Semi-Analytical Modeling and Active Control of Nonlinear Vibrations in Membrane Structures with Non-Classical Boundaries

The nonlinear dynamic behavior of spacecraft membrane structures necessitates accurate response prediction and effective vibration suppression. To address the limitations of previous models assuming simplified boundary conditions, this study develops a semi-analytical modeling and control framework for membranes with non-classical boundaries. First, the dynamic model is formulated based on Hamilton's principle and discretized using the assumed mode method combined with Gram-Schmidt characteristic orthogonal polynomials, which efficiently capture the effects of complex boundary constraints. The incremental harmonic balance (IHB) method is then employed to solve the nonlinear steady-state response, with results validated using the fourth-order Runge-Kutta (RK4) method. Numerical simulations reveal pronounced nonlinear frequency responses characterized by stiffness hardening and modal coupling, leading to multiple turning points and bubble-shaped responses. Parametric analyses demonstrate the effects of excitation amplitude, pre-tension, and damping on the primary resonance, while bifurcation diagrams, phase portraits, and Poincaré maps further elucidate the transition mechanism of the global dynamic behavior. Finally, an active vibration control scheme based on negative velocity feedback (NVF) is incorporated to establish the closed-loop model. The influences of actuator/sensor placement, quantity, and control gain on suppressing nonlinear primary resonance are systematically explored. Results show that using four optimally placed actuator/sensor pairs achieves more than 75% vibration attenuation in the primary resonance regions. The proposed framework provides a comprehensive semi-analytical basis for response prediction, nonlinear vibration suppression, and stability enhancement in large-scale membrane-based space systems.