Zeliang Liu

To address the insufficient convergence robustness, strong dependence on the initial guess, and computational-efficiency bottlenecks of the incremental harmonic balance (IHB) method for strongly nonlinear systems, this paper develops an adaptive framework that integrates a trust-region strategy with a hybrid Hessian matrix. The framework reformulates the harmonic-balance iteration as a constrained nonlinear least-squares optimization problem, constructs a symmetric hybrid Hessian matrix by blending the Gauss–Newton and Newton directions, and uses a trust-region algorithm to adaptively regulate the step size, thereby jointly optimizing the iterative path and convergence behavior. Numerical results show that the proposed method significantly enhances convergence robustness, reduces sensitivity to initial guesses, and improves computational efficiency while maintaining high accuracy. It also captures both stable and unstable periodic solutions in strongly nonlinear systems.

The nonlinear vibrations developing in sheets and strips during rolling are of practical technological importance and of theoretical interest. However, the existing numerical methods adopted in research are plagued by problems such as computational inefficiency, difficulty in dealing with bifurcations, slow convergence and frequency cancellations. This study develops an analytical method for solving the problem of rolled strip quasi‐periodic motions in the resonant state. First, the conventional incremental harmonic balance method is used to analyze the periodic response of a strip. Its nonlinear frequency response curve is obtained and the type of bifurcation along with its location is determined. Second, the quasi‐periodic motion spectrum of the strip is characterized utilizing the fact that the adjacent frequencies in the side bands are equidistant from each other and contain two incommensurable fundamental frequencies. The fundamental frequencies of the strip quasi‐periodic motion, as well as all of the frequency components and their corresponding amplitudes, are accurately resolved, especially the quasi‐periodic motion response near the critical point. Finally, several numerical examples of the nonlinear vibrations exhibited by the rolled strip in the presence of internal resonance are investigated in depth to validate the proposed method performance and accuracy. These examples are concerned with the internal resonance with an even inherent frequency ratio of 1:6, which may arise during the rolling process, but remains relatively unexplored. The newly proposed method will add to the arsenal of available analytical techniques, expanding the capabilities of both researchers and practicing engineers.

Creating lightweight and impact-resistant box structures has been an enduring pursuit among researchers. A new energy-absorbing structure consisting of a bionic gradient lattice-enhanced thin-walled tube is presented in this article. The gradient lattice and thin-walled tube were prepared using selective laser melting (SLM) and wire-cutting techniques, respectively. To analyze the effects of gradient pattern, mass ratio, diameter range and impact speed on structural crashworthiness, low-speed impact at 4 m/s and finite element simulation experiments were conducted. The study demonstrates that the design of inward radial gradient lattice-reinforced thin-walled tubes can effectively enhance structure’s energy-absorption efficiency and provide a more stable mode of deformation. It also shows a 17.44% specific energy-absorption advantage over the uniformly lattice-reinforced thin-walled tubes, with no significant overall gain in peak crushing force. A complex scale evaluation method was used to determine the optimum structure and the structure type with the best crashworthiness was found to be a gradient lattice-filled tube with a thickness of 0.9 mm and a slope index of 10. The gradient lattice-reinforced thin-walled tube suggested in this investigation offers guidance for designing a more efficient thin-walled energy-absorption structure.