平动弹性梁的刚-柔耦合动力学

本文建立了作大范围平动弹性梁的刚-柔耦合动力学控制方程。分析了大范围平动对弹性梁变形运动动力学性质的影响，发现了大范围平动与变形运动之间的耦合动力学与大范围转动与变形运动之间的耦合动力学存在显著的差异。大范围平动使弹性梁的刚度降低，同时使系统阻尼增加；而大范围转动使弹性梁的刚度增加，同时使系统产生了能量转换的陀螺效应。因此，柔性多体系统刚-柔耦合动力建模中必须包括大范围平动与柔性体变形运动之间的耦合动力学效应。     

Dynamics Investigation of Three Coupled Rods with a Horizontal Barrier

This report is a part of the larger project of non-linear dynamics investigation of three coupled physical pendulums with damping and with arbitrary situated barriers, and externally driven. The set of differential equations and the set of algebraic inequalities (representing a barrier) governing the motion of three coupled rods are presented in the non-dimensional form. The system of governing equations is integrated between two successive impacts, and the discontinuity points are detected (by halving time step until a required precision is obtained). In each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analyzed system exhibiting discontinuities. The fundamental matrices are used during calculation of Lyapunov exponents, during stability analysis of periodic solutions (Floquet multipliers) and in shooting method applied to detect and trace periodic orbits. Some examples for three coupled identical rods with horizontal barrier are reported.     

Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations ${(-\Delta )^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}}$ with an isolated singularity, where ${\sigma\in (0,1)}$ . We prove that all the solutions are asymptotically radially symmetric. When σ = 1, these have been proved by Caffarelli et al. (Comm Pure Appl Math 42:271–297, 1989).     

Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators

This paper reports the analysis of a model of pulse-coupled oscillators with global inhibitory coupling, inspired by experiments on colonies of bacteria-embedded synthetic genetic circuits. Populations are represented by one-dimensional profiles and their time evolution is governed by a singular differential equation. We address the initial value problem and characterize the dynamics’ main features. In particular, we prove that all trajectory behaviors are asymptotically periodic, with asymptotic features only depending on the population cluster distribution and on the model parameters. A criterion is obtained for the existence of attracting periodic orbits, which reveals the existence of a sharp transition as the coupling parameter is increased. The transition separates a regime where any cluster distribution can be obtained in the limit of large times, to a situation where only trajectories with sufficiently large groups of synchronized oscillators perdure.     

Multi-Field Coupled Dynamics for a Micro Beam

This paper proposes a multi-field coupled dynamics equation for a micro beam. The natural frequencies and the amplitude–frequency relationship of the micro beam in the coupled fields are investigated. Changes in the natural frequencies of the micro beam along with time, bias voltage, and dynamic viscosity of gas are discussed. The effects of the system parameters on the amplitude–frequency relationship are investigated. A number of useful results are obtained. These results are useful in the sensitivity design of resonant micro gas sensors excited by the electrostatic force.     

The Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling

We investigate the dynamics of a system of twovan der Pol oscillators with delayed velocity coupling.We use the method of averaging to reduce the problem to the studyof a slow-flow in three dimensions.We study the steady state solutions of this slow-flow, with specialattention given to the bifurcations accompanying their change innumber and stability. We compare these stability results with numericalintegration of the original equations and show that the two sets of resultsare in excellent agreement under certain parameter restrictions.Our interest in this system is due to its relevance to coupled laseroscillators.     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.     

Global Dynamics of a Coupled Chemotaxis-Fluid Model on Bounded Domains

This paper is concerned with the long-time behavior of large amplitude classical solutions to an initial-boundary value problem of a coupled chemotaxis-fluid model which describes the so-called “chemotactic Boycott effect” arising from the interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions. The result is proved via energy method.     

Component-Mode Synthesis Method Variants in the Dynamics of Coupled Structures

The component-mode synthesis method is usually adopted in order to reduce the number of degrees-of-freedom of structures composed of two or more substructures without loosing the main physical characteristics of the whole structure. Many approaches of this method have been proposed in the literature. These approaches differ from each other for the boundary conditions which are imposed at the interface of the two substructures. In this paper four variants of interface boundary conditions are examined. For each set of conditions a suitable coordinate transformation, compatible with the conditions at the boundary degrees-of-freedom between the two substructures, is presented. Moreover, in the numerical applications a comparison between the four component-mode synthesis variants here proposed with respect to the same variants proposed in the literature is presented. The better accuracy of the proposed approach is shown.     

Local Well-Posedness of Dynamics of Viscous Gaseous Stars

We establish the local-in-time well-posedness of strong solutions to the vacuum free boundary problem of the compressible Navier–Stokes–Poisson system in the spherically symmetric and isentropic motion. Our result captures the physical vacuum boundary behavior of the Lane–Emden star configurations for all adiabatic exponents g < \frac65{\gamma < \frac{6}{5}} .     

Gradient Bounds for Elliptic Problems Singular at the Boundary

Let Ω be a bounded smooth domain in \({{R}^N, N \geqq 2}\), and let us denote by d(x) the distance function d(x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model is

$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $

where f belongs to \({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at \({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field \({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies \({B\cdot \nu\geqq \alpha}\) at ?Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.

