Scholarlink

<p>This paper investigates the nonlinear flutter in the vicinity of the linear flutter speed of an airfoil with cubic nonlinearity in plunge and multiple freeplays in pitch and control surface by Galerkin averaging-incremental harmonic balance (EGA-IHB). The model investigated in the paper is highly non-smooth, which contains multiple nonlinearities to simulate the actual problems. In order to track both stable and unstable periodic solutions, the EGA-IHB method is applied to the system for flutter analysis. The EGA-IHB method adopts tensor contraction, fast Fourier transform (FFT) and sparse matrix multiplication to decrease the complexity of computation and programming in IHB. In the EGA-IHB method, Galerkin averaging on the Jacobian matrix of differential equations is divided into two parts by tensor contraction. One part is a constant tensor on time integration, which depends only on the selected trigonometric bases. This part is only calculated once at the beginning of the procedure and repeatedly used in iterations. The other part is the Fourier coefficient vectors of elements in the Jacobian matrix, which can be calculated by FFT. Only this part needs to be updated in Newton iterations. Galerkin averaging is efficiently calculated using sparse matrix multiplication on the two parts. In this way, the differential equations to be solved are decoupled with trigonometric bases. The arc-length method is embedded in EGA-IHB to track bifurcation paths and the Floquet theory is adopted for stability analysis. Results from EGA-IHB agree well with those from the fourth-order Runge–Kutta (RK4). A complete parametric analysis for each nonlinear parameter is carried out. Multiple freeplays can produce multiple coexisting stable and unstable limit cycles, multiple bifurcations in the vicinity of the linear flutter speed. These bifurcations induced by multiple freeplays, including grazing, symmetry breaking and saddle-node bifurcations, will lead to shifting of limit cycles between different branches and jump phenomena in bifurcation diagrams. In contrast, limit cycle oscillations (LCOs) around the linear flutter speed of the present system are hardly affected by hardening and softening properties of plunge since these LCOs are caused by the freeplay nonlinearities. Based on the parametric discussion, some guidelines are made, which may contribute to the aeroelastic design of airfoils.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Nonlinear flutter analysis of an airfoil with cubic nonlinearity and multiple freeplays by the efficient Galerkin averaging-incremental harmonic balance method

  • 期刊论文

  • 发表日期：

    2025年6月16日

查看全文
  • Senkai Mo,
  • Kaiping Yu,
  • Rui Zhao,
  • Jinze Li

摘要

This paper investigates the nonlinear flutter in the vicinity of the linear flutter speed of an airfoil with cubic nonlinearity in plunge and multiple freeplays in pitch and control surface by Galerkin averaging-incremental harmonic balance (EGA-IHB). The model investigated in the paper is highly non-smooth, which contains multiple nonlinearities to simulate the actual problems. In order to track both stable and unstable periodic solutions, the EGA-IHB method is applied to the system for flutter analysis. The EGA-IHB method adopts tensor contraction, fast Fourier transform (FFT) and sparse matrix multiplication to decrease the complexity of computation and programming in IHB. In the EGA-IHB method, Galerkin averaging on the Jacobian matrix of differential equations is divided into two parts by tensor contraction. One part is a constant tensor on time integration, which depends only on the selected trigonometric bases. This part is only calculated once at the beginning of the procedure and repeatedly used in iterations. The other part is the Fourier coefficient vectors of elements in the Jacobian matrix, which can be calculated by FFT. Only this part needs to be updated in Newton iterations. Galerkin averaging is efficiently calculated using sparse matrix multiplication on the two parts. In this way, the differential equations to be solved are decoupled with trigonometric bases. The arc-length method is embedded in EGA-IHB to track bifurcation paths and the Floquet theory is adopted for stability analysis. Results from EGA-IHB agree well with those from the fourth-order Runge–Kutta (RK4). A complete parametric analysis for each nonlinear parameter is carried out. Multiple freeplays can produce multiple coexisting stable and unstable limit cycles, multiple bifurcations in the vicinity of the linear flutter speed. These bifurcations induced by multiple freeplays, including grazing, symmetry breaking and saddle-node bifurcations, will lead to shifting of limit cycles between different branches and jump phenomena in bifurcation diagrams. In contrast, limit cycle oscillations (LCOs) around the linear flutter speed of the present system are hardly affected by hardening and softening properties of plunge since these LCOs are caused by the freeplay nonlinearities. Based on the parametric discussion, some guidelines are made, which may contribute to the aeroelastic design of airfoils.