Nonlinear Dynamics of a Pendulum with Periodically Varying Length
This paper presents an analytical and numerical investigation into the dynamics of a pendulum with a periodically varying length (PPVL), offering significant insights into its stability characteristics and chaotic motion. The authors, Anton BELYAKOV and Alexander SEYRANIAN, explore the traditional problem of oscillations, drawing parallels with a simple swing model. Historically, while the PPVL concept is well acknowledged, analytical results have been sparse, claiming reference in works dating back to the mid-20th century.
The paper systematically develops asymptotic expressions for the boundaries of instability domains near resonance frequencies. The authors provide a comparative analysis between analytical and numerical results, identifying domains for various motion types - oscillation, rotation, and combined oscillation-rotation. Noteworthily, the paper conducts a numerical investigation into two distinct types of transitions to chaos.
The pendulum's motion is characterized using a quasi-linear approach under the assumption of minimal problem parameters, including excitation amplitude, damping coefficient, and frequency. This assumption allows deriving higher-order approximations utilizing the averaging method, thus facilitating the exploration of both regular and chaotic dynamics.
Instability Domains: The research analytically determines the boundaries of instability within a three-dimensional parameter space, demonstrating that domains for the lower vertical position form semi-conal structures. However, the second domain remains unoccupied. This theoretical result is verified through comprehensive numerical simulations, indicating a strong agreement between theoretical predictions and computational results, particularly near the first resonance domain.
Rotational Dynamics: Employing the averaging method, the authors address resonance rotations, highlighting the dependency of such rotations on specific excitation amplitudes. Regular rotations are explored, and domains of varying relative rotational velocities are mapped, validated numerically. The paper reveals the complexity of motion patterns within specified domain boundaries.
Chaos Transition: Two major transition types leading to chaos are delineated - namely, period-doubling (PD) bifurcation and subcritical Andronov-Hopf (AH) bifurcation. The research thoroughly maps the basins of attraction in Poincaré sections, elucidating the conditions under which various attractors coexist. These observations are interpreted within the context of Lyapunov exponents, indicating the regime shift from regular to chaotic motion.
Implications and Future Directions: The paper elucidates both the theoretical and practical implications of understanding PPVL dynamics. The implications extend to mechanical applications, potentially influencing design and stability assessments in engineering domains. This detailed exploration of instability and chaotic transitions contributes to the broader understanding of nonlinear dynamical systems.
Looking forward, such methodologies could advance research in areas encompassing advanced systems exhibiting complex dynamical behaviors. The results herein underscore the necessity for further refinements in modeling periodic systems, which may eventually lead to improved predictive capabilities, thereby enhancing stability characterization methods in mechanical structures.
Overall, the paper contributes robust analytical, numerical, and conceptual foundations to the paper of pendulum systems with periodically varying lengths, setting the stage for future explorations into nonlinear dynamics and chaos theory.
