On primary resonances of weakly nonlinear delay systems with cubic nonlinearities

We implement the method of multiple scales to investigate primary resonances of a weakly nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear delay terms with small coefficients (Hu et al. in Nonlinear Dyn. 15:311, 1998; Ji and Leung in Nonlinear Dyn. 253:985, 2002), in this effort, we propose a modified approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified approach lumps the delay state into unknown linear damping and stiffness terms that are functions of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the method of multiple scales to match that obtained directly by solving the forced linear problem. We examine the validity of the modified procedure by comparing its results to solutions obtained via a harmonic balance approach. Several examples are discussed demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses. Analytical results also reveal that the system can exhibit responses with different nonlinear characteristics near its multiple delay frequencies.     

Vibration control of piezoelectric beam-type plates with geometrically nonlinear deformation

This paper presents a wavelet-based approach of deformation identification and vibration control of beam-type plates with geometrically nonlinear deflection using piezoelectric sensors and actuators. The identification is performed by transferring the nonlinear equations of identifying deflection into a system of solvable nonlinear algebraic equations in terms of the measurable electric charges and currents on piezoelectric sensors. After that, a control law of negative feedback of the identified signals of deflection and velocity is employed, and the weighted residual method is chosen to determine control voltages applied on the piezoelectric actuators. Due to that the scaling function transform is like a low-pass filter which can automatically filter out high-order signals of vibration or disturbance from the measurement and the controller employed here, this control approach does not lead to the undesired phenomenon of control instability which is generated by the spilling over of high-order signals. Finally, some numerical simulations are carried out to show the efficiency of the proposed approach.     

Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities

Micro/nanomechanical resonators often exhibit nonlinear behaviors due to their small size and their ease to realize relatively large amplitude oscillation. In this work, we design a nonlinear micromechanical cantilever system with intentionally integrated geometric nonlinearity realized through a nanotube coupling. Multiple scales analysis was applied to study the nonlinear dynamics which was compared favorably with experimental results. The geometrically positioned nanotube introduced nonlinearity efficiently into the otherwise linear micromechanical cantilever oscillator, evident from the acquired responses showing the representative hysteresis loop of a nonlinear dynamic system. It was further shown that a small change in the geometry parameters of the system produced a complete transition of the nonlinear behavior from hardening to softening resonance.     

Analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method

In this paper a boundary element formulation for analysis of shear deformable plates with combined geometric and material nonlinearities by boundary element method is presented. The dual reciprocity method is used in dealing with the geometric nonlinearity and domain discretization is implemented in dealing with material nonlinearity. The material is assumed to undergo large deflection with small strains. The von Mises criteria is used to evaluate the plastic zone and an elastic perfectly plastic material behaviour is assumed. An initial stress formulation is used to formulate the boundary integral equations. A total incremental method is applied to solve the nonlinear boundary integral equations. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed method.     

薄板几何非线性中的精化元方法及膜闭锁问题

基于放松单元间协调条件的大变形变分原理和全局拉格朗日方法，推导了几何非线性精化三角形薄板单元。对几何刚度矩阵，通过引入特殊的单元位移函数，有效地消除了薄板弯曲问题中伴生的膜闭锁现象。数值结果表明该单元在几何非线性分析中既能消除膜闭锁又具有较高精度。     

非线性压电效应下压电弯曲执行器的动力分析

研究压电弯曲执行器在强电场作用下的非线性动力行为．考虑电致伸缩和电致弹性的非线性压电效应,导出了压电悬臂执行器变刚度的弯曲振动控制方程．利用非定常振动的渐近理论,讨论了弯曲压电执行器的动力特征．根据目前的非线性模型可以计算压电悬臂执行器的固有共振频率与电场的变化关系．结果表明压电执行器端头挠度谐振幅度随作用电场振幅的增大而增大,以及力学品质因数随电场振幅的增大而减少,并且与实验结果非常吻合．通过数值比较得到在电场频率随时间变化非常缓慢的情况下非定常振动问题可以近似地用定常振动来处理．     

细长结构几何非线性分析的子结构方法

将细长结构沿长度方向划分为多个子结构,并在每个子结构上建立一个随结构一起运动的连体基,则结构内任意点的位移可分解为连体基的转动和相对于连体基的小位移。利用细长结构这样的变形特征,本文详细讨论了连体基的转动,给出了与连体基选择方式相协调的节点位移及其虚变分表达式,并将子结构内部位移凝聚到了边界节点上。在此基础上,提出了一种细长结构几何非线性分析的子结构方法,可在不损失计算精度的前提下大幅度降低求解规模,从而提高了计算效率。数值算例验证了所提方法的有效性。     

Linear and nonlinear analysis of stiffened plates

An analysis is presented for the large deflection of clamped laterally loaded skew plates with stiffeners parallel to the skew directions. The governing nonlinear differential equations are derived taking into account the eccentricity of the stiffeners. A numerical procedure involving the use of integral equations of beams and the Newton-Raphson method is employed to get the solution. Numerical work has been done. The effect of variation of skew angle and size of stiffener on the behaviour of the stiffened skew plate has been studied.     

Energy harvesting from the secondary resonances of a nonlinear piezoelectric beam under hard harmonic excitation

Meccanica - This paper investigates the dynamical response of a nonlinear piezoelectric energy harvester under a hard harmonic excitation and assesses its output power. The system is composed of a...     

