On the perturbed equations of motion of aerosol particles in regions of strong curvature of the fluid streamlines

Asymptotic forms are obtained and solved for the equations of motion of aerosol particles in various flow fields. The flows chosen to be treated include regions of strong curvature of the fluid streamlines, and regions where the initial motion of the particles differ from that of the undisturbed flow. It is shown that regular perturbations cannot yield the correct details of the trajectories of the particles in such regions, and that singular perturbations must be applied. The parameters whose magnitude control the phenomena come out to be the Stokes numbers, the curvature of the fluid streamlines, and the stopping distance of the aerosol articles. In principle this analysis is analogous to the derivation of the classical boundary layer equations. However, the classical boundary layer equations are field equations, which describe the flow in an Eulerian fashion, and therefore apply to a certain domain, geographically fixed in the field. The aerosol equations of motion considered here are Lagrangian, and as the particles move they carry with them the equations whose perturbed form must thus be modified by the fluid flow encountered along the trajectories traced by the particles. The forms assumed by the perturbed equations of motion are, therefore, different in the various cases treated. Analytical solutions are obtained for two cases which until now were handled only numerically, i.e., particle motion in viscous stagnation flow, and injection of particles into a laminar boundary layer on a flat plate. The comparison with numerical solutions shows that this new method gives rather good approximations.     

Numerical solution of dynamical systems by direct application of Hamilton's principle

In this paper, a method is described which allows a direct derivation of a set of first-order finite difference equations to numerically compute the motion of any conservative or non-conservative dynamic system with a finite number of degrees of freedom. The derivation of the method is based on an application of Lagrangian multipliers to a functional form of Hamilton's equations, and reduces the work required to obtain the most desirable form for numerical integration from the standpoint of computational efficiency and accuracy. For systems with many degrees of freedom, the required matrix inversions produce first derivatives of the co-ordinates, instead of second derivatives, thus eliminating a potential source of error in numerical integration. Two examples are given to illustrate the method.     

On the instability of a spinning projectile due to a nonlinear Magnus moment

A nonlinear differential equation representing the initial nutational vibrations of a spinning projectile in the presence of a nonlinear Magnus moment has been approximated by a perturbed Duffing’s equation and the quantitative and qualitative agreement of the approximation brought out. Then the combined motion in nutation and precession has been examined and sufficient conditions for the Magnus instability of the normal motion establishedvia Lyapunov’s second method.     

直齿圆柱齿轮啮合耦合振动系统参数振动研究

齿轮副啮合耦合振动系统是一个多自由度参数振动系统。该文考虑啮合刚度时变性,传动轴、轴承和箱体等支撑刚度和阻尼,轮齿传动误差以及输入转矩非线性等因素的影响,建立了直齿圆柱齿轮副啮合耦合动力学模型。将动力学方程转换到正则模态下,利用多尺度法对其进行动力稳定性分析,推导出主共振和亚谐共振条件下系统的组合共振频率以及稳定性边界。数值模拟系统非参数和参数共振响应,与摄动法结果吻合较好。结果表明:当轮齿啮合频率接近和型共振频率时,系统发生参数共振,存在着不收敛的无界解。系统的非参数共振响应为概周期响应,包含着多种组合频率成分。     

Dynamic stability of linear parametrically excited twisted Timoshenko beams under periodic axial loads

In this paper, the parametric instability of twisted Timoshenko beams with various end conditions and under an axial pulsating force is studied. The equations of motion in the twisted frame are derived using a finite element method. Based on Bolotin??s method, a set of second-order ordinary differential equations with periodic coefficients of Mathieu?CHill type is formed to determine the instability regions for twisted Timoshenko beams. A dynamic instability index is defined and used as an instability measure to study the influence of various parameters. The effects of beam length, inertia ratio, pre-twist angle, dynamic component of axial force and restraint condition on the instability regions and dynamic instability index of the twisted beam are investigated and discussed.     

Lyapunov stability of columns, pipes and rotating shafts under time-dependent excitation

Structural or mechanical systems governed by non-autonomous partial differential equations are considered. The systems are such that they would be conservative in the absence of dissipation and time variation of the loading parameter. They possess an equilibrium state, and sufficient conditions for its stability are obtained with the use of Lyapunov's direct method. Three problems are treated: a column with a time-varying axial load, a pipe conveying fluid with time-varying velocity, and a rotating shaft with time-varying angular velocity. These excitations appear in coefficients of the equations of motion, and the stability conditions involve the excitations and their time rates of change     

