附磁阶梯变厚度悬臂梁压电俘能器的理论建模及分析

论文建立了一种附磁阶梯变厚度压电悬臂梁的动力学模型并分析了系统的俘能特性。基于Euler-Bernoulli梁理论分段建立系统能量函数并引入非线性磁势能,利用Lagrange方程建立了系统机电耦合动力学方程;利用数值方法分析了磁间距对系统振动特性的影响,此外还研究了系统单稳态和双稳态响应,探讨了厚度比、长度比、磁间距和外激励幅值对系统动力学响应和俘能特性的影响。结果表明,磁间距是影响系统势能的主要因素,调节磁间距可使系统产生单稳态和双稳态响应,从而有效提高俘能器俘能特性;与传统等截面悬臂梁压电俘能器相比,通过优化结构参数,附磁阶梯变厚度悬臂梁压电俘能器能够发生明显的非线性振动现象,实现宽频带振动能量采集。     

Accurate modeling,comparative analysis,and performance enhancement of broadband piezoelectric energy harvesters with single and dual magnetic forces

The accurate electromechanical modeling and effectiveness of a piezoelectric energy harvester having dual magnetic forces at its tip is evaluated for low-frequency excitation purposes. To accurately represent the magnetic force, the exact finite element magnetic force is accurately fitted with a fifth-order polynomial representation in order to use it for low spacing distances between the two magnets. A comparative study is then carried out between the performances of piezoelectric energy harvesters having single and dual magnetic forces. It is indicated that hardening nonlinear behaviors take place in the dual attractive magnets compared to softening behaviors in the single magnet design. Static and eigenvalue problem analyses are also performed in order to determine the impacts of the spacing distance between the magnets on the static deflection and fundamental natural frequency of the energy harvester. The results show that the inclusion of a second magnetic force with an attractive interaction results in a delay in the static pull-in and a decrease in the fundamental natural frequency for same spacing distances. After that, a nonlinear distributed-parameter model is derived using the Galerkin discretization and used to study the performance and effectiveness of a piezoelectric energy harvester with dual attractive magnetic forces. The results show that the attractive dual magnets with same spacing distances lead to the presence of broadband resonance regions when the spacing distance between the magnets decreases to lower values. In addition, unlike the dual magnetic forces configuration which has only cubic nonlinearity, it is demonstrated that the single magnetic force configuration results in the presence of the quadratic nonlinearity which allows the energy harvester to have a softening behavior.     

自由端受磁力吸引悬臂梁的非线性振动

研究了含分式函数的非线性磁力系统的受迫振动。磁铁吸引力横向作用在悬臂梁自由端处。根据含有非线性边界的连续体模型,给出了非平凡静平衡位形。以稳定位形作坐标变换后,计算了相应的固有频率,并求得微幅受迫振动稳态响应的近似解析解。研究了不同磁铁间距与频响曲线的关系,结果表明磁力方向和大小对它们均有较大影响。数值结果与解析结果吻合得很好。     

线形?拱形组合梁式三稳态压电俘能器动力学特性研究

利用振动能量俘获技术将设备工况振动能转化为电能, 为实现煤矿井下无线监测节点自供电提供了新的思路. 通过引入非线性磁力设计了一种线形?拱形组合梁式三稳态压电俘能器, 分析了磁铁水平间距、垂直间距和激励加速度对动力学特性的影响规律. 利用磁偶极子法建立磁力模型, 通过实验测量线形?拱形组合梁的恢复力, 并采用多项式拟合得到恢复力模型, 基于欧拉?伯努利梁理论和拉格朗日方程建立系统的动力学模型, 从时域角度仿真分析了磁铁水平间距、垂直间距和激励加速度对系统动力学特性的影响规律. 研制线形?拱形组合梁式三稳态压电俘能器样机并搭建实验平台进行实验研究, 通过采集组合梁末端响应速度数据, 验证了理论分析的正确性. 研究表明: 引入非线性磁场能够使系统势能呈现单势阱、双势阱或三势阱, 激励一定时, 调整磁铁水平间距和垂直间距能够使系统实现单稳态、双稳态或三稳态运动, 且在三稳态运动时响应位移较大, 增大激励水平有利于系统越过势垒实现大幅响应. 研究为线形?拱形组合梁式三稳态压电俘能器的设计提供了理论指导.      

