基于θ1方法的多体动力学数值算法研究

将结构动力学领域的\theta_1方法拓展到数值求解多体系统运动方程------微分--代数方 程(DAEs), 分别求解指标-3 DAEs形式的运动方程和指标-2超定DAEs (ODAEs)形式的运动方程. 通过数值算例验证了方法的有效性, 并得到\theta _1 方法中参数\theta _1的选取与数值耗散量之间的关系. 数值算例还说明对于同 一个多体系统, 采用指标-3的DAEs 描述时存在速度违约, 用指标-2的ODAEs描述时, 从计算机精度上讲, 位置和速度约束方程 同时满足, 并且\theta_1方法在求解非保守系统DAEs和ODAEs形式的运动方程时 都具有2阶精度. 最后\theta_1 方法与其他直接积分法求解DAEs和ODAEs形式运 动方程的CPU时间进行了比较.     

分析力学初值问题的一种变分原理形式

明确了分析力学初值问题的控制方程,按照广义力和广义位移之间的对应关系,将 各控制方程卷乘上相应的虚量,代数相加,进而在 原空间中建立了分析力学初值问题的一种变分原理形式,即建立了分析力学初值问题的卷积 型变分原理和卷积型广义变分原理. 推导了分析力学初值问题卷积型变分原理和卷积型广义 变分原理的驻值条件. 在建立分析力学初值问题的一种变分原理形式的同时, 将变积方法推广为卷变积方法.     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

Particle swarm optimization-based algorithm of a symplectic method for robotic dynamics and control

Multibody system dynamics provides a strong tool for the estimation of dynamic performances and the optimization of multisystem robot design. It can be described with differential algebraic equations(DAEs). In this paper, a particle swarm optimization(PSO) method is introduced to solve and control a symplectic multibody system for the first time. It is first combined with the symplectic method to solve problems in uncontrolled and controlled robotic arm systems. It is shown that the results conserve the energy and keep the constraints of the chaotic motion, which demonstrates the efficiency, accuracy, and time-saving ability of the method. To make the system move along the pre-planned path, which is a functional extremum problem, a double-PSO-based instantaneous optimal control is introduced. Examples are performed to test the effectiveness of the double-PSO-based instantaneous optimal control. The results show that the method has high accuracy, a fast convergence speed, and a wide range of applications.All the above verify the immense potential applications of the PSO method in multibody system dynamics.     

Localized Basis Function Method for Computing Limit Cycle Oscillations

An alternate approach to the standard harmonic balance method (based on Fourier transforms) is proposed. The proposed method begins with an idea similar to the harmonic balance method, i.e. to transform the initial set of differential equations of the dynamics to a set of discrete algebraic equations. However, as distinct from previous harmonic balance techniques, the proposed method uses a set of basis functions which are localized in time and are not necessarily sinusoidal. Also as distinct from previous harmonic balance methods, the algebraic equations obtained after the transformation of the differential equations of the dynamics are solved in the time domain rather than the frequency domain. Numerical examples are provided to demonstrate the performance of the method for autonomous and forced dynamics of a Van der Pol oscillator.     

多体系统动力学动态最优化设计与灵敏度分析

基于多体系统的动态最优化设计过程包括传统的多体系统仿真分析、系统设计灵敏度分析、 系统最优化设计等过程, 针对多体系统运动学、用二阶常微分方程和微分代数方程描述 的动力学,基于含设计参数的通用数学模型及通用的积分型目标函数,采用高效的系统灵 敏度分析伴随变量方法及易于实施的惩罚函数最优设计方法,建立了多体系统最优设计数学 模型和算法. 通过双摆系统、曲柄-滑块系统、弹簧/阻尼器-滑块系统3个算例对上述 算法的有效性进行了验证.     

带约束非线性多体系统动力学方程数值分析方法

Lagrange方法是建立带约束多体系统动力学方程的普遍方法之一 ,其方程的形式为微分 代数方程组 ,数值计算与数值分析是研究多体系统动力学特性的重要方法。本文利用缩并法给出了带约束多体系统动力学方程的隐式数值计算方法和Lyapunov指数的计算方法。将数值仿真、Lya punov指数计算和Poincare映射有机结合 ,分析非线性多体系统动力学行为。通过一个算例 ,说明该方法的有效性     

Simulation of electrically driven jet using Chebyshev collocation method

The model of electrically driven jet is governed by a series of quasi 1D dimensionless partial differential equations (PDEs). Following the method of lines, the Chebyshev collocation method is employed to discretize the PDEs and obtain a system of differential-algebraic equations (DAEs). By differentiating constrains in DAEs twice, the system is transformed into a set of ordinary differential equations (ODEs) with invariants. Then the implicit differential equations solver “ddaskr” is used to solve the ODEs and post-stabilization is executed at the end of each step. Results show the distributions of radius, linear charge density, stretching ratio and also the horizontal velocity at a time point. Meanwhile, the spiral and expanding projections to X-Y plane of the jet centerline suggest the occurring of bending instability.     

Numerical Methods for Constrained Equations of Motion in Mechanical System Dynamics

ABSTRACT

ABSTRACT A new class of numerical methods for solving equations of motion of constrained mechanical systems is presented, the framework of which is based on manifold theoretic methods. Rewriting the system of differential-algebraic equations (DAEs) that describe constrained motion is ordinary differentia] equations (ODEs) on a constraint manifold, the theoretical framework for solving equations of motion is constructed, using a local     

A Legendre wavelet spectral collocation technique resolving anomalies associated with velocity in some boundary layer flows of Walter-B liquid

A Legendre wavelet spectral collocation method is proposed here to solve three boundary layer flow problems of Walter-B fluid namely the stagnation point flow, Blasius flow and Sakiadis flow. In the proposed method, we first transform the boundary value problems into initial value problems using shooting method. We then split the semi infinite domain into subintervals and the governing initial value problems are transformed to system of algebraic equations in each subinterval. The solutions of these algebraic equations yield an approximate solution of the differential equation in each subinterval. The overshoot in the velocity profile associated with the stagnation point and Blasius flows and undershoot in the Sakiadis flow is controlled. Physically realistic solutions are presented for both weakly and strongly viscoelastic parameters. The residual error validates the correctness, convergence and accuracy of the obtained solutions.     

Nonlinear dynamics and performance analysis of modified snap-through vibration energy harvester with time-varying potential function

Vibration energy harvesting has emerged as a promising method to harvest energy for small-scale applications. Enhancing the performance of a vibration energy harvester(VEH) incorporating nonlinear techniques, for example, the snap-through VEH with geometric non-linearity, has gained attention in recent years. A conventional snap-through VEH is a bi-stable system with a time-invariant potential function, which was investigated extensively in the past. In this work, a modified snap-through VEH wit...     

微分求积方法对于桩基大变形分析的应用

本文基于大变形的理论,采用弧坐标首先建立了具有初始位移的桩基的非线性数学模型,一组强非线性的微分-积分方程,其中,地基的抗力采用了Winkeler模型;其次,引入变数变换将微分-积分方程转化为一组非线性微分方程,并用微分求积方法离散了方程组,得到一组离散化的非线性代数方程;最后用Newton-Raphson迭代方法对离散化方程进行了求解,得到了桩基变形前后的构形、弯矩和剪力.计算中选取了两种不同类型的初始位移,并考察了它们对桩基大变形力学行为的影响.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic theory of neutral stability of the Couette flow of a vibrationally excited gas

An asymptotic theory of the neutral stability curve for a supersonic plane Couette flow of a vibrationally excited gas is developed. The initial mathematical model consists of equations of two-temperature viscous gas dynamics, which are used to derive a spectral problem for a linear system of eighth-order ordinary differential equations within the framework of the classical linear stability theory. Unified transformations of the system for all shear flows are performed in accordance with the classical Lin scheme. The problem is reduced to an algebraic secular equation with separation into the “inviscid” and “viscous” parts, which is solved numerically. It is shown that the thus-calculated neutral stability curves agree well with the previously obtained results of the direct numerical solution of the original spectral problem. In particular, the critical Reynolds number increases with excitation enhancement, and the neutral stability curve is shifted toward the domain of higher wave numbers. This is also confirmed by means of solving an asymptotic equation for the critical Reynolds number at the Mach number M ≤ 4.     

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.     

Constructing generalized Cesàro formulas for finite plane strains

The problem of finding the displacement vector from a system of nonlinear differential equations which includes displacement gradient components is studied. Expressions on the right side of this system for certain parameter values have the kinematic sense of Lagrange and Euler finite strain tensors. The task is to construct generalized Cesàro formulas for finite strains. The construction of the solution consists of two stages (algebraic and differential), and the second is performed for space whose dimension is greater than or equal to two. An algorithm for the inversion of the original system is proposed, and analytical constructions for the case of two-dimensional space are performed. The problem is solved at the first (algebraic) stage, i.e., an exact analytical expression for the displacement vector components is derived through the known finite strain tensor and an unknown scalar function having the kinematic sense of rotation. Necessary conditions for the existence of this relationship are formulated.     

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.     

Dimensional Reduction for Nonlinear Time-Delayed Systems Composed of Stiff and Soft Substructures

This paper presents a new approach, based on the center manifoldtheorem, to reducing the dimension of nonlinear time-delay systemscomposed of both stiff and soft substructures. To complete the reductionprocess, the dynamic equation of a delayed system is first formulated asa set of singular perturbed equations that exhibit dynamic behaviorevolving in two different time scales. In terms of the fast time scale,the dynamic equation of system can be converted into the standard formof a functional differential equation in critical cases, namely, to aform that can be treated by means of the center manifold theorem. Then,the approximated center manifold is determined by solving a series ofboundary-value problems. The center manifold theorem ensures that thedominant dynamics of the system is described by a set of ordinarydifferential equations of low order, the dimension of which is identicalto that of the phase space of slowly variable states. As an applicationof the proposed approach, a detailed stability analysis is made for aquarter car model equipped with an active suspension with a time delaycaused by a hydraulic actuator. The analysis shows that the dimensionalreduction is surprisingly effective within a wide range of the systemparameters.     

DIFFERENTIAL-ALGEBRAIC APPROACH TO COUPLED PROBLEMS OF DYNAMIC THERMOELASTICITY

An efficient numerical approach for the general thermomechanical problems was developed and it was tested for a two-dimensional thermoelasticity problem. The main idea of our numerical method is based on the reduction procedure of the original system of PDEs describing coupled thermomechanical behavior to a system of Differential Algebraic Equations (DAEs) where the stress-strain relationships are treated as algebraic equations. The resulting system of DAEs was then solved with a Backward Differentiation Formula (BDF) using a fully implicit algorithm. The described procedure was explained in detail, and its effectiveness was demonstrated on the solution of a transient uncoupled thermoelastic problem, for which an analytical solution is known, as well as on a fully coupled problem in the two-dimensional case.