Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

A hierarchical finite element for geometrically non‐linear vibration of doubly curved,moderately thick isotropic shallow shells

A p‐version, hierarchical finite element for doubly curved, moderately thick, isotropic shallow shells is derived and geometrically non‐linear free vibrations of panels with rectangular planform are investigated. The geometrical non‐linearity is due to large displacements, and the effects of the rotatory inertia and transverse shear are considered. The time domain equations of motion are obtained by applying the principle of virtual work and the d'Alembert's principle. These equations are mapped to the frequency domain by the harmonic balance method, and are finally solved by a predictor–corrector method. The convergence properties of the element proposed and the influence of several parameters on the dynamic response are studied. These parameters are the shell's thickness, the width‐to‐length ratio, the curvature‐to‐width ratio and the ratio between curvature radii. The first and higher order modes are analysed. Some results are compared with results published or calculated using a commercial finite element package. It is demonstrated that with the proposed element low‐dimensional, accurate models are obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

Aeroelastic forces and dynamic response of long‐span bridges

In this paper a time domain approach for predicting the non‐linear dynamic response of long‐span bridges is presented. In particular the method that leads to the formulation of aeroelastic and buffeting forces in the time domain is illustrated in detail, where a recursive algorithm for the memory term's integration is properly developed. Moreover in such an approach the forces' expressions, usually formulated according to quasi‐static theory, have been substituted by expressions including the frequency‐dependent characteristics. Such expressions of aeroelastic and buffeting forces are made explicit in the time domain by means of the convolution integral that involves the impulse functions and the structural motion or the fluctuating velocities. A finite element model (FEM) has been developed within the framework of geometrically non linear analysis, by using 3‐d degenerated finite element. The proposed procedure can be used to analyze both the flutter instability phenomenon and buffeting response. Moreover, working in the geometrically non‐linearity range, it verifies the possibility of strongly flexible structures of actively resisting the wind loading. Copyright © 2004 John Wiley & Sons, Ltd.     

Wave propagation analysis in inhomogeneous piezo‐composite layer by the thin‐layer method

The thin‐layer method (TLM) is used to study the propagation of waves in inhomogeneous piezo‐composite layered media caused by mechanical loading and electrical excitation. The element is formulated in the time‐wavenumber domain, which drastically reduces the cost of computation compared to the finite element (FE) method. Fourier series are used for the spatial representation of the unknown variables. The material properties are allowed to vary in the depthwise direction only. Both linear and exponential variations of elastic and electrical properties are considered. Several numerical examples are presented, which bring out the characteristics of wave propagation in anisotropic and inhomogeneous layered media. The element is useful for modelling ultrasonic transducers (UT) and one such example is given to show the effect of electric actuation in a composite material and the difference in the responses elicited for various ply‐angles. Further, an ultrasonic transducer composed of functionally graded piezoelectric materials (FGPM) is modelled and the effect of gradation on mechanical response is demonstrated. The effect of anisotropy and inhomogeneity is shown in the normal modes for both displacement and electric potential. The element is further utilized to estimate the piezoelectric properties from the measured response using non‐linear optimization, a strategy that is referred to as the pulse propagation technique (PPT). Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical analysis of an elasto‐piezoelectric problem with damage

In this paper, we analyze an algorithm for the quasistatic evolution of the mechanical state of an elasto‐piezoelectric body with damage. Both damage, caused by the development and the growth of internal microcracks, and piezoelectric effects, are included in the model. The mechanical problem is expressed as an elliptic system for the displacement field coupled with a non‐linear parabolic partial differential equation for the damage field and a linear partial differential equation for the electric potential. The variational formulation leads to a coupled system composed of two linear variational equations for the displacement field and the electric potential, and a non‐linear parabolic variational equation for the damage field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some numerical simulations are performed, in one, two and three dimensions, to demonstrate the accuracy of the scheme and the behaviour of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Time and frequency domain identification and analysis of a permanent magnet synchronous servo motor

