Multiple Time-Scales Analysis to Predict the Quasiperiodic Oscillatory Response of a Thin-Walled Beam Subjected to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2725pt" id="M1" height="11.4049pt" version="1.1" viewBox="-0.0657574 -11.1324 46.2811 11.4049" width="46.2811pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g187-213" d="M345 0V33C260 38 258 46 258 122V635C174 612 105 594 35 583V554L87 550C124 546 130 539 130 494V122C130 46 127 38 42 33V0H345Z"/></g><g transform="matrix(.017,0,0,-0.017,11.167,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/></g><!-- ipunc --><g transform="matrix(.017,0,0,-0.017,19.874,0)"><use xlink:href="#g187-213"/></g><g transform="matrix(.017,0,0,-0.017,31.041,0)"><use xlink:href="#g113-59"/></g><!-- ipunc --><g transform="matrix(.017,0,0,-0.017,39.748,0)"><use xlink:href="#g187-213"/></g> Simultaneous Resonance - Kandil, Ali; Hamed, Y. S.; Awrejcewicz, Jan; Saeed, Nasser A. (Shock and Vibration, 2023)
This paper introduces a study on the horizontal and vertical deflections of the cross section of a thin-walled rotating beam. These deflections are governed by a system of two ordinary differential equations in order to describe their Cartesian directions. Based on multiple time-scales analysis, truncated asymptotic expansions are assumed to be approximate solutions to the given problem. Furthermore, an extracted autonomous system of differential equations governs the change rate of the amplitudes and phases of the beam deflections. The beam’s rotation speed is adjusted to be in the neighborhood of both of the natural frequencies of the deflections such that the beam is subjected to 1:1:1 simultaneous resonance. A stability test is conducted according to the first method of Lyapunov in order to determine whether the equilibrium point is asymptotically stable or not. The beam’s deflections turn unstable once its speed is in the neighborhood of its modal natural frequencies. There exists a multistable solution at some values of the beam’s speed depending on the hysteresis manner of the model according to forward or backward sweeping of this speed. Furthermore, a range of centrifugal forces of the rotating hub can make the beam’s deflections exhibit quasiperiodic responses which are confirmed by time response, orbital map, and amplitude spectrum. Eventually, some remarks are recommended for the external excitation frequency in order that the beam stays in the periodic behavior.