凸肩叶片的非线性振动特性与运动分岔

以凸肩叶片作为研究模型, 建立了考虑凸肩摩擦力, 几何大变形与阻尼的非线性振动方程.采用Galerkin法对振动方程离散化, 应用平均法对离散后模态方程组的非线性响应进行解析分析, 得到了非线性幅频特性曲线, 与数值解比较验证了解析解, 并讨论了系统周期解的稳定性. 用非线性振动理论详细研究了平均方程组的运动分岔现象, 揭示了平均方程组周期解的变化过程及其具有的非线性动力学性质. 解析结果表明, 凸肩之间的摩擦对系统第二阶非线性振动特性影响很大. 由于凸肩之间摩擦力方向的不断改变, 系统第二阶非线性幅频特性曲线不连续, 出现两个共振频域. 随着时间的推移, 系统振动的幅值会以$T/ 4$为周期在两个频域的幅频曲线上来回跳动, 这会使叶片的振动响应大幅降低.      

静载荷作用下柔韧圆板的大幅度振动

本文首先给出了均布载荷作用下柔韧圆板的大幅度振动方程，按文中给出的时间模态假设导了该问题的非线性耦合的代数和微分特征方程组，利用修正迭代法求出了该方程组的近似解析解，得到了柔韧圆振动的幅频－载荷特征关系，讨论了静载荷对其振动性态的影响。     

非线性振动系统的多项式向量方法

对于常微分方程描述的非线性振动系统,当采用摄动方法求近似解时,先是给出满足各阶近似解的二阶常微分方程组,继而依次对每一个常微分方程进行求解,以致多自由度非线性振动系统的求解过程相当繁琐.文章针对常微分方程表示的非线性振动系统,提出了一种求解非线性振动系统近似解的多项式向量方法,该方法将二阶常微分方程组表示成一阶状态方程组,将非线性部分写成常数矩阵和多项式向量之积的形式.然后,采用直接摄动方法,获得每个幂次近似解所满足的一组状态方程,此时状态方程的非线性部分成为常数矩阵和前一幂次近似解作为元素组成的多项式向量的乘积.进一步,借助Toeplitz矩阵将多项式向量之乘法表示成矩阵形式,以解决多项式相乘带来的幂次方系数的确定问题,再根据一阶非齐次方程组的求解方法,获得状态方程组的全部近似解析解.多项式向量方法将二阶常微分描述的非线性振动求解过程转换为一阶非齐次状态方程组的求解问题,计算过程主要是矩阵和向量之间乘法运算,提高了计算效率和程序化水平.     

轴向移动局部浸液单向板的1:3内共振分析

考虑单向板的轴向速度、轴向张力、流固耦合作用以及阻尼等因素, 基于由 von Kármán薄板大挠度方程得到的轴向移动局部浸液单向板的非线性振动方程, 研究了外激励作用下单向板在1:3内共振情况时的非线性振动特性. 首先利用Galerkin法对非线性振动方程离散化, 然后分别应用数值法和近似解析法对离散后模态方程组进行求解, 获得了系统内共振情况下复杂的幅频特性曲线, 并讨论了周期解的稳定性. 最后研究了1:3内共振系统平均方程组的运动分岔现象.     

悬索桥非共振情况下的振动

本文在连续膜假设条件下,建立了新的能描述吊索变形和松弛影响的悬索桥横向振动非线性偏微分方程组.该方程组的不等式定解条件反映出吊索松弛与否情况.在假设吊索不松弛的条件下,对上述方程组进行简化后得到一组只含双侧约束的非线性偏微分方程组.此方程组的定解条件是用等式表示的双侧约束条件.通过Galerkin方法把双侧约束的偏微分方程组离散为时域上非线性常微分方程组.用多尺度法求得了非线性常微分方程组非共振情况下的一次近似解析解.通过比较数值解和解析解发现,解析解有良好的精度.同时数值和解析的结果指出,在非共振情况下悬索桥的加劲梁和主缆的振幅都是有限值并正比于激励的幅值.     

损伤弹性中厚板的动力学行为

建立了一组关于损伤弹性中厚板的非线性偏微分方程组。为了方便求解方程组,首先利用伽辽金法对原方程组进行简化,得到一组非线性常微分方程;然后利用Matlab软件进行数值模拟,考察了载荷参数、板的几何参数、损伤对中厚板振动的影响。数值结果表明增大板的厚度,有利于增强结构运动的稳定性,而损伤会降低结构运动的稳定性。     

一种求解非线性振动系统渐近解的新方法——计算向量场Normal Form系数的简单方法

利用非线整变换,本文推导出了一种只需通过简单代数运算即可算出Hopf分叉Normal Form系数的简单方法,用这种方法求解非线性振动问题,只需把原方程变换成本文讨论的典则形式的一阶微分方程组,然后进行简单的代数运算即可得到原非线性振动方程的解,这种方法简单方便容易掌握。     

