Optimal design of vibrating circular plates

Georgian Mathematical Journal, 2017

In this paper, we consider an eigenvalue problem for the biharmonic operator that describes the transverse vibrations of the plate. Under the imposed boundary conditions, the eigenvalues of this operator are indeed eigenfrequencies of the clamped plate. The domain of the plate is taken variable and the domain functional, involving an eigenfrequency, is studied. A new formula for an eigenfrequency is proved, the first variation of the functional with respect to the domain is calculated, and the necessary condition for an optimal shape is derived. New explicit formulas are obtained for the eigenfrequency in the optimal domain in some particular cases.

Journal of Applied Mathematics and Mechanics, 1985

An approximate thickness optimization of a rectangular Kirchhoff-Love plate with variable stiffness under uniform load is performed in this paper. The authors propose an original method for formulating problems of optimal design for plate structures of variable thickness. Partial discretization, which is described in this paper, reduces the number of independent variables in the problem formulation to only one, making the problem possible to solve via application of the Pontryagin’s minimum principle. The optimization problem relates to the search for the optimal plate thickness distributions, which provides the minimum structural volume of the material used while simultaneously meeting all constraint conditions. The optimal design task is formulated as a control theory problem, maintaining the formal structure of the minimum principle, and then is transformed into a two-point boundary value problem. Such an approximate solution, meeting all necessary optimality conditions, is found by using Dircol software for a chosen illustrative example.

Journal of the Franklin Institute, 2005

A maximum principle is formulated and validated for the vibration control of an annulus plate with the control forces acting on the boundary. In addition, the maximum principle can be applied to plates with multiply connected domains. The performance index is specified as a quadratic functional of displacement and velocity along with a suitable penalty term involving the control forces. Using this index an explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions together with the proposed maximum principle. The implementation of the theory is presented and the effectiveness of the boundary control is investigated by a numerical example.

Nonlinear Analysis: Theory, Methods & Applications, 1997

A large number of exact boundary controllability/stabilizability results for various plate models have appeared in the literature [see e.g. Refs. 2,3]. However, there are relatively few results on the optimal boundary control of vibrating plates [see e.g. Refs. 1.61. In the present study a maximum principle is formulated for the active control of vibrating plate of general shape with the control forces acting on the boundary. The applicability of the maximum principle is shown for plates which can have multiply connected domains. The performance index is specified as a quadratic functional of displacement and velocity with a suitable penalty term involving the control forces. An explicit control law is derived with the help of an adjoint variable satisfying the adjoint differential equation and certain terminal conditions and the proposed maximum principle.

Archive for Rational Mechanics and Analysis, 2011

The natural way to find the most compliant design of an elastic plate, is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotic of such compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in [18]. Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in [16, 17]. Finally, some explicit computation of optimal plates are given.

Journal of Sound and Vibration, 1999

Sādhanā, 2019

In this study, optimal design of the transversely vibrating Euler-Bernoulli beams segmented in the longitudinal direction is explored. Mathematical formulation of the beams in bending vibration is obtained using transfer matrix method, which is later coupled with an eigenvalue routine using the ''fmincon solver'' provided in Matlab Optimization Toolbox. Characteristic equations, namely frequency equations, for determining natural frequencies of the segmented beams for all end conditions are obtained and for each case, square of this equation is selected as a fitness function together with constraints. Due to the explicitly unavailable objective functions for the natural frequencies as a function of segment length and volume fraction of the materials, especially for the beams made of a large number of segments, initially, prescribed value is assumed for the natural frequency and then the variables minimizing objective function and satisfying the constraints are searched. Clamped-free, clamped-clamped, clamped-pinned and pinned-pinned boundary conditions are considered. Among the end conditions, maximum increment in the fundamental natural frequency is more pronounced for the case of clamped-clamped end condition and for this case, maximum increment up to 17.3274% is attained. Finally, the beam configurations maximizing fundamental natural frequencies will be presented.

Periodica Polytechnica Civil Engineering, 2014

The plated structures are one of the most frequently used engineering structures. The object of this research work is the optimal design of curved folded plates. This work is an ongoing investigation. There are various solution methods to analyze this type of structures. Here the finite strip method is used. At first single load condition is considered, but later the multiple load conditions are used for the design. The base formulation is a minimum volume design with displacement constraint what is represented by the compliance. For the multiple loading two equivalent topology optimization algorithms can be elaborated: minimization of the maximum strain energy with respect to a given volume or minimization of the volume of the structure subjected to displacement constraints. The numerical procedures are based on iterative formulas which is formed by the use of the first order optimality condition of the Lagrangian-functions. The application is illustrated by numerical examples. Keywords optimization • multiple loading • curved plate • curved folded plate • optimal layout • optimality criteria method • optimal design

SIAM Journal on Control and Optimization, 2007

We revisit a classic design problem for a plate of variable thickness under the model of Kirchhoff. Our main contribution has two goals. One is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants. The other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold. In both situations, the cost functional can be pretty general.