壁板非线性气动弹性颤振及稳定性研究
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摘要
壁板颤振是发生在高速飞行器上的一种典型的超音速气动弹性现象。壁板颤振对飞行器结构的疲劳寿命、飞行性能甚至飞行安全带来不利的影响。本文基于气动力活塞理论、Von Karman大变形几何应变-位移关系以及伽辽金离散方法,对壁板热气动弹性系统非线性响应的稳定性进行了深入的研究,分析各种参数对系统颤振临界动压与极限环幅值的影响以及系统发生分叉失稳的复杂运动形态。主要体现在以下几个方面:
1、将二维壁板非线性颤振系统的高阶运动微分方程化为四维一阶微分方程,针对系统在平衡点处的Jacobi矩阵,应用特征值理论和Hopf分叉代数判据解析推导了系统发生静态叉式分又及动态Hopf分叉的临界条件。在参数平面内讨论了该系统在各个区域内平衡点的个数和稳定性。
2、运用准定常的热应力公式,基于一阶气动力活塞理论建立了考虑温度效应的几何非线性无限展长二维壁板颤振方程。采用伽辽金方法对方程进行离散处理,基于Matlab编程对系统进行了数值仿真分析。结果表明温度载荷降低了系统的颤振临界动压,系统极限环响应幅值随温度的升高而增大。位移响应的分又图表明,系统随温度的变化表现出各种周期振动行为,且在特定的参数区间内存在混沌响应。热效应导致的材料非线性使得系统的颤振临界动压进一步减小,在分叉参数范围内系统表现出更为丰富的非线性动力学行为。
3、基于三阶活塞气动力理论,采用伽辽金方法建立了热环境下三维壁板在超音速流作用下的颤振方程。采用龙格-库塔法对系统进行数值积分,分析模态截取阶数、温度、非线性气动力以及壁板几何尺寸对壁板颤振的影响规律。结果表明,热应力是降低壁板稳定性的一个重要因素,随着温度的升高,壁板颤振临界动压降低。以温度为可变参量,给出了壁板振动的分叉图。结果表明,系统不仅存在简单的极限环颤振,而且会出现周期2、周期4等复杂的运动形态,甚至会出现混沌运动。系统通向混沌的道路不都是周期倍化分叉,还有拟周期道路等。继续升高温度,系统还会产生热屈曲失稳。
4、以热环境下三维壁板为研究对象,研究大气紊流作用下系统的动力响应。将大气紊流速度分解为平均速度和脉动速度两部分,采用Von Karman紊流谱结合随机场三角级数合成法得到时域脉动速度的表达式。基于一阶活塞理论求解气动力,用随机理论对壁板均方根响应值进行分析。着重考查了紊流场的几个主要参数对均方根响应值的影响。结果表明,当来流平均动压接近和超过颤振临界动压时,均方根响应值随平均动压和温度的增大而增加。均方根响应值仅在平均动压接近颤振临界动压时随紊流强度变化显著,但对紊流尺度的变化不敏感。
5、基于Kelvin粘弹模型建立了三维粘弹壁板非线性颤振方程。通过对粘弹阻尼、面内力、壁板长宽比和来流速度变参分析,研究上述参数变化对粘弹壁板颤振的影响规律。在无量纲动压—面内力的二维参数平面内分析粘弹壁板颤振系统的运动特性,给出了系统存在不同响应特性的分区图。结果表明,颤振临界动压随粘弹阻尼的增大先减小后增大。混沌响应区域随粘弹阻尼系数的增大快速减小,说明粘弹阻尼有抑制混沌响应的作用。而屈曲区域却基本不受影响。通过分叉图观察到随着动压的变化,粘弹性壁板系统由混沌经一系列的倒分叉进入简单的极限环颤振状态。
6、以面内力作用的壁板模型为对象,研究边界松驰条件下壁板的复杂响应。首先采用坐标变换结合伽辽金方法建立非理想边界三维壁板几何非线性颤振方程,再通过对非线性系统的数值积分,分析不同参数下该系统的周期振动以及进入各种复杂分叉的过程。结果表明,随边界约束的放松,颤振临界动压减小。分别以边界松驰因子、来流动压及面内压力为可变参数,研究系统的分叉现象。结果表明,系统存在Hopf分叉、极限环叉式分叉、混沌以及混沌中周期3窗口等。另外,在一些特殊参数区域内,同一动压下,系统存在两个稳定的极限环。根据初始条件的不同,系统分别收敛到不同的极限环上。
The panel flutter is a classical supersonic aeroelastic phenomenon on the outer skin of high-speed aerospace vehicles, which always leads to the fatigue failure of the skin panels and has a harmful influence on the flight perfermance and even flight safety. Based on the piston theory of supersonic aerodynamics, Von Karman large deformation strain-displacement relation and the Galerkin method, the aerothemoelastic stability of the non-linear panel is systematically investigated by numerical method. The main works are as follows:
1. The higher order motion differential equations of the two-dimensional panel flutter system are rewritten as the lower order differential equations. Applying the eigenvalue theory and Hopf bifurcation criterian to analyze the Jacobi matrix of the equilibrium points, the analytical solutions of critical conditions of pitchfork bifurcations and Hopf bifurcations are obtained in a two-dimensional parameter plane. The number and the stabilities of the equilibrium points in the different regions of the two-dimension parameter plane are analyzed.
