuous emission of a slow magnetosonic wave. Possible types of shocks which are formed by two types of existing soliton solutions (solitons and generalized solitary waves) are examined in the context of such solutions.     

Study on rectangular ultrasonic radiator of flexural resonant vibration in high modes

Based on the theory of coupling vibration and flexural vibration of thin rod of rectangular cross section, the fiexural vibration of rectangular thin plate was studied. The frequency equation was derived under the condition of freeboundaries. The normal modes and the relation between the normal modes and the resonant frequency were obtained. Experiments showed that the calculated resonant frequencies agree well with the measured results, and the rectangular thin plate in flexural vibration has abundant resonant frequencies. The radiator of flexural vibration used in ultrasonic cleaning and other techniques has the advantages of large acoustic radiating area, uniform acoustic field and easy adjustment of resonant frequencies, proving that it is a promising ultrasonic source.     

Wave propagation in carbon nanotube intramolecular junctions: finite element calculations

The dispersion of longitudinal and transverse waves in (n,0)–(2n,0) intramolecular junctions (IMJs) are investigated using an atomistic finite element method (FEM). The transient responses of IMJs with different connection types subjected to harmonic incident wave were modelled using three-dimensional elastic beams of carbon bonds and point masses. The linkage between the force-field constants of molecular mechanics and input parameters of beam and mass elements was established through the molecular structural mechanics approach. The wave dispersion simulated by FEM shows good agreement with that of the non-local elastic model in a wide frequency range up to the terahertz region. It is shown that both the microstructure of conical part (connection part) and the coupling of longitudinal vibration and transverse vibration brought by the conicity play important roles in the dispersion of longitudinal and transverse wave in a single-walled IMJ. The amplitude decay of longitudinal wave depended on the distance propagating; the wavelength and the structure in connection part are examined. The results show that the dispersion of the decay of the wave amplitude in IMJ with less pentagon–heptagon defects has a better agreement with analytical results of macroscopic conical shell.     

多振子梁弯曲振动中的局域共振带隙

从梁的弯曲振动方程出发,利用传递矩阵法,给出了无限周期结构的一维多振子声子晶体梁的弯曲振动能带结构,并利用有限元方法计算了有限周期结构梁的弯曲振动频率响应.建立了多振子声子晶体梁的简化模型,推导出带隙起始截止频率公式.结果表明:一维多振子声子晶体梁具有比单振子声子晶体梁更宽更丰富的振动带隙,可应用于呈倍频关系的减振降噪中;振动在带隙频率范围内频率响应具有明显的衰减;所建立的简化模型与理论模型结果符合较好.研究工作为梁类结构的减振提供一种新的思路.     

Generation of elastic stress waves at a T-junction of square rods

Fourier techniques have been used to predict transmitted and reflected waves at a T-joint in rods of square cross-section for an arbitrary longitudinal impulse approaching the joint in the terminating rod. Elementary and Timoshenko beam theory were used for longitudinal and flexural wave motions respectively. Good agreement is obtained between the predicted and experimental results for an input pulse with large wavelengths compared with the lateral dimensions of the rod. The results demonstrate the similarity to transverse impact of an infinite rod and the unidirectional translation of the joint as a valid model.     

Wave propagation in beams with anti-symmetric piezoelectric shunting arrays

Piezoelectric shunting arrays are employed to control the wave propagation in flexible beams. Contrary to conventional symmetric configuration, a substrate beam with anti-symmetric shunting arrays is investigated by adapted transfer matrix method. Compared with symmetric scheme, the anti-symmetric one demonstrates some distinctive characteristics.Primarily, the longitudinal and flexural waves are coupled, so they are correlated and must be considered simultaneously.Moreover, the attenuation of flexural wave is much stronger in anti-symmetric scenario, while the longitudinal wave demonstrates the converse side. As a result, the anti-symmetric scheme can be utilized to improve the vibration isolation capability of shunting arrays. Finally, the theoretical analyses are validated by finite element simulations.     

Equilibrium distribution of the wave energy in a carbyne chain

The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh–Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.     

LCR分流电路下压电声子晶体智能材料的带隙

将带有LCR分流电路的压电陶瓷片对贴在铝和环氧树脂组成的声子晶体结构中.使智能材料的机械振动与压电陶瓷的压电效应耦合起来,推导出机械振动在压电陶瓷片上的等效附加应力;使LCR分流电路中的电磁振荡效应和声子晶体的能带特性有机结合,计算了在分流电路作用下智能材料扭转和弯曲振动的带隙特性,研究了电阻、电感、电容元件的改变对压电声子晶体智能材料带隙的影响.研究结果表明:在合理尺寸下,随着分流电路中电阻值的增大,带隙的频率范围变宽,但衰减幅值有所降低;电感和电容值的增大都可以使带隙向低频移动,带隙的衰减幅值随着电感值的增大而升高,但随着电容值的增大而降低.从而给压电声子晶体智能材料减震降噪的控制提供了一种新思路.     

