Geometric Nonlinear Finite Element and Genetic Algorithm Based Vibration Energy Harvesting from Functionally Graded Nonprismatic Piezolaminated Beams

Energy harvesting technology has the ability to create autonomous, self-powered systems which do not rely on the conventional battery for their operation. The term energy harvesting is the process of converting the ambient energy surrounding a system into some useful electrical energy using certain materials. Among several energy conversion techniques, the conversion of ambient vibration energy to electrical energy using piezoelectric materials has great deal of importance which encompasses electromechanical coupling between mechanical and electrical domains. The energy harvesting systems are designed by incorporating the piezoelectric materials in the host structure located in vibration rich environment. The work presented in this dissertation focuses on upgrading the concept of energy harvesting in order to engender more power than conventional energy harvesting designs. The present work deals with first the finite element (FE) formulation for coupled thermo-electro-mechanical analysis of vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated cantilever beam. A two noded beam element with two degrees of freedom (DOF) at each node has been used in the FE formulation. The FG material (i.e. non-homogeneity) in the axial direction has been considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) have been modelled using the Euler-Bernoulli beam theory and Hamilton‘s principle is used to solve the governing equation of motion. The simultaneous variation of tapers (both width and height in length directions) is incorporated in the mathematical formulation. The FE formulation developed in the present work has been compared with the analytical solutions subjected to mechanical, electrical, thermal and thermo-electro-mechanical loading. Results obtained from the present work shows that the axially FG nonprismatic beam generates more output power than the conventional energy harvesting systems. Further, the work has been focussed towards the nonlinear vibration energy harvesting from an axially FG non-prismatic piezolaminated cantilever beam. Geometric nonlinear based FE formulation using Newmark method in conjunction with Newton-Raphson method has been formulated to solve the obtained governing equation. Moreover, a real code GA based constrained optimization technique has also been proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of the beam and voltage of the PZT sensor. It is observed that more output power can be obtained based on the present optimization formulation within the allowable limits of stress and voltage than that of selection of design variables by trial and error in FE modelling

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

A new continuum based non-linear finite element formulation for modeling of dynamic response of deep water riser behavior

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The principal objective of this investigation is to develop a nonlinear continuum based finite element formulation to examine dynamic response of flexible riser structures with large displacement and large rotation. Updated Lagrangian incremental approach together with the 2nd Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor is employed to derive the nonlinear finite element formulation. The 2nd Piola-Kirchhoff stress and the Green-Lagrange strain tensors are energy conjugates. These two Lagrangian tensors are not affected by rigid body rotations. Thus, they are used to describe the equilibrium equation of the body independent of rigid rotations. While the current configuration in Updated Lagrangian incremental approach is unknown, the resulting equation becomes strongly nonlinear and has to be modified to a linearized form. The main contribution of this work is to obtain a modified linearization method during development of incremental Updated Lagrangian formulation for large displacement and large rotation analysis of riser structures. For this purpose, the Green-Lagrange strain and the 2nd Piola-Kirchhoff stress tensors are decomposed into two second-order six termed functions of through-thethickness parameters. This decomposition makes it possible to explicitly account for the nonlinearities in the direction along the riser thickness, as well. It is noted that using this linearization scheme avoids inaccuracies normally associated with other linearization schemes. The effects of buoyancy force, riser-seabed interaction as well as steady-state current loading are considered in the finite element solution for riser structure response. An efficient riser problem fluid-solid interaction Algorithm is also developed to maintain the quality of the mesh in the vicinity of the riser surface during riser and fluid mesh movements. To avoid distortions in the fluid mesh two different approaches are proposed to modify fluid mesh movement governing elasticity equation matrices values; 1) taking the element volume into account 2) taking both element volume and distance between riser centre and element centre into account. The formulation has been implemented in a nonlinear finite element code and the results are compared with those obtained from other schemes reported in the literature

Least-Squares, Continuous Sensitivity Analysis for Nonlinear Fluid-Structure Interaction

A least-squares, continuous sensitivity analysis method is developed for transient aeroelastic gust response problems to support computationally efficient analysis and optimization of aeroelastic design problems. A key distinction between the local and total derivative forms of the sensitivity system is introduced. The continuous sensitivity equations and sensitivity boundary conditions are derived in local derivative form which is shown to be superior for several applications. The analysis and sensitivity problems are both posed in a first-order form which is amenable to a solution using the least-squares finite element method. Several example and validation problems are presented and solved, including elasticity, fluid, and fluid-structure interaction problems. Significant contributions of the research include the first sensitivity analysis of nonlinear transient gust response, a local derivative formulation for shape variation that requires parameterizing only the boundary, and statement of sufficient conditions for using nonlinear black box software to solve the sensitivity equations. Promising paths for future investigation are presented and discussed

Hp-spectral Methods for Structural Mechanics and Fluid Dynamics Problems

We consider the usage of higher order spectral element methods for the solution of problems in structures and fluid mechanics areas. In structures applications we study different beam theories, with mixed and displacement based formulations, consider the analysis of plates subject to external loadings, and large deformation analysis of beams with continuum based formulations. Higher order methods alleviate the problems of locking that have plagued finite element method applications to structures, and also provide for spectral accuracy of the solutions. For applications in computational fluid dynamics areas we consider the driven cavity problem with least squares based finite element methods. In the context of higher order methods, efficient techniques need to be devised for the solution of the resulting algebraic systems of equations and we explore the usage of element by element bi-orthogonal conjugate gradient solvers for solving problems effectively along with domain decomposition algorithms for fluid problems. In the context of least squares finite element methods we also explore the usage of Multigrid techniques to obtain faster convergence of the the solutions for the problems of interest. Applications of the traditional Lagrange based finite element methods with the Penalty finite element method are presented for modelling porous media flow problems. Finally, we explore applications to some CFD problems namely, the flow past a cylinder and forward facing step