一种求解非线性振动系统渐近解的新方法——计算向量场Normal Form系数的简单方法

利用非线整变换,本文推导出了一种只需通过简单代数运算即可算出Hopf分叉Normal Form系数的简单方法,用这种方法求解非线性振动问题,只需把原方程变换成本文讨论的典则形式的一阶微分方程组,然后进行简单的代数运算即可得到原非线性振动方程的解,这种方法简单方便容易掌握。     

鞍形膜结构在冲击载荷作用下的振动特性

以鞍形膜结构为研究对象,研究了冲击载荷作用下鞍形膜结构的振动问题。首先推导了冲击载荷作用下的结构振动非线性微分方程,该方程中考虑了膜材的正交异性,空气阻尼以及膜材变形的几何非线性。然后假设一小球以某确定的冲击速度差冲击膜面,模拟冲击荷载。最后利用随机摄动法求解方程,得到鞍形膜结构振动响应的位移函数,通过该函数分别对膜面预张力、拱跨比以及小球速度差进行参数分析。同时,对鞍形膜结构的振动进行数值模拟,并与解析理论作对比分析。结果表明鞍形膜结构振动位移会随着膜面预张力或拱跨比的增大而减小,同时也会随着速度差的增大而增大。由此可知该理论模型研究鞍形膜结构在冲击载荷作用下的振动特性是可行的,得到的结果能够准确地预测膜结构响应规律。     

复合材料冲击损伤非线性振动声调制检测技术的试验研究

复合材料因其优良性能在航空航天领域应用广泛,但其受低速冲击后强度与结构完整性会严重下降。非线性振动声调制技术作为一种检测范围广、检测精度高的无损检测新技术,能对结构中的接触缺陷进行有效检测。本文制作了含冲击损伤试样,通过选取合适的高频超声激励频率,分别在不同的低频振动幅值及模态频率下对含冲击损伤复合材料进行检测。研究结果表明,非线性振动声调制技术能对含冲击缺陷试样与参考试样进行有效区分,调制系数与低频激励幅值呈近似线性关系。检测时需选取合适的频率及幅值。缺陷处振动不剧烈且低频激励幅值较小时,较小冲击能量损伤试样与参考试样不易区分;缺陷处振动剧烈时,输入较低的低频幅值即可较易分辨参考试样与损伤试样。     

General Relativistic Shock Waves that Extend the Oppenheimer-Snyder Model

In earlier work we constructed a class of spherically symmetric, fluid dynamical shock waves that satisfy the Einstein equations of general relativity. These shock waves extend the celebrated Oppenheimer-Snyder result to the case of non-zero pressure. Our shock waves are determined by a system of ordinary differential equations that describe the matching of a Friedmann-Robertson-Walker metric (a cosmological model for the expanding universe) to an Oppenheimer-Tolman metric (a model for the interior of a star) across a shock interface. In this paper we derive an alternate version of these ordinary differential equations, which are used to demonstrate that our theory generates a large class of physically meaningful (Lax-admissible) outgoing shock waves that model blast waves in a general relativistic setting. We also obtain formulas for the shock speed and other important quantities that evolve according to the equations. The resulting formulas are important for the numerical simulation of these solutions. (Accepted January 19, 1996)     

Viscous shock layer near the stagnation point on a rotating body with unsteady injection and surface cooling

Unsteady supersonic flow regimes in the neighborhood of a stagnation point are investigated on the basis of a system of viscous shock layer equations [10] containing all the terms of the Euler equations and the boundary layer equations. An analytic solution of the unsteady equations valid near the surface of the body is found in the case of strong injection. The unsteady equations of the viscous shock layer are solved numerically on the basis of a divergent implicit scheme of the second order of approximation across the shock layer, using Newtonian linearization and vector sweep methods with allowance for the boundary relations on the surface of the body and at the isolated bow shock. Certain calculation results illustrating the effect of injection, surface cooling, the swirl of the external flow and the angular velocity of the body on the structure of the steady and unsteady viscous shock layer are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–122, September–October, 1987.     

One-dimensional model for microscale shock tube flow

A one-dimensional model for the numerical simulation of transport effects in small-scale, i.e., low Reynolds number, shock tubes is presented. The conservation equations have been integrated in the lateral directions and three-dimensional effects have been introduced as carefully controlled sources of mass, momentum and energy, into the axial conservation equations. The unsteady flow of gas behind the shock wave is reduced to a quasi-steady flow by choosing a coordinate system attached to the shock. The boundary layer problem is thereby reduced to a laminar solution, similar to the Blasius solution, with the exception that the wall velocity can be nonzero. The resulting one-dimensional equations are then solved numerically using a two-step Lax-Wendroff/ MacCormack scheme with flux correction transport. For validation purposes, comparisons are performed against previously published shock structure and low Reynolds number shock tube experiments; good agreement is observed. The model has been used to predict the performance of a 10μm shock tube and the result of this simulation shows the possibility of shock wave disappearance at lower pressure ratios for a micro-scale shock tube.      

