Geometrically non-linear oscillations of composite laminated plates by the hierarchical finite element method

Tese de mestrado. Métodos Computacionais em Ciências e Engenharia. Faculdade de Engenharia. Universidade do Porto. 200

Nonlinear oscillations and chaotic dynamics of a supported FGM rectangular plate system under mixed excitations

In this paper, we investigated the nonlinear oscillations, vibrations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) through a temperature field subjected to mixed excitations. The rectangular FGM plate system described by a coupled of nonlinear differential equations (two degree of freedom) including the quadratic and cubic nonlinear terms. The mathematical solutions of the governing equations of motion for the FGM plate derived using the perturbation method based on the power series expansion up to and including the second order approximation. The numerical simulations investigated using Runge-Kutta of fourth order using MATLAB and MAPLE programs. All different resonance cases reported and studied numerically. We applied both frequency response equations and phase-plane technique and also Lyapunov’s first method near the worst resonance cases to analyze the stability of the steady state solution of vibrating system. The effects of the different parameters of the rectangular plate system studied numerically. Results compared to previously published work. In the future work, we can deal with the same system subjected multi-external and tuned and parametric excitation forces. Also, the system can be studied at another worst different resonance cases, active and passive controller

Modeling and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

This paper investigates the multipulse heteroclinic bifurcations and chaotic dynamics of a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. According to the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. The method of multiple scales and Galerkin’s approach are applied to the partial differential governing equation. Then, the four-dimensional averaged equation is obtained for the case of 1 : 3 internal resonance and primary parametric resonance. The extended Melnikov method is used to study the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multipulse chaotic dynamics are analytically obtained. From the investigation, the geometric structure of the multipulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multipulse chaotic motions can occur. To sum up, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists for the laminated composite piezoelectric rectangular plate

Nonlinear Finite Element Analysis of Laminated Composite Beams Subjected to Harmonic Excitations Using a 20 DOF Beam Element

The purpose of this Thesis is to study the nonlinear analysis of antisymmetrically laminated composite beams including shear deformation subjected to harmonic excitation, using a 20-degree of freedom finite element beam. The beam has 10 degrees of freedom at each node: The axial displacements, the transverse deflection due to bending and transverse shear, the twisting angle, the in-plane shear rotation, and their derivatives along the axial direction. In this study, the effect of different parameters such as damping, shear deformation and different edge conditions on the steady-state frequency-responce will be investigated. The analysis was based on the use of finite element methodology for composite laminated beam structures. The harmonic force matrix represents the externally applied force in matrix form, instead of a vector form. Thus the analysis of nonlinear forced vibration can be performed efficiently to get a converged solution. The analysis was also based on the nonlinear stiffness matrix and both in-plane longitudinal, and transverse deflections are included in the formulation. The amplitude-frequency ratios for different boundary conditions, lamination angles, number of plies and thickness to length ratios are presented. The finite element results are compared with available approximate continuum solutions

Nonlinear Flutter of Curved Panels Under Yawed Supersonic Flow Using Finite Elements

In the extensive published literature on panel flutter, a large number of papers are dedicated to investigation of flat plates in the supersonic flow regime. Very few authors have extended their work to flutter of curved panels. The curved geometry generates a pre-flutter behavior, triggering a static deflection due to a static aerodynamic load (SAL) over the panel as well as dynamic characteristics unique to this geometry. The purpose of this dissertation is to provide new insights in the subject of flutter of curved panels. Finite element frequency and time domain methods are developed to predict the pre/post flutter responses and the flutter onset of curved panels under a yaw flow angle. The first-order shear deformation theory, the Marguerre plate theory, the von Karman large deflection theory, and the quasi-steady first-order piston theory appended with SAL are used in the formulation. The principle of virtual work is applied to develop the equations of motion of the fluttering system in structural node degrees of freedom. In the frequency domain method, the Newton-Raphson method is used to determine the panel static deflection under the SAL, and an eigen-value solution is employed for the determination of the stability boundary margins at different panel height-rises and yaw flow angles. Pre-flutter static deflection shape, flutter coalescence frequency, and damping rate of various cylindrical panels are thoroughly investigated. The main results revealed that the pre-flutter static response of cylindrical panels is fundamentally different from the one associated with flat plates. It is shown that curvature has a detrimental effect for 2-dimensional (2-D) curved panels, and is beneficial for 3-D components at an optimum height-rise. In the time domain method, the system equations of motion are transformed into modal coordinates, and solved by a fourth-order Runge-Kutta numerical scheme. Time history responses, phase plots, power spectrum density plots, and bifurcation diagrams uncovered the pre/post flutter responses of cylindrical panels. The computed stability boundary margins and onset frequencies matched very well with the ones computed by the frequency domain method. Bifurcation diagrams revealed limit-cycles oscillations (LCO) and chaotic motion. It was found that 2-D cylindrical panels settle in a multiplicity of LCO as the height-rise of the panel increases, whereas chaotic motion characterize the dynamic behavior of 3-D cylindrical panels at high height-rises

