共查询到20条相似文献,搜索用时 15 毫秒
1.
A.T. Il’ichev 《Journal of Applied Mathematics and Mechanics》2008,72(3):323-330
The spectral stability of compression solitons in non-linear elastic rods with respect to perturbations of the flexural mode of the oscillations of the rod is investigated. The system of equations of the isotropic theory of elasticity, taking account of the non-linear corrections corresponding to the interaction being studied, is used to describe the interaction of longitudinal and flexural waves in the rod. This system of equations describes long longitudinal-flexural waves of small but finite amplitude. It is shown that trapped flexural modes exist, which propagate together with a compression soliton. It is established that these modes, which are the least stable, do not increase with time. 相似文献
2.
A. E. Ordanovich 《Journal of Mathematical Sciences》2007,146(3):5820-5825
The Euler problem on the stability of a thin elastic rod under compressing forces is widely known. In a number of papers this
problem is generalized to finding the shape of such a rod under the simultaneous action of a compressing force and a torsional
moment. The shape is determined by solving a complex nonlinear boundary value problem by numerical methods. In this paper,
an approach providing a full analytical solution of the problem for a broad range of external conditions is discussed.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 35–42, 2005. 相似文献
3.
N. I. Vlasov 《Journal of Mathematical Sciences》1991,57(5):3413-3416
We study the dynamics of a building with a nonlinear seismic insulation system whose motion is described by the equation of transverse oscillations of a rod. We obtain the connection between the displacement of the lower section of the rod and the forces and moments in the section. We propose a numerical procedure for solving a nonlinear Volterra integral equation of second kind with dry friction damping. We determine the region of variation of the parameters of the building and the seismic insulation system in which it is possible to use the rigidbody model of the building. Two figures.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 76–80. 相似文献
4.
Sergey A. Nazarov Andrey S. Slutskij Jari Taskinen 《Mathematical Methods in the Applied Sciences》2020,43(10):6396-6415
We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4. 相似文献
5.
The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated. 相似文献
6.
K. Sh. Mkrtchyan 《Journal of Applied Mathematics and Mechanics》1999,63(6):989-992
The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1]. 相似文献
7.
8.
The buckling of a pin-ended slender rod subjected to a horizontal end load is formulated as a nonlinear boundary value problem. The rod material is taken to be governed by constitutive laws which are nonlinear with respect to both bending and compression. The nonlinear boundary value problem is converted to a suitable integral equation to allow the application of bounded operator methods. By treating the integral equation as a bifurcation problem, the branch points (critical values of load) are determined and the existence and form of nontrivial solutions (buckled states) in the neighborhood of the branch points is established. The integral equation also affords a direct attack upon the question of uniqueness of the trivial solution (unbuckled state). It is shown that, under certain conditions on the material properties, only the trivial solution is possible for restricted values of the load. One set of conditions gives uniqueness up to the first branch point. 相似文献
9.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have to connect a one-dimensional model with director variables to a three-dimensional model without directors. Two alternative coupling conditions are proposed, which correspond to two different configuration trace spaces. For both, we show existence of solutions of the coupled problems, using the direct method of the calculus of variations. From the first-order optimality conditions, we also derive the corresponding conditions for the dual variables. These are then interpreted in mechanical terms. 相似文献
10.
Anca Ignat Jürgen Sprekels Dan Tiba 《Mathematical Methods in the Applied Sciences》2002,25(10):835-854
We indicate a new approach to the deformation of three‐dimensional curved rods with variable cross‐section. The model consists of a system of nine ordinary differential equations for which we prove existence and uniqueness via the coercivity of the association bilinear form. From the geometrical point of view, we are using the Darboux frame or a new local frame requiring just a C1‐parameterization of the curve. Our model also describes the deformation occurring in the cross‐sections of the rod. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
11.
In this paper, we prove exact observability, boundary and internal exact controllability results for a nonsimple elastic rod. By the spectral analysis method, it is shown that the considered problem has a sequence of eigenfunctions, which forms a Riesz basis for the state space. The exact observability is proved by application of a modified version of Ingham’s Theorem. Then, via the Hilbert Uniqueness method, we prove a boundary controllability result with only one controller and an internal controllability result. 相似文献
12.
13.
《Communications in Nonlinear Science & Numerical Simulation》2010,15(9):2617-2633
We perform system identification and modelling of the strongly nonlinear modal interactions in a system composed of a linear elastic rod with an essentially nonlinear attachment at its end. Our method is based on slow/fast decomposition of the transient dynamics of the system, combined with empirical mode decomposition (EMD) and Hilbert transforms. The derived reduced order models (ROMs) are in the form of sets of uncoupled linear oscillators (termed intrinsic modal oscillators – IMOs), each corresponding to a basic frequency of the dynamical interaction and forced by transient excitations that represent the nonlinear modal interactions between the rod and the attachment at each of these basic frequencies. A main advantage of our proposed technique is that it is nonparametric and multi-scale, so it is applicable to a broad range of linear as well as nonlinear dynamical systems. Moreover, it is computationally tractable and conceptually meaningful, and it leads to reduced order models of rather simple form that fully capture the basic strongly nonlinear resonant interactions between the subsystems of the problem. 相似文献
14.
