Rocking Blocks: Phase Shift Effects under Sinusoidal Forcing

Abstract

The dynamic response of rocking rigid blocks subjected to sinusoidal base excitation exhibits rich nonlinear behavior, with strong sensitivity to initial conditions and external parameters that often leads to chaotic dynamics. Classical analyses have identified bands of forcing frequencies that appear favorable for sustained rocking, whereas others are considered unfavorable due to overturn. This work revisits that assumption by explicitly controlling the initial phase of a second finite-duration sinusoidal pulse. Using an autonomous embedding that enforces phase scheduling and systematic sweeps over the full range ϕ∈[0,2π), we show that many frequencies previously deemed unfavorable can sustain bounded oscillations when phase alignment is chosen appropriately. Phase acts not only as an initial-condition knob but also through the implicit time factor in the excitation, reshaping the immediate torque balance right after activation. We quantify these effects via long-term response maps in the (tr, ϕ) plane (reaction time tr) and a complementary dimensionless momentum-impulse metric, revealing windows of stability within the “saw-like” diagrams reported when phase is uncontrolled. The results suggest phase-based control as a simple and effective strategy for mitigating overturn in rocking blocks, refine our understanding of chaotic transitions, and motivate amplitude-phase-frequency co-design for robust stabilization.