Piezoelectric Control of Structures Prone to Instabilities

Thin-walled structures such as stiffened panels fabricated out of high strength materials are ubiquitous in aerospace structures. These are prone to buckle in a variety of modes with strong possibility of adverse interaction under axial compression and/or bending. Optimally designed stiffened panels, at an appropriate combination of axial compression and suddenly applied lateral pressure undergo large amplitude oscillations and may experience divergence. Under aerodynamic loading, they can experience flutter instability with the amplitudes of oscillations attaining a limit: LCO) or escalating without any limit. Control of structures prone to these forms of instability using piezo-electric actuators is the theme of this dissertation. Issues involved in the control of stiffened panels under axial compression and liable to buckle simultaneously in local and overall modes are studied. The analytical approach employs finite elements in which are embedded periodic components of local buckling including the second order effects. It is shown that the adverse effects of mode interaction can be counteracted by simply controlling the overall bending of the stiffener by piezo-electric actuators attached its tips. Control is exercised by self-sensing actuators by direct negative feedback voltages proportional to the bending strains of the stiffener. In a dynamic loading environment, where vibrations are triggered by suddenly applied lateral pressure, negative velocity feedback is employed with voltages proportional to the bending strain-rate. The local plate oscillations are effectively controlled by a piezo-electric actuators placed along the longitudinal center line of the panel. The problem of flutter under aerodynamic pressure of stiffened panels in the linear and post-critical regimes is studied using modal analysis and finite strips. The analysis, control and interpretation of the response are facilitated by identification of two families of characteristic modes of vibration, viz. local and overall modes and by a classification of the local modes into two distinct categories, viz. symmetric and anti-symmetric modes respectively. The symmetric local modes interact with overall modes from the outset, i.e. in the linear flutter problem whereas both the sets of local modes interact with overall modes in the post-critical range via cubic terms in the elastic potential. However the effects of interaction in the flutter problem are far less dramatic in comparison to the interactive buckling problem unless the overall modes are activated, say by dynamic pressure on the plate. Control of the panel is exercised by piezo-electric patches placed on the plate at regions of maximum curvature as well as on the stiffener. Two types of control strategies were investigated for the panel subject to fluttering instability. The first is the direct negative velocity feedback control using a single gain factor for each of the sets of plate patches and stiffener patches respectively. A systematic method of determining the gains for the patches has been developed. This is based on the application of LQR algorithm in conjunction with a linearized stiffness matrix of the uncontrolled structure computed at a set of pre-selected times. This type of control was successful till the aerodynamic pressure coefficient reaches up to about six times its critical value, where after it simply failed. The second type of control is the multi-input and multi-output full state feedback control. The LQR algorithm and the linearized stiffness matrix are invoked again, but the gain matrix is computed at the beginning of every time step in the analysis and immediately implemented to control the structure. This type of control proved very effective the only limitation stemming from the maximum field strength that can be sustained by the piezo-electric material employed

Research in Structures, Structural Dynamics and Materials, 1990

The Structural Dynamics and Materials (SDM) Conference was held on April 2 to 4, 1990 in Long Beach, California. This publication is a compilation of presentations of the work-in-progress sessions and does not contain papers from the regular sessions since those papers are published by AIAA in the conference proceedings

The Analysis of Stress and Deformation

This book was prepared for a course in the mechanics of deformable bodies at the authors' institution, and is at a level suitable for advanced undergraduate or first-year graduate students. It differs from the traditional treatment by going more deeply into the fundamentals and giving less emphasis to the design aspects of the subject. In the first two chapters the principles of stress and strain are presented and a sufficient introduction is given to the theory of elasticity so that the student can see how exact solutions of problems can be derived, and can appreciate the nature of the approximations embodied in some commonly used simplified solutions. The third chapter is devoted to the bending of beams, and the fourth chapter treats the instability of elastic systems. Applications to axially symmetric problems, curved beams, and stress concentrations are discussed in Chapter 5; applications to torsion problems are discussed in Chapter 6; applications to problems of plates and shells are discussed in Chapter 7. Applications to problems involving viscous and plastic behavior are treated in Chapter 8, and problems of wave propagation are treated in Chapter 9. An introduction to numerical methods of solving problems is given in Chapter 10. An introduction to tensor notation by means of the equations of elasticity is given in Appendix I. Experimental methods of determining stresses by means of strain gages, brittle coatings, and photoelasticity are described in Appendices II and III. A brief introduction to variational methods is presented in Appendix IV. The material in the book is laid out so that a short course can be based on Chapters 1, 2, 3, 4, and 8 and Appendices II and III. Some of the special aspects of the subject and some of the details of the derivations are left to the problems; the assignment of homework should be made with this in mind. To indicate to the student the nature of the more advanced parts of the subject, some topics are included that would not necessarily be covered in the formal course work. The book is aimed primarily at those students who will pursue graduate work, and it is intended to give a good preparation for advanced studies in the field. It should also give a good foundation to students primarily interested in design who would cover the more applied aspects of the subject in courses on design. The authors wish to express their appreciation to those members of the California lnstitute of Technology community whose suggestions and efforts have helped to bring our lecture notes into book form