含界面脱粘压电复合材料层合板的非线性动力稳定性分析

基于Reddy提出的板高阶剪切变形简化理论,研究了含界面脱粘损伤压电复合材料层合板非线性动力稳定性问题.首先,建立了分层模型,推导了考虑几何非线性、阻尼效应、纵向惯性力和力-电耦合效应的Mathieu方程,并且给出了该方程解的解析表达式.其次,通过典型算例讨论了界面脱粘损伤以及反馈控制力对该层合板动力不稳定区域、纵向、横向共振频率和最大"牵引"深度的影响.由典型算例讨论可知:随着层合板界面脱粘损伤的扩大,其动力稳定性能逐渐减弱,其中在损伤较小时,反馈控制力对智能结构几乎没有影响;而在损伤比较大的情况下,反馈控制力将能有效地减少动力不稳定区域重合面积.     

Thermal instabilities in melt spinning of viscoelastic fibers

Nonisothermal melt spinning of viscoelastic fibers for which the viscosity varies in a step-like manner with respect to temperature is studied in this work. A set of one-dimensional equations based on the slender-jet approximation and the upper convected Maxwell model is used to describe the melt spinning process. The process is characterized by the force required to pull the fiber, the strength of external heating, and the draw ratio, the square of the ratio of the fiber diameter at the spinneret to that at the take-up roller. For low levels of elasticity and sufficiently strong external heating, there can be three pulling forces consistent with the same draw ratio, similar to the Newtonian case studied by Wylie et al. [31]. For higher levels of elasticity, the process exhibits a draw ratio plateau where the draw ratio hardly changes with the pulling force, reflecting a competition between thermal and elastic effects. As in the Newtonian case, external heating introduces a new instability – termed thermal instability – that is absent in isothermal systems. Linear stability analysis reveals that external heating improves stability for low levels of elasticity, but can worsen stability for higher levels of elasticity, which is again a consequence of the interplay between thermal and elastic effects. Nonlinear simulations indicate that the predictions of linear stability analysis carry over to the nonlinear regime, and show that unstable systems exhibit limit-cycle behavior. The results of the present work demonstrate a possible mechanism through which external heating can stabilize the melt spinning of viscoelastic fibers.     

Stability analysis in spatial mode for channel flow of fiber suspensions

Different from previous temporal evolution assumption, the spatially growing mode was employed to analyze the linear stability for the channel flow of fiber suspensions. The stability equation applicable to fiber suspensions was established and solutions for a wide range of Reynolds number and angular frequency were given numerically . The results show that, the flow instability is governed by a parameter H which represents a ratio between the axial stretching resistance of fiber and the inertial force of the fluid. An increase of H leads to a raise of the critical Reynolds number, a decrease of corresponding wave number, a slowdown of the decreasing of phase velocity , a growth of the spatial attenuation rate and a diminishment of the peak value of disturbance velocity. Although the unstable region is reduced on the whole, long wave disturbances are susceptible to fibers.     

受限亚音速气流中倒置悬臂壁板静气弹稳定性的理论及实验研究

板壳结构在航空航天、高速列车、能量采集等诸多工程领域已经得到了广泛应用.将悬臂壁板倒置于轴向气流中并在壁板周围流场中设置刚性壁面可有效地调控壁板的失稳速度,是俘能器优化设计的重要措施之一.但针对刚性壁面作用下亚音速气流中倒置悬臂壁板的失稳机制仍需要开展深入研究.本文以受限亚音速气流中倒置的二维悬臂壁板为对象,以理论分析及风洞实验为手段,研究了单侧刚性壁面效应对倒置悬臂壁板静态失稳特性的影响规律.在理论分析中,首先应用镜像函数法来处理壁面约束条件,基于算子理论研究获得了以Possio积分方程为表征的壁板气动力,壁面效应实际表征为一包含移位Tricomi算子的复合算子;然后将壁板失稳方程的求解问题转化为定区间上的函数逼近问题;最后,依据Wererstrass定理并利用最小二乘法求解该最优函数,以获得系统的失稳临界参数.在试验研究中依据压杆稳定原理设计了壁板静态失稳的测试方法并完成了风洞实验.理论分析结果表明,壁板会发生发散(静气动弹性)失稳,临界动压随壁板与壁面间距的增加而增大并最终趋于稳定(无壁面情况);通过理论与风洞实验结果的对比分析,验证了本文气动力及理论分析的适用性及准确性.针对倒...     

