INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.     

Bifurcation and chaos of the circular plates on the nonlinear elastic foundation

According to the large amplitude equation of the circular plate on nonlinear elastic foundation , elastic resisting force has linear item , cubic nonlinear item and resisting bend elastic item. A nonlinear vibration equation is obtained with the method of Galerkin under the condition of fixed boundary. Floquet exponent at equilibrium point is obtained without external excitation. Its stability and condition of possible bifurcation is analysed. Possible chaotic vibration is analysed and studied with the method of Melnikov with external excitation . The critical curves of the chaotic region and phase figure under some foundation parameters are obtained with the method of digital artificial.     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Quasi-periodic and chaotic behaviour of a two-degree-of-freedom impact system in a strong resonance case

A two-degree-of-freedom system contacting a single stop is considered. Quasi-periodic and chaotic behavior of the system in a strong resonance case is investigated by theoretical analysis and numerical simulation This work was supported by the National Natural Science Foundation of China(19672052).     

Dynamic analysis of two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flow

The nonlinear aeroelastic response of a two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flows is investigated. The second-order piston theory is used to analyze a double-wedge airfoil. Then, the fold bifurcation and the amplitude jump phenomenon are detected by the averaging method and the multi-variable Floquet theory. The analytical results are further verified by numerical simulations. Finally, the influence of the freeplay parameters on the aeroelastic response is analyzed in detail.     

Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ₃2₂ and ₂2₁ to a harmonic excitation of the third mode, where the _m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.     

一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉

通过理论分析和数值仿真,研究了一类二维冲击振动系统在一种强共振条件下的Hopf分叉与亚谐分叉。分析并证实了该类系统在此共振条件下可由稳定的周期1 1振动分叉为周期4 4振动或概周期振动,讨论了亚谐振动和概周期振动向混沌运动的演化过程。     

The post-Hopf-bifurcation response of an airfoil in incompressible two-dimensional flow

A bifurcation analysis of a two-dimensional airfoil with a structural nonlinearity in the pitch direction and subject to incompressible flow is presented. The nonlinearity is an analytical third-order rational curve fitted to a structural freeplay. The aeroelastic equations-of-motion are reformulated into a system of eight first-order ordinary differential equations. An eigenvalue analysis of the linearized equations is used to give the linear flutter speed. The nonlinear equations of motion are either integrated numerically using a fourth-order Runge-Kutta method or analyzed using the AUTO software package. Fixed points of the system are found analytically and regions of limit cycle oscillations are detected for velocities well below the divergent flutter boundary. Bifurcation diagrams showing both stable and unstable periodic solutions are calculated, and the types of bifurcations are assessed by evaluating the Floquet multipliers. In cases where the structural preload is small, regions of chaotic motion are obtained, as demonstrated by bifurcation diagrams, power spectral densities, phase-plane plots and Poincaré sections of the airfoil motion; the existence of chaos is also confirmed via calculation of the Lyapunov exponents. The general behaviour of the system is explained by the effectiveness of the freeplay part of the nonlinearity in a complete cycle of oscillation. Results obtained using this reformulated set of equations and the analytical nonlinearity are in good agreement with previously obtained finite difference results for a freeplay nonlinearity.     

Bifurcation analysis of a double pendulum with internal resonance

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Application of wavelet transform to bifuraation and chaos study

The response of a nonlinear vibration system may be of three types, namely, periodic, quasiperiodic or chaotic, when the parameters of the system are changed. The periodic motions can be identified by Poincare map, and harmonic wavelet transform (HWT) can distinguish quasiperiod from chaos, so the existing domains of different types of motions of the system can be revealed in the parametric space with the method of HWT joining with Poincar'e map. Communicated Zhang Ruqing Project supported by the National Natural Science Foundation of China     

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

Grazing Bifurcation in the Response of Cracked Jeffcott Rotor

A cracked rotor is modeled by a piecewise linear system due to thebreath of crack in a rotating shaft. The differential equations ofmotion for the nonsmooth system are derived and solved with thenumerical integration method. From the simulation results, it isobserved that a grazing bifurcation exists in the response. Thegrazing bifurcation can give rise to jumps between periodic motions,quasi-periodic motions from the periodic ones, chaos, and intermittentchaos.     

