Fractional Euler-Lagrange modeling of a pendulum with a vertically vibrating pivot

This study develops classical and fractional variational models for a simple pendulum whose pivot undergoes prescribed vertical harmonic motion. The nonlinear classical equation of motion and its small-angle, Mathieu-type reduction are first derived from the Euler-Lagrange equation. Memory is then introduced through a dimension-preserving Caputo-type operator of order <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>α</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , and the corresponding left-right fractional Euler-Lagrange equation is obtained under the adopted endpoint convention. The small-angle, linearized fractional problem is recast as a coupled system consisting of a left-sided state equation and a right-sided auxiliary equation. Numerical solutions are computed over the finite interval <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> using an L1 discretization of the left- and right-sided Caputo derivatives. The discretization produces a coupled sparse linear boundary-value system that is solved directly for several values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> and the excitation frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> </mml:math> . The results demonstrate that the finite-time response depends strongly on both parameters. For the selected conditions, the cases <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>&gt;</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> produce quantitatively and qualitatively different trajectories. Maximum amplification, RMS response, deviation from the corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> solution, and a Hamiltonian-like diagnostic are used to supplement the time histories and parametric trajectories. In addition, as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the fractional equations and their numerical solutions approach the corresponding integer-order boundary-value problem. The formulation, therefore, provides a reproducible framework for studying the interaction between hereditary effects and prescribed base excitation.