界面初始扰动形态影响RM不稳定过程的数值研究

采用Navier-Stokes (NS)方程对激波诱导扰动界面的Richtmyer-Meshkov (RM)不稳定增长过程进行了二维数值模拟,分析了初始扰动条件对反射波前后界面扰动增长的影响,并与已有的模型预测结果进行了对比和分析。研究结果发现:单模初始的扰动振幅和扰动波长都直接影响反射波前后的界面增长速率;多模随机初始的扰动振幅和扰动波长对于反射波前后界面增长的影响没有单模扰动明显,其扰动增长行为均和单模大波长扰动的增长行为相似。     

端部约束悬臂输流管道的动力学特性

根据梁模型横向弯曲振动模态函数一般表达式,由边界约束条件确定其模态函数的一般表达式,采用Galerkin法将运动方程在模态空间内展开,利用动力学分析方法,分析端部受线性弹簧支承和扭转弹簧约束的端部约束悬臂管道从非保守系统逐渐变为保守系统过程中的固有特性和稳定性。数值仿真结果表明,这种特殊边界输流管道具有复杂变化的动力学特性,支承和约束刚度系数的变化对系统固有特性和稳定性产生很大的影响:随着弹簧刚度的增大,系统的固有频率上升,管道失稳方式从颤振变为屈曲,并且影响系统其他参数对管道动力学特性的作用。     

Oscillating two-phase flow in a prestressed thick elastic tube

Summary The pulsating flow of a fluid with dusty particles in a prestressed thick walled elastic tube has been studied. The tube, subjected to a static inner pressure P_i and an axial stretch λ, is taken to be an incompressible, isotropic, elastic material. The fluid with particles is treated as incompressible Newtonian. Employing the theory of small deformation superimposed on large initial deformations, for an axially symmetric perturbed motion the governing equations are obtained in cylindrical polar coordinates. The analytical solutions of the equations of motion for the dust and the fluid have been obtained. Because of the variable character of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.     

不同边界条件下转动锥壳的参激失稳特性分析

采用Haar小波方法结合Floquet指数法对不同边界条件下转动锥壳的参激振动稳定性进行了分析。基于Love一阶近似壳体理论,给出了周期性载荷作用下转动锥壳的动力学控制微分方程,采用Haar小波离散方法将其转化为具有周期性时变系数的Mathieu-Hill型常微分方程组。考虑到Bolotin法不能应用于陀螺系统的参激失稳特性分析,以及多尺度法受限于小参数情形的事实,该研究采用了对参激系统普遍适用的Floquet指数法对转动锥壳的参激振动稳定性进行分析。通过与其他文献结果的对比,验证了所采用模型及稳定性分析方法的正确性。在此基础上,讨论了固支-固支、简支-简支、固支-简支和简支-固支等几种不同边界条件下转速和半顶角对转动锥壳不稳定区的影响。     

Finite element method applied to the problem of stability of a non-conservative system

The finite element method is applied to the stability analysis of structural systems subject to non-conservative forces. The development of the method is general, but the specific application considered here is the stability of thin-walled members subject to follower forces. The method predicts the type of instability, whether it be buckling or flutter. Example problems, for which exact solutions are known, illustrate the accuracy and convergence characteristic of the finite element formulation.     

Parametric instability of conical shells by the Generalized Differential Quadrature method

The parametric instability of truncated conical shells of uniform thickness under periodic edge loading is examined. The material considered is homogeneous and isotropic. This is the first instance that the Generalized Differential Quadrature (GDQ) method is used to study the effects of boundary conditions on the parametric instability in shells. The formulation is based on the dynamic version of Love's first approximation for thin shells. A formulation is presented which incorporates the GDQ method in the assumed‐mode method to reduce the partial differential equations of motion to a system of coupled Mathieu–Hill equations. The principal instability regions are then determined by Bolotin's method. Assumptions made in this study are the neglect of transverse shear deformation, rotary inertia as well as bending deformations before instability. Copyright © 1999 John Wiley & Sons, Ltd.     

Chaotic analyses of weakly damped parametrically excited cross waves with surface tension

Summary. The Wiggins-Holmes extension of the Generalized Melnikov Method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to non-autonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to demonstrate that cross waves are chaotic. The Luke Lagrangian density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates (or, equivalently, three degrees of freedom) that are the time-dependent components of three velocity potentials that represent three standing waves. The generalized momenta are computed from the Lagrangian, and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains both autonomous and non-autonomous components that must be suspended by applying an extension of the Herglotz algorithm for non-autonomous transformations in order to apply the Kolmogorov-Arnold-Moser (KAM) averaging operation and the GMM. Three canonical transformations are applied to (i) eliminate cross product terms by a rotation of axes; (ii) to transform to action-angle canonical variables and to eliminate two degrees of freedom; and (iii) to suspend the non-autonomous terms and to apply the Hamilton-Jacobi transformation. The system of nonlinear non-autonomous evolution equations determined from Hamiltons equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic separatrices are computed from the unperturbed autonomous system. The non-dissipative perturbed Hamiltonian system with surface tension satisfies the KAM non-degeneracy requirements, and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic separatrix; consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents.     