Computation of the viscous shock layer on blunted cones

A numerical method is described for solving the equations of the compressible viscous shock layer on smooth spherically blunted axisymmetric cones at zero angle of attack and flow of a perfect gas. Effective use is made of the scheme of separating the original system of equations into parabolic (second order) and inviscid (first order) subsystems, which are solved by intrinsic methods. The results of the computations are presented. The method is capable of natural generalization to the case of nonequilibrium physical and chemical processes and diffusion. In most published papers dealing with computation of the compressible shock layer, the authors examine either the vicinity of the stagnation point or a certain region of spherical blunting [1–5]. In all the papers except [4, 5], a number of simplified assumptions have been made regarding the flow picture. Very few papers [6–8] have calculated the viscous shock layer on the forward surface of blunted bodies. In [6, 7] an approximate examination was made only of hyperboloids and paraboloids of revolution, which have very favorable geometry. Reference [8] used a approximate Karman—Polhausen integral method for a very simple system of equations. The method proposed here is essentially an accurate numerical method for solution of the viscous shock layer equations.     

Shock-induced two-phase flows in an aligned baffle system filled with suspended particles

This paper describes the shock propagation through a dilute gas-particle suspension in an aligned baffle system. Numerical solution to two-phase flows induced by a planar shock wave is given based on the two-continuum model with interphase coupling. The governing equations are numerically solved by using high-resolution schemes. The computational results show the shock reflection and diffraction patterns, and the shock-induced flow fields in the 4-baffle system filled with the dusty gas.     

A coupled problem of thermoelastic vibrations of a circular plate with exact boundary conditions

Thermoelastic vibrations of a free supported and clamped circular plate caused by a thermal shock upon the plate surface have been analyzed The system of partial differential equations of the coupled system has been reduced to Volterra's first and second kind integral equations in the time domain. In both cases the solutions are given in the.form 4f series of Bessel functions of the first kind     

The Riemann problem for nonlinear degenerate wave equations

This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity. Since the stress function is neither convex nor concave, the shock condition is degenerate. By introducing a degenerate shock under the generalized shock condition, the global solutions are constructively obtained case by case.     

Analysis of one-dimensional horizontal two-phase flow in geothermal reservoirs

In the absence of capillarity the single-component two-phase porous medium equations have the structure of a nonlinear parabolic pressure (equivalently, temperature) diffusion equation, with derivative coupling to a nonlinear hyperbolic saturation wave equation. The mixed parabolic-hyperbolic system is capable of substaining saturation shock waves. The Rankine-Hugoniot equations show that the volume flux is continuous across such a shock. In this paper we focus on the horizontal one-dimensional flow of water and steam through a block of porous material within a geothermal reservoir. Starting from a state of steady flow we study the reaction of the system to simple changes in boundary conditions. Exact results are obtainable only numerically, but in some cases analytic approximations can be derived. When pressure diffusion occurs much faster than saturation convection, the numerical results can be described satisfactorily in terms of either saturation expansion fans, or isolated saturation shocks. At early times, pressure and saturation profiles are functionally related. At intermediate times, boundary effects become apparent. At late times, saturation convection dominates and eventually a steady-state is established. When both pressure diffusion and saturation convection occur on the same timescale, initial simple shock profiles evolve into multiple shocks, for which no theory is currently available. Finally, a parameter-free system of equations is obtained which satisfactorily represents a particular case of the exact equations.     

Self-refraction of small amplitude shock waves

A theory is developed for self-refraction of small amplitude shock waves, which includes the effects of a nonlocal nonlinearity. A complete system of self-refraction equations describing the two-dimensional motion of a shock-wave front is suggested. The system involves equations describing the amplitude variation at the front and the cross sectional area of a ray tube when a wave propagates along rays orthogonal to the front. Coupling equations are relations of nonlinear geometrical acoustics, defining the amplitude at the front through the ray tube cross section and the amplitude gradient along a ray. A particular form of the system of equations describing the self-refraction of triangular pulses is analysed and automodel solutions are given.     

Study into the Interaction of Cracks in an Elastic Half-Space under a Shock Load by Means of Boundary Integral Equations

The effect of a shock load on the interaction of circular cracks in an elastic half-space is studied. In the space of Fourier time transforms, the problem is reduced to a system of two-dimensional boundary integral equations in the form of the Helmholtz potential with unknown densities characterizing the discontinuities in the displacements of the opposite crack faces. Discrete analogs of those equations are constructed. As an example, two cracks are considered whose faces are under the action of shock tensile loads varying in time as the Heaviside function. The time dependences of the dynamic stress intensity factors are obtained. Their dependence on the relative position of the cracks in the half-space is analyzed.