On oscillation modes and basins of attraction of a harmonically excited 2-DOF cart model oscillating over an inclined surface under 1:1 internal resonance

The nonlinear dynamics of a 2-DOF cart model system oscillating over an inclined surface are explored in this work. The considered structure consists of two subsystems: Cart-1, which acts as the primary system and is modeled as a hard-spring Duffing oscillator subjected to near-resonant harmonic excitation, and Cart-2, another Duffing oscillator with either a hard or soft spring, supported by Cart-1 and oscillating over its surface at an inclination angle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> relative to the horizontal direction. Using the Lagrange formulation, the mathematical model of the entire system is derived, and the corresponding normalized equations of motion are obtained. Utilizing multiple time-scale techniques, the autonomous slow-flow dynamical system governing the modulated amplitudes and phases of the 2-DOF cart model is extracted. By applying the continuation algorithm, the evolution of oscillation amplitudes as functions of the excitation force and frequency is analyzed, revealing various response curves. In terms of Poincaré return maps, bifurcation diagrams, basins of attraction, the <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:mn>1</mml:mn> </mml:mrow> </mml:math> chaos test, and stability charts, all established response curves are validated numerically. The analysis reveals that the studied system exhibits six oscillation modes, depending on both the excitation force and frequency. These modes include monostable, bistable, and tristable periodic solutions, period-n solution, a mono-quasiperiodic solution, and the coexistence of monostable periodic and mono-quasiperiodic solutions. Furthermore, it is found that Cart-2 functions as an effective vibration absorber under linear coupling with Cart-1. However, softening nonlinear stiffness coupling between the two carts induces unbounded oscillations, which could ultimately compromise the structural integrity of the coupled subsystems. This study contributes to the fields of passive vibration control and transportation system safety and stability, offering valuable insights for the design and optimization of coupled oscillatory systems with varying orientations in diverse engineering applications.