2. Based on the quasi-steady theory of thermal stress and the first order piston theory of supersonic aerodynamics, the flutter differential equations of a panel with the thermal effect and the geometrical nonlinearity are established. The Galerkin approach is applied to simplify the equations into the discrete forms, which are solved by the fourth order Ronger-Kutta method. The calculated results show that the critical dynamic pressure of the panel is descended by the temperature loads and the amplitude of limite cycle flutter is raised with the increase of temperature. The bifurcation curve indicates that there are some different period responses in the flutter system when the temperature is changed, especially, the chaotic responses are observed in some special parameter region. When the thermal effect of material is considered, the critical dynamic pressure of the panel is further reduced and the more complex dynamic behaviors in the flutter system are observed.
3 The third order piston theory is employed to calculate the aerodynamic load acting on the panel. A nonlinear aeroelastic system of a three-dimensional panel with the thermal effect is modelled by Galerkin method. The fourth order Runge-Kutta method is utilized to analyze the bifurcation forms of the nonlinear system. The effects of mode number, temperature, nonlinear aerodynamic terms and the length /width ratio of panel on the flutter of the panel are investigated. The results show that the thermal stress is an important parameter for the flutter stability of the panel and with the rise of temperature the critical dynamic pressure of the panel is descended. The bifurcation curve of the system response vs temperature indicates that with the increase of temperature, the flutter system undertakes period 1, period 2, period 4, chaos, quasi-period and so on. The routes to chaos involve period-doubling bifurcation and quasi-period in the system. When the temperature is higher, the system loses its stability into the thermal buckling.
4 The dynamical responses of the three-dimensional panel with the thermal effect and geometrical nonlinearity in turbulence are studied. The velocity of atmosphere turbulence is divided into two parts, the mean velocity and the fluctuating velocity. Von Karman's spectrum of atmosphere turbulence and the composition method of trigonometric series are used to calculate the the fluctuating velocity in time domain. The first order piston theory of supersonic aerodynamics is used to calculate the aerodynamic forces. The mean square response of the panel is calculated by random theory. The effects of some main parameters of the turbulence on the mean square roots of response are analyzed, the results of which show that the mean square roots of the response increase with the rise of the mean pressure or temperature when the mean pressure is close to and over the flutter critical pressure. The root mean square response has a pronounce change to the strength of atmosphere turbulence only when the mean pressure is close to the flutter critical pressure, but it is insensitive to the change of integral measure of atmosphere turbulence.
5 Based on the Kelvin's viscoelastic damping model, the flutter equations of a three-dimensional panel made of viscoelastic material are set up. The effects of viscoelastic damping, in-plane loads, length-to-width ratios of panel and flow velocity on the flutter of the panel are analyzed. In a two-dimensional parameter plane, the characteristics of the responses of the panel are discussed, the results of which show that with the rise of viscoelastic damping the chaotic region fleetly decreases, so, the viscoelastic damping can control the chaotic response of the panel. But the viscoelastic damping has almost no effect on the buckling region. The bifurcation curve of response vs dynamical pressure shows that the flutter system of the viscoelastic panel may represent complex dynamics characteristics with variation of dynamic pressure. With the increase of dynamical pressure, the response will change into the simple limits cycle oscillation from the chaotic oscillation through a series of bifurcations.
6. The nonlinear dynamical characteristics of a three-dimensional panel with boundary conditions relaxation are studied. The conventional boundary value problem of the panel involves time-dependent boundary conditions which are converted to an autonomous form using a special coordinate transformation. The flutter equations of the panel are set up by using Galerkin method. The complex dynamic behaviors in the regions close to the bifurcation points of nonlinear system are simulated with numerical method. Taking the dynamic pressure or the relaxation parameter or the in-plane loading as control parameters, the bifurcation diagrams are drawn, the results of which indicate that with variation of bifurcation parameters there are various bifurcation behaviors in the system, such as hopf bifurcation, pitchfork bifurcation, and the period-3 windows in chaos zone. In addition, in some special parameters region, there exist two stable limit cycles in the system at a same dynamic pressure, depending on different initial conditions respectively.
