Vibrations of a nonlinear string with a nonlinear quasi-zero stiffness system as a boundary condition
J. van de Velde (TU Delft - Electrical Engineering, Mathematics and Computer Science)
W.T. van Horssen – Mentor (TU Delft - Mathematical Physics)
S. Jain – Graduation committee member (TU Delft - Numerical Analysis)
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Abstract
Vibrations in engineering structures can lead to severe instabilities, especially under low-frequency excitations that traditional linear isolators cannot effectively suppress. To address this, quasi-zero stiffness (QZS) vibration isolators, known for their high-static-low-dynamic stiffness properties, have gained increasing attention. This report investigates the reflection and absorption characteristics of a nonlinear string with a QZS mechanism applied as a boundary condition. The model is considered, and the governing equations are derived and nondimensionalized. Using regular perturbation methods and the method of multiple time scales, analytical solutions are obtained and evaluated. The analysis distinguishes between cases where the oblique springs are extended or compressed. It is found that with compressed springs, when the vertical damping coefficient is below unity, the system is counterintuitively stable. Furthermore, the inclusion of oblique dampers leads to unphysical energy growth. These phenomena are attributed to the singular nature of the system’s dynamics and the limitations of the chosen multiple time scale method. The results indicate that the current model does not fully capture the effects of the oblique springs and dampers, underscoring the need for further investigation into the system’s asymptotic expansions. Moreover, exploring second-order dynamics and external forcing could provide a deeper understanding of the system’s complex behaviour.