端部切向载荷作用下的压杆失稳分析及其应用

细直杆件在压应力作用下会产生横向屈曲即失稳．直杆撞击刚性平面或拉断卸载后将形成压缩波，因承载压缩载荷的长度增加可以引起失稳．冲击速度转换的压应力沿着杆件切线方向，该处弯矩和剪力为零；而众多文献设定的失稳段固支边界条件并不准确．基于精确的杆件变形曲率方程得到端部载荷指向杆件中固定点时的受压失稳条件，得到其极限状态即载荷沿杆端切向作用时失稳长度相当于两端简支的1.5 倍．对于钢丝绳拉断形成的冲击失稳，载荷恒定而长度增加，可以产生高阶屈曲即在侧向出现多次曲折，并基于尼龙-橡胶带的模拟试验给出了定性说明．     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

On the non-linear instability of fiber suspensions in a Poiseuille flow

The non-linear instability characteristics of fiber suspensions in a plane Poiseuille flow are investigated. The evolution equation of the perturbation amplitude analogous to Landau equation is formulated and solved numerically for different fiber parameters. It is found that the equilibrium amplitude decreases with the increase of the fiber aspect ratio and volume fraction, i.e. the addition of the fibers reduces the amplitude of the perturbation, and leads to the reduction of the flow instability. This phenomenon becomes significant for large volume concentration and aspect ratio. The mechanism of the reduction of the flow instability is also analyzed by taking into account of the modification of the mean flow and the energy transfer from the mean flow to the perturbation flow.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

双气圈动力学模型的建立与求解

在建立自由气圈的动力学模型的基础上,进一步提出了双气圈的动力学模型。该模型整体化求解双气圈纱线的形态和张力的方法,既改进了以往忽略控制环的解析方法,又避免了求解中由边界条件的不确定性带来的计算上的繁琐和近似。因微分方程中加入了切向空气阻力、纱线的卷取速度及重力等微影响因素,使模型更为严密、可靠,同时拓宽了模型的使用范围。     

Non-isothermal analysis of extrusion film casting using multi-mode Phan-Thien Tanner constitutive equation and comparison with experiments

Extrusion film casting (EFC) is an industrially important process which produces thousands of tons of polymer films, sheets, and coating used for various industrial as well as household applications. In this paper, we focus on an instability which occurs during certain polymer processing operations operating under predominantly elongational flow, such as extrusion film casting and fiber spinning. This instability, called the draw resonance, occurs in the form of sustained periodic fluctuations in the film dimensions. It appears when the process goes beyond the critical line speed of the EFC process. In this work, a conventional linear stability analysis is carried out for nonisothermal EFC process to determine the onset of the draw resonance. The polymer rheology is modeled by the Phan-Thien Tanner (PTT) multi-mode constitutive equation. For the implementation, a conventional shooting method approach is used. Extrusion film casting experiments were also carried out using a conventional linear low-density polyethylene (LLDPE) by varying process parameters such as draw ratio and aspect ratio, to observe the effect on the stability of the process. Linear stability analysis results under non-isothermal conditions are compared and validated with existing results from literature and with our own experimental data. This work displays the effect of multiple relaxation modes as well as the temperature influence on the stability of EFC process. Finally, results also indicate that the temperature highly affects the stability of the EFC process and cannot be ignored from modeling of EFC process.     

Characteristic solutions of scalar newton equations in one space dimension

Newton equations are dynamical systems on the space of fields. The solutions of a given equation which are curves of characteristic fields for its force are planar and have constant angular momentum. Separable solutions are characteristic with angular momentum equal to zero. A Newton equation is separable if and only if its characteristic equation is homogeneous. Separable equations correspond to invariants of homogeneous ordinary differential equations, and those associated with a given homogenous equation correspond to its generalized dilation symmetries. A Newton equation is compatible with the characteristic condition if and only if its characteristic equation is linear. Such equations correspond to invariants of linear ordinary differential equations. Those associated with a given linear equation correspond to the central force problems on its solution space. Regardless of compatibility, any Newton equation with a plane of characteristic fields has non-separable characteristic solutions.     

求解矩形夹层板在集中载荷下的稳定问题

在考虑了横向切应力和横向正应力对夹层板稳定影响的情况下,给出了矩形夹层板结构屈曲失稳的控制方程、基本解以及边界条件。应用功的互等定理求解了在均布载荷作用下的矩形夹层板的屈曲失稳问题。     

裂纹端部细短纤维的应力分析

基于裂纹端部存在与其裂纹面相垂直的二相细短纤维分析模型，采用叠加原理推导了求解纤维表面应力分布函数的积分方程，通过简化得到了该方程的解析表达显式，该积分方程的特征值方程是纤维几何参数，材料常数以及纤维相对于裂纹位置的相关函数，当材料参数不满足特征方程时，积分方程将具有唯一解；并借助数值方法，给出了纤维剪应力分布算例，和纤维对应力强度因子的影响。     

Reduced-order models for microelectromechanical rectangular and circular plates incorporating the Casimir force