多稳态俘能系统的准确磁力建模方法

混沌和分岔使得多稳态俘能系统的非线性动力学响应对系统结构参数非常敏感, 导致了系统的非线性特性正向设计比较困难. 为了定量地表征非线性恢复力与结构参数的关系, 提出了一种多稳态俘能系统的准确磁力建模方法. 推导了多稳态俘能系统端部磁铁和外部磁铁的相对距离和转角位置, 并采用磁荷理论建立了多稳态系统的非线性磁力模型. 通过搭建实验平台测量了不同结构参数条件下多稳态系统的非线性磁力, 并对比了本方法与传统方法和实验测量的结果. 结果表明: 本方法的磁力计算结果与实验测量值吻合较好, 双稳态系统和三稳态系统的磁力峰值误差分别仅为4.3%和6.49%, 验证了本方法计算多稳态系统非线性磁力的有效性. 此外, 基于本方法探究了多稳态系统结构参数对系统势阱的影响机理, 获取了多稳态系统的稳态临界位置, 研究了双稳态和三稳态系统在不同结构参数下的响应电压规律. 参数优化结果表明, 双稳态系统在竖直距离为34 mm时, 均方电压最大为10.22 V; 三稳态系统在竖直距离为28 mm且水平距离为8 mm时, 均方电压最大为12.7 V. 该研究提出的模型以期为多稳态系统的输出性能优化设计提供借鉴.      

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam

Grazing behavior in soft impact dynamics of a harmonically based excited flexible cantilever beam is investigated. Numerical and experimental methods are employed to study the dynamic behavior of macro- and micro-scale cantilever beam–impactor systems. For off-resonance excitation at two and a half times the fundamental frequency, the response of the oscillating cantilever experiences period doubling as the separation distance or clearance between the beam axis and the contact surface is decreased. The nonlinear phenomenon is studied by using phase portraits, Poincaré sections, and spectral analysis. Motivated by atomic force microscopy, this general dynamic behavior is studied as a means to locating the separation distance corresponding to grazing where the contact force is minimized.     

Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions

In this article, nonsmooth dynamics of impacting cantilevers at different scales is explored through a combination of analytical, numerical, and experimental efforts. For off-resonance and harmonic base excitations, period-doubling events close to grazing impacts are experimentally studied in a macroscale system and a microscale system. The macroscale test apparatus consists of a base excited aluminum cantilever with attractive and repulsive tip interactions. The attractive force is generated through a combination of magnets, one located at the cantilever structure??s tip and another attached to a high-resolution translatory stage. The repulsive forces are generated through impacts of the cantilever tip with the compliant material that covers the magnet on the translatory stage. The microscale system is an atomic force microscope cantilever operated in tapping mode. In this mode, this microcantilever experiences a long-range attractive van der Waals force and a repulsive force as the cantilever tip comes close to the sample. The qualitative changes observed in the experiments are further explored through numerical studies, assuming that the system response is dominated by the fundamental cantilever vibratory mode. In both the microscale and macroscale cases, contact is modeled by using a quadratic repulsive force. A reduced-order model, which is developed on the basis of a single mode approximation, is employed to understand the period-doubling phenomenon experimentally observed close to grazing in both the macroscale and microscale systems. The associated near-grazing dynamics is examined by carrying out local analyses with Poincaré map constructions to show that the observed period-doubling events are possible for the considered nonlinear tip interactions. In the corresponding experiments, the stability of the observed grazing periodic orbits has been assessed by constructing the Jacobian matrix from the experimentally obtained Poincaré map. The present study also sheds light on the use of macroscale systems to understand near-grazing dynamics in microscale systems.     

