Dynamic stability of piles with fractional damping foundation

This paper investigates the dynamic stability and vibration response of a slender pile embedded in soil, both theoretically and numerically. The pile is modeled as a column resting on a Winkler-type elastic foundation with fractional damping characteristics. Investigation of the pile loaded axially leads to a fractional Mathieu differential equation of motion. To analyze stability, the Bolotin method is employed to derive approximate instability boundaries. Additionally, a numerical approach based on block-pulse functions is developed to construct detailed instability diagrams, which also serve to validate the theoretical results obtained using the Bolotin method. It is found that higher order instability regions generally need higher order infinite determinants. Finally, a numerical case study is presented to explore the effects of key parameters—such as the fractional order, foundation stiffness, damping coefficient, and static and dynamic load components—on the pile’s dynamic stability. The dynamic stability of a pile subjected to a real earthquake is determined.