A novel methodology for a time-delayed controller to prevent nonlinear system oscillations

Abstract

The paper investigates the nonlinear transversal vibrations of a cantilever beam structure in the primary resonance case. A time-delayed position-velocity control is suggested to reduce the nonlinear vibrations of the structure under consideration. A non-perturbative method (NPM) is used to get an equivalent analogous linear differential equation (DE) to the original nonlinear one. For the benefit of the readers, a comprehensive description of the NPM method is provided. The theoretical findings are validated through a numerical comparison carried out by employed the Mathematica Software. Both the numerical solutions and the theoretical outcomes showed excellent agreement. As well-known, all classic perturbation techniques use Taylor expansion, when the restoring forces are present, to expand these forces and therefore lessen the difficulty of the given problem. Under the NPM, this weakness is no longer present. Furthermore, one may examine the stability examination of the issue with the NPM something that was not possible with prior traditional techniques. The controlled linear equivalent model is examined using the multiple-scales homotopy method. The amplitude-phase modulation equations which control the dynamics of the structure at the various resonance circumstances are established. The loop-delay stability diagrams are analyzed. It is looked at how the different controller parameters impact the oscillation behaviors of the system. The obtained theoretical outcomes showed that the loop delay has an important impact on the effectiveness of the control. Therefore, the ideal loop-delay values are given and used to develop the enactment of the organized control. The completed analytical results are also numerically validated, to reveal their good correlation with the achieved theoretical new results.

Short remarks on the paper: A novel methodology for a time-delayed controller to prevent nonlinear system oscillations

“…It raises a pertinent question: why do authors pursue approximate solutions when an exact solution, as presented in equation (10), is readily available? Typically, this approach is adopted to evaluate a new approximation method by contrasting it with the 14) and the numerical solution of equation (1). For the numerical values: ω¼ 3; μ¼ 0:1; 14) and the numerical solution of equation (1).…”

“…Typically, this approach is adopted to evaluate a new approximation method by contrasting it with the 14) and the numerical solution of equation (1). For the numerical values: ω¼ 3; μ¼ 0:1; 14) and the numerical solution of equation (1). For the numerical values: ω¼ 3; μ¼ 0:1; a 1 ¼ 0:1; a 2 ¼ 0:1; a 3 ¼ 0:1; f ¼ 1:2; β 1 ¼ 0; β 2 ¼ 0; A¼ 1; V ¼ 0:5.…”

“…For the numerical values: ω¼ 3; μ¼ 0:1; a 1 ¼ 0:1; a 2 ¼ 0:1; a 3 ¼ 0:1; f ¼ 1:2; β 1 ¼ 0; β 2 ¼ 0; A¼ 1; V ¼ 0:5. 14) and the numerical solution of equation (1). For the numerical values: ω¼ 3; μ¼ 0:1; a 1 ¼ 0:1; a 2 ¼ 0:1; a 3 ¼ 0:1; f ¼ 1:2;…”

“…In this concise note, we elucidate a fundamental error present in the results published in1, arising from the use of the natural frequency instead of the linear frequency in the proposed trial solution. We provide a corrective solution, as detailed in (6), 14) and the numerical solution of equation (1). For the numerical values: ω¼ 3; μ¼ 0:1; and address the necessary adjustments throughout the entire article to rectify this mistake.…”

“…In the study outlined in Ref 1, the focus is on a Duffing-type equation representing a forced delayed nonlinear oscillator. The approach taken is a non-perturbative method aimed at approximating a solution for equation (1). This method's cornerstone is the transformation of the nonlinear oscillator equation into an equivalent linearized version.…”