Garbuzov F. E. Beltukov Y. M. Khusnutdinova K. R. 《Theoretical and Mathematical Physics》2020,202(3):319-333
Theoretical and Mathematical Physics - We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section in the framework of general weakly nonlinear elasticity... 相似文献
15.
I. V. Savenkov 《Computational Mathematics and Mathematical Physics》2011,51(12):2155-2161
The pressure-driven Poiseuille flow of a viscous incompressible fluid between two parallel plates is considered. Within the
framework of the triple-deck theory, the elasticity of the walls is shown to have a stabilizing effect on the flow. 相似文献
16.
John J. McCoy 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1964,15(5):456-465
Zusammenfassung Das Problem eines unendlichen kreiszylindrischen Halbstabes, dessen Endquerschnitt mit beliebigen Torsionsspannungen belastet ist, wird auf Grund der strengen dreidimensionalen Elastizitätstheorie gelöst. Die Lösungsmethode besteht aus einer Integraltransformationstechnik mit asymptotischer Inversion für eine frequenzmodulierte Stufenfunktion. Die Ergebnisse zeigen, dass die Gültigkeit eines «dynamischen Saint-Venant-Prinzips» von der betreffenden Frequenzkomponente abhängt.
This work was initiated while the author was still employed by the IBM Corp. at Endicott, New York. 相似文献
This work was initiated while the author was still employed by the IBM Corp. at Endicott, New York. 相似文献
17.
《Journal of Applied Mathematics and Mechanics》2014,78(5):468-479
Optimal controls of the rotation of a mechanical system consisting of two rigid bodies, joined by an elastic rod, through a specified angle about an axis passing through the centre of mass of one of the bodies are constructed. The problem of the optimal control of the rotation of the system through a given angle with complete suppression of the oscillations of the elastic rod at the minimum of the energy functional of the control moment and the problem of time-optimality for a specified constraint on the energy functional of the control moment are solved. 相似文献
18.
Yu. I. Dorogov 《Journal of Applied Mathematics and Mechanics》2013,77(3):338-345
The stability of an unattached column consisting of an elastic rod with stiff flanges on its ends under longitudinal compression is investigated. The load under which the plane of the flange surface is tilted from the plane of the support surface is found. This tilting is accompanied by considerable rotation (reversing) of the flanges and corresponding bending of the rod axis. Abrupt replacement of the rectilinear or bent equilibrium shape by an equilibrium shape that is non-contiguous to it occurs. It is established that the columns behave differently when this a change in the equilibrium shape occurs, depending on the ratio of the length of the rod to the length of the flanges. 相似文献
19.
I.L. Antonov 《Journal of Applied Mathematics and Mechanics》2013,77(2):225-229
The stability of the vertical position of an elastic rods, hinged at the ends and loaded at the upper end with a time-dependent longitudinal force containing a stochastic component, is considered. A homogeneous Markov chain is used to model the stochastic component. An inequality is obtained that depends on the characteristics of the stochastic force and the damping factor of the bar and, depending on whether this inequality is satisfied, the above equilibrium position will either be attracting or unstable. 相似文献
20.
Damoder P. Reddy Jan D. Achenbach 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1968,19(3):473-485
Zusammenfassung Ein halb-unendlicher dünner elastischer Stab wird einer monoton ansteigenden Störung am freien Ende unterworfen. Die unstetige Wellenbewegung für endliche Verzerrungen wird untersucht, und die Bedingungen zur Bildung von Stössen werden aufgestellt. Eine implizite Lösung für die Wellenbewegung im Gebiete einfacher Wellen wird mittels der Methode der Charakteristiken erhalten. Eine andere und explizite Lösung in diesem Gebiete wird gefunden durch Anwendung der Theorie sich fortpflanzender Diskontinuitätsflächen, als eine Taylor-Reihe um die Ankunftszeit der Wellenfront. Es wird gezeigt dass der Koeffizient des ersten Gliedes in der Entwicklung eine Lösung für eine homogene nichtlineare gewöhnliche Differentialgleichung ecster Ordnung darstellt. Die übrigen Koeffizienten befriedigen lineare, jedoch inhomogene gewöhnliche Differentialgleichungen mit veränderlichen Koeffizienten. Für den Fall einer rampenartigen Belastung und für Stossbildung an der Wellenfront wird für schwache Stösse durch Anwendung einer Methode von Friedrichs die Vereinigung von einfachen Wellen untersucht. Als Beispiel wird die Entlastung eines Stabes betrachtet, der einer anfänglichen Dehnung von 50% unterworfen ist. 相似文献