Natural Mode Method Research Articles

An accurate deicing method by utilizing the piezoelectric materials based on the active mode control theory

Icing has a great influence on the dynamic characteristics of structures. One of an effective deicing method is to utilize the shear stresses caused by the transverse vibration of the structure, and this is usually realized by exciting the proper natural vibration mode according to the icing position. However, the natural mode cannot always be consistent with the icing position, and on the other hand, although there is vibration at the icing position, it will also cause vibration at other positions of the structure. To solve these problems, an accurate and efficient deicing method is proposed in this study. In this method, piezoelectric materials are used. They are bonded on the surface of the structure to act as the actuators. According to the icing positions, an appropriate mode shape is assumed, the peak value of which is consistent with the position of the ice. To achieve this goal, it is essential to obtain the input voltages of the piezoelectric actuators. The genetic algorithm is introduced, through which the optimal control voltage of each actuator can be found. As a result, the actual vibration of the structure will tend to the designed deicing vibration mode. Experimental and simulation results show that the present active mode control method possesses better deicing effect compared with the former natural vibration mode method. In addition, there is almost no vibration where there is no ice, which ensures the stability of the structure.

Scattering acoustic from one-layered cylindrical shell: Estimation of the radius ratio using reduced cutoff frequencies of circumferential waves

The obtained results from previous studies done in the acoustic scattering of a plane wave by an elastic cylindrical shell, show that the acoustic resonances of a cylindrical shell are related to its physical and geometrical properties. In order to estimate the radius ratio of an airfield immersed cylindrical shell, using reduced cutoff frequencies of circumferential waves that propagate around it. In this work, an approach based on the artificial neural networks was proposed. This approach uses the reduced cutoff frequencies of circumferential waves, extracted using modal isolation plan representation us the analyzing data, in the object to obtain the value of the radius ratio corresponding to the studied cylindrical shell. The proposed approach allows us to estimate accurately the values of the radius ratio of a copper cylindrical shell, and can help us to resolve other problems related to scattering acoustic. Furthermore, it can be used to estimate various parameters of a cylindrical shell starting from the characteristics of which it is disposed. The proposed approach used in this study does not present any approximation as in the case of the natural mode method which assimilate the cylindrical shell to the plate with the same thickness.

Some Explicit Beam, Plate and Shell Finite Elements

The present article presents some new beam, plate and shell finite elements based on explicit theory. The new elements are considered further developments of the Natural-Mode Method. We start from basic concepts of strain energy and based on concepts of modes or interpolations we provide a five matrix transformation on which the elemental stiffness matrix is based. Subquently, beam, plate and shell finite elements are described. New features include the introduction of sinus shape functions and surfaces. Some of the elements can be considered expressions of Fourier series. Explicit elemental matrices are developed and all integrations are in close form. The theory is suited for efficient and fast computer implementation. Numerical examples are provided

A 3D fibre beam-column element with shear modelling for the inelastic analysis of steel structures

A force-based fiber beam-column element is proposed for the capacity assessment of frame structures under high shear. The proposed element is suitable for the performance assessment of large scale steel structures, which are not flexure-dominated. The element formulation follows the assumptions of the Timoshenko beam theory, while its kinematics are obtained through the natural-mode method. The element state-determination phase, instead of uniaxial material laws, typically associated with fiber elements, is based on a three-dimensional law taking into consideration the interaction between axial, bending, shear and torsion. Numerical examples are presented confirming the accuracy and the computational efficiency of the proposed formulation under monotonic, cyclic and dynamic/seismic loading. Compared to experimental results and the results of detailed finite element models, excellent agreement is achieved.

Nonlinear dynamic analysis of shells with the triangular element TRIC

TRIC is a facet triangular shell element, which is based on the natural mode method. It has been shown that the TRIC shell element satisfies the individual element test and in the framework of the nonconsistent formulation the convergence requirements are fulfilled, while it has been proved to be very efficient in linear and nonlinear static problems. Moreover, another major advantage in the formulation of this element is the incorporation of the transverse shear deformations in a way that defies the shear-locking phenomenon. In this work the derivation of the consistent and lumped mass matrices of the TRIC element is presented so that it can be used in linear and nonlinear dynamic problems. Both translational and rotational inertia are included in the consistent mass matrix, which is conceived, using kinematical and geometrical arguments consistent with the assumed natural rigid body and straining modes of the element. All the kinematical and geometrical arguments that are invoked for the derivation of the consistent mass matrix are briefly presented. Moreover, two formulations of the lumped mass matrix of TRIC are derived. The first formulation is based entirely on geometrical considerations whereas the second is based on lumping the consistent mass matrix of TRIC. Finally, the element’s robustness and accuracy will be shown by applying it to properly selected benchmark examples of nonlinear shell dynamics, while its computational efficiency will be demonstrated by comparing the CPU performance of the element with the other available shell elements.

