Semi-analytical framework for nonlinear vibration analysis of hard-magnetic soft beams

Hard-magnetic soft beams (HMSB) have emerged as foundational components for magnetic soft continuum robots, where resonant responses under periodic magnetic excitations govern bio-inspired locomotion modes such as crawling and swimming. However, the inherently strong geometric nonlinearities induced by large deformations lead to complex dynamic phenomena—including bifurcations, amplitude jumps, and multiple solutions—that challenge conventional transient dynamics frameworks. To address this, we propose a semi-analytical nonlinear dynamic framework of HMSB integrating three key advancements: (1) A geometrically exact kinematic model based on angular coordinates to capture large deformations; (2) An incremental harmonic balance (IHB) method enhanced by arc-length continuation for efficiently tracing stable/unstable periodic branches; (3) Parametric analysis of magnetic field amplitude, particle volume fractions, and nonuniform magnetization patterns. The framework is validated through numerical method and experimental data, first revealing the nonlinear dynamic characteristics of HMSB in both the primary and secondary resonance regions. In the primary resonance region, amplitude-frequency curves exhibit hardening behavior modulated by particle volume fraction φ, with a 40 % amplitude enhancement (compared to uniform φ = 20 %) and a 65 % reduction (compared to uniform φ = 40 %) in amplitude achieved via nonuniform magnetization pattern design. In the secondary resonance region, small amplitude and high-frequency oscillations are dominated by large damping, reducing nonlinear effects. This framework bridges the gap between nonlinear dynamics theory and magnetoactive soft robotic design, offering predictive tools for tailoring resonance-driven locomotion in soft robots.