On an elastic dissipation model for a cantilevered beam

Horssen, Wim T. van; Zarubinskaya, M.A.

doi:10.1090/qam/1999837

Cited by 8 publications

(8 citation statements)

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“…They are elaborated for a 2-dimensional cylindrical coordinate system, assuming plane strain conditions and using polar coordinates for the circular domain. The general closed form analytical solution is then obtained using an extended version of the method of separation of variables developed by Van Horssen [8], [9]. The uniqueness criterion is derived for the Dirichlet-type of boundary conditions.…”

Section: List Of Symbols Amentioning

confidence: 99%

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On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

2007

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Summary. The uniqueness and mathematical stability of the Dirichlet boundary value problem of linear elastostatics is studied. The problem is posed as a set of partial differential equations in terms of displacements and Dirichlet-type of boundary conditions (displacements) for arbitrary bounded domains. Then for the circular interior domain the closed form analytical solution is obtained, using an extended version of the method of separation of variables. This method with corresponding complete solution allows for the derivation of a necessary and sufficient condition for uniqueness. The results are compared with existing energy and uniqueness criteria. A parametric study of the elastic characteristics is performed to investigate the behaviour of the displacement field and the strain energy distribution, and to examine the mathematical stability of the solution. It is found that the solution for the circular element with hourglass-like boundary conditions will be unique for all v 6 ¼ 0:5, 0:75, 1:0 and will be mathematically stable for all v 6 ¼ 0:75. Locking of the circular element occurs for v ¼ 0:75 as the energy tends to infinity.

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Section: List Of Symbols Amentioning

confidence: 99%

“…It can be solved using an extended version of the method of separation of variables. A description can be found in [8], and an application can be found in [9], [26]. First, suppose a separated product form of the solution of (13) …”

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confidence: 99%

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

2007

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“…This rectangular plate model is one of the simplest models to describe a suspension bridge. For the linear rectangular plate model the relationship between the plate parameters and the frequencies has been obtained by using an adapted version of the method of separation of variables (see [10]). This result is important to investigate the wind-induced oscillations of a rectangular plate.…”

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On Aspects of Asymptotics for Plate Equations

Zarubinskaya

Horssen

2005

Nonlinear Dyn

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It is well-known that flexible structures like suspension bridges or overhead power transmission lines can be subjected to oscillations due to windforces or other causes. Simple models for such oscillations are described with second-and fourth-order partial differential equations. Usually asymptotic methods can be used to construct approximations for the solutions of these wave, beam or plate equations. For a long time initial boundary value problems for weakly nonlinear wave equations have been studied. The analysis becomes more compliceted for beam equations [1], [2]. Even less is known about weakly nonlinear plate equations. However, plates of various geometries, i.e. circular, rectangular etc are extensively used in engineering applications. These plates are widely used in modern aerospace technology, aircraft structures and flexible structures like suspension bridges. The majority of the literature deals with classical boundary conditions representing clamped, simply supported or free edges, and only a small number deals with edges which are restrained against translation or/and rotation, or with other nonclassical boundary conditionsA simple way to model the behavior of a suspension bridge is to describe it as a vibrating onedimensional beam with simply supported edges [3]- [4]. However, to study wind-induced oscillations of suspension bridges one can of course use plate equations to describe the displacement of the deck of the bridge [9].The deck of the bridge is modeled as a rectangular plate, and the cabels are modeled as springs. Such a model can be described by the following initially boundary value problemwhere p 2 represents the linear restoring force of the springs, is a small parameter, d and l are width and length of the plate correspondently. The initial displacement and the initial velocity of the plate in z-direction are given by u 0 (x, y) and u 1 (x, y) respectively.In this paper the following subproblems will be studied in detail• a linear problem with f (x, t, u, u t ) = 0, and g(u, u t ) = 0;• a nonlinear problem for u tt + u xxxx + 2u xxyy + u yyyy + p 2 u = u 2 , where p 2 u − u 2 represents the restoring force of the springs;• a linearized problem with f (x, t, u, u t ) = u t , and g(u, u t ) = αu t , where is α-damping parameter.

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“…Russell showed in [3] that it is, in general, not true that the energy decreases monotonically. In [5] it has been shown for the cantilevered beam problem as formulated in [1], [2] that the first (the lowest) vibration mode is unstable. So, for this mode there certainly is no energy dissipation.…”

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“…In Section 2 of this paper it will be indicated how this dissipation model can be obtained. By using the recently developed, adapted version of the method of separation of variables (see [5], [6]), we will study the dissipation model in Section 3 of this paper. It will turn out that three cases have to be distinguished, that is, δ = 2, δ > 2, and 0 < δ < 2.…”

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On an improved elastic dissipation model for a cantilevered beam

Zarubinskaya

Horssen

2005

Quart. Appl. Math.

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Abstract. In this paper we will study an improved elastic dissipation model for a cantilevered beam, where the damping is assumed to be proportional to the bending rate of the beam. For an earlier formulated dissipation model for the cantilevered beam it has been recently shown that damping will not always be generated. However, for the improved dissipation model it will be shown in this paper that damping will always be generated.

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On an elastic dissipation model for a cantilevered beam

Cited by 8 publications

References 10 publications

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

On Aspects of Asymptotics for Plate Equations

On an improved elastic dissipation model for a cantilevered beam

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