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．     

非线性压电效应下压电弯曲执行器的弯曲

研究压电弯曲执行器在强电场作用下的非线性弯曲行为。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自由端挠度或激励力和作用电场之间的非线性关系。结果表明,考虑非线性压电效应在很大的电场范围内都与实验结果吻合得很好,而线性压电效应只适合于低电场的情况。     

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.     

Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators

The active vibration control of a rectangular plate either partially or fully submerged in a fluid was investigated. Piezoelectric sensors and actuators were bonded to the plate, and the assumed mode method was used to derive a dynamic model for the submerged plate. The properties of the piezoelectric actuators and sensors, as well as their coupling to the structure, were used to derive the corresponding equations of their behaviour. The fluid effect was modelled according to the added virtual mass obtained by solving the Laplace equation. The natural vibration characteristics of the plate both in air and in water were obtained theoretically and were found to be consistent with the experimental results, and the changes in the natural frequencies resulting from submersion in fluid can be accurately predicted. A multi-input, multi-output positive position feedback controller was designed by taking the natural vibration characteristics into account and was then implemented by using a digital controller. The experimental results show that piezoelectric sensors and actuators along with the control algorithm can effectively suppress the vibration of a rectangular plate both in air and submerged in a fluid.     

Nonlinear integral resonant controller for vibration reduction in nonlinear systems

A new nonlinear integral resonant controller (NIRC) is introduced in this paper to suppress vibration in nonlinear oscillatory smart structures. The NIRC consists of a first-order resonant integrator that provides additional damping in a closed-loop system response to reduce high-amplitude nonlinear vibration around the fundamental reso-nance frequency. The method of multiple scales is used to obtain an approximate solution for the closed-loop system. Then closed-loop system stability is investigated using the resulting modulation equation. Finally, the effects of different control system parameters are illustrated and an approximate solution response is verified via numerical simulation results. The advantages and disadvantages of the proposed controller are presented and extensively discussed in the results. The controlled system via the NIRC shows no high-amplitude peaks in the neighboring frequencies of the resonant mode, unlike conventional second-order compensation methods. This makes the NIRC controlled system robust to excitation frequency variations.     

A comparative study of piezoelectric unimorph and multilayer actuators as stiffness sensors via contact resonance

Piezoelectric bar-shaped resonators were proposed to act as hardness sensors in the 1960 s and stiffness sensors in the 1990 s based on the contact impedance method.In this work, we point out that both multilayer and unimorph(or bimorph) piezoelectric actuators could act as stiffness/modulus sensors based on the principle of mechanical contact resonance. First, the practical design and the performance of a piezoelectric unimorph actuator–based stiffness sensor were presented. Then the working principle of piezoelectric multilayer actuator–based stiffness sensors was given and verified by numerical investigation. It was found that for these two types of resonance-based sensors, the shift of the resonance frequency due to contact is always positive, which is different from that of the contact impedance method. Further comparative sensitivity study indicated that the unimorph actuator–based stiffness sensor is very suitable for measurement on soft materials, whereas the multilayer actuator–based sensor is more suitable for hard materials.     

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method

In this study, the out-of-plane free vibration analysis of a double tapered Euler–Bernoulli beam, mounted on the periphery of a rotating rigid hub is performed. An efficient and easy mathematical technique called the Differential Transform Method (DTM) is used to solve the governing differential equation of motion. Parameters for the hub radius, rotational speed and taper ratios are incorporated into the equation of motion in order to investigate their effects on the natural frequencies. Calculated results are tabulated in several tables and figures and are compared with the results of the studies in open literature where a very good agreement is observed.     

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.     

Nonlinear bending analysis of ring-stiffened circular and annular general angle-ply laminated plates with various boundary conditions

In this study, elastic large deflection analysis of axisymmetric ring-stiffened circular and annular general angle-ply laminated plates subjected to transverse uniform load is studied. Based on first order shear deformation theory (FSDT) and large deflection von-Karman relations, the governing equations are derived. The dynamic relaxation (DR) method in conjunction with the central finite difference discretization technique is used to solve the nonlinear equilibrium equations. A detailed parametric study is carried out to investigate the influences of plate thicknesses, stiffener width, stiffener depth, fiber orientation, stacking sequence and different types of boundary conditions. Also, some linear and nonlinear analysis is provided to consider the effect of nonlinearity on the results.     

考虑弯扭形变的薄壁箱形梁的非线性后屈曲

在大位移和扭转的前提下,通过一中等弯曲扭转的位移场描述了薄壁箱形梁在偏心载荷作用下的静稳定性问题.该非线性公式可用于分析简支薄壁箱形梁在不同载荷作用下的屈曲和后屈曲行为.采用伽辽金方法将非线性微分系统离散,并通过牛顿-拉普森增量迭代法求解得代数方程组.数值计算结果表明,当前屈曲位移不可忽略时,经典的横向屈曲预测是保守的...     

Exact closed-form solutions for the static analysis of multi-cracked gradient-elastic beams in bending

Cracks and other forms of concentrated damage can significantly affect the performance of slender beams under static and dynamic loads. The computational model for such defects often consists of a localised reduction in the flexural stiffness, which is macroscopically equivalent to a beam where the undamaged parts are hinged at the position of the crack, with a rotational spring taking into account the residual stiffness (“discrete spring” model). It has been recently demonstrated that this model is equivalent to an inhomogeneous Euler–Bernoulli beam in which a Dirac’s delta is added to the bending flexibility at the position of each damage (“flexibility crack” model). Since these models concentrate the increased curvature at a single abscissa, a jump discontinuity appears in the field of rotations. This study presents an improved representation of cracked slender beams, based on a general class of gradient elasticity with both stress and strain gradient, which allows smoothing the singularities in the flexibility crack model. Exact closed-form solutions are derived for the static response of slender gradient-elastic beams in flexure with multiple cracks, and the numerical examples demonstrate the effects of the nonlocal mechanical parameters (i.e. length scales of the gradient elasticity) in this context.     