The TRIC shell element: theoretical and numerical investigation

The TRIC facet triangular shell element, which is based on the natural mode method, is seen under the light of the non-consistent formulation proposed by Bergan and co-workers. Under this formulation, the convergence requirements are fulfilled even with relaxed conditions on the conformity demands of the displacement shape functions. The intrinsic connection between the non-consistent formulation and the natural mode method is demonstrated, establishing thus a rigorous theoretical foundation for the TRIC element. Under the perspective of the non-consistent formulation, TRIC's convergence characteristics are established by satisfying apriori the patch test due to its inherent properties and thus guaranteeing convergence to the exact solution. Furthermore, the element's accuracy, robustness and efficiency are tested in a number of judiciously selected numerical examples on benchmark plate and shell problems, while a CPU time comparison with a pure displacement-based isoparametric shell element demonstrates its computational merits.

Combined steady-state nonlinear heat transfer/thermal postbuckling computations in unstiffened and stiffened laminated composite plates and shells

A computational scheme for the solution of the decoupled (i.e. applied stepwise sequentially) nonlinear steady-state heat transfer and geometrically nonlinear thermoelastic problem is presented for unstiffened and stiffened multilayered composite plates and shells. More specifically, a two-step formulation is conceived; first, the nonuniform temperature field resulting from applied heat fluxes is estimated by considering the three modes of heat transfer, namely nonlinear conduction, convection and radiation; second, the resulting temperatures are used as input to a stress analysis code which performs geometrically nonlinear analysis of composite panels with emphasis on thermal postbuckling computations. Two triangular elements are used for the computational experiments. For the solution of the heat transfer problem a 3-node triangular shell element is adopted which estimates the temperatures based on a first-order thermal lamination theory by employing primarily Cartesian notation. The element uses exact integrations for all nonlinear conduction, convection and radiation matrices [1,2] and accomplishes this by using extensive symbolic algebra techniques. The nonlinear stress analysis problem is solved using a shallow shell multilayered triangular element of varying and adaptable curvature which can accomodate the dependence of the material properties on temperature and also utilizes only exact integrations [5] made possible by employing once again symbolic computation; the latter element is developed using the principles of the natural mode method. The algorithms used for the solution of the two-stage problem are discussed. Numerical examples are presented which show the efficiency of the formulation and the interest of the thermophysical problem in hand.

Natural Mode Method: A Practicable and Novel Approach to the Global Analysis of Laminated Composite Plates and Shells

An overview of the natural mode method is presented, as applied in the linear and geometrically nonlinear analysis of multilayered composite plates and shells of arbitrary geometry. The method is particularly suitable for the finite element analysis of thin and moderately thick composite shell structures using simple 3-node triangular plate and shallow shell elements which are based on the definition of total strain, as well as physical decomposition principles, and involve only exact integrations. The triangular elements comprise 12 degrees of freedom on the natural coordinate. Numerical examples have shown that they address a well-balanced trade-off between accuracy and economy; as such they are intended for the analysis of large and complex composite shell structures and for use in interdisciplinary topics such as efficient sensitivity analysis and design optimization where complexity and economy must be considered in the computations. A brief history of the evolution of the natural mode technique is also given. Areas of application include static, vibration, buckling, postbuckling, and nonlinear dynamic and chaotic response topics under both load and temperature. This review article includes 36 references.

Postbuckling of composite laminates under compressive load and temperature

Here we study the thermo-mechanical postbuckling response of fibrous composite laminates. This study is based on the Natural Mode Method and invokes rigid-body and straining modes of deformation that are assigned to the edges of flat and shallow shell laminated triangular finite elements. Some new ideas on the subject are introduced and implemented in the conceived nonlinear algorithm, most notably the incorporation of temperature-dependent material properties (cubic variation), the decomposition of the total strain energy into its invariant natural components, assessing also their effect on failure initiation, the combined effect of load and temperature and the influence of initial imperfections on the overall behaviour, and finally comments on some of the intricate differences revealed for some problems by using the flat and shallow shell triangular elements. Comparisons with reported analytical and experimental results is attempted where available. Overall, the computational experiments substantiate the developed methodology and show its potential to treat larger and more complex panels and also contribute to their design.