磁耦合式双稳态宽频压电俘能器的设计和俘能特性

为了同时提高振动能量俘获系统的效率和实用性, 俘能器主结构的振动特性与环境振动特性的匹配度显得尤为重要. 非线性系统复杂的动力学行为为设计高效的俘能器奠定了基础, 但结构一旦被设计、生成出来, 其工作频率往往是固定的, 无法根据环境中的振动而发生相应的改变. 本文利用可移动铰支座和非线性磁力设计了一种具有双稳态特性的宽频压电俘能器, 通过拓宽压电俘能器的工作频带, 来匹配环境中较宽的振动频率. 为了保证系统低频宽带的俘能效果, 详细分析了结构的长度比、磁间距、负载阻抗、外激励频率和幅值等对系统线性刚度、非线性刚度以及动力学行为的影响, 并进行了实验验证. 首先将系统简化为欧拉-伯努力梁, 利用拉格朗日方程建立系统的非线性动力学方程, 并利用谐波平衡法进行求解. 针对理论分析给出的不同外激励频率下的最优长度比, 搭建了实验平台进行验证. 理论和实验的结果表明: 非线性磁力的引入使系统呈现负刚度特性, 使俘能器能够在单稳态和双稳态之间的变换, 实现低频俘能效果; 通过调节可移动铰支座的位置, 改变系统的长细比, 能够实现从0到16 Hz的宽频俘能效果.      

Nonlinear gravity-capillary waves instability in superposed magnetic fluids influenced by a vertical magnetic field and time-dependent acceleration

This paper deals with the effect of a periodic forcing on nonlinear modulation of interfacial gravity-capillary waves propagating between two magnetic fluids of infinite depth under the influence of a constant vertical magnetic field. Based on the method of multiple scales expansion for a small amplitude of periodic force, two parametric nonlinear Schrödinger equations with explicit expressions of coefficients are derived in the resonance case. A classical nonlinear Schrödinger equation is derived in the non-resonance case. The stability of the uniform time-dependent solution is analyzed. Theoretical analysis and numerical calculations show that the resonance point is affected by the magnetic field and the applied frequency. The linear stability shows that the periodic force has a destabilizing influence in the stability criterion. It is observed that the vertical field plays the same role, and that the acceleration frequency plays a dual role in the nonlinear stability criterion. Instability was revealed in the system for large values of the applied magnetic field, but the small values of the field redistribute the stable areas.     

Effect of boundary condition nonlinearities on free large-amplitude vibrations of rectangular plates

The effect of boundary condition nonlinearities on free nonlinear vibrations of thin rectangular plates is analyzed. The method for analysis of the plate vibrations with geometrical nonlinearity and the boundary condition nonlinearity is suggested. The nonlinear boundary conditions for membrane forces are transformed into linear ones using the in-plane stress function. Additional boundary conditions for the in-plane displacements vanishing on the clamped edge of the plate are imposed on the stress function. Simply supported and cantilever plates are analyzed. The backbone curves obtained by satisfying linear and nonlinear boundary conditions are compared. It is shown that the results of the calculations with nonlinear boundary conditions differ essentially from the data obtained without these boundary conditions.     

Analytic expression of magnetic field distribution of rectangular permanent magnets

From the molecular current viewpoint, an analytic expression exactly describing magnetic field distribution of rectangular permanent magnets magnetized sufficiently in one direction was derived from the Biot-Savart’s law. This expression is useful not only for the case of one rectangular permanent magnet bulk, but also for that of several rectangular permanent magnet bulks. By using this expression, the relations between magnetic field distribution and the size of rectangular permanent magnets as well as the magnitude of magnetic field and the distance from the point in the space to the top ( or bottom ) surface of rectangular permanent magnets were discussed in detail. All the calculating results are consistent with experimental ones. For transverse magnetic field which is a main magnetic field of rectangular permanent magnets, in order to describe its distribution, two quantities, one is the uniformity in magnitude and the other is the uniformity in distribution of magnetic field, were defined. Furthermore, the relations between them and the geometric size of the magnet as well as the distance from the surface of permanent magnets were investigated by these formulas. The numerical results show that the geometric size and the distance have a visible influence on the uniformity in magnitude and the uniformity in distribution of the magnetic field.     