     

Nonlinear Dynamics of a Flexible Spinning Disc Coupled to a Precompressed Spring

This paper analyses the nonlinear transverse vibrations of a rotating, clamped-free, flexible disc coupled to a precompressed spring. This is representative of a large class of loadings in rotating disc systems such as air jet and electromagnetic excitation commonly used in experiments. Such a loading induces a simultaneous critical speed resonance and parametric instability. The disc is modelled as a Von Kármán plate, and the equations of motion are discretised by a Galerkin projection onto a pair of 1:1 internally resonant modes. The large amplitude wave motions and their stabilities are studied using the averaging method and via numerical continuation techniques. The analysis is carried out in a co-rotating as well as a ground-fixed frame. Numerical simulations are used to verify the above analyses. The response predicted by these analyses is substantially different from that arising from a critical speed resonance or of a parametric instability alone. As many as five equilibrium solutions can coexist at supercritical speed. Two distinct regimes of large amplitude response appear to exist depending on the relationship between the strength of the parametric excitation and the damping. The existence of these regimes underscores the subtle competition between critical speed resonance and parametric instability that is likely to be observed in experiments near critical speed in such systems.Contributed by Prof. A.K. Bajaj.     

Coupled CFD-DEM simulation of hydrodynamic bridging at constrictions

This paper presents a coupled CFD-DEM approach to simulate the flow of particulate suspensions in the intermediate concentration regime where solid volume concentration is 1% < ϕ < 50%. In particular, hydrodynamic multi-particle bridging during flow through a single constriction in a rectangular channel is studied. It is shown that for neutrally buoyant, monodispersed particulate suspensions, the probability of jamming increases with the particle concentration. There also exists a critical particle concentration (ϕ*) for spontaneous bridging, which depends on the ratio of pore size to particle size, the flow velocity, the particle-fluid density contrast, and the flow geometry leading to the constriction. The ϕ* has a strong dependence on the outlet-to-particle relative size (R_o). For 1.5 ≤ R_o ≤ 2.5, a direct transition from a flowing state to a jammed state was observed. For R_o ≥ 3, the flowing state typically transitioned to a dense state characterized by the accumulation of particles near the constriction before jamming. Increasing the inlet-to-particle relative size (R_ip) lowers ϕ* by increasing the number of particles arriving at the constriction simultaneously. The effect of changing R_ip is more pronounced at high R_o when the probability of bridging is lower. A high fluid velocity increases particle interactions near the constriction and accelerates the onset of bridging. However, no distinct effect of velocity on ϕ* was observed in this study. A higher particle-to-fluid density ratio (ρ_p/ρ_f) increases the probability of bridging and leads to a lower ϕ* in a given constriction geometry. The effect saturates at higher ρ_p/ρ_fwhen gravitational forces completely dominate over viscous drag forces. ϕ* is also found to decrease with increasing angle of constriction convergence (θ) for θ < 30°, but increases beyond that at $\theta ={60}^{\circ }$.     

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.     

Dynamics of vorticity at a sphere

The article discusses the two-dimensional flow of an incompressible liquid between two infinitely close concentric spheres, due to an initial distribution of the vorticity differing from zero. The concept of point singularities (vortices, sources, and sinks) at a sphere is introduced. Equations of motion are obtained for point vortices, as well as invariants of the motion, known for the plane case [1]. The simplest case of the mutual motion of a pair of vortices is considered. Equations are obtained for the motion of point vortices at a rotating sphere. Integral invariants for the continuous distribution of the vorticity are obtained, having the dynamic sense of the total kinetic energy and the momentum of the liquid at the sphere. The effect of the topology of the sphere on the dynamics of the vorticity is noted, and a comparison is made with the plane case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 57–65, November–December, 1977.     

Application of Nonlinear Normal Mode Analysis to the Nonlinear and Coupled Dynamics of a Floating Offshore Platform with Damping

The nonlinear dynamics of ships and floating offshore platforms hasattracted much attention over the last several years. However the topicof multiple-degrees-of-freedom systems has almost been completely ignoredwith very few exceptions. This is probably due to the complexity ofanalyzing strongly nonlinear and coupled systems. It turns out thatcoupling may be particularly important for certain critical dynamicssuch as the dynamics of a floating offshore platform about its diagonalaxis. In a previous work, Kota et al. [1] applied the recently developed nonlinearnormal mode technique to analyze the coupled nonlinear dynamics of afloating offshore platform. Although this previous work was restrictedto unforced and undamped systems, in this work a comparison of the twoalternative nonlinear normal mode analysis techniques was completed.Considering the relative practical importance of damping versus externalforcing for this system, in the present work, we utilize just one of thetwo major techniques available [2] to analyze damped multiple-degrees-of-freedom nonlinear dynamics. Specifically, we investigate the effect ofnonlinearity, and non-proportionate damping. Our results show that thistechnique allows one to simply consider the effect of nonlinearity andgeneral damping on the resulting normal modes. This technique isparticularly powerful because it allows one to visualize the modes in ageometric fashion using the invariant manifold concept from dynamicalsystems.     

两种椭圆翼型高速气动特性实验研究

新概念旋转机翼飞机的主机翼既能高速旋转作为旋翼,又可锁定作为固定翼,所以只能使用特殊的前后对称翼型。针对主机翼翼型的这一特殊要求,对16％相对厚度,相对弯度分别为0％和3％的两种椭圆翼型的高速气动特性进行了风洞实验研究,试验分别在中国空气动力研究发展中心FL-21风洞和荷兰代尔夫特大学TST-27风洞进行,采用表面测压和尾排型阻测量技术。试验结果的对比分析表明,有弯度椭圆翼型的升力和力矩特性优于无弯度椭圆翼型,而阻力特性和最大升阻比劣于无弯度椭圆翼型。试验结果为旋转机翼飞机主机翼翼型的选取提供了参考。