A practicable and locking-free laminated shallow shell triangular element of varying and adaptable curvature

The concerted effort aimed at the conception and evolution of simple triangular finite elements for the analysis of laminated composite structures continues in the present discourse with the presentation of a 3-node (18 degrees of freedom) multilayered anisotropic shallow shell triangular element. Essentially the triangular edges are represented by cubic polynomials thus allowing for linear curvatures which during the course of a geometrically nonlinear deformation are automatically modified (by the natural modes) so that the element adapts to the new shell geometry. The formulation is based on kinematical and geometrical arguments in combination with subtle physical lumping principles and basic assumptions of shallow shell theory - all elements of our ARTE shallow shell formulation which is specifically oriented towards finite element analysis. The Natural Mode Method provides the element's kinematical field through rigid-body and straining modes of deformation. The straining modes are assigned to the triangular edges and implicitly provide for the complete kinematical field. The 3-node composite triangular element combines accuracy and economy (it only necessitates the computation of a 12 × 12 natural stiffness matrix) and all numerical experiments show that is free from the usual deficiencies of isoparametric displacement shell elements (i.e. locking, spurious modes, excessive stiffness and reduced quadrature etc.). Throughout the formulation emphasis is placed on issues of economy, efficiency and practicality. Numerical examples for linear and nonlinear deformation of isotropic and composite shells demonstrate the accuracy of the theory and substantiate the element's physical, geometrical and mathematical bases. A full laminated cylinder comprising 1504 degrees of freedom is studied to show the potential of the present element in the analysis of real-type practical structures. The major advantage of the developed shallow triangular element is expected in the nonlinear analysis of large and complex composite shells.

An efficient and locking-free flat anistropic plate and shell triangular element

The present study discusses, in a unified manner, the application of generalised decomposition principles and physical lumping techniques in finite element analysis of isotropic and laminated plates and shells. Using physical ideas and basic relations of mechanics, as well as geometrical arguments, it attempts to reduce the apparent complexity of the kinematical behaviour of such structures by means of assigning rigid-body and straining modes of deformation to an assembly of elements which comprise a compatible kinematical idealization. Subsequently, it applies these assertions to study a flat three-node 18 degrees-of-freedom triangular shell element applicable to the analysis of isotropic and fibre-reinforced laminated composite structures. Following the presentation of the general thermoelastic lamination theory, the Natural Mode Method is employed to form the elemental natural and Cartesian stiffness matrices. Although the elemental Cartesian stiffness matrix is of the order 18 × 18 and the initial Cartesian thermal load vector of the order 18 × 1, the triangular element essentially incorporates 12 generalized freedoms (the natural straining modes), and therefore neccesitates the computation of a 12 × 12 (not fully populated) natural stiffness matrix, and a 12 × 1 thermal (initial) vector, which to our knowledge make the present element the most inexpensive flat shell element available. The features of economy, simplicity and accuracy, facilitate the application of the present element in the non-linear range, in CPU intensive applications such as optimisation and failure — in general in the analysis of large and complex shell structures. The triangular finite element is supported by significant postprocessing features including the computation of all through-the-thickness stresses being equally important for structures composed from composite materials. Numerical examples are presented for both isotropic and laminated composite structures, and comparisons with analytic solutions are attempted where available. All computational experiments illustrate the accuracy of the conceived simple triangular element and substantiate its physical and theoretical bases.

High-temperature bending, buckling, and postbuckling of laminated composite plates using the natural mode method

In the present study, the behaviour of laminated composite plates under thermally induced loads is examined. A natural thermoelastic theory is developed, based on a linear through-the-thickness temperature variation. The material properties are assumed independent of temperature, but this assumption in no way restricts the generality of the formulation and the developed computer program. The theory is implemented on a model three-node triangular facet finite element which accounts for transverse shear deformation. The underlying principles of the developed methodology lie in the Natural Mode method which is a physically inspired and mathematically consistent method which was conceived with the intention of analyzing large and complex structures. The triangular element neccesitates the computation of a 12 × 12 natural stiffness matrix, and a 12 × 1 thermal (initial) load vector, which makes it probably one of the most inexpensive shell elements available. The effects of large displacements are included in our theory through the geometrical stiffness. In this regard, an Eulerian scheme conceived for the solution of geometrically nonlinear thermoelastic deformation is discussed. The methodology is validated with numerical examples which show the response of multilayered composite plates on thermally or thermomechanically induced bending, buckling, and postbuckling. All composite plates examined have shown remarkable resistance against high-temperature.

