Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions

The dynamic analysis of a three-layered symmetric sandwich beam with magnetorheological elastomer (MRE) embedded viscoelastic core and conductive skins subjected to a periodic axial load have been carried out under various boundary conditions. As the skins of the sandwich beam are conductive, magnetic loads are applied to the skins during vibration. Due to the field-dependent shear modulus of MRE material, the stiffness of the MRE embedded sandwich beam can be changed by the application of magnetic fields. Using extended Hamilton’s principle along with generalized Galarkin’s method the governing equation of motion has been derived. The free vibration analysis of the system has been carried out and the results are compared with the published experimental and analytical results which are found to be in good agreement. The parametric instability regions of the sandwich beam have been determined for various boundary conditions. Here, recently developed magnetorheological elastomer based on natural rubber containing iron particles and carbon blacks have been used. The effects of magnetic field, length of MRE patch, core thickness, percentage of iron particles and carbon blacks on the regions of parametric instability for first three modes of vibration have been studied. These results have been compared with the parametric instability regions of the sandwich beam with fully viscoelastic core to show the passive and active vibration reduction of these structures using MRE and magnetic field. Also, the results are compared with those obtained using higher order theory.     

Stationary response of nonlinear magneto-piezoelectric energy harvester systems under stochastic excitation

Recent years have shown increasing interest of researchers in energy harvesting systems designed to generate electrical energy from ambient energy sources, such as mechanical excitations. In a lot of cases excitation patterns of such systems exhibit random rather than deterministic behaviour with broad-band frequency spectra. In this paper, we study the efficiency of vibration energy harvesting systems with stochastic ambient excitations by solving corresponding Fokker-Planck equations. In the system under consideration, mechanical energy is transformed by a piezoelectric transducer in the presence of mechanical potential functions which are governed by magnetic fields applied to the device. Depending on the magnet positions and orientations the vibrating piezo beam system is subject to characteristic potential functions, including single and double well shapes. Considering random excitation, the probability density function (pdf) of the state variables can be calculated by solving the corresponding Fokker-Planck equation. For this purpose, the pdf is expanded into orthogonal polynomials specially adapted to the problem and the residual is minimized by a Galerkin procedure. The power output has been estimated as a function of basic potential function parameters determining the characteristic pdf shape.     

Vibration suppression of rotating beams using time-varying internal tensile force

A new strategy for vibration suppression of a rotating beam using a time-increasing internal tensile force is proposed in this paper. Nonlinear coupled longitudinal and bending equations of motion are derived in non-dimensional form using the Hamilton principle. The first-order analytical solution of the equations of motion is obtained using the Galerkin technique combined with the multiple scales method (MSM). Numerical simulations are then performed for various increasing rates of the internal tensile force and performance of the vibration suppression strategy is studied. A very close agreement between the simulation results obtained by the numerical integration and the first-order analytical solution is achieved. Forced vibrations of the system for input excitations of either a sinusoidal or a random function with white noise time history are considered. The simulation results and dynamic performance of the suppressed system for an externally excited rotating beam show an interesting phenomenon of the form of remarkable effectiveness of the proposed vibration reduction strategy.     

Effect of Cyclic Deformation on Magnetorheological Elastomers

采制备了铁颗粒含量分别是60%,70%和80%的三种材料,在应变幅值为50%,75%和100%的循环加载条件下,利用改进后的动态力学分析仪测试了循环加载后材料的储能模量和损耗模量. 结果显示铁颗粒含量为70%和80%的样品,其储能模量和损耗模量都和循环加载的幅值以及循环加载的次数有关,铁颗粒含量为80%样品受加载条件的影响尤其明显. 但铁颗粒含量为60%的样品,其性能却与循环加载的幅值和次数无关. 为了更好的研究磁流变弹性体受循环加载后的性能,样品还进行了准静态拉伸测试,并用扫描电镜对测试的样品进行了原位观察.     

Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads

The vibration of an Euler-Bernoulli beam, resting on a nonlinear Kelvin-Voight viscoelastic foundation, traversed by a moving load is studied in the frequency domain. The objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response. The parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient. The solution is obtained by using the Galerkin method in conjunction with the multiple scales method (MSM). The governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations. Subsequently, the solution is calculated for different harmonics by using the MSM as one of the powerful perturbation techniques. The steady-state responses of the main harmonic as well as its two super-harmonics are then obtained. As a case study, a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics. Moreover, a thorough stability analysis of the system is carried out.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

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.     

非对称夹芯板隔声量的解析计算模型及影响因素

提出了一种计算上下面板非对称的三明治夹芯板隔声性能的方法.通过对非对称夹芯梁表观抗弯曲刚度的计算,得到对应夹芯板随频率变化的表观抗弯刚度,代入4阶的控制方程,应用模态展开法可以方便地计算简支非对称夹芯板的隔声量.对4种定制的3层非对称碳纤维夹芯板进行了理论计算和实验测试对比,在频率范围100～3150Hz内,计权隔声量...     

Stabilization of an axially moving web via regulation of axial velocity

In this paper, a novel control algorithm for suppression of the transverse vibration of an axially moving web system is presented. The principle of the proposed control algorithm is the regulation of the axial transport velocity of an axially moving beam so as to track a profile according to which the vibration energy decays most quickly. The optimal control problem that generates the proposed profile of the axial transport velocity is solved by the conjugate gradient method. The Galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the axially moving beam into a set of ordinary differential equations (ODEs). For control design purposes, these ODEs are rewritten into state-space equations. The vibration energy of the axially moving beam is represented by the quadratic form of the state variables. In the optimal control problem, the cost function modified from the vibration energy function is subjected to the constraints on the state variables, and the axial transport velocity is considered as a control input. Numerical simulations are performed to confirm the effectiveness of the proposed control algorithm.     

On the nonlinear primary resonances of a piezoelectric laminated micro system under electrostatic control voltage

In this article, a comprehensive nonlinear analysis for a piezoelectric laminated micro system around its static deflection is presented. This static deflection is created by an electrostatic DC control voltage through an electrode plate. The micro system beam is assumed as an elastic Euler-Bernoulli beam with clamped-free end conditions. The dynamic equations of this model have been derived by using the Hamilton method and considering the nonlinear inertia, curvature, piezoelectric and electrostatic terms. The static and dynamic solutions have been achieved by using the Galerkin method and the multiple-scales perturbation approach, respectively. The results are compared with numerical and other existing experimental results. By studying the primary resonance excitation, the effects of different parameters such as geometry, material and excitations voltage on the system?s softening and hardening behaviors are evaluated. In a piezoelectrically actuated micro system it was showed that because of existence of curvature and inertia nonlinear terms a small change in excitation amplitude can lead to the formation and expansion of nonlinear response. In this paper, it is demonstrated that by applying an electrostatic DC control voltage, these nonlinearities can be controlled and altered to a linear domain. This model can be used to design a nano or micro-scale smart device.     

Free vibration and buckling analysis of clamped skew sandwich plates by the Galerkin method

Galerkin's variational method has been used in the past by several investigators [1–3] to solve bending problems of clamped skew plates. In this paper the suitability of the Galerkin method for solution of problems of buckling under the action of in-plane forces and of free vibration of skew plates is studied. The method is first applied to investigate the problems for clamped rectangular sandwich plates. After the validity of the method has been established, the method is then extended to analyze similar problems for clamped skew sandwich plates. The governing differential equations for the skew sandwich plates are obtained by transforming the corresponding differential equations in Cartesian coordinates into skew co-ordinates. The parameters considered herein for the buckling and free vibration behaviour of the skew sandwich plates are the aspect ratio of the plate, Poisson's ratio, skew angle and various shearing stiffnesses of the core. Simplicity and quick convergence is the advantage of the method in comparison with other much more laborious numerical methods requiring extensive computer facilities.     