We consider the von Kármán nonlinearity and the Casimir force to develop reduced-order models for prestressed clamped rectangular and circular electrostatically actuated microplates. Reduced-order models are derived by taking flexural vibration mode shapes as basis functions for the transverse displacement. The in-plane displacement vector is decomposed as the sum of displacements for irrotational and isochoric waves in a two-dimensional medium. Each of these two displacement vector fields satisfies an eigenvalue problem analogous to that of transverse vibrations of a linear elastic membrane. Basis functions for the transverse and the in-plane displacements are related by using the nonlinear equation governing the plate in-plane motion. The reduced-order model is derived from the equation yielding the transverse deflection of a point. For static deformations of a plate, the pull-in parameters are found by using the displacement iteration pull-in extraction method. Reduced-order models are also used to study linear vibrations about a predeformed configuration. It is found that 9 basis functions for a rectangular plate give a converged solution, while 3 basis functions give pull-in parameters with an error of at most 4%. For a circular plate, 3 basis functions give a converged solution while the pull-in parameters computed with 2 basis functions have an error of at most 3%. The value of the Casimir force at the onset of pull-in instability is used to compute device size that can be safely fabricated.     

Elongational viscosity by fiber spinning

Isothermal melt, fiber-spinning was recently analyzed by means of a nonlinear, integral, constitutive equation that incorporates shear history effects, spectrum of relaxation times, shear-thinning, and extension thinning or thickening when either the drawing force or the draw ratio is specified. The predictions agreed with experimental data on spinning of polystyrene, low-density polyethylene, and polypropylene melts. The predicted apparent elongational viscosity along the threadline (which, as shown in this work, must be identical to that measured experimentally by fiber spinning type of elongational rheometers) is compared with the true elongational viscosity predicted by the same constitutive equation under well-defined experimental conditions of constant extension rate, independent of any strain history. It is concluded that the apparent elongational viscosity, as measured by fiber-spinning, approaches the true elongational viscosity at low Weissenberg numbers (defined as the product of the liquid's relaxation time multiplied by the extension rate). At moderate Weissenberg numbers, the two viscosities may differ by an order of magnitude and their difference grows even larger at high Weissenberg numbers.     

Symmetry and ordinary differential constraints

The representative generalized symmetries of any ordinary differential equation are described in terms of its invariants. This identifies the evolution equations compatible with a given constraint. The restriction of the flow of a compatible equation to the solution space of the constraint is generated by the corresponding internal symmetry. This reduces the evolution equation to a finite dimensional system of first-order ordinary differential equations. The Euler–Lagrange equation of any conserved density of a given evolution equation yields such a reduction. Other examples include the generalized method of separation of variables, the characterization of separable evolution equations, and the characterization of equations with complete families of wave solutions. A Newton equation is compatible with an ordinary differential constraint if and only if the constraint is affine, with force field symmetry, in which case the equation reduces to a finite-dimensional dynamical system. Newton equations with complete families of characteristic solutions reduce to central force problems on solution spaces of linear constraints.     

A form of the solution to the Mathieu equation

A special linear transformation is introduced to express the general solution to a second-order differential equation with a periodic coefficient in terms of a particular solution to an auxiliary second-order nonlinear system with a periodically perturbed right-hand side. It is numerically shown that there exist periodic solutions to the auxiliary system outside the instability regions of the solutions to the Mathieu equation. The estimates obtained for the instability regions are in agreement with known results.     

The instability of jets of arbitrary exit geometry

This paper describes a calculation technique to determine the linear instability characteristics of jets of arbitrary exit geometry. In particular, elliptic and rectangular jets are considered. The numerical procedure involves both a conformal transformation between the computational domain and the physical plane and a solution of the transformed stability equation in the computational domain. Modern, efficient, conformal mappings are used for both simply and doubly connected domains. The numerical solution is based on a hybrid finite difference/pseudospectral discretization of the stability equation. The technique is validated by comparison with previous stability calculations for circular and elliptic jets. Calculations are performed for the stability characteristics of elliptic and rectangular jets of aspect ratio 2:1. Growth rates, phase velocities, and pressure eigenfunctions are presented.     

Non-linear vibration of bimaterial magneto-elastic cantilever beam with thermal loading

Formulas and numerical results are studied for the transient vibration and dynamic instability of a bimaterial magneto-elastic cantilever beam which is subjected to alternating magnetic field and thermal loading. Materials are assumed isotropic, and the physical properties are assumed to have unique values in each layer. The governing equation of motion is derived by the extended Hamilton's principle, in which the damping factor, the electromagnetic force, the electromagnetic torque, and the thermal load are considered. The solution of thermal effect is obtained by superposing certain fundamental linear elastic stress states which are compatible with the Euler–Bernoulli beam theory. The axial stresses results are found to be in good agreement with some known numerical solutions. Using Galerkin's method, the equation of motion is reduced to a time-dependent Mathieu equation. The numerical results of the regions of dynamic instability are determined by the incremental harmonic balance (IHB) method, and the transient vibratory behaviors are presented by the fourth-order Runge–Kutta method. The results show that the responses of the transient vibration and dynamic instability of the system are influenced by the magnetic field, the thickness ratio, the excitation frequency, but not by the temperature increase in this study.