A Novel Inspection of a Time-Delayed Rolling of a Rigid Rod (European Journal of Pure and Applied Mathematics, 2024)
The work examines the stability configuration of a rolling rigid rod in the presence of a time-delayed (TD) in a square position as well as velocity. Examining time-delayed rolling rigid rod bridges presents real engineering challenges and raises significant theoretical questions, making it a desirable problem in applied and theoretical contexts. It is recommended to use thenon-perturbative approach (NPA) to find an equivalent linearized differential equation. Actually, the NPA is based on the He’s frequency formula (HFF). The Mathematica Software (MS) is used to compare and assess of this suitability. Using the appropriate numerical methodology (NM), a matching between the strong nonlinear ordinary differential equation (ODE) and the equivalent analytical linear one is obtained. This matching has revealed a very significant agreement for different criteria. Stated differently, the new performance appears powerful, promising, and beneficial, and it may be applied to different classes of nonlinear oscillators. It is well-precision, flexible, and convenient. The new approach has many advantages in contrast to all other perturbed methods. It avoids the usage of the Taylor expansion in expanding the restoring forces; particularly in the topic of the dynamical systems. Therefore, the stability analysis is analyzed and the current work no longer incorporates this shortcoming. Furthermore, the temporal histories of the obtained novel outcomes and their various stable zones are accomplished. It is possible to examine the relevance of the employed parameter and demonstrate the precision of the outcomes through an exploration of the data. It is found that periodic solutions provide reliable and predictable behavior in a system, whereas phase plane diagrams offer a visual and quantitative comprehension of dynamics and stability. This appreciation is vital for ensuring safe operation under different conditions.
Keywords: Nonlinear vibrations; Square position-velocity time delay; Non-perturbative approach; Stability configuration