Numerical integration of non-linear elastic multi-body systems

This paper is concerned with the modelling of nonlinear elastic multi-body systems discretized using the finite element method. The formulation uses Cartesian co-ordinates to represent the position of each elastic body with respect to a single inertial frame. The kinematic constraints among the various bodies of the system are enforced via the Lagrange multiplier technique. The resulting equations of motion are stiff, non-linear, differential-algebraic equations. The integration of these equations presents a real challenge as most available techniques are either numerically unstable, or present undesirable high frequency oscillations of a purely numerical origin. An approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system, whereas the forces of constraint are discretized so that the work they perform vanishes exactly. The combination of these two features of the discretization guarantees the stability of the numerical integration process for non-linear elastic multi-body systems. Examples of the procedure are presented.     

Dynamic buckling of limit-point systems under step loading

The nonlinear dynamic buckling response of discrete systems under step loading of infinite duration is thoroughly discussed by using one-degree-of-freedom models. The analysis is based on the exact nonlinear differential equations of motions and refers to those systems which when subjected to the same loading applied statically exhibit a limit-point instability. It is found that an unbounded motion may start for the smallest step load which forces the system to pass through an unstable equilibrium state of the postbuckling path with zero total potential energy. This leads to dynamic buckling criteria which allow the determination of exact dynamic buckling loads without solving the corresponding nonlinear differential equations of motion. A comparison of dynamic buckling estimates of previous works with those obtained herein shows that the latter are exact regardless of the magnitude of the initial imperfection. Moreover, some additional results provide a better insight into the actual mechanism of nonlinear dynamic buckling associated with the foregoing type of loading.     

Dynamic buckling of limit-point systems under step loading

The nonlinear dynamic buckling response of discrete systems under step loading of infinite duration is thoroughly discussed by using one-degree-of-freedom models. The analysis is based on the exact nonlinear differential equations of motions and refers to those systems which when subjected to the same loading applied statically exhibit a limit-point instability. It is found that an unbounded motion may start for the smallest step load which forces the system to pass through an unstable equilibrium state of the postbuckling path with zero total potential energy. This leads to dynamic buckling criteria which allow the determination of exact dynamic buckling loads without solving the corresponding nonlinear differential equations of motion. A comparison of dynamic buckling estimates of previous works with those obtained herein shows that the latter are exact regardless of the magnitude of the initial imperfection. Moreover, some additional results provide a better insight into the actual mechanism of nonlinear dynamic buckling associated with the foregoing type of loading.     

具有局部分层的对称铺设层合板在非保守力作用下的屈曲分析

利用弹性非保守系统自激振动的拟固有频率变分原理,推导出复合材料矩形板受非保守随从力作用的变分方程,进而导出复合材料层合板含分层损伤的有限元基本方程及求解临界荷载力和固有频率的特征方程。用载荷增量法计算了不同边长比的复合材料矩形板在面内受随从力作用且含有分层的临界载荷,分析了不同角铺设方向对临界载荷的影响。分析表明,分层对临界载荷有一定影响,复合材料层合板的角铺设方向对临界载荷有较大影响。     

Dynamic stability analysis of laminated composite cylindrical shells subjected to conservative periodic axial loads

The dynamic stability of thin, laminated cylindrical shells under combined static and periodic axial forces is studied here using Love's theory for thin shells. A system of Mathieu–Hill equations is obtained by a normal-mode expansion of the equations of motion, the stability of which is examined by Bolotin's method. The dynamic instability regions are investigated for different lamination schemes. The effects of the length-to-radius and thickness-to-radius ratios of the cylinder on the instability regions are also examined.     

Unsteady second grade aligned MHD fluid flow

Summary Exact solutions are established for the equations of a class of an unsteady, plane, second grade, electrically conducting, MHD aligned fluid which undergoes isochoric motion. This class consists of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream.     

一类周期系数力学系统的Hopf-Flip分岔

研究了一类周期系数力学系统因周期运动失稳而产生Hopf-Flip分岔的问题.首先根据拉格朗日方程给出了该力学系统的运动微分方程,并确定其周期运动的具有周期系数的扰动运动微分方程,再根据周期系数系统的稳定性理论建立了其给定周期运动的Poincaré映射,进一步根据该系统的特征矩阵的特征值穿越单位圆情况分析判断该Poincaré映射不动点失稳后将发生Hopf-Flip分岔,并用数值计算加以验证.结果表明,非共振条件下,系统的周期运动可通过Hopf-Flip分岔,进而演变成次谐运动,而三阶强共振条件下系统周期运动失稳后形成不稳定的次谐运动.