Linear and geometrically nonlinear bending of isotropic and multilayered composite plates by the natural mode method

In the present study we apply the Natural Mode Method and formulate an appropriate linear and geometrically nonlinear theory which is implemented on a model three-node multilayered triangular facet element which incorporates the effect of transverse shear deformation and is intended for the nonlinear analysis of large and complex isotropic and composite structures. In this context, we discuss the derivation of the elastic and geometrical elemental stiffnesses which are based on appropriate kinematic as well as physical idealizations, which constitute basic elements of the method of Intelligent Physical Lumping. Subsequently, we present the ARIBAN scheme (accumulation of the rigid-body and natural modes) for geometrically nonlinear analysis. It is emphasized that both the triangular facet element and the nonlinear algorithm are developed for the analysis of large and complex structures for which higher order formal finite elements cannot provide answers inexpensivelly. In addition, the structure of the algorithm and its matrix language are ideally suited for modern parallel computers. We conclude our study with numerical examples for isotropic and composite plates. All the numerical experiments show the accuracy and potential of the developed methodology for extensive structural engineering computations.

Nonlinear free vibrations of composite plates

The evolution of the Natural Mode Method (NMM) for finite element analysis of complex composite structures continues in the present study by applying its principles and formulating the kinematically consistent matrix of a model three-node multilayered triangular element. Both translational and rotational inertia are included in the mass matrix which is conceived using kinematical and geometrical arguments consistent with the assumed natural rigid-body and straining modes of the element. Linear eigenfrequencies are validated with experimental and finite element solutions. Subsequently the large-amplitude nonlinear undamped free vibration of composite plates is investigated. A computational scheme is conceived, whereby the structure is initially displaced by conducting a full geometrically nonlinear static analysis, and subsequently set to transient nonlinear free oscillations by removing the static loading. In this regard, we discuss the two structured Eulerian (convective) computational schemes employed namely, the ARIBAN scheme (accumulation of rigid-body and natural modes) for static nonlinear analysis, and the cubic Hermitian scheme (CUHERM) for large-displacement transient deformation. Numerical examples demonstrate the efficiency of the formulation and the potential of the Natural Mode Method to deal vigorously with intricate nonlinear time-dependent phenomena, as well as its potential to provide answers for larger and more complex structures. The study also indicates the efficiency and tailoring flexibility offered by advanced composite structural systems.

Transient response of a thin wire to a moving charged particle

The transient current on a thin wire induced by a charged particle moving parallel to the wire axis is studied. An explicit analytical solution for the induced current is obtained by utilizing the natural-mode method in conjunction with the method of asymptotic expansion. It is found that the solution can be split into two parts: one part consists of damped sinusoidal oscillations and the other exhibits quasistatic-like behavior. Numerical results are presented and compared with some selected results obtained directly by numerically solving a spacetime domain integral equation for the induced wire current.

A simple triangular facet shell element with applications to linear and non-linear equilibrium and elastic stability problems

This paper has two main objectives. First to describe a very simple facet triangular plate and shell finite element called TRUMP which includes, if required, transverse shear deformation and is based on physical lumping ideas with a simple mechanical interpretation [ 1,2,4,5 ]. Second to give an account of some non-trivial numerical examples of large deflection and post-buckling of shells. There are two types of non-linear structural problems which give rise to particularly delicate numerical experimentation. They are those involving deflections of the order of the structural dimensions, such as three-dimensional elastica, and the instability phenomena of the type leading to dynamic snapthrough, e.g. in cylindrical panels. To tackle such problems using a highly sophisticated shell element such as SHEBA is neither easy nor inexpensive. It is shown that the TRUMP element with only 18 displacement and rotation degrees of freedom is relatively economical to use and yet capable of engineering accuracy. The paper makes use of the theory of simplified geometrical stiffness based on the natural mode method which has been described fully in previous publications [1,2].