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Effects of linear and nonlinear acceleration feedback on the impact dynamics of a cantilever beam using the incremental harmonic balance method
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DOI：10.1007/s11071-026-12795-z.png)
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Abstract
En 中文
Impact phenomena occur in many engineering systems—such as gear transmissions with backlash, robotic manipulators, and jib cranes—where structural components often behave like cantilever beams with distinct piecewise-linear or nonlinear properties. Traditional perturbation-based analyses relying on a few harmonics often fail to capture the complex and strongly nonlinear dynamics associated with discontinuities in the equation of motion. A thorough understanding of these dynamics, along with effective control strategies, is therefore crucial for ensuring performance and safety. This study provides a detailed frequency-domain analysis of the fundamental and multi-periodic resonances of a highly nonlinear cantilever beam with a tip mass, subjected to one-sided and two-sided elastic impact constraints. The analysis uses the incremental harmonic balance (IHB) method, which accurately characterizes steady state responses of the beam with strong nonlinearity and discontinuities. For the one-sided impact case, both subcritical and supercritical flip bifurcations are identified, leading to period-doubling cascades and chaotic responses. To reduce unwanted multi-periodic resonances and chaos, the study introduces time-delayed linear and nonlinear acceleration feedback control schemes—an area rarely explored with the IHB framework. Control gains and delay parameters are chosen based on the linearized stability of the static equilibrium. The proposed controllers effectively prevent jump phenomena and bi-stability, producing small-amplitude, mono-stable responses. Chaotic motions caused by period-doubling bifurcations can be stabilized to a low-amplitude period-1 response through suitable gain–delay settings. Time-domain simulations, including time histories, phase portraits, Fourier spectra, bifurcation diagrams, and basins of attraction, strongly agree with steady-state IHB results for both uncontrolled and controlled cases. Basin analysis shows that increasing the negative linear acceleration feedback reduces the likelihood of convergence to higher-harmonic attractors. Overall, the findings offer new insights into analyzing and controlling impact-driven systems and demonstrate the effectiveness of time-delayed acceleration feedback within the IHB framework.
Keywords:
Impact dynamics
Piecewise nonlinear system
Delayed acceleration feedback control
Cantilever beam
Incremental harmonic balance
Period-doubling route to chaos
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6
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1.4W
Citations:
4.1W
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