Nonlinear vibrations of an extensional beam with tip mass in slewing motion
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RECENT ADVANCES IN NONLINEAR DYNAMICS AND VIBRATIONS
Nonlinear vibrations of an extensional beam with tip mass in slewing motion Jerzy Warminski
. Lukasz Kloda
. Stefano Lenci
Received: 20 January 2020 / Accepted: 28 August 2020 Ó The Author(s) 2020
Abstract Dynamics of a rotor composed of a flexible beam attached to a slewing rigid hub is presented in the paper. Dynamics of the structure is studied for a slender beam model, based on extended Bernoulli–Euler theory, which takes into account a nonlinear curvature, coupled transversal and longitudinal oscillations and non-constant angular velocity of the hub. Moreover, to demonstrate a general case for dynamical boundary conditions, lumped mass fixed at the beam tip is added. The partial differential equations (PDEs) are derived from Hamilton principle of the least action. The analytical solutions of the PDEs are obtained by the multiple time scale method applied directly to PDEs. Forced vibrations around selected resonance zones are studied and the influence of beam rotation, preset angle, hub radius, tip mass is presented. Hardening and softening phenomena,
J. Warminski (&) L. Kloda Department of Applied Mechanics, Faculty of Mechanical Engineering, Lublin University of Technology, Lublin, Poland e-mail: [email protected] L. Kloda e-mail: [email protected] S. Lenci Department of Civil and Buildings Engineering, and Architecture, Polytechnic University of Marche, Ancona, Italy e-mail: [email protected]
respectively for the first and the second mode, are obtained for various angular velocity values. Keywords Extensional slewing beam Nonlinear beam model Nonlinear vibrations Analytical solutions
1 Introduction and motivation Rotating structures play an important role in mechanical or aerospace engineering, wind turbines or helicopter blades may serve as classical examples. Linear models of such structures are very useful in a case of relatively stiff systems, performing small oscillations. The development of composite materials technology gives a new possibility to create rotating blades having new properties and enable to perform large deformations. The methodology of determining equations of motion of composite beams or blades considered as a thin-walled structures was presented in books [13, 20, 22]. The rotating thin-walled composite blades model which takes into account composite beam layout and also hub dynamics was derived on the basis of Hamilton principle in [15]. The model enabled to study complex 3 D deformations of the blade, including warping effect, for various configurations of placement of the reinforcing fibres. The elaborated model took into account varied angular speed, coupled hub and beam dynamics
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and complex deformations of the thin-walled beam. But the proposed model gave reliable results for relatively small deformations of the beam. For flexible systems large deformations may occur. Therefore, nonlinear models have to be developed to predict properly dynamics
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