On the plastic buckling of curved carbon nanotubes

This research, for the first time, predicts theoretically static stability response of a curved carbon nanotube(CCNT) under an elastoplastic behavior with several boundary conditions. The CCNT is exposed to axial compressive loads. The equilibrium equations are extracted regarding the Euler–Bernoulli displacement field by means of the principle of minimizing total potential energy.The elastoplastic stress-strain is concerned with Ramberg–Osgood law on the basis of deformation and flow theories of plasticity. To seize the nano-mechanical behavior of the CCNT, the nonlocal strain gradient elasticity theory is taken into account. The obtained differential equations are solved using the Rayleigh–Ritz method based on a new admissible shape function which is able to analyze stability problems. To authorize the solution, some comparisons are illustrated which show a very good agreement with the published works. Conclusively, the best findings confirm that a plastic analysis is crucial in predicting the mechanical strength of CCNTs.     

A modeling and vibration analysis of a piezoelectric micro-pump diaphragm

The vibration analysis of a micro-pump diaphragm is presented. A piezoelectric micro-pump is studied. For this purpose, a dynamic model of the micro-pump is derived. The micro-pump diaphragm is modeled as circular double membranes, a piezoelectric one as actuator and a silicon one for representing the membrane for pumping action. The damping effect of the fluid is introduced into the equations. Vibration analysis is established by explicitly solving the dynamic model. The natural frequencies and mode shapes are calculated. The orthogonality conditions of the system are discussed. To verify the results, the finite-element micro-pump model is developed in ANSYS software package. The results show that the two methods are well comparable.     

An efficient solution methodology to study the response of a beam on viscoelastic and nonlinear unilateral foundation: Static response

Many engineering materials and foundations such as soils demonstrate nonlinear and viscoelastic behaviour. Yet, it is challenging to develop static and dynamic models of systems that include these materials and are able to predict the behaviour over a wide range of loading conditions. This research is focused on a specific example: a pinned–pinned beam interacting with polyurethane foam foundation. Two cases, when the foundation can react in tension and compression as well as only in compression, are considered. The model developed here is capable of predicting the response to static as well as dynamic forces, whether concentrated or distributed. Galerkin’s method is used to derive modal amplitude equations. In the tensionless foundation case, the contact region changes with beam motion and the estimation of the co-ordinates of the lift-off points is embedded into the solution procedure. An efficient solution technique is proposed that is capable of handling cases where there are multiple contact and non-contact regions. Depending on the loading profiles a high number of modes may need to be included in the solution and to speed up computation time, a convolution method is used to evaluate the integral terms in the model. The adaptability of the solution scheme to complicated loading patterns is demonstrated via examples. The solution approach proposed is applicable to dynamic loadings as well and in these cases the automated treatment of complicated response patterns makes the convolution approach particularly attractive. The influence of various parameters on the static response is discussed.     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene

In the present article, the governing nonlinear nonlocal elastic equations are obtained for a monolayer graphene with an initial curvature and the related softening and hardening bending stiffness is analytically calculated. The effects of large deformation, initial curvature, discreteness and direction of chiral vector on the bending stiffness of the monolayer graphene are discussed in detail. A behavior more complex than previously reported in the literature emerges. It is found that the bending stiffness of graphene strongly depends on the initial configuration, showing not obvious maxima and minima, and suggesting the possibility of a smart tuning.     

Pull-in voltage analysis of electrostatically actuated MEMS with piezoelectric layers: A size-dependent model

A size-dependent model for electrostatically actuated microbeam-based MEMS (micro-electro-mechanical systems) with piezoelectric layers attached is developed based on a modified couple stress theory. By using Hamilton's principle, the nonlinear differential governing equation and boundary conditions of the MEM structure are derived. In the newly developed model, the residual stresses, fringing-field and axial stress effects are considered for the fixed–fixed microbeam with piezoelectric layers. The results of the present model are compared with those from the classical model. The results show the size effect becomes prominent if the beam dimension is comparable to the material length scale parameter (MLSP). The effects of MLSP, the residual stresses and axial stress on the pull-in voltage are also studied. The study may be helpful to characterize the mechanical and electrostatic properties of small size MEMS, or guide the design of microbeam-based devices for a wide range of potential applications.     

Redundant actuators to achieve minimal vibration trajectory tracking of flexible multibodies: Theory and application

We address the problem of inverse dynamics for flexible multibodies, which arises, in trajectory tracking control of flexible multibodies such as space manipulators and articulated flexible structures. Previous research has resolved this trajectory tracking problem by computing the system inputs for feedforward control of actuators at the joints. Recently, the use of distributed actuators like electro-strictive actuators in flexible structures has introduced a new dimension to this trajectory tracking problem. In this paper we optimally utilize such actuators to aid joint actuators for tracking control, and introduce a new inverse dynamics scheme for simultaneously (1) tracking a prescribed trajectory and (2) minimizing ensuing elastic deflections. We apply this scheme for trajectory tracking of a two-link two-joint planar manipulator with joint motors and distributed electro-strictive actuators. Experimental results are presented to contrast our new scheme with other existing methods.     

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.