Abstract

This study employed the approach of non‐linear autoregressive moving average model with exogenous inputs (NARMAX) to analyze the dynamics of a Permanent Magnet Synchronous Motor (PMSM). The non‐linearity in PMSM including cogging force, reluctance force and force ripple is difficult to estimate. By using the NARMAX approach, thrust‐speed relationship and thrust‐position relationship could be analyzed by identifying both time and frequency domain models of the system. The frequency domain analysis is studied by mapping the discrete‐time NARMAX models into generalized frequency response functions (GFRFs) to reveal the non‐linear coupling between the various input spectral components and the energy transfer mechanisms in the system. From the results, the interpretation of the higher‐order GFRFs has been comprehensively studied and non‐linear effects have been related to the physical models of the systems.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

加层压电半空间界面裂纹力电联合冲击动态响应

基于线性压电弹性理论,研究电绝缘和电接触两种电边界条件下,任意多个任意排列的压电加层半空间界面裂纹在力电联合冲击作用下的动态响问题。通过引进Laplace变换和 Fourier 变换及位错密度函数,将问题首先转化为Cauchy 奇异积分方程,进而转化为代数方程进行数值求解。数值算例表明,电载荷、加层厚度及材料参数对动能量释放率具有重要的但不同的影响。研究结果对结构设计及结构失效的预防具有理论和应用价值。     

Efficient non‐linear reanalysis of skeletal structures using combined approximations

Non‐linear reanalysis of large‐scale structures usually involves much computational effort, because the set of non‐linear equations must be solved repeatedly during the solution process. Various approximations that are often used for linear reanalysis are not sufficiently accurate for non‐linear problems. In this study, solution procedures based on the combined approximations approach are developed and compared in terms of efficiency and accuracy. Various path‐independent non‐linear analysis and reanalysis problems are considered, including material non‐linearity, geometric non‐linearity and buckling analysis. Numerical examples demonstrate the effectiveness of the procedures presented. It is shown that in various cases accurate results can be achieved efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized coupled thermoelasticity of functionally graded cylindrical shells

In this paper, the response of a circular cylindrical thin shell made of the functionally graded material based on the generalized theory of thermoelasticity is obtained. The governing equations of the generalized theory of thermoelasticity and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. Thermoelasticity with second sound effect in cylindrical shells based on the Lord–Shulman model is compared with the Green–Lindsay model. A second‐order shear deformation shell theory, that accounts for the transverse shear strains and rotations, is considered. Including the thermo‐mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain is used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The effects of temperature field for linear and non‐linear distributions across the shell thickness are examined. The results are validated with the known data in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

In this paper, we present a non‐linear finite element formulation for piezoelectric shell structures. Based on a mixed multi‐field variational formulation, an electro‐mechanical coupled shell element is developed considering geometrically and materially non‐linear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference in the electric potential in the thickness direction of the shell. Incorporating non‐linear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner–Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three‐dimensional transversal isotropic material law, thus the kinematic in the thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.     

On the design of energy–momentum integration schemes for arbitrary continuum formulations. Applications to classical and chaotic motion of shells

The construction of energy–momentum methods depends heavily on three kinds of non‐linearities: (1) the geometric (non‐linearity of the strain–displacement relation), (2) the material (non‐linearity of the elastic constitutive law), and (3) the one exhibited in displacement‐dependent loading. In previous works, the authors have developed a general method which is valid for any kind of geometric non‐linearity. In this paper, we extend the method and combine it with a treatment of material non‐linearity as well as that exhibited in force terms. In addition, the dynamical formulation is presented in a general finite element framework where enhanced strains are incorporated as well. The non‐linearity of the constitutive law necessitates a new treatment of the enhanced strains in order to retain the energy conservation property. Use is made of the logarithmic strain tensor which allows for a highly non‐linear material law, while preserving the advantage of considering non‐linear vibrations of classical metallic structures. Various examples and applications to classical and non‐classical vibrations and non‐linear motion of shells are presented, including (1) chaotic motion of arches, cylinders and caps using a linear constitutive law and (2) large overall motion and non‐linear vibration of shells using non‐linear constitutive law. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large‐amplitude multi‐mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non‐linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees‐of‐freedom (DOF) are transformed into modal co‐ordinates, and the non‐linear stiffness matrices are transformed into non‐linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap‐through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap‐through between the two BEPs, and (iii) non‐linear large‐amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.     