Passive effects of rare-earth permanent magnets on flexible conductive structures

The paper presents a theoretical and experimental study of vibrating structures where paramagnetic or diamagnetic systems interact with rare-earth passive magnets.The theoretical model of the system is focused on the damping properties of permanent magnets and on their interactions with the dynamic behaviour of an Euler–Bernoulli beam. In particular, the magnetic model is based on the analogy of the equivalent currents method in a quasi-static open-circuit-type configuration and it is used to determine the influence of eddy currents on the dynamic behaviour of conducting material structures. The magnetic effects are characterised by a viscous-type damping and by a stiffening dynamic effect of the structure, called “phantom effect”.The authors present the experimental outcomes for uniform cantilever clamped-free beams of different kinds of paramagnetic or diamagnetic conducting materials. It appears that the system frequency response can be modified by the presence of a pair of concordant or discordant permanent magnets of high residual induction settled at the free end.Through the comparison between theoretical and experimental results, the paper demonstrates the validity of the model, that is able to describe both the above mentioned effect of dynamic stiffening of the structure and the considerable localised damping properties in paramagnetic or diamagnetic materials having low electric resistivity.     

Analytical solutions of the frequency shifts of several modes in AFM scanning an inclined surface,subjected to the Lennard-Jones force

The role of higher cantilever modes is important to obtain some material contrast. The analysis of AFM subjected to a short-range force can improve greatly the studies of surface topography and interaction energies and interaction forces, especially for chemistry and biology materials. When the tip-sample distance is in the order of inter-atomic spacing, the short-range tip-sample force is usually simulated by the Lennard-Jones model. In typical AFM experiments the cantilever is inclined at an angle to the substrate surface. In this study, the analytical method to determine the frequency shift of AFM scanning the relative inclined surface, subjected to the Lennard-Jones force is proposed. The inclined angle may be the angle between the probe and the scanning table or due to the inclined plane and the roughness of sample surface. The closed-form solution of the partial differential equation with a nonlinear boundary condition is derived and then the corresponding frequency shifts of higher modes can be determined easily. The perturbation method transforming a continuous system into the lumped-masses one for determining the frequency shift will lead to an error. Although the above disadvantage exists, the lumped-masses model is simple and intuitive. Therefore, using the principle of dynamic strain energy, the conventional perturbation method is revised to determine the frequency shifts of higher modes in scanning an relative inclined surface. Further, the ratio relation among the frequency shifts is discovered. Moreover, it is found that the effect of the relative inclined angle on the first three frequency shifts is significant.     

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M ₃) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

嵌入薄膜传感器悬臂梁三向力测量技术研究

以薄膜传感器悬臂梁作为等效模型,通过传感器的应变效应对三向力测量技术进行了研究。为提高薄膜传感器的应变输出响应,对悬臂梁上布放薄膜传感器的位置加设弹性结构,研究了三向力测力模型输出电压与传感器所在位置应变的关系;分析了受力位置对测力模型输出响应的影响关系,结合实验验证了其工作原理、测力模型应变输出响应与可控尺寸参数的关系。研究表明:该测力模型可实现三向力测量,各个方向最大测量误差均在9%以内,悬臂梁宽度方向x和高度方向z的交叉干涉误差分别为2.84%和3.37%;当悬臂梁自由端受力位置发生变化时,测力模型输出响应只在梁长度方向y上发生变化。     

Bifurcation and pull-in voltages of primary resonance of electrostatically actuated SWCNT cantilevers to include van der Waals effect

In this work the voltage response of primary resonance of electrostatically actuated single wall carbon nano tubes (SWCNT) cantilevers over a parallel ground plate is investigated. Three forces act on the SWCNT cantilever, namely electrostatic, van der Waals and damping. While the damping is linear, the electrostatic and van der Waals forces are nonlinear. Moreover, the electrostatic force is also parametric since it is given by AC voltage. Under these forces the dynamics of the SWCNT is nonlinear parametric. The van der Waals force is significant for values less than 50 nm of the gap between the SWCNT and the ground substrate. Reduced order model method (ROM) is used to investigate the system under soft excitation and weak nonlinearities. The voltage-amplitude response and influences of parameters are reported for primary resonance (AC near half natural frequency).     

亚音速气流中复合材料悬臂板的非线性振动响应研究

随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和 Kutta--Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯$\cdot$卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.