Bifurcation characteristics of a two-dimensional structurally non-linear airfoil in turbulent flow

The dynamics of a structurally non-linear two-dimensional airfoil in turbulent flow is investigated numerically using a Monte Carlo approach. Both the longitudinal and vertical components of turbulence, corresponding to parametric (multiplicative) and external (additive) excitation, respectively, are modelled. The properties of the airfoil are chosen such that the underlying non-excited, deterministic system exhibits binary flutter; the loss of stability of the equilibrium point due to flutter then leads to a limit cycle oscillation (LCO) via a supercritical Hopf bifurcation. For the random system, the results are examined in terms of the probability structure of the response and the largest Lyapunov exponent. The airfoil response is interpreted from the point of view of the concepts of D- and P-bifurcations, as defined in random bifurcation theory. It is found that the bifurcation is characterized by a change in shape of the response probability structure, while no discontinuity in the variation of the largest Lyapunov exponent with airspeed is observed. In this sense, the trivial bifurcation obtained for the deterministic airfoil, where the D- and P-bifurcations coincide, appears only as a P-bifurcation for the random case. At low levels of turbulence intensity, the Gaussian-like bell-shaped bi-dimensional PDF bifurcates into a crater shape; this is interpreted as a random fixed point bifurcating into a random LCO. At higher levels of turbulence intensity, the post-bifurcation PDF loses its underlying deterministic LCO structure. The crater is transformed into a two-peaked shape, with a saddle at the origin. From a more universal point of view, the robustness of the random bifurcation scenario is critiqued in light of the relative importance of the two components of turbulent excitation.     

Numerical simulation of thin airfoil stall by using a modified DES approach

The detached‐Eddy simulation (DES) method was applied to calculate pre‐ and post‐stall aerodynamic characteristics of airfoil stall. A discrepancy between numerical and experimental data was observed near the stall regime for the airfoil NACA64_A‐ 006 which is a thin airfoil stall type. The reason of this discrepancy and one possible way for improvement of the numerical model are discussed here. It is shown that the use of the Baldwin–Lomax model in the RANS region improves the DES results in this case. If the relevant factors (grid density, time step, turbulence model, etc.) are appropriately taken into account, the DES approach could reliably predict stall aerodynamical characteristics. Copyright © 2006 John Wiley & Sons, Ltd.     

On the response of a preloaded mechanical oscillator with a clearance: Period-doubling and chaos

The dynamic behavior of a harmonically excited, preloaded mechanical oscillator with dead-zone nonlinearity is described quantiatively. The governing strongly nonlinear differential equation is solved numerically. Damping coefficient-force ratio maps for two different values of the excitation frequency have been formed and the boundaries of the regions of different motion types are determined. The results have been compared with the results of the forced Duffing's equation available in the literature in order to identify the differences between cubic and dead-zone nonlinearities. Period-doubling bifurcations, which take place with a change of any of the system parameters, have been found to be the most common route to chaos. Such bifurcations follow the scaling rule of Feigenbaum. b half length of the clearance.     

On CML model for study of spatiotemporal chaos

A new coupled map lattic(CML)model is given by using some stability analysis for the related difference equations.Numerical results show that the new model is.an effective one of studying spatiotemporal chaos,especially for strongly coupled systems.     

二维Logistic映射的分岔与分形

理论分析了二维Logistic映射的分岔,并采用相图、分岔图、功率谱、Lyapunov指数和分维数计算的方法,揭示出：二维Logistic映射可按倍周期分岔和Hopf分岔走向混沌;在倍周期分岔过程中,系统在参数空间和相空间中都表现出自相似性和尺度变换下的不变性．对二维Logistic映射的吸引盆及其Mandelbrot-Julia集(简称M-J集)的研究表明：吸引盆中周期和非周期区域之间的边界是分形的,这意味着无法预测相平面上点运动的归宿;M-J集的结构由控制参数决定,且它们的边界是分形的．