Gangan Prathap

Galerkin finite element method for non-linear beam vibrations

A Galerkin finite element method is presented, for studying non-linear vibrations of beams describable in terms of moderately large bending theory. The transverse displacement term w alone is used, although several previous attempts to do the same with a Ritz element have failed. This, together with certain assumptions regarding the nature of the vibration, allows an eigenvalue-like quantity characteristic of non-linear vibration to be defined and computed for various amplitudes of vibration. The solution to the non-linear eigenvalue problem is effected in two ways. In one, the exact mode shape and the frequency corresponding to the reference amplitude of vibration are determined by solving iteratively a series of eigenvalue problems until the required convergence is obtained. In the second approach, one assumes that the mode shape does not change with the amplitude and, by a virtual work type approach in which the linear mode shape is used as the weighting vector, reduces the problem to that of a single degree of freedom system. The eigenvalue corresponding to the chosen mode is determined in a manner similar to but subtly different from Rayleighs method. The accuracy and applicability of this approximate method is critically examined. Numerical results are presented to demonstrate that the governing differential equations of the problem do not admit variables separable solutions (time and space) in the clamped-clamped and simply supported-clamped cases.

Active vibration control of composite sandwich beams with piezoelectric extension-bending and shear actuators

We have used quasi-static equations of piezoelectricity to derive a finite element formulation capable of modelling two different kinds of piezoelastically induced actuation in an adaptive composite sandwich beam. This formulation is made to couple certain piezoelectric constants to a transverse electric field to develop extension-bending actuation and shear-induced actuation. As an illustration, we present a sandwich model of three sublaminates: face/core/face. We develop a control scheme based on the linear quadratic regulator/independent modal space control (LQR/IMSC) method and use this to estimate the active stiffness and the active damping introduced by shear and extension-bending actuators. To assess the performance of each type of actuator, a dynamic response study is carried out in the modal domain. We observe that the shear actuator is more efficient in actively controlling the vibration than the extension-bending actuator for the same control effort.

The second frequency spectrum of Timoshenko beams

Abstract A second spectrum of frequencies was reported in early analytical work on the vibrations of Timoshenko beams. However, in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded. In fact, from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged-hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies. In this paper, a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately. Guidelines are provided which help to identify and classify the frequencies into two separate spectra.

BEAM ELEMENTS BASED ON A HIGHER ORDER THEORY-I. FORMULATION AND ANALYSIS OF PERFORMANCE

The flexure of deep beams, thick plates and shear flexible (e.g. laminated composite) beams and plates is often approached through a finite element formulation, based on the Lo-Christensen-Wu (LCW) theory. This paper is a systematic analytical evaluation of the use of the LCW higher order theory for finite element formulation. The accuracy and other features of the computational model are evaluated by comparing finite element method (FEM) results with available closed form classical and elasticity solutions. Wherever possible, errors are predicted by an a priori analysis using these solutions and concepts from an understanding of what the finite element method does.

Field-consistency analysis of the isoparametric eight-noded plate bending element

The eight-node isoparametric plate bending element based on the serendipity shape functions behaves very poorly even after reduced integration of the shear strain energy. It has therefore been the subject of considerable study using various devices to improve it—mixed methods, enforcing of constraints, tensorial transformations, etc. In this paper, we shall proceed from the field-consistency paradigm to understand why the original element and even the element modified by the 2 ∗ 2 Gaussian rule cannot achieve, consistently, the true shear strain constraints in the penalty limit of thin plate behaviour. We then derive the optimal shear strain definitions that leave the element free of all problems in the rectangular form, for most sets of practical boundary suppressions. From this, we next determine the optimum manner of co-ordinate transformation that preserves the true constraints even in the form of a general quadrilateral. This is achieved within the context of iso-P Jacobean transformations and without having to bring in tensorial or base vector definitions and transformations. This should be the simplest displacement type version of this element.

The Energy---Exergy---Entropy (or EEE) sequences in bibliometric assessment

Bibliometric research assessment has matured into a quantitative phase using more meaningful measures and analogies. In this paper, we propose a thermodynamic analogy and introduce what are called the energy, exergy and entropy terms associated with a bibliometric sequence. This can be displayed as time series (variation over time), or in event terms (variation as papers are published) and also in the form of phase diagrams (energy–exergy–entropy representations). It is exergy which is the most meaningful single number scalar indicator of a scientist’s performance while entropy then becomes a measure of the unevenness (disorder) of the publication portfolio.

A C0 continuous four-noded cylindrical shell element

Abstract A C 0 continuous four-noded cylindrical shell element with independent interpolations for in-plane displacements u and v ; transverse displacement w and face rotations θ x , and θ y are made efficient by using substitute smoothed shape functions for w in membrane strain evaluation. This removes “membrane locking”, making it the simplest efficient quadrilateral cylindrical shell element available.

The large amplitude vibration of hinged beams

The large amplitude free vibrations of a simply-supported beam with ends kept a constant distance apart is studied using the actual nonlinear equilibrium equations (i.e. specification of loads in terms of the deformed coordinates of the beam) and the exact nonlinear expression for curvature in addition to the nonlinearity arising from the axial force. A variable separable assumption, together with certain assumptions as to the behaviour of the time function defines an eigenvalue characteristic of the vibration. A numerically exact successive integration and iterative technique establishes the dependence of this quantity on the amplitude of vibrations. The hardening effect of nonlinearity is then interpreted in terms of the variation of this quantity with the amplitude of vibration. This new criteria to define nonlinearity, is compared with several existing in the literature. The present analysis allows the separation of the effects of stretching and large deflection equations on the nonlinear behaviour and the conclusion can be made, based on numerical evidence, that the predominant nonlinearity is due to stretching. The axial force at any station in the beam and the bending stress can also be computed in a numerically exact sense, at the point of maximum amplitude.

Thermally induced vibration control of composite plates and shells with piezoelectric active damping

A coupled piezoelectric field is modelled with an expansion strain in the numerical formulation to analyse piezohygrothermoelastic laminated plates and shells. Finite element actuator and sensor equations are derived using a nine-noded field consistent shallow shell element. Thermally induced vibration control is attempted using piezoelectrically developed active damping. The influence of piezoelectric anisotropy on active damping is evaluated, adopting a simple modelling technique. With 40% reduced actuation capability in the lateral direction, the directionally active lamina is observed to be equally efficient in controlling the vibration. In general, the directionally active lamina is efficient if the primary actuation direction is oriented along the fibre direction or in the direction of bending. The directional actuation appeared to be more effective in the velocity feedback control for cantilevered plates and shells. However, in the simply supported case, a balanced actuation effort is required to provide better controllability, which can be achieved by tailoring the directional actuation. The importance of geometric curvature for the actuator performance is also highlighted.

Consistent thermal stress evaluation in finite elements

Abstract The computation of thermal stresses in displacement finite elements through a formal theoretical basis using the minimum potential principle leads to oscillating stress predictions. Many finite element packages have therefore used an average temperature in simple elements; thermal stresses were computed only at the centroids of such elements to avoid these problems. In this paper we trace this difficulty to a consistency requirement—the requirement that stress fields derived from temperature fields (or initial strains) must be consistent with the total strain field interpolations used in the finite element formulation. The principle governing the problem is developed from the minimum total potential theorem and the Hu-Washizu theorem. This gives it a formal rational basis and leads to an orthogonality condition that provides the procedure for determining consistent, thermal stresses in a variationally correct manner. The princple is demonstrated using some simple problems. A four-noded laminated plane-shell element is also considered to prove the rigour and generality of the approach presented here.