Long distance real-time measurement of multi-points micro-vibration in region by digital holography

Many applications require micro-vibration measurement, especially multi-points detection at long distance in real-time. In this paper, a micro-vibration measurement approach based on digital holographic interferometry is proposed for middle-low frequency detection. It can be used to monitor irregular frequency/amplitude vibration in selected region over 10 m away simultaneously and synchronously. A series of experiments were conducted including real-time measurement of 300 Hz, 1 kHz, 2 kHz and 3 kHz constant frequency/amplitude periodic vibration, precision and frequency response tests with calibration of LDV, 1 kHz irregular amplitude vibration, irregular frequency/amplitude vibration as well as the real-time measurement and simultaneous display of multi-points vibration. The experimental results demonstrate the feasibility of the proposed method and reveal its unique advantages.     

Linear and nonlinear vibrations analysis of viscoelastic sandwich beams

In this paper, numerical models are proposed for linear and nonlinear vibrations analyses of viscoelastic sandwich beams with various viscoelastic frequency dependent laws using the finite element based solution. Real and various complex eigenmodes approaches are investigated as Galerkin bases. Based on harmonic balance method, simplified and general approaches are developed for nonlinear vibration analysis. Analytical frequency-amplitude and phase-amplitude relationships are elaborated based on the numerically computed complex eigenmodes. The equivalent loss factors and frequencies as well as the forced harmonic response and phase curves are performed for sandwich beams with various boundary conditions and frequency dependent viscoelastic laws.     

Galerkin method for steady-state response of nonlinear forced vibration of axially moving beams at supercritical speeds

The present paper investigates the steady-state periodic response of an axially moving viscoelastic beam in the supercritical speed range. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. It is assumed that the excitation of the forced vibration is spatially uniform and temporally harmonic. Under the quasi-static stretch assumption, a nonlinear integro-partial-differential equation governs the transverse motion of the axially moving beam. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. For a beam constituted by the Kelvin model, the primary resonance is analyzed via the Galerkin method under the simply supported boundary conditions. Based on the Galerkin truncation, the finite difference schemes are developed to verify the results via the method of multiple scales. Numerical simulations demonstrate that the steady-state periodic responses exist in the transverse vibration and a resonance with a softening-type behavior occurs if the external load frequency approaches the linear natural frequency in the supercritical regime. The effects of the viscoelastic damping, external excitation amplitude, and nonlinearity on the steady-state response amplitude for the first mode are illustrated.     

Vibration analysis of a partially treated multi-layer beam with magnetorheological fluid

Semi-active vibration control based on magnetorheological (MR) materials offers excellent potential for high bandwidth control through rapid variations in the rheological properties of the fluid under varying magnetic field. Such fluids may be conveniently applied to partial or more critical components of a large structure to realize more efficient and compact vibration control mechanism with variable damping. This study investigates the properties and vibration responses of a partially treated multi-layer MR fluid beam. The governing equations of a partially treated multi-layered MR beam are formulated using finite element method and Ritz formulation. The validity of the proposed finite element formulations is demonstrated by comparing the results with those obtained from the Ritz formulation and the experimental measurements. The properties of different configurations of a partially treated MR-fluid beam are evaluated to investigate the influences of the location and length of the MR-fluid for different boundary conditions. The properties in terms of natural frequencies and loss factors corresponding to various modes are evaluated under different magnetic field intensities and compared with those of the fully treated beams. The effect of location of the fluid treatment on deflection mode shapes is also investigated. The forced vibration responses of the various configurations of partially treated MR sandwich beam are also evaluated under harmonic force excitations. The results suggest that the natural frequencies and transverse displacement response of the partially treated MR beams are strongly influenced not only by the intensity of the applied magnetic field, but also by the location and the length of the fluid pocket. The application of partial treatment could also alter the deflection pattern of the beam, particularly the location of the peak deflection.     

