Viscous Damping Stabilization and Destabilization of Resonant Self-Tuning Variable-Length Structures

The effect of viscous damping on the nonlinear planar dynamics of a paradigmatic variable-length structural system is analyzed. The system comprises a lumped mass connected to a variable-length elastic structure, as constrained by a frictionless sliding sleeve. Although in a gravitational field parallel to the sliding direction, when the sleeve oscillates transversely and harmonically, the system can achieve a resonant self-tuning response that enables the lumped mass to overcome its fall through a configurational reaction at the sliding sleeve's exit. This response is achieved through sustained oscillatory motion around a finite length, which changes by varying the frequency and amplitude of the sleeve's oscillations. Using an experimentally validated viscous dissipation model, the influence of transverse and longitudinal damping on the self-tuning length and first-order limit cycles is examined analytically through a perturbation approach. The analysis reveals the potential for unique or multiple periodic responses, depending on system parameters and damping levels. Stability analysis via Floquet theory identifies conditions for monostable or bistable dynamic responses, showing that viscous damping can stabilize or, somewhat unexpectedly, destabilize the system, akin to other dynamic instability problems. These findings are finally validated by numerical integration of the fully nonlinear version of the equations of motion. The proposed framework facilitates the design of innovative devices incorporating bistable elements and metamaterials, enhancing their self-tuning capabilities over a broader frequency range. Additionally, it establishes a basis for future investigations under different dissipation sources and guides a mechanical design approach influenced by both the type and amount of dissipation.