Finite Element Modal Formulation for Panel Flutter at Hypersonic Speeds and Elevated Temperatures

A finite element time domain modal formulation for analyzing flutter behavior of aircraft surface panels in hypersonic airflow has been developed and presented for the first time. Von Karman large deflection plate theory is used for description of the structural nonlinearity and third order piston theory is employed to account for the aerodynamic nonlinearity. The thermal loadings of uniformly distributed temperature and temperature gradients across the panel thickness are incorporated into the finite element formulation. By applying the modal reduction technique, the number of governing equations of motion is reduced dramatically so that the computational time of direct numerical integration is dropped significantly. All possible types of panel behavior, including flat, buckled but dynamically stable, limit cycle oscillation (LCO), periodic motion, and chaotic motion can be observed and analyzed. As examples of the applications of the proposed methodology, flutter responses of isotropic, specially orthotropic and laminated composite panels are investigated. Special emphasis is put on the boundary between LCO and chaos, as well as the routes to chaos. A systematic mode filtering procedure that helps mode selection without specific knowledge of the complex mode shapes is presented and illustrated. Influences of aerodynamic parameters, including aerodynamic damping and Mach number, on the panel flutter responses are studied. The importance of nonlinear aerodynamic terms is examined in detail. The supporting conditions and panel aspect ratio on the onset condition of chaos are also investigated as an illustration of optimization among different design options. Several mathematical tools, including the time history, phase plane plot, Poincaré map, and bifurcation diagram are employed in the chaos study. The largest Lyapunov exponent is also evaluated to assist in detection of chaos. It is found that at low or moderately high nondimensional dynamic pressures, the fluttering panel typically takes a period-doubling route to evolve into chaos, whereas at high nondimensional dynamic pressure, the route to chaos generally involves bursts of chaos and rejuvenations of periodic motions. Various bifurcation behaviors, such as the Hopf bifurcation, pitchfork bifurcation, and transcritical bifurcation, are observed. On the basis of the successful applications presented, the proposed finite element time domain modal formulation and the mode filtering procedure have proven to be an efficient and practical design tool for designers of hypersonic vehicle

Vibration, Stability, and Resonance of Angle-Ply Composite Laminated Rectangular Thin Plate under Multiexcitations

An analytical investigation of the nonlinear vibration of a symmetric cross-ply composite laminated piezoelectric rectangular plate under parametric and external excitations is presented. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations describing the system up to and including the second-order approximation. All possible resonance cases are extracted at this approximation order. The case of 1 : 1 : 3 primary and internal resonance, where Ω3≅ω1, ω2≅ω1, and ω3≅3ω1, is considered. The stability of the system is investigated using both phase-plane method and frequency response curves. The influences of the cubic terms on nonlinear dynamic characteristics of the composite laminated piezoelectric rectangular plate are studied. The analytical results given by the method of multiple time scale is verified by comparison with results from numerical integration of the modal equations. Reliability of the obtained results is verified by comparison between the finite difference method (FDM) and Runge-Kutta method (RKM). It is quite clear that some of the simultaneous resonance cases are undesirable in the design of such system. Such cases should be avoided as working conditions for the system. Variation of the parameters μ1, μ2, α7,β8, ω1, ω2, f1, f2 leads to multivalued amplitudes and hence to jump phenomena. Some recommendations regarding the different parameters of the system are reported. Comparison with the available published work is reported

Vibration analysis of cracked aluminium plates

This research is concerned with analytical modelling of the effects of cracks in structural plates and panels within aerospace systems such as aeroplane fuselage, wing, and tail-plane structures, and, as such, is part of a larger body of research into damage detection methodologies in such systems. This study is based on generating a so-called reduced order analytical model of the behaviour of the plate panel, within which a crack with some arbitrary characteristics is present, and which is subjected to a force that causes it to vibrate. In practice such a scenario is potentially extremely dangerous as it can lead to failure, with obvious consequences. The equation that is obtained is in the form of the classical Duffing equation, in this case, the coefficients within the equation contain information about the geometrical and mass properties of the plate, the loading and boundary conditions, and the geometry, location, and potentially the orientation of the crack. This equation has been known for just over a century and has in the last few decades received very considerable attention from both the analytical dynamics community and also from the dynamical systems researchers, in particular the work of Ueda, Thompson, in the 1970s and 1980s, and Thomsen in the 1990s and beyond. An approximate analytical solution is obtained by means of the perturbation method of multiple scales. This powerful method was popularized in the 1970s by Ali H.Nayfeh, and discussed in his famous books, ‘Perturbation Methods’ (1974) and ‘Nonlinear Oscillations’ (1979, with D.T.Mook), and also by J.Murdock (1990), and M.P.Cartmell et al. (2003) and has been shown to be immensely useful for a wide range of nonlinear vibration problems. In this work it is shown that different boundary conditions can be admitted for the plate and that the modal natural frequencies are sensitive to the crack geometry. Bifurcatory behaviour of the cracked plate has then been examined numerically, for a range of parameters. The model has been tested against experimental work and against a Finite Element model, with good corroboration from both. In all events, this is a significant new result in the field and one that if implemented within a larger damage detection strategy, could be of considerable practical use