Experimental investigation of vibration power flow in thin technical orthotropic plates by the method of vibration intensity

Vibration intensity technique is used to measure vibration power transmission in thin single layer technical orthotropic plates for flexural waves. Measurement of flexural wave power is carried out in far-field conditions. All measurements are undertaken in the frequency domain using the cross-spectra of acceleration signals, facilitating the use of FFT analyzer. The two-transducer technique applicable to these plates is used for these measurements. Technical orthotropic (rectangular corrugation) plates of steel are used for the measurements. One isotropic plate of steel is also considered for comparison. Method of elastic equivalence technique is used. Both input power and vibration power transmission through the plates are estimated. Far-field power is normalized with the input power for flexural wave. Influence of flexural rigidity on vibration energy transfer is also investigated.     

Time reversal of flexural waves in a beam at audible frequency

There has been very limited work on the application of time reversal to the propagation of audible frequency waves in mechanical structures. The present work concentrates on the application of time reversal to the focusing of audible range, flexural waves in an infinite beam, and to the detection of local heterogeneity in such a beam. Practical applications of time reversal of flexural waves in structures include vibration energy focusing, detection of vibratory or acoustic sources, and detection of defects in mechanical structures. An analytical model of flexural wave propagation in the beam as well as sensing and emission using piezoelectric transducers is presented. Time reversal experiments are conducted and compared to the model results in either a homogeneous beam or a beam with point mass heterogeneities. In the various situations tested, it is shown that time reversal effectively compensates the spreading in time of the impulse due to the dispersive propagation of flexural waves. One interesting aspect of this property is the generation of large amplitude impulsive responses in the beam using remote actuators. Finally, the "Decomposition de l'Operateur de Retournement Temporel" approach is examined to detect and localize point mass scatterers in the beam.     

Elastic Waves in Generalized Thermo-Piezoelectric Transversely Isotropic Circular Bar Immersed in Fluid

In this paper, a mathematical model is developed to study the wave propagation in an infinite, homogeneous, transversely isotropic thermo-piezoelectric solid bar of circular cross-sections immersed in inviscid fluid. The present study is based on the use of the three-dimensional theory of elasticity. Three displacement potential functions are introduced to uncouple the equations of motion and the heat and electric conductions. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied based on Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity. The frequency equations of the coupled system consisting of cylinder and fluid are developed under the assumption of perfect-slip boundary conditions at the fluid-solid interfaces, which are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 material bar immersed in fluid. The computed non-dimensional frequencies are compared with Lord-Shulman, Green-Lindsay and Classical theory theories of thermo elasticity for longitudinal and flexural modes of vibrations. The dispersion curves are drawn for longitudinal and flexural modes of vibrations. Moreover, the dispersion of specific loss and damping factors are also analyzed for longitudinal and flexural modes of vibrations.     

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation.In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.     

考虑弯曲和轴向振动模态的一维梁压电阻抗模型及试验研究

同时考虑一维梁结构的弯曲和轴向振动,对其压电阻抗模型进行建模分析和试验验证。在0.02～42 kHz频段内区分并标记了一维钢梁弯曲振动模态前18阶及轴向振动模态前3阶。结果表明:在0.02～7.5kHz频段内,数值计算和试验结果中谐振峰对应频率的相对误差较大:11.7%～16.5%,其原因可能是低频时振动能量较低且波的传播受结构阻尼、边界条件及环境噪音等因素影响较为明显;在7.5～42kHz范围内,两者谐振峰位置符合良好,相对误差较小:0.11%～2.31%,表明该模型在高频段具有较好的适用性;轴向振动模态对应频率大于弯曲振动模态。本研究为结构健康监测过程中检测频段的选取及损伤信息的提取提供参考。     

Wave dispersion relation of two-dimensional plasma crystals in a magnetic field

The wave dispersion relation in a two-dimensional strongly coupled plasma crystal is studied by theoretical analysis and molecular dynamics simulation taking into account a constant magnetic field parallel to the crystal normal. The expression for the wave dispersion relation clearly shows that high-frequency and low-frequency branches exist as a result of the coupling of longitudinal and transverse modes due to the Lorenz force acting on the dust particles. The high-frequency and the low-frequency branches are found to belong to right-hand and left-hand polarized waves, respectively.     

A study of modal characteristics and the control mechanism of finite periodic and irregular ribbed plates

An analytical solution is presented in this paper to investigate the control mechanism and modal characteristics of finite periodic and irregular ribbed plates. Peak responses of a finite periodic ribbed plate were examined where they were grouped into two sets of propagation zones according to the coupling mechanism at beam/plate interfaces. Details of modal characteristics in pass bands of the periodic ribbed plate were elucidated and the control mechanism was discussed. Modes in each pass band that are governed by shear force couplings were characterized by one of the beam flexural modes whose modal responses could be represented approximately by those of the corresponding orthotropic plate modes. Modes in the second set of pass bands were found to retain the resonance frequencies of the corresponding modes of the unribbed base plate. Higher order orthotropic plate modes were also identified, which could not be grouped into any pass bands defined by the classical periodic theory. The control mechanism leading to vibration confinement in disordered and irregular ribbed plates was also discussed. It was found that beam spacing irregularity attributes to localization of the group of modes associated with flexural wave couplings but not the group of modes associated with moment couplings.     

A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh's Quotient and Rayleigh's Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.