Three-to-one internal resonance in multiple stepped beam systems

Abstract

In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales (a perturbation technique). The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and frequencyresponse curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.

Helicopter flap/lag energy exchange study

“…The nonlinear equations obtained represent an advantage with respect to the work presented in Chen [2] and Tekin et al [7], as it finds analogies with nonlinear systems, allowing to predict some behaviour on the system under study. This is an advantage with respect to Santos et al [6] and Tekin et al [7]. In addition, in order to test the impact of the rotor angular speed and the damping coefficient on the blade, the results proposed by Yan et al [8] are considered.…”

“…The nonlinear dynamical equations of the system -obtained via VehicleSim-will be analysed by studying an equivalent spring pendulum system that has two coupled degrees of freedom whose dynamical characteristics can resemble those of the coupled flap/lag motions. The aim therefore is to identify particular terms in the nonlinear equations of motion that can shed on light energy exchange being this something that should be considered as proposed by Saved et al [5] and Tekin et al [7]. The nonlinear equations obtained represent an advantage with respect to the work presented in Chen [2] and Tekin et al [7], as it finds analogies with nonlinear systems, allowing to predict some behaviour on the system under study.…”

“…The aim therefore is to identify particular terms in the nonlinear equations of motion that can shed on light energy exchange being this something that should be considered as proposed by Saved et al [5] and Tekin et al [7]. The nonlinear equations obtained represent an advantage with respect to the work presented in Chen [2] and Tekin et al [7], as it finds analogies with nonlinear systems, allowing to predict some behaviour on the system under study. This is an advantage with respect to Santos et al [6] and Tekin et al [7].…”

“…Superharmonic vibration associated to the dynamics of the flexible rotating parts could be identified by using the full nonlinear model. On the other hand, in Tekin et al [7], the nonlinear vibrations of multiple EulerBernoulli stepped beams were analysed. A 3:1 internal resonance case was studied for the system.…”

Non-Linear Transverse Vibrations and 3:1 Internal Resonances of a Tensioned Beam on Multiple Supports

“…These studies include restrained beams [4,5], in-span support [6,7], different boundary conditions for nonlinear vibrations [8][9][10], stepped beam systems using artificial neural networks [9] and finite element methods [12], single, multiple mass on simply supports [13][14][15][16], and non ideal support cases for three different simply supported beams [16], infinite mode analysis was performed [17]. Nonlinear vibrations and 3:1 internal resonances on multiple supports were investigated and excitation frequency-frequency response curves drawn for different support numbers [18] and Tekin et al studied on three-to one resonance in multi stepped beam systems [19]. For slightly curved beams with stretching, one may refer to Rehfield [20].…”