1、将二维壁板非线性颤振系统的高阶运动微分方程化为四维一阶微分方程,针对系统在平衡点处的Jacobi矩阵,应用特征值理论和Hopf分叉代数判据解析推导了系统发生静态叉式分又及动态Hopf分叉的临界条件。在参数平面内讨论了该系统在各个区域内平衡点的个数和稳定性。
2、运用准定常的热应力公式,基于一阶气动力活塞理论建立了考虑温度效应的几何非线性无限展长二维壁板颤振方程。采用伽辽金方法对方程进行离散处理,基于Matlab编程对系统进行了数值仿真分析。结果表明温度载荷降低了系统的颤振临界动压,系统极限环响应幅值随温度的升高而增大。位移响应的分又图表明,系统随温度的变化表现出各种周期振动行为,且在特定的参数区间内存在混沌响应。热效应导致的材料非线性使得系统的颤振临界动压进一步减小,在分叉参数范围内系统表现出更为丰富的非线性动力学行为。
3、基于三阶活塞气动力理论,采用伽辽金方法建立了热环境下三维壁板在超音速流作用下的颤振方程。采用龙格-库塔法对系统进行数值积分,分析模态截取阶数、温度、非线性气动力以及壁板几何尺寸对壁板颤振的影响规律。结果表明,热应力是降低壁板稳定性的一个重要因素,随着温度的升高,壁板颤振临界动压降低。以温度为可变参量,给出了壁板振动的分叉图。结果表明,系统不仅存在简单的极限环颤振,而且会出现周期2、周期4等复杂的运动形态,甚至会出现混沌运动。系统通向混沌的道路不都是周期倍化分叉,还有拟周期道路等。继续升高温度,系统还会产生热屈曲失稳。
4、以热环境下三维壁板为研究对象,研究大气紊流作用下系统的动力响应。将大气紊流速度分解为平均速度和脉动速度两部分,采用Von Karman紊流谱结合随机场三角级数合成法得到时域脉动速度的表达式。基于一阶活塞理论求解气动力,用随机理论对壁板均方根响应值进行分析。着重考查了紊流场的几个主要参数对均方根响应值的影响。结果表明,当来流平均动压接近和超过颤振临界动压时,均方根响应值随平均动压和温度的增大而增加。均方根响应值仅在平均动压接近颤振临界动压时随紊流强度变化显著,但对紊流尺度的变化不敏感。
5、基于Kelvin粘弹模型建立了三维粘弹壁板非线性颤振方程。通过对粘弹阻尼、面内力、壁板长宽比和来流速度变参分析,研究上述参数变化对粘弹壁板颤振的影响规律。在无量纲动压—面内力的二维参数平面内分析粘弹壁板颤振系统的运动特性,给出了系统存在不同响应特性的分区图。结果表明,颤振临界动压随粘弹阻尼的增大先减小后增大。混沌响应区域随粘弹阻尼系数的增大快速减小,说明粘弹阻尼有抑制混沌响应的作用。而屈曲区域却基本不受影响。通过分叉图观察到随着动压的变化,粘弹性壁板系统由混沌经一系列的倒分叉进入简单的极限环颤振状态。
6、以面内力作用的壁板模型为对象,研究边界松驰条件下壁板的复杂响应。首先采用坐标变换结合伽辽金方法建立非理想边界三维壁板几何非线性颤振方程,再通过对非线性系统的数值积分,分析不同参数下该系统的周期振动以及进入各种复杂分叉的过程。结果表明,随边界约束的放松,颤振临界动压减小。分别以边界松驰因子、来流动压及面内压力为可变参数,研究系统的分叉现象。结果表明,系统存在Hopf分叉、极限环叉式分叉、混沌以及混沌中周期3窗口等。另外,在一些特殊参数区域内,同一动压下,系统存在两个稳定的极限环。根据初始条件的不同,系统分别收敛到不同的极限环上。
The panel flutter is a classical supersonic aeroelastic phenomenon on the outer skin of high-speed aerospace vehicles, which always leads to the fatigue failure of the skin panels and has a harmful influence on the flight perfermance and even flight safety. Based on the piston theory of supersonic aerodynamics, Von Karman large deformation strain-displacement relation and the Galerkin method, the aerothemoelastic stability of the non-linear panel is systematically investigated by numerical method. The main works are as follows:
1. The higher order motion differential equations of the two-dimensional panel flutter system are rewritten as the lower order differential equations. Applying the eigenvalue theory and Hopf bifurcation criterian to analyze the Jacobi matrix of the equilibrium points, the analytical solutions of critical conditions of pitchfork bifurcations and Hopf bifurcations are obtained in a two-dimensional parameter plane. The number and the stabilities of the equilibrium points in the different regions of the two-dimension parameter plane are analyzed.
2. Based on the quasi-steady theory of thermal stress and the first order piston theory of supersonic aerodynamics, the flutter differential equations of a panel with the thermal effect and the geometrical nonlinearity are established. The Galerkin approach is applied to simplify the equations into the discrete forms, which are solved by the fourth order Ronger-Kutta method. The calculated results show that the critical dynamic pressure of the panel is descended by the temperature loads and the amplitude of limite cycle flutter is raised with the increase of temperature. The bifurcation curve indicates that there are some different period responses in the flutter system when the temperature is changed, especially, the chaotic responses are observed in some special parameter region. When the thermal effect of material is considered, the critical dynamic pressure of the panel is further reduced and the more complex dynamic behaviors in the flutter system are observed.
3 The third order piston theory is employed to calculate the aerodynamic load acting on the panel. A nonlinear aeroelastic system of a three-dimensional panel with the thermal effect is modelled by Galerkin method. The fourth order Runge-Kutta method is utilized to analyze the bifurcation forms of the nonlinear system. The effects of mode number, temperature, nonlinear aerodynamic terms and the length /width ratio of panel on the flutter of the panel are investigated. The results show that the thermal stress is an important parameter for the flutter stability of the panel and with the rise of temperature the critical dynamic pressure of the panel is descended. The bifurcation curve of the system response vs temperature indicates that with the increase of temperature, the flutter system undertakes period 1, period 2, period 4, chaos, quasi-period and so on. The routes to chaos involve period-doubling bifurcation and quasi-period in the system. When the temperature is higher, the system loses its stability into the thermal buckling.
4 The dynamical responses of the three-dimensional panel with the thermal effect and geometrical nonlinearity in turbulence are studied. The velocity of atmosphere turbulence is divided into two parts, the mean velocity and the fluctuating velocity. Von Karman's spectrum of atmosphere turbulence and the composition method of trigonometric series are used to calculate the the fluctuating velocity in time domain. The first order piston theory of supersonic aerodynamics is used to calculate the aerodynamic forces. The mean square response of the panel is calculated by random theory. The effects of some main parameters of the turbulence on the mean square roots of response are analyzed, the results of which show that the mean square roots of the response increase with the rise of the mean pressure or temperature when the mean pressure is close to and over the flutter critical pressure. The root mean square response has a pronounce change to the strength of atmosphere turbulence only when the mean pressure is close to the flutter critical pressure, but it is insensitive to the change of integral measure of atmosphere turbulence.
5 Based on the Kelvin's viscoelastic damping model, the flutter equations of a three-dimensional panel made of viscoelastic material are set up. The effects of viscoelastic damping, in-plane loads, length-to-width ratios of panel and flow velocity on the flutter of the panel are analyzed. In a two-dimensional parameter plane, the characteristics of the responses of the panel are discussed, the results of which show that with the rise of viscoelastic damping the chaotic region fleetly decreases, so, the viscoelastic damping can control the chaotic response of the panel. But the viscoelastic damping has almost no effect on the buckling region. The bifurcation curve of response vs dynamical pressure shows that the flutter system of the viscoelastic panel may represent complex dynamics characteristics with variation of dynamic pressure. With the increase of dynamical pressure, the response will change into the simple limits cycle oscillation from the chaotic oscillation through a series of bifurcations.
6. The nonlinear dynamical characteristics of a three-dimensional panel with boundary conditions relaxation are studied. The conventional boundary value problem of the panel involves time-dependent boundary conditions which are converted to an autonomous form using a special coordinate transformation. The flutter equations of the panel are set up by using Galerkin method. The complex dynamic behaviors in the regions close to the bifurcation points of nonlinear system are simulated with numerical method. Taking the dynamic pressure or the relaxation parameter or the in-plane loading as control parameters, the bifurcation diagrams are drawn, the results of which indicate that with variation of bifurcation parameters there are various bifurcation behaviors in the system, such as hopf bifurcation, pitchfork bifurcation, and the period-3 windows in chaos zone. In addition, in some special parameters region, there exist two stable limit cycles in the system at a same dynamic pressure, depending on different initial conditions respectively.
引文
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