Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation

Abstract

We present results of the investigations on flexural-flexural-torsional nonlinear vibrations of twisted rotating beams described by the system of three nonlinear integro-differential equations expressed in partial derivatives. Cross-sectional deplanation of beams is adjusted to the equations and it is assumed that centre of gravity and shear centre are located at different points. Vibration systems are expressed as a row by free forms of the linear problem. Free vibrations are studied with the help of the Show-Pierre nonlinear normal mode.Keywords: flexural-flexural-torsional vibrations, nonlinear normal mode method, backbone curves. Introduction.Rotating beams are the components of the helicopter blades, manipulators, gas and steam turbine blades and propeller blades. In service, such beam structures often make oscillations and cause fatigue damage. Such vibrations are difficult for studying because their cross sections are asymmetric. As the cross section is asymmetric, the centre of gravity and shear centre of a beam cross-section are not assumed to coincide.The attempts to study nonlinear vibrations of beams with asymmetric cross section have been already undertaken. Timoshenko [1] has derived the equations of linear flexural-torsional vibrations of direct untwisted beams with asymmetric cross section. Vorob'ev and Shorr [2, 3] obtained the equations of linear flexural-flexuraltorsional-longitudinal vibrations of twisted rotating beams with cross-sectional deplanation for displacements and under torsion. System of the equations expressed in partial derivatives which describes geometrically nonlinear flexural-flexural-torsional-longitudinal vibrations of rotating beams is presented in [4]. In these equations, an assumption is made that centre of gravity of the cross section and shear centre are located at the same point. Crespo da Silva [5, 6] obtained the equations which describe flexural-flexural-torsional vibrations of beams with allowance for inextensibility of the median line, assuming that centre of gravity and shear centre are located at the same point. The theory of flexible beams is described in [7]. We introduce the system of three nonlinear integro-differential equations expressed in partial derivatives which describe flexural-flexural-torsional vibrations of rotating beam with allowance for cross-sectional deplanation. In deriving the equation, it was assumed that center of gravity and shear centre do not coincide. Discrete nonlinear dynamic system which approximates the indicated vibrations has been obtained. Motion of system is studied with the help of a nonlinear normal mode method. The effect of cross-sectional deplanation on vibrations is analyzed.Vibration Equations. A rotating beam with constant angular velocity W is considered (Fig.

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

“…al. [33] analyzed flexural-flexural-torsional free vibrations of twisted rotating beams employing nonlinear normal modes. Rozmarynowski and Szymczak [34] and Sapountzakis and Tsipiras [35] focus to the problem of nonlinear torsional vibrations.…”

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

“…Machado and Cortinez in [13] presented also a FEM solution to the linear free vibration analysis of bisymmetric thin-walled composite beams of open cross section, taking into account initial stresses and deformations considering geometrical nonlinearity only for the static loading and presenting examples of bars subjected to special boundary conditions. Avramov et al [14][15] studied the free flexural-torsional vibrations of beams and obtained nonlinear normal modes by expansion of the equations of motion employing the Galerkin technique and neglecting the cross section warping. Lopes and Ribeiro [16] studied also the nonlinear flexural-torsional free vibrations of beams employing a FEM solution and neglecting the longitudinal and rotary inertia as well as the cross-section warping.…”

Nonlinear Effects in Elastic Flexural-Torsional Vibrations of Beams of Arbitrary Cross Section

“…al. [14][15] studied the free flexural-torsional vibrations of beams and obtained nonlinear normal modes by expansion of the equations of motion employing the Galerkin technique and neglecting the crosssection warping. Lopes and Ribeiro [16] studied also the nonlinear flexural-torsional free vibrations of beams employing a FEM solution and neglecting the longitudinal and rotary inertia as well as the cross-section warping.…”