Vibration analysis of FG annular sector in moderately thick plates with two piezoelectric layers

Free vibration of functionally graded(FG) annular sector plates embedded with two piezoelectric layers is studied with a generalized differential quadrature(GDQ)method. Based on the first-order shear deformation(FSD) plate theory and Hamilton's principle with parameters satisfying Maxwell's electrostatics equation in the piezoelectric layers, governing equations of motion are developed. Both open and closed circuit(shortly connected) boundary conditions on the piezoelectric surfaces, which are respective conditions for sensors and actuators, are accounted for. It is observed that the open circuit condition gives higher natural frequencies than a shortly connected condition. For the simulation of the potential electric function in piezoelectric layers, a sinusoidal function in the transverse direction is considered. It is assumed that properties of the FG material(FGM) change continuously through the thickness according to a power distribution law.The fast rate convergence and accuracy of the GDQ method with a small number of grid points are demonstrated through some numerical examples. With various combinations of free, clamped, and simply supported boundary conditions, the effects of the thicknesses of piezoelectric layers and host plate, power law index of FGMs, and plate geometrical parameters(e.g., angle and radii of annular sector) on the in-plane and out-of-plane natural frequencies for different FG and piezoelectric materials are also studied. Results can be used to predict the behaviors of FG and piezoelectric materials in mechanical systems.     

A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.     

Incremental virtual work equation for geometric nonlinear analysis

In this paper, an incremental virtual work equation is derived. It is suitable for geometric nonlinear analysis in finite element method. The effect of truncation errors is considered in the incremental virtual work equation.     

Nonlinear combination resonances in cantilever composite plates

We experimentally investigated nonlinear combination resonances in two graphite-epoxy cantilever plates having the configurations (90/30/-30/-30/30/90)_s and (-75/75/75/-75/75/-75)_s. As a first step, we compared the natural frequencies and modes shapes obtained from the finite-element and experimental-modal analyses. The largest difference in the obtained frequencies for both plates was 6%. Then, we transversely excited the plates and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaIYaaabeaakiab% gUcaRiabeM8a3naaBaaaleaacaaI3aaabeaaaaa!45C9!\[\Omega \approx \omega _2 + \omega _7 \] in the quasi-isotropic plate and the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGa% aiikaiabeM8a3naaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3n% aaBaaaleaacaaI1aaabeaakiaacMcaaaa!4AAD!\[\Omega \approx (1/2)(\omega _2 + \omega _5 )\] and the internal combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaI4aaabeaakiab% gIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGaaiikaiabeM8a3n% aaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaI% XaGaaG4maaqabaGccaGGPaaaaa!4FDC!\[\Omega \approx \omega _8 \approx (1/2)(\omega _2 + \omega _{13} )\] in the ±75 plate, where the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeM8a3naaBaaaleaacaWGPbaabeaaaaa!3F16!\[\omega _i \] are the natural frequencies of the plate and is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.     

Dynamic stability analysis of finite element modeling of piezoelectric composite plates

The dynamic stability of negative-velocity feedback control of piezoelectric composite plates using a finite element model is investigated. Lyapunov’s energy functional based on the derived general governing equations of motion with active damping is used to carry out the stability analysis, where it is shown that the active damping matrix must be positive semi-definite to guarantee the dynamic stability. Through this formulation, it is found that imperfect collocation of piezoelectric sensor/actuator pairs is not sufficient for dynamic stability in general and that ignoring the in-plane displacements of the midplane of the composite plate with imperfectly collocated piezoelectric sensor/actuator pairs may cause significant numerical errors, leading to incorrect stability conclusions. This can be further confirmed by examining the complex eigenvalues of the transformed linear first-order state space equations of motion. To overcome the drawback of finding all the complex eigenvalues for large systems, a stable state feedback law that satisfies the second Lyapunov’s stability criteria strictly is proposed. Numerical results based on a cantilevered piezoelectric composite plate show that the feedback control system with an imperfectly collocated PZT sensor/actuator pair is unstable, but asymptotic stability can be achieved by either bonding the PZT sensor/actuator pair together or changing the ply stacking sequence of the composite substrate to be symmetric. The performance of the proposed stable controller is also demonstrated. The presented stability analysis is of practical importance for effective design of asymptotically stable control systems as well as for choosing an appropriate finite element model to accurately predict the dynamic response of smart piezoelectric composite plates.     

Analysis of an electromechanical energy harvester system with geometric and ferroresonant nonlinearities

Enhancing the performance of vibrating energy harvesting systems has been the backbone of several research contributions for the last few years, and it is considered in this paper. Specifically, an electromechanical energy harvester is analyzed, and the effects of geometric and ferroresonant nonlinearities on the electric power are discussed. The geometric nonlinearity includes the small- and high-order terms in Euler internal force while the ferroresonant nonlinearity is included by assuming different levels of saturation in the circuit. Our results reveal regions in the parameter space where nonlinear stiffness is better than linear stiffness and vice versa. Similarly, increasing the saturation parameter can be used to enhance the electric power.     

Coupled buckling and postbuckling analysis of active laminated piezoelectric composite plates

A theoretical framework for analyzing the pre- and postbuckling response of composite laminates and plates with piezoactuators and sensors is presented. The mechanics include nonlinear effects due to large rotations and stress stiffening, and are incorporated into a coupled mixed-field piezoelectric laminate theory. Using the previous mechanics, a nonlinear finite element method and an incremental-iterative solution are formulated for the analysis of nonlinear adaptive plate structures subject to in-plane electromechanical loading. A novel eight-node nonlinear plate finite element is also developed. Evaluation cases predict the buckling and postbuckling response of adaptive composite beams and plates with piezoelectric actuators and sensors. The case of piezoelectric buckling and postbuckling induced by the actuators is addressed and quantified. Finally, the possibility to actively mitigate the mechanical buckling and postbuckling response of adaptive piezocomposite plates is illustrated.