On the Stability of a 3DOF Vibrating System Close to Resonances

PurposeIn the current work, the motion of a three degrees-of-freedom (DOF) dynamical system as a vibrating model is examined. The proposed system is of high importance in vibration engineering applications, such as the analysis of the control of flexible arm robotics, flexible arm vibrational motion as a dynamic system, pump compressors, transportation devices, rotor dynamics, shipboard cranes, and human or walking analysis robotics.MethodsLagrange's equations (LE) are used to derive the equations of motion of the controlling system. The analytic solutions (AS) are obtained utilizing the multiple-scales method (MSM) up to the third order.ResultsThe framework for removing secular terms provides the requirements for the solvability of this problem. Various resonance scenarios are categorized and the modulation equations (ME) are constructed. To graphically demonstrate the beneficial impacts of the distinct parameters of the problem, the time histories (TH) of the approximate solutions as well as the resonance curves (RC) are depicted. The Runge-Kutta algorithm (RKA) is employed to obtain the numerical solutions (NS) of the regulating system.ConclusionA comparison of the AS and NS reveals the accuracy of the perturbation approach. The stability/instability zones are studied using Routh-Hurwitz criteria (RHC), and then they are examined using a steady-state situation. Basically, the used perturbation method is considered a traditional method that is applied to solve a new dynamical system. Then, the achieved results are considered new because they weren't obtained previously, which indicates the novelty of this work.