Nonlinear free vibrations of multispan beams on elastic supports

Annular plates are commonly found in the fields of engineering. The present study is concerned with the integral equation method for the free vibrations of annular plates with elastic supports. A set of complete systems of orthogonal functions, which is constructed by Bessel functions of the first and the second kind is used to construct the Green's function of annular plates. The eigenvalue problem of free vibration of annular plates with Elastic Supports is transformed into the eigenvalue problem of integral equation. And then, the problem of integral equation is transformed into a standard eigenvalue problem of a matrix with infinite order. Numerical example shows the significant advantages of the present method.

Download Full-text

SOME PROBLEMS OF OPTIMIZATION AND CONTROL OF THE NATURAL FREQUENCIES OF AN ELASTICALLY SUPPORTED RIGID BODY

Mechanics And Mathematical Methods ◽

10.31650/2618-0650-2021-3-2-88-102 ◽

2021 ◽

Vol 3 (2) ◽

pp. 88-102

Author(s):

S. Bekshaev ◽

Keyword(s):

Rigid Body ◽

Fundamental Frequency ◽

Natural Frequencies ◽

Optimization Problems ◽

Free Vibrations ◽

The Body ◽

Natural Vibrations ◽

Natural Oscillations ◽

Optimal Position ◽

Elastic Supports

The article analytically investigates the behavior of the frequencies and modes of natural vibrations of a rigid body, based on point elastic supports, when the position of the supports changes. It is assumed that the body is in plane motion and has two degrees of freedom. A linear description of body vibrations is accepted. The problems of determining such optimal positions of elastic supports at which the fundamental frequency of the structure reaches its maximum value are considered. Two groups of problems were studied. The first group concerns a body supported by only two supports. It was found that in the absence of restrictions on the position of the supports to maximize the fundamental natural frequency, these supports should be positioned so that the basic natural vibrations of the body are translational. Simple analytical conditions are formulated that must be satisfied by the corresponding positions of the supports. In real practical situations, these positions may be unreachable due to the presence of various kinds of restrictions due to design requirements. In this paper, optimization problems are considered taking into account a number of restrictions on the position of supports, typical for practice, expressed analytically by equations and inequalities. For each of the considered types of constraints, results are obtained that determine the optimal positions of the supports and the corresponding maximum values of the main natural frequencies. The approach applied allows us to consider other types of restrictions, which are not considered in the article. In the second group of problems for a body resting on an arbitrary number of supports, the optimal position of an additional elastic support introduced in order to maximize the fundamental frequency in fixed positions and the stiffness coefficients of the remaining supports was sought. It was found that this position depends on the value of the stiffness coefficient of the introduced support. Results are obtained that qualitatively and quantitatively characterize this position and the corresponding frequencies and modes of natural oscillations, including taking into account practically established limitations. The research method uses a qualitative approach, systematically based on the well-known Rayleigh theorem on the effect of imposing constraints on the free vibrations of an elastic structure.

Download Full-text

Recent research advances in the dynamic behavior of shells: 1989–2000, Part 2: Homogeneous shells

Applied Mechanics Reviews ◽

10.1115/1.1483078 ◽

2002 ◽

Vol 55 (5) ◽

pp. 415-434 ◽

Cited By ~ 143

Author(s):

Mohamad S Qatu

Keyword(s):

Finite Elements ◽

Dynamic Response ◽

Initial Stress ◽

Laminated Composite ◽

Free Vibrations ◽

Review Article ◽

Analysis Method ◽

Elastic Supports ◽

Added Masses ◽

Laminated Composite Shells

Abstract Shell-like structures are used in various engineering applications including civil, aerospace, mechanical, marine, and automotive engineering. This article reviews most of the recent research done in the field of dynamic response of homogeneous shells with special attention to free vibrations. Literature on dynamics of laminated composite shells was reviewed in Part 1 published in the July 2002 issue of AMR. Emphasis is given to the theory being applied (thin, thick, 3D, nonlinear,…), the shell geometries that were subject to dynamics research (cylindrical, conical, spherical,…), the analysis method (exact, Ritz, finite elements,…), and the various complicating effects (initial stress, imperfection, added masses and springs, elastic supports, rotating shells, interaction with fluids, and others). This review article contains 606 references.

Download Full-text