Some aspects of destabilization in reversible dynamical systems with application to follower forces

This paper examines the destabilization of the equilibria of reversible dynamical systems which is induced by the addition of irreversible perturbations. Attention is restricted to reversible dynamical systems which have frequently appeared in the literature on elastic stability. There they are often referred to as follower force problems. The destabilization phenomenon is linear in nature and explicit criteria are established to determine the particular eigenvalue splittings. The post-destabilization dynamics are also examined using the appropriate normal forms for two specific cases, one where the eigenvalues are non-resonant and the other where the eigenvalues are in a strong one-to-one resonance. Finally, the destabilization criteria and certain features of the post-destabilization dynamics are illustrated using two examples of follower force systems.     

Vibrations of Circular Sandwich Plates under Resonance Loads

The paper studies axisymmetric resonance vibrations of an elastic circular sandwich plate under local periodic surface loads of rectangular, sinusoidal, and parabolic forms. The hypotheses of broken normal are used to describe the kinematics of the plate, which is asymmetric in thickness. The core is assumed to be light. The initial–boundary-value problems are solved analytically. The solutions are analyzed     

非线性模态构造方法与机电耦合系统Hopf分岔

大型汽轮发电机组轴系与电网耦合次同步谐振（ＳＳＲ）现象是在某种参数条件下机电耦合系统产生Ｈｏｐｆ分岔的结果。在文献［１］中，作者提出了分析这种系统Ｈｏｐｆ分岔的非线性模态方法，得出了在固定参数下分岔解的结果。本文针对高维非线性动力系统（包括奇数维），提出新的非线性模态构造方法，并给出了机电耦合次同步振荡系统在辅助参数变化条件下分岔解的的变化规律。     

LOCAL BIFURCATION ANALYSIS OF STRONGLY NONLINEAR DUFFING SYSTEM

ＬＯＣＡＬＢＩＦＵＲＣＡＴＩＯＮＡＮＡＬＹＳＩＳＯＦＳＴＲＯＮＧＬＹＮＯＮＬＩＮＥＡＲＤＵＦＦＩＮＧＳＹＳＴＥＭＢｉＱｉｎｓｈｅｎｇ（毕勤胜）；ＣｈｅｎＹｕｓｈｕ（陈予恕）；ＷｕＺｈｉｑｉａｎｇ（吴志强）Ａｂｓｔｒａｃｔ：Ｂｙｕｓｉｎｇｃｏｏｒｄｉｎ...     

Normal Forms for Control Systems at Singular Points

A normal form for open loop control systems is provided, based on their interpretation as skew product flows and on normal forms for nonautonomous differential equations.     

Analysis on Double Hopf Bifurcation Using Computer Algebra with the Aid of Multiple Scales

In this paper, a perturbation method based on multiple scales is used forcomputing the normal forms of nonlinear dynamical systems. The approach,without the application of center manifold theory, can be used to systematicallyfind the explicit normal form of a system described by a general n-dimensionaldifferential equation. The attention is focused on the dynamic behaviour ofa system near a critical point characterized by two pairs of purely imaginaryeigenvalues without resonance. The method can be easily formulated and implemented using a computer algebra system. Maple programs have been developedwhich can be `automatically' executed by a user without knowing computeralgebra. Examples chosen from mathematics, electrical circuits, mechanics andchemistry are presented to show the applicability of the technique and theconvenience of using computer algebra.     

QUALITATIVE ANALYSIS OF SPHERICAL CAVITY NUCLEATION AND GROWTH FOR INCOMPRESSIBLE GENERALIZED VALANIS-LANDEL HYPERELASTIC MATERIALS

I. INTRODUCTION In practice, cavity formulation in materials is recognized as precursors to failure. Thus void nucleationand growth in solid materials have a great in?uence on failure mechanism. Gent and Lindley[1] have observed experimentally the phe…     

DYNAMIC RESPONSE OF SATURATED POROUS ELASTIC FOUNDATION UNDER POROSITY ANISOTROPY$^{\bf 1)}$

由于不同的沉积条件和应力状态, 天然土体通常表现出一定的各向异性特征. 文章研究地基上表面受温度载荷和机械载荷时, 孔隙率各向异性参数变化对饱和多孔弹性地基热-水-力耦合动力响应问题的影响. 基于Lord-Shulman广义热弹性理论, 结合孔隙率各向异性基本假设, 建立了孔隙率各向异性饱和多孔弹性地基热-水-力耦合动力响应模型, 利用正则模态法推导出无量纲竖向位移、超孔隙水压力、竖向应力和温度分布的解析表达式并加以图示. 正则模态法是一种利用加权残差求得解析解的方法, 相较于其他方法能快速求解偏微分方程. 当孔隙率各向异性参数为1时, 可将该各向异性耦合动力响应模型退化为热-水-力耦合动力响应模型验证该地基模型的合理性. 着重分析了孔隙率各向异性参数变化对不同物理量的影响. 结果表明: 孔隙率各向异性参数变化对物理量均有一定影响. 在地基上表面受温度载荷作用时, 对超孔隙水压力和竖向应力影响最为明显; 在地基上表面受机械载荷作用时, 对超孔隙水压力和温度影响明显. 整体而言, 无论地基上表面受何种载荷, 随着各向异性参数增大, 峰值逐渐减小, 在地基深度增加方向峰值所在位置向靠近地基上表面方向移动.     

Second-Order Shell Kinematics Implied by Rotation Constraint-Equation

The paper presents a general methodology of introducing the shell-type variables which is based on the rotation constraint-equation (RC-equation). The RC-equation is proven to be equivalent to the polar decomposition of the deformation gradient formula, and the rotations which it yields are interpreted in terms of rotations of vectors of an ortho-normal basis. The deformation function and rotations are assumed as polynomials of the thickness coordinate ζ, and in this form used in the RC-equation. Solving this equation, we can express the coefficients of the quadratic deformation function in terms of the following shell-type variables: (a) the mid-surface position x ₀, (b) the constant rotation Q ₀, (c) the rotation vector ψ ^* for the ζ-dependent rotations, and (d) the normal components U ₃₃ ⁰ and U ₃₃ ¹ of the right stretching tensor. This new methodology (i) ensures that all shell kinematical variables are consistent with the RC-equation, which is justified on 3D grounds, (ii) provides a general framework from which various Reissner-type hypotheses can be obtained by suitable assumptions. As an example, two generalized Reissner hypotheses are derived: one with two normal stretches, and the other with the in-plane twist and the bubble-like warping parameters. This revised version was published online in June 2006 with corrections to the Cover Date.     

Resonant Hopf–Hopf Interactions in Delay Differential Equations

A second-order delay differential equation (DDE) which models certain mechanical and neuromechanical regulatory systems is analyzed. We show that there are points in parameter space for which 1:2 resonant Hopf–Hopf interaction occurs at a steady state of the system. Using a singularity theoretic classification scheme [as presented by LeBlanc (1995) and LeBlanc and Langford (1996)], we then give the bifurcation diagrams for periodic solutions in two cases: variation of the delay and variation of the feedback gain near the resonance point. In both cases, period-doubling bifurcations of periodic solutions occur, and it is argued that two tori can bifurcate from these periodic solutions near the period doubling point. These results are then compared to numerical simulations of the DDE.