Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Boundary-value problems for differential equations unresolved with respect to the derivative in the space of bounded number sequences

We study a nonlinear countable-point boundary-value problem for a differential equation unresolved with respect to the derivative. This equation and a nonlinear boundary condition are defined in the Banach space of bounded number sequences. We study the reducibility of the posed problem to a multipoint boundary-value problem in a finite-dimensional space. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 391–415, July–September, 2007.     

Scattering of SH-wave by cracks originating at an elliptic hole and dynamic stress intensity factor

The method of complex function and the method of Green‘s function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamicstress intensity factor at the crack tip was given. A Green‘s function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.     

轴对称载荷下正交异性圆环壳的一般解

本文导出了在Kirchhoff假设下受轴对称载荷时、材料正交异性的圆环壳的复变量方程,给出了该方程的一般解,该解可用于α=a/R<1,其中a为环壳截面的半径,R为环壳的总体半径,文中并举例说明其应用,结果表明本文所提出的解是很有效的。     

电磁轴承系统的刚度阻尼特性分析

分析了电磁轴承系统的刚度阻尼特性与系统结构参数及其控制器频响特性之间的关系。给出了系统控制电流的相位对系统稳定性和刚度阻尼特性的影响。提出了电磁轴承系统的复阻尼概念，以及部分刚度阻尼特性指标的实际意义和计算方法。     

Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

Exact solutions of the axisymmetric equations of motion of a viscous heat-conducting perfect gas described by systems of ordinary differential equations

All invariant solutions of rank 1 of the two-dimensional equations of motion of a heat-conducting perfect gas with a polytropic equation of state are described. A sufficient condition for reducibility of regular, partially invariant solutions of rank 1 and defect 1 to invariant solutions is given. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 51–54, September–October, 1999.     

A numerical model for the flooding and drying of irregular domains

A numerical technique for the modelling of shallow water flow in one and two dimensions is presented in this work along with the results obtained in different applications involving unsteady flows in complex geometries. A cell‐centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the flow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. It is shown that this modification reproduces exactly steady state of still water in configurations with strong variations in bed slope and contour. The applications presented are mainly related with unsteady flow problems. The scheme is capable of handling complex flow domains as will be shown in the simulations corresponding to the test cases that are going to be presented. Comparisons of experimental and numerical results are shown for some of the tests. Copyright © 2002 John Wiley & Sons, Ltd.     

Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic GL(d,\mathbb R){GL(d,\mathbb R)} cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.     

An algorithm of finite element grid generation for three-dimensional complex connected domain

An automated quasi three-dimensional finite element grid generation method is presented for particular three-dimensional complex connected domain, across which some are simply connected two-dimensional.regions and some are multiply connected two-dimensional regions. A subdivision algorithm based on the variational principle has been developed to ascertain the smoothness and orthogonality of the generated grid in any cross sections. Smooth transition between the simply and multiply connected regions is maintained. For illustration, the method is applied to generate a finite element three-dimensional grid for human above knee stump.     

磁控溅射沉积高承载、低摩擦MoS_2/Ti复合薄膜

采用磁控溅射技术制备了不同Ti含量的Mo S2/Ti复合薄膜,利用SEM、AFM、纳米压痕仪、XRD和CSM摩擦试验机分析了复合薄膜的结构、力学和摩擦学性能.结果表明:复合薄膜结构致密,表面光滑平整,且具有较高的硬度,Ti含量较低的Mo S2/Ti复合薄膜呈现以(002)基面为主的择优取向;在大气环境下,赫兹接触应力为2.5 GPa的摩擦工况下,Ti含量较低的Mo S2/Ti复合薄膜的摩擦系数低至0.02,磨损率低至10-17m3/(N·m)数量级,呈现出高承载、低摩擦、耐磨损的优异摩擦学性能.这是由于Ti的掺杂一方面提高了复合薄膜的力学和抗氧化性能,另一方面复合薄膜的(002)基面取向对其摩擦磨损性能发挥了重要作用.     

Numerical analysis of bifurcation buckling for rotationally periodic structures under rotationally periodic loads

By considering the characteristics of deformation of rotationally periodic structures under rotationally periodic loads, the periodic structure is divided into some identical substructures in this study. The degrees-of-freedom (DOFs) of joint nodes between the neighboring substructures are classified as master and slave ones. The stress and strain conditions of the whole structure are obtained by solving the elastic static equations for only one substructure by introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get the bifurcation buckling load and mode for the whole rotationally periodic structure by solving the eigenvalue problem for only one substructure without introducing any additional approximation. The finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is derived. Different measures of tackling internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the high efficiency and validity of this method.     

Design framework for DEM-supported prototyping of grabs including full-scale validation

The design of machinery for handling granular materials relies mainly on empirical methods and in-house engineering knowledge. This traditional approach provides incremental improvements that are often limited. Advancements in simulation and optimization can offer a promising alternative approach. Most of the research involved in improving or optimizing equipment design does not include the realistic performance of the new prototype and as such it is uncertain that the predicted performance is also guaranteed in practice.In this study, a design framework for a new generation of machinery handling granular materials, grabs, has been established that includes a full-scale validation step. This has been proven to lead to a breakthrough in equipment design. This design framework uses a co-simulation between Discrete Element Method (DEM) and Multi Body Dynamics (MBD), thus, capturing operational conditions in full-scale. The DEM simulation supported design step integrated as the main step to generate new prototypes. The performance of the prototype is evaluated by conducting full-scale experiments, thus validating the adequacy of the new design as well as the accuracy of the co-simulation. Through this a full design cycle has been fulfilled and a validated model has been achieved that is independent of specific design configurations.     