Rapid acquisition technique for MR elastography of the liver

Magnetic resonance elastography (MRE) of the liver is a novel noninvasive clinical diagnostic tool to stage fibrosis based on measured stiffness. The purpose of this study is to design, evaluate and validate a rapid MRE acquisition technique for noninvasively quantitating liver stiffness which reduces by half the scan time, thereby decreasing image registration errors between four MRE phase offsets. In vivo liver MRE was performed on 16 healthy volunteers and 14 patients with biopsy-proven liver fibrosis using the standard clinical gradient recalled echo (GRE) MRE sequence (MREs) and a developed rapid GRE MRE sequence (MREr) to obtain the mean stiffness in an axial slice. The mean stiffness values obtained from the entire group using MREs and MREr were 2.72 ± 0.85 kPa and 2.7 ± 0.85 kPa, respectively, representing an insignificant difference. A linear correlation of R² = 0.99 was determined between stiffness values obtained using MREs and MREr. Therefore, we can conclude that MREr can replace MREs, which reduces the scan time to half of that of the current standard acquisition (MREs), which will facilitate MRE imaging in patients with inability to hold their breath for long periods.     

Transient vibrations of a beam/mass system fixed to a rotating body

Transient flexural vibrations of a beam/mass system fixed to a rotating body are investigated. The rotating body is driven so as to have a velocity profile of trapezoidal shape. The governing ordinary differential equations of the beam mass system are derived by use of the extended Galerkin method, and the transient response is obtained by the Laplace transformation. Then the effects of the flexibility of the beam and the rotational period of the rotating body upon the flexural vibrations are investigated.     

Analysis of the sandwich piezoelectric ultrasonic transducer in coupled vibration

The coupled vibration of the sandwich piezoelectric transducer with a large cross-section is analyzed using an approximate analytic method. The resonance frequency equations of the transducer are derived and the effect of the geometrical dimensions on the resonance frequency is studied. It is illustrated that when the radial vibration in the transducer is considered, the vibration of the sandwich transducer becomes more complex. Apart from the longitudinal resonance frequency, the radial resonance frequency can also be obtained. For comparison, numerical methods are also used to simulate the coupled vibration; the resonance frequency and the vibrational displacement distribution are computed. Compared with one-dimensional longitudinal theory, the radial dimensions of the transducer are no longer limited because the coupled vibration is considered. Compared with numerical methods, the physical meaning of the analytic method is concise. It is illustrated that the resonance frequencies obtained from the coupled resonance frequency equations are in good agreement with those from numerical methods, and they are in better agreement with the measured results than those from one-dimensional theory. Since the radial and the coupled vibration are considered in the analysis, more resonance frequencies can be obtained. Therefore, using the coupled resonance frequency equations, the sandwich transducer with multifrequency or wide frequency bandwidth can be designed and used in ultrasonic cleaning, ultrasonic sonochemistry and other applications.     

On the performance and resonant frequency of electromagnetic induction energy harvesters

This paper investigates the linear response of an archetypal energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. In contrast with most prior works, the influence of the circuit inductance is not assumed negligible. Instead, we highlight parameter regimes where the inductance can alter resonance and derive an expression for the resonant frequency.The governing equations consider the case of a vibratory generator directly powering a resistive load. These equations are non-dimensionalized and analytical solutions are obtained for the system's response to single harmonic, periodic, and stochastic environmental excitations. The presented analytical solutions are then used to study the power delivered to an electrical load.     

Influence of composition of carbonyl iron particles on dynamic mechanical properties of magnetorheological elastomers

Magnetorheological elastomers (MRE) are known as smart materials. However, the magnetorheological (MR) effect of MRE is not high enough at present, which limits its engineering applications. Prior studies have shown that magneto-induced shear storage modulus and MR effect were mainly determined by the performance of the ferromagnetic particles. In this paper, MRE samples were prepared by carbonyl iron particles (CIP) of different compositions based on silicon rubber under external magnetic field. Their microstructures were observed using an optical digital microscope and a scanning electron microscope. The dynamic mechanical properties of MRE samples were measured using a modified dynamic mechanical analyzer under varying magnetic field strength and frequency. The results show that the carbon content of CIP have a greater impact on the dynamic mechanical properties of MRE. The magneto-induced shear storage modulus and MR effect can be increased by selecting CIP of low carbon content. In addition, the damping property is also significantly influenced by the carbon content of the CIP. This study is expected to provide guidance for fabrication of high performance MRE.