Large amplitude free vibration Analysis of composite plates by finite element method

Most of the structural components are generally subjected to dynamic loadings in their working life. Very often these components may have to perform in severe dynamic environment where in the maximum damage results from the resonant vibrations. Susceptibility to fracture of materials due to vibration is determined from stress and frequency. Maximum amplitude of the vibration must be in the limited for the safety of the structure. Hence vibration analysis has become very important in designing a structure to know in advance its response and to take necessary steps to control the structural vibrations and its amplitudes. The non-linear or large amplitude vibration of plates has received considerable attention in recent years because of the great importance and interest attached to the structures of low flexural rigidity. These easily deformable structures vibrate at large amplitudes. The solution obtained based on the lineage models provide no more than a first approximation to the actual solutions. The increasing demand for more realistic models to predict the responses of elastic bodies combined with the availability of super computational facilities have enabled researchers to abandon the linear theories in favor of non-linear methods of solutions. In the present investigation, large amplitude free vibration analyses of composite Mindlin’s plates have been carried out using a C0 eight noded Langragian element by finite element method. The formulation is based on “First order shear deformation theory”. The large deformation effect on plate structures has been taken care by the dynamic version of von Karman’s field equation. The effects of variations in the Poisson’s ratio, amplitude ratio, thickness parameter & plate aspect ratio on the non-linear frequency ratio has also been included in the research. Chapter 1 includes the general introduction and the scope of present investigation. The review of literature confining to the scope of the study has been presented in the Chapter 2. The general methods of analysis of the laminated composite plates have been briefly addressed in this chapter. The chapter 3 presents some information about the theoretical background of finite element method and composite materials. The Chapter 4 comprises the mathematical formulation of the finite elements. The elastic stiffness and the mass matrices for the plate element have been formulated. The boundary conditions have been implemented by eliminating the constrained degrees of freedom from the global stiffness matrix. The Chapter 5 briefly describes the computer program implementation of the theoretical formulation presented in Chapter 4. The different functions and the associated variables which have been used in writing the codes in MATLAB have been presented in brief. A few numbers of flow-chart of the computer program has been illustrated. Several numerical examples which include “large amplitude free vibration analysis” have been presented in the Chapter 6 to validate the formulation of the proposed method. The Chapter 7 sums up and concludes the present investigation. An account of possible scope of extension to the present study has been appended to the concluding remarks. Some important publications and books referred during the present investigation have been listed in the References section

Vibration analysis of a plate with an arbitrarily orientated surface crack

This research presents a vibration analysis for a thin isotropic plate containing an arbitrarily orientated surface crack. The work has been motivated by the well known applicability of various vibrational techniques for structural damage detection in which the detection and localisation of damage to thin plate structures at the earliest stage of development can optimise subsystem performance and assure a safer life, and is intended to be an enhancement to previous work on cracked plates for which the orientation of the crack angle was not included. The novelty of this research activity has been in the assimilation of a significantly enhanced crack model within the analytical model of the plate, in modal space, and taking the form of a specialised Duffing equation. The governing equation of motion of the plate model with enhanced crack modelling is proposed to represent the vibrational response of the plate and is based on classical plate theory into which a developed crack model has been assimilated. The formulation of the angled crack is based on a simplified line-spring model, and the cracked plate is subjected to transverse harmonic excitation with arbitrarily chosen boundary conditions. In addition, the nonlinear behaviour of the cracked plate model is investigated analytically from the amplitude-frequency equation by use of the multiple scales perturbation method. For both cracked square and rectangular plate models, the influence of the boundary conditions, the crack orientation angle, crack length, and location of the point load is demonstrated. It is found that the vibration characteristics and nonlinear characteristics of the cracked plate structure can be greatly affected by the orientation of the crack in the plate. The dynamics and stability of the cracked plate model are also examined numerically using dynamical systems tools for representing the behaviour of this system for a range of parameters. Finally the validity of the developed model is shown through comparison of the results with experimental work and finite element analysis in order to corroborate the effect of crack length and crack orientation angle on the modal parameters, as predicted by the analysis. The results show excellent predictive agreement and it can be seen that the new analytical model could constitute a useful tool for subsequent investigation into the development of damage detection methodologies for generalised plate structures