A maximum principle in spectral optimization problems for elliptic operators subject to mass density perturbations

We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the eigenvalues upon variation of the mass density and we prove a maximum principle for extremum problems related to mass density perturbations which preserve the total mass

On the stability of a nonlinear nonhomogeneous multiply hinged beam

The paper deals with a nonlinear evolution equation describing the dynamics of a nonhomogeneous multiply hinged beam, subject to a nonlocal restoring force of displacement type. First, a spectral analysis for the associated weighted stationary problem is performed, providing a complete system of eigenfunctions. Then, a linear stability analysis for bimodal solutions of the evolution problem is carried out, with the final goal of suggesting optimal choices of the density and of the position of the internal hinged points in order to improve the stability of the beam. The analysis exploits both analytical and numerical methods; the main conclusion of the investigation is that nonhomogeneous density functions improve the stability of the structure

Symmetry in the composite plate problem

In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$\inf_{\rho \in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_{\Omega}(\Delta u)^2}{\int_{\Omega} \rho u^2},$$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(\Omega)$ for Dirichlet boundary conditions and $\mathcal W= H^2(\Omega) \cap H^1_{0}(\Omega)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $\Delta^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,\rho)$. In particular, we prove existence and regularity and we find the explicit expression of $\rho$. When $\Omega$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $\rho$.Comment: 26 page

Dynamic stability and buckling of viscoelastic plates and nanobeams subjected to distributed axial forces.

Doctor of Philosophy in Mechanical Engineering. University of KwaZulu-Natal, Durban 2016.Plates and beams are typical examples of structures that must be analyzed and understood. Buckling and vibration represent for such structures a potential source of fatigue and damage. Damage and fatigue are often caused by axial forces. The current research uses differential quadrature method to study the stability of viscoelastic plate subjected to follower forces in one hand, and the Rayleigh-Ritz method to analyze the buckling of Carbone nanotubes subjected to point and axial load in other hand. For plate, the 3D relation of viscoelastic is used to derive the equation of vibration of viscoelastic rectangular plate subjected to follower force. This equation is solved numerically by differential quadrature method, then the dynamic stability analysis is done by plotting the eigenvalues versus the follower force. We employ the Euler Bernoulli beam theory and the nonlocal theory to derive the equation of equilibrium of Carbone nanotubes subjected to point and axial loads. Rayleigh-Ritz method is used to calculate buckling loads, and the effects of equation's parameters on that buckling loads are analysed properly. Frequencies of vibration of viscoelastic plates and critical load obtained by using differential quadrature method are compared to other results with good satisfaction. The same satisfaction is observed when the buckling load values of Carbone nanotubes obtained using the Rayleigh-Ritz methods are compared to those existing in the literature. The cantilever viscoelastic plate undergoes flutter instability only and the delay time appears to influence that instability more than other parameters. The SFSF plate undergoes divergence instability only. The both types of instability are observed CSCS plate subjected to uniformly follower load but the flutter instability disappears in presence of triangular follower load. The values of the mentioned critical loads increase with triangular follower load for all boundary conditions. The aspect ratio has a large influence on the divergence and flutter critical load values and little influence on the instability quality. The laminar friction coefficient of the flowing fluid increases the critical fluid velocity but its effect on the stability of viscoelastic plate behavior is minor. The nonlocal parameter appears to decrease buckling load considerably. Buckling is more sensitive to the magnitude of the tip load for the clamped-free boundary conditions. The application of the present theory to a non-uniform nanocone shows that the buckling loads increases with radius ratio and decreases with small scale constants

Transient flexural wave propagation in beams with discontinuities of cross section

An investigation was carried out to determine the transient response of finite beams with discontinuities of cross section subiected to eccentric longitudinal impact. Experiments were performed on several stepped beams with increased and reduced cross section and with various end conditions. The analysis was based on the Timoshenko beam theory which takes into account the effects of shear deformation and rotatory inertia. The governing equations were solved as a system of two second order hyperbolic partial differential equations. The numerical solution was obtained by the method of characteristics and theoretical predictions were in excellent agreement with experimental observations at several monitoring positions along the various test beams. The agreement between theoretical and experimental results verified the adequacy of the Timoshenko theory and its numerical solution for describing the flexural wave propagation in beams with discontinuities of cross section. The effect of the discontinuity of beam cross section on the bending moment time distribution showed the importance of reflections in estimating the level of stresses and strains in structural elements when subjected to transient dynamic loading. The computer program developed in this work can be used to obtain numerical solutions for. a wide range of flexural wave propagation problems in beams with discontinuities of cross section with various end conditions and loading configurations

Vibration, Control and Stability of Dynamical Systems

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Proceedings of the Workshop on Identification and Control of Flexible Space Structures, Volume 2

The results of a workshop on identification and control of flexible space structures held in San Diego, CA, July 4 to 6, 1984 are discussed. The main objectives of the workshop were to provide a forum to exchange ideas in exploring the most advanced modeling, estimation, identification and control methodologies to flexible space structures. The workshop responded to the rapidly growing interest within NASA in large space systems (space station, platforms, antennas, flight experiments) currently under design. Dynamic structural analysis, control theory, structural vibration and stability, and distributed parameter systems are discussed

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach