航天器贮箱液体非线性晃动动力学的多维模态分析方法
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摘要
晃动是指贮箱中液体自由液面的运动,它是由于对部分充液贮箱施加扰动而引起的。运动或静止贮箱中的液体晃动问题在航空航天、民用、海洋工程、水陆运输等领域都受到极大的关注。本文将多维模态理论应用到求解航天领域中的圆柱贮箱液体非线性晃动问题中,在得到一般形式的模态系统之后,系统地研究了液体非线性自由晃动、液体横向受迫共振晃动的瞬态响应和稳态响应,分析了高阶模态的影响,并且推导出一个适合于工程应用的计算液体晃动力的公式,主要研究工作分为如下四部分:
第一部分通过压力积分变分原理和模态展开的方法将描述液体非线性晃动的自由边界值问题转化为了无穷维模态系统。所得到的模态系统具有一般形式,适合于贮箱作任意三维运动。相对于原始的复杂非线性边界值问题,求解此模态系统要简单得多,同时其物理意义清晰,将为求解各种类型液体非线性晃动问题奠定基础。
第二部分研究了圆柱贮箱内液体的非线性自由晃动问题。在无穷维模态系统基础之上,推导出描述液体非线性自由晃动的有限维模态系统,最后利用Runge-Kutta方法对此模态系统进行数值积分,揭示出液体作自由晃动时的各种非线性现象,从而验证公式推导的正确性和将多维模态理论应用到圆柱贮箱液体非线性晃动问题中的正确性和有效性。
第三部分由第4、5、6章组成,是本文工作的重点。这部分针对贮箱作水平横向运动,首先推导出描述液体非线性共振晃动的有限维模态系统,然后系统地研究了液体的瞬态响应和稳态响应,并分析了被忽略的高阶模态的影响。
第4章通过直接对模态系统进行数值积分,定性地揭示出一些重要的非线性现象,如“拍”现象、节径移动、非平面运动等,并证明了次模态在液体运动描述中的不可忽略性。
第5章则是对模态系统进行更深入的稳定性分析,首先求出主模态函数的稳态周期解,然后应用Floquet-Lyapunov方法研究它们的稳定性和稳定区间,从而定量地揭示出液体横向受迫共振晃动的不同晃动波形式—稳定“平面”波、稳定“旋转”波和不稳定“混沌”波,并得出一些有意义的结论。最后将理论分析结果与实验结果进行了对比,发现两者有很好的吻合,从而验证了稳态响应分析的正确性。
第6章分析了次模态对稳态响应的影响和由于次共振造成的高阶模态在液体运动描述中的不可忽略性,指出Narimanov-Moiseev三阶渐近假设关系的适用范围和局限性,对指导将来的工作有着重要意义。
第四部分面向工程应用,在前几章工作的基础之上,推导出部分充液圆柱贮箱中液体产生非线性晃动时其作用于贮箱壁的力的公式,此公式物理意义清晰,可用于工程实际中液体晃动力的估算。
Sloshing means motion of the liquid free surface inside its container. It is caused by disturbance to partially filled liquid containers. The problem of liquid sloshing in moving or stationary containers remains of great concern to aerospace, civil, ocean engineering, designers of road tankers and ship tankers. The multidimensional modal theory is applied to solve liquid nonlinear sloshing in circular cylindrical tank used in aerospace engineering in this paper. After deriving the modal system in general form, the liquid nonlinear free sloshing, the transient response and steady response of liquid transverse forced resonant sloshing are investigated systematically and the influences of higher order modes are analyzed. A formula of liquid sloshing force, which is fit for engineering application, is also derived. The major works of this paper include four parts as follows:
The free boundary value problem describing liquid nonlinear sloshing is translated into infinite dimensional modal system through the pressure integral variational principle and modes expanding method in the first part. The modal system takes on general form and is suitable for arbitrary three dimensional motion of tank. In comparing with the original complicated nonlinear boundary value problem, solving the modal system is easy greatly. At the same time, the physical meaning of the modal system is manifest. So, it is can be taken as the basis of solving all kinds of liquid nonlinear sloshing problem.
The liquid nonlinear free sloshing in circular cylindrical tank is investigated in the second part. Based on the infinite dimensional modal system, the finite dimensional modal system describing liquid nonlinear free sloshing is derived. The modal system is integrated by Runge-Kutta method. By discovering many nonlinear phenomena when liquid is in free sloshing, the correctness of formula derivation and the validity of applying multidimensional modal theory to liquid nonlinear sloshing in circular cylindrical tank are proved.
The third part is composed of chapter 4, chapter 5 and chapter 6, which are the emphases of the paper. The finite dimensional modal system describing liquid nonlinear resonance sloshing when the tank is in translation transverse motion is derived first in this part and the liquid transient response and steady response are investigated systematically. The influences of higher modes which are ignored are analyzed too.
In Chapter 4, many important nonlinear phenomena, such as beating phenomenon, shift of nodal diameter and nonplanar motion are discovered qualitatively by direct numerical integration of the modal system and the non-ignoring of secondary modes in the describing of liquid motion are proved.
Chapter 5 is the stability analysis of the modal system. The steady periodic solutions of the primary modal functions are gained first. The stability and the stable zones of the solutions are investigated by Floquet-Lyapunov method. So the different liquid transverse forced resonance sloshing wave forms—stable planar wave, stable rotary wave and unstable chaotic wave are revealed quantitatively and some significative conclusions are gained. At last, the theory results are compared with the experimental results and they agree well. So the validity of steady response analysis is proved.
The influence of secondary modes on steady response and non-ignoring of higher modes in the describing of liquid motion due to secondary resonance are analyzed in Chapter 6. So the scope and localization of Narimanov-Moiseev third order asymptotic hypothesis are disclosed, which can direct future work.
The fourth part faces the engineering application. Based on the work of former chapters, a formula of liquid force acted on the wall of partially filled circular cylindrical tank when the liquid produces nonlinear sloshing is derived. The physical meaning of this formula is manifest and can be used in the estimating of liquid sloshing force in engineering.
第一部分通过压力积分变分原理和模态展开的方法将描述液体非线性晃动的自由边界值问题转化为了无穷维模态系统。所得到的模态系统具有一般形式,适合于贮箱作任意三维运动。相对于原始的复杂非线性边界值问题,求解此模态系统要简单得多,同时其物理意义清晰,将为求解各种类型液体非线性晃动问题奠定基础。
第二部分研究了圆柱贮箱内液体的非线性自由晃动问题。在无穷维模态系统基础之上,推导出描述液体非线性自由晃动的有限维模态系统,最后利用Runge-Kutta方法对此模态系统进行数值积分,揭示出液体作自由晃动时的各种非线性现象,从而验证公式推导的正确性和将多维模态理论应用到圆柱贮箱液体非线性晃动问题中的正确性和有效性。
第三部分由第4、5、6章组成,是本文工作的重点。这部分针对贮箱作水平横向运动,首先推导出描述液体非线性共振晃动的有限维模态系统,然后系统地研究了液体的瞬态响应和稳态响应,并分析了被忽略的高阶模态的影响。
第4章通过直接对模态系统进行数值积分,定性地揭示出一些重要的非线性现象,如“拍”现象、节径移动、非平面运动等,并证明了次模态在液体运动描述中的不可忽略性。
第5章则是对模态系统进行更深入的稳定性分析,首先求出主模态函数的稳态周期解,然后应用Floquet-Lyapunov方法研究它们的稳定性和稳定区间,从而定量地揭示出液体横向受迫共振晃动的不同晃动波形式—稳定“平面”波、稳定“旋转”波和不稳定“混沌”波,并得出一些有意义的结论。最后将理论分析结果与实验结果进行了对比,发现两者有很好的吻合,从而验证了稳态响应分析的正确性。
第6章分析了次模态对稳态响应的影响和由于次共振造成的高阶模态在液体运动描述中的不可忽略性,指出Narimanov-Moiseev三阶渐近假设关系的适用范围和局限性,对指导将来的工作有着重要意义。
第四部分面向工程应用,在前几章工作的基础之上,推导出部分充液圆柱贮箱中液体产生非线性晃动时其作用于贮箱壁的力的公式,此公式物理意义清晰,可用于工程实际中液体晃动力的估算。
Sloshing means motion of the liquid free surface inside its container. It is caused by disturbance to partially filled liquid containers. The problem of liquid sloshing in moving or stationary containers remains of great concern to aerospace, civil, ocean engineering, designers of road tankers and ship tankers. The multidimensional modal theory is applied to solve liquid nonlinear sloshing in circular cylindrical tank used in aerospace engineering in this paper. After deriving the modal system in general form, the liquid nonlinear free sloshing, the transient response and steady response of liquid transverse forced resonant sloshing are investigated systematically and the influences of higher order modes are analyzed. A formula of liquid sloshing force, which is fit for engineering application, is also derived. The major works of this paper include four parts as follows:
The free boundary value problem describing liquid nonlinear sloshing is translated into infinite dimensional modal system through the pressure integral variational principle and modes expanding method in the first part. The modal system takes on general form and is suitable for arbitrary three dimensional motion of tank. In comparing with the original complicated nonlinear boundary value problem, solving the modal system is easy greatly. At the same time, the physical meaning of the modal system is manifest. So, it is can be taken as the basis of solving all kinds of liquid nonlinear sloshing problem.
The liquid nonlinear free sloshing in circular cylindrical tank is investigated in the second part. Based on the infinite dimensional modal system, the finite dimensional modal system describing liquid nonlinear free sloshing is derived. The modal system is integrated by Runge-Kutta method. By discovering many nonlinear phenomena when liquid is in free sloshing, the correctness of formula derivation and the validity of applying multidimensional modal theory to liquid nonlinear sloshing in circular cylindrical tank are proved.
The third part is composed of chapter 4, chapter 5 and chapter 6, which are the emphases of the paper. The finite dimensional modal system describing liquid nonlinear resonance sloshing when the tank is in translation transverse motion is derived first in this part and the liquid transient response and steady response are investigated systematically. The influences of higher modes which are ignored are analyzed too.
In Chapter 4, many important nonlinear phenomena, such as beating phenomenon, shift of nodal diameter and nonplanar motion are discovered qualitatively by direct numerical integration of the modal system and the non-ignoring of secondary modes in the describing of liquid motion are proved.
Chapter 5 is the stability analysis of the modal system. The steady periodic solutions of the primary modal functions are gained first. The stability and the stable zones of the solutions are investigated by Floquet-Lyapunov method. So the different liquid transverse forced resonance sloshing wave forms—stable planar wave, stable rotary wave and unstable chaotic wave are revealed quantitatively and some significative conclusions are gained. At last, the theory results are compared with the experimental results and they agree well. So the validity of steady response analysis is proved.
The influence of secondary modes on steady response and non-ignoring of higher modes in the describing of liquid motion due to secondary resonance are analyzed in Chapter 6. So the scope and localization of Narimanov-Moiseev third order asymptotic hypothesis are disclosed, which can direct future work.
The fourth part faces the engineering application. Based on the work of former chapters, a formula of liquid force acted on the wall of partially filled circular cylindrical tank when the liquid produces nonlinear sloshing is derived. The physical meaning of this formula is manifest and can be used in the estimating of liquid sloshing force in engineering.
引文
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