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Linear vibrations of periodic Timoshenko and Rayleigh beams

Świątek M., Jędrysiak J., Domagalski Ł.

Vibrations in Physical Systems

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2018

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Vol. 29

art. no. 2018035

EN

Elastic periodic structures with variable material and geometrical properties exhibit dynamic characteristics that are investigated in this contribution. The paper is devoted to analysis of geometrically linear vibrations of Rayleigh and Timoshenko beams with cross-sections and material properties periodically varying along the longitudinal axis. The period of inhomogeneity is assumed to be sufficiently small when compared to the beam length. Equations of motion in both beam theories under consideration have highly-oscillating coefficients. In order to derive the averaged model equations with constant coefficients for vibrations, the tolerance averaging approach is applied. The method of averaging differential operators with rapidly varying coefficients is applied to obtain averaged governing equations with constant coefficients. An assumed tolerance and indiscernibility relations and the definition of slowly varying function found the applied technique. Numerical results from the tolerance Rayleigh and Timoshenko beam model equations are compared.

2

Finite Transition Elements for Cst to 3-Node Timoshenko Beam Elements

Hać M.

Machine Dynamics Research

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2018

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Vol. 42, No. 3

119--125

EN

Effective finite element analysis of mechanical structures needs to use proper finite elements for modeling machine parts. The selection of finite elements is very important for the accuracy and reliability of the solution. If machine components are of complex shapes, there often appears a need to use different types of finite elements for modeling sub-areas of the considered element. These various types of finite elements are to be connected by properly selected transition finite elements. In the paper the application of transition finite elements for connection of plane CST (constant strain triangular) and 3-node Timoshenko beam elements is presented. The use of the transition elements enables to couple structural and continuum elements without using constraint equations. As an example a cantilever beam is considered which is fixed by one transitional element to the base. The numerical simulation’s results showed that it is very important to choose the appropriate finite element adapted to the corresponding sub-areas as well as the proper design of the transition element.

3

Non-linear vibration of Timoshenko beams by finite element method

Rakowski J., Guminiak M.

Journal of Theoretical and Applied Mechanics

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2015

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Vol. 53 nr 3

731—743

EN

The paper is concerned with free vibrations of geometrically non-linear elastic Timoshenko beams with immovable supports. The equations of motion are derived by applying the Hamilton principle. The approximate solutions are based on the negligence of longitudinal inertia forces but inclusion of longitudinal deformations. The Ritz method is used to determine non-linear modes and the associated non-linear natural frequencies depending on the vibration amplitude. The beam is discretized into linear elements with independent displacement fields. Consideration of the beams divided into the regular mesh enables one to express the equilibrium conditions for an arbitrary large number of elements in form of one difference equation. Owing to this, it is possible to obtain an analytical solution of the dynamic problem although it has been formulated by the finite element method. Some numerical results are given to show the effects of vibration amplitude, shear deformation, thickness ratio, rotary inertia, mass distribution and boundary conditions on the non-linear natural frequencies of discrete Timoshenko beams.