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Computer‐aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce ‘mega‐sized’, arbitrary‐shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis (FEA) and reduced engineering manpower requirement for final quality assurance. A method, denoted as c‐type, has been proposed by constructively defining a finite element space whereby the above hurdles may be overcome with a minimal number of hyper‐sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B‐splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter‐element continuity on demand unifies modelling and finite element method. The c‐type method may introduce additional control, namely, an inter‐element continuity condition to the existing h‐type and p‐type methods. Adaptation of the c‐type method to existing commercial and general‐purpose computer programs based on a conventional displacement‐based finite element method is straightforward. The c‐type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non‐linear formulation for the two‐dimensional curved beams/arches is presented. Several beam problems ranging from truly three‐dimensional tortuous linear curved beams to geometrically extremely non‐linear two‐dimensional arches are solved to establish numerical efficiency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non‐linearity and to treat general material non‐linearity. Copyright © 2003 John Wiley & Sons, Ltd.     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

Electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers

A piezoelectric transformer is a power transfer device that converts its input and output voltage as well as current by effectively using electrical and mechanical coupling effects of piezoelectric materials. Equivalent-circuit models, which are traditionally used to analyze piezoelectric transformers, merge each mechanical resonance effect into a series of ordinary differential equations. Because of using ordinary differential equations, equivalent circuit models are insufficient to reflect the mechanical behavior of piezoelectric plates. Electromechanically, fully coupled governing equations of Rosen-type piezoelectric transformers, which are partial differential equations in nature, can be derived to address the deficiencies of the equivalent circuit models. It can be shown that the modal actuator concept can be adopted to optimize the electromechanical coupling effect of the driving section once the added spatial domain design parameters are taken into account, which are three-dimensional spatial dependencies of electromechanical properties. The maximum power transfer condition for a Rosen-type piezoelectric transformer is detailed. Experimental results, which lead us to a series of new design rules, also are presented to prove the validity and effectiveness of the theoretical predictions.     

Decoupling joint and elastic accelerations in deformable mechanical systems using non‐linear recursive formulations

The aim of this paper is to develop non‐linear recursive formulations for decoupling joint and elastic accelerations, while maintaining the non‐linear inertia coupling between rigid body motion and elastic deformation in deformable mechanical systems. The inertia projection schemes used in most existing recursive formulations for the dynamic analysis of deformable mechanisms lead to dense coefficient matrices in the equations of motion. Consequently, there are strong dynamic couplings between the joint and elastic coordinates. When the number of elastic degrees of freedom increases, the size of the coefficient matrix in the equations of motion becomes large. Consequently, the use of these recursive formulations for solving the joint and elastic accelerations becomes less efficient. In this paper, the non‐linear recursive formulations have been used to decouple the elastic and joint accelerations in deformable mechanical systems. The relationships between the absolute, elastic and joint variables and generalized Newton–Euler equations are used to develop systems of loosely coupled equations that have sparse matrix structure. By using the inertia matrix structure of deformable mechanical systems and the fact that joint reaction forces associated with elastic coordinates do represent independent variables, a reduced system of equations whose dimension is dependent of the number of elastic degrees of freedom is obtained. This system can be solved for the joint accelerations as well as for the joint reaction forces. The use of the approaches developed in this investigation is illustrated using deformable open‐loop serial robot and closed‐loop four‐bar mechanical systems. Copyright © 2007 John Wiley & Sons, Ltd.