On Invertibility of Elastic Single-Layer Potential Operator

The question of unique solvability of the boundary integral equation of the first kind given by the single-layer potential operator is studied in the case of plane isotropic elasticity. First, a sufficient condition of the positivity, and hence invertibility, of this operator is presented. Then, considering a scale transformation of the domain boundary, the well known formula for scaling the Robin constant in potential theory is generalized to elasticity. Subsequently, an explicit equation for evaluation of critical scales for a given boundary, when the single-layer operator fails to be invertible, is deduced. It is proved that there are either two simple critical scales or one double critical scale for any domain boundary. Numerical results, obtained applying a symmetric Galerkin boundary element code, confirm the propositions of the theory developed for both single and multi-contour boundaries.     

用虚拟激励法求解非比例阻尼线性体系的非平稳随机地震响应

应用复振型分解方法,将非比例阻尼线性体系在地震作用下的动力方程求解问题转化为若干个广义复振子的求解与叠加问题。通过假定地震地面运动为一零均值的非平稳随机激励,应用虚拟激励法原理,推导得到了广义复振子动力坐标计算的一般公式,进而得到了非比例阻尼线性体系非平稳随机地震响应计算的一般解答。由于可以选择少量共轭复振型的影响进行计算,对于大型复杂非比例阻尼结构,其随机地震响应计算工作量可以大幅度减小。算例证实了这种方法的可靠性及可行性。     

Conjugated laminar forced convection in ducts with periodic variation of inlet temperature

Conjugated laminar forced convection with parabolic velocity is studied for flow inside both parallel-plate and circular ducts subjected to a periodically varying inlet temperature. The analysis of this classs of problems lead to a Sturm-Liouville type complex eigenvalue problem for which no known solution is available. In this work, a new methodology is presented for a direct solution of such complex eigenvalue problems in the complex domain using the shooting method along with the Runge-Kutta method. The methodology is applicable for solving both laminar and turbulent flow problems; here we consider only the laminar flow with a parabolic velocity profile. The results clearly show that the slug-flow assumption overestimates the phase lag and the Nusselt number. Some benchmark results are also presented for the eigenquantities in tabular form.     

复合材料动态粘弹性能的细观研究

利用细观力学的Eshelby等效夹杂方法研究了颗粒增强复合材料的动态粘弹性力学性能,分析了材料复模量随夹杂体积分数、载荷频率之间的变化规律,给出了许多有意义的结论,为复合材料结构的优化设计及应用提供了理论基础。     

Euler solutions for aerodynamic inverse shape design

Contributions to the aerodynamics development have to be involved to achieve an increase in quality, reducing time and computer costs. Therefore, this work develops an optimization method based on the finite volume explicit Runge–Kutta multi‐stage scheme with central spatial discretization in combination with multigrid and preconditioning. The multigrid approach includes local time‐stepping and residual smoothing. Such a method allows getting the goal of compressible and almost incompressible solution of fluid flows, having a rate of convergence almost independent from the Mach number. Numerical tests are carried out for the NACA 0012 and 0009 airfoils and three‐dimensional wings based on NACA profiles for Mach‐numbers ranging from 0.8 to 0.002 using the Euler equations. These calculations are found to compare favorably with experimental and numerical data available in the literature. Besides, it is worth pointing out that these results build on earlier ones when finding appropriate new three‐dimensional aerodynamical geometries. Copyright © 2004 John Wiley & Sons, Ltd.     

Parametrically Excited Non-Linear Traveling Beams with and without External Forcing

The effects of parametric excitation on a traveling beam, both with and without an external harmonic excitation, have been studied including the non-linear terms. Non-linear, complex normal modes have been used for the response analysis. Detailed numerical results are presented to show the effects of non-linearity on the stability of the parametrically excited system. In the presence of both parametric and external harmonic excitations, the response characteristics are found to be similar to that of a Duffing oscillator. The results are sensitive to the relative strengths of and the phase difference between the two forms of excitations.     

The study of equivalent material parameters in a hyperelastic model

We study a hyperelastic model of some biological soft tissues with emphasis on the problem of its matching with the material parameters acquired by experimental mechanical tests. First, we study the polyconvexity property of the hyperelastic model. Then, we explore the notion of equivalent sets of material parameters. We perform a numerical study of the regions of equivalent material parameters characterizing the curves predicted by the hyperelastic model that are close, within a prefixed tolerance, to those given by the experimental data. In the numerical study we use the quadratic variation and the Hausdorff distance. The study suggests that a qualitative knowledge of shape and volume of the regions of equivalent material parameters can provide both a criterion for the optimal match between the model with the experimental data and an indication on the reducibility of the number of parameters used in the model.