Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

Unilateral Multibody Dynamics

Contact processes may be represented by local discretization, by a rigid body approach or by a mixed method using both ideas. For the dynamics of mechanical systems a rigid body approach is described achieving good results also for multiple contact problems. This paper considers mainly contacts in multi-body systems where the corresponding contact constraints vary with time thus generating structure-variant systems. The equations of motion for dynamical systems with such an unilateral behavior are discussed, solution methods and applications are presented.     

Steady-state analysis of multibody systems with reference to vehicle dynamics

This paper presents a systematic methodology and formulation for determining the steady-state response of multibody systems. The equations of motion for a general multibody system are described in terms of a set of relative joint accelerations. Then, the differential equations of motion are converted to a set of algebraic equations for the steady-state response. These equations are derived based upon a set of conditions that must exist for the steady state. The application of this formulation in determining the steady-state response of a vehicle moving in a circular path is shown. The multibody model of the vehicle for two- or four-wheel steering is presented. The results of the steady-state simulation are compared with those obtained from a transient dynamic analysis.     

Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates

A wide variety of mechanical and structural multibody systems consist ofvery flexible components subject to kinematic constraints. The widelyused floating frame of reference formulation that employs linear modelsto describe the local deformation leads to a highly nonlinear expressionfor the inertia forces and can be applied to only small deformationproblems. This paper is concerned with the formulation and computerimplementation of spatial joint constraints and forces using the largedeformation absolute nodal coordinate formulation. Unlike the floatingframe of reference formulation that employs a mixed set of absolutereference and local elastic coordinates, in the absolute nodalcoordinate formulation, global displacement and slope coordinates areused. The nonlinear kinematic constraint equations and generalized forceexpressions are expressed in terms of the absolute global displacementsand slopes. In particular, a new formulation for the sliding jointbetween two very flexible bodies is developed. A surface parameter isintroduced as an additional new variable in order to facilitate theformulation of this sliding joint. The constraint and force expressionsdeveloped in this paper are also expressed in terms of generalizedCholesky coordinates that lead to an identity inertia matrix. Severalexamples are presented in order to demonstrate the use of theformulations developed in the paper.     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

Hamiltonian dynamics of an elastica and the stability of solitary waves

A method is presented for deriving unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems. The method is applied to the theory of planar, finite-amplitude motions of inextensible and unshearable elastic rods. The constraints of inextensibility and unshearability become integrals of motion in the Hamiltonian formulation.It is known that in the theory of uniform, inextensible, unshearable rods of infinite length there arise solitary-wave solutions with the property that each profile can move at arbitrary speed. The Hamiltonian formulation is exploited to analyze the stability properties of these solitary waves. The wave profiles are first characterized as critical points of an appropriate time-invariant functional. It is then shown that for a certain range of wave speeds the solitary-wave profiles are actually nonisolatedminimizers of the functional, a fact with implications for nonlinear stability.     

Multiple impacts with friction in rigid multibody systems

An impact model for two-dimensional contact situations is developed which contains the main physical effects of a compliance element in the normal direction and a series of a compliance and Coulomb friction elements in the tangential direction. For systems with multiple impacts a unilateral formulation based on Poisson's hypothesis is used to describe the impulses which are transferred in the normal direction. The event of an impact is divided into two phases. The phase of compression ends with vanishing approaching velocity if normal impulses are transferred and is equivalent to a completely inelastic collision. The phase of expansion allows the bodies to separate under the action of the normal impulses whenever they are large enough. The absolute values of the tangential impulses are bounded by the magnitudes of the normal impulses, due to the Coulomb friction relationship on the impulse level. One part of the transferred tangential impulse during compression is assumed to be partly reversible which may be regarded as an application of Poisson's law. The remaining part is completely irreversible and considered friction. This formulation contains the special case of completely elastic tangential impacts as well as the situation when only Coulomb friction acts. It is proven that the presented impact model is always dissipative or energy preserving. The evaluation of the problem is done by solving one set of complementarity conditions during compression and a nearly identical set of equations during expansion. The theory is applied to some basic examples which demonstrate the difference between Newton's and Poisson's hypotheses.     

含非理想约束多柔体系统递推建模方法

基于多体系统中邻接物体运动学递推关系,可以证明树状多体系统中末端物体的作用体现为传递给其内接物体的惯性和外力. 由于闭环系统切断铰约束反力和非理想约束反力可看作为系统外力,任何复杂系统都可以转化为等效的树系统,并且系统约束方程中所涉及的广义加速度可以系统化地用描述约束反力的拉氏乘子替换. 基于以上结果,提出了针对含非理想约束多柔体系统递推建模方法. 利用该方法可以将复杂多体系统动态减缩为单个物体,从而在求解系统加速度时不需对整个系统的质量矩阵进行求逆运算,同时大幅度地降低了非理想约束反力方程的维数. 通过一个算例具体说明了所提方法的求解过程,算例结果与现有商业软件所得结果一致.      

刚柔耦合约束多体系统的动力分析

本文提出了描述柔性多体系统的牵连坐标系统。该系统由惯性参考系,牵连坐标系,物体坐标系及单元坐标系组成,实现了对刚体平动,刚体转动及弹性运动的连续分解,最大限度地消除了由于刚体大角度转动导致的非线性特性。以有限元法为基础,应用拉格朗日方程建立了在该坐标下的刚柔耦合约束多体系统的动力学控制方程。该方程具有耦合程度小、易于推导、编程及求解等优点,为大规模约束多体系统的动力分析提供了新的途径。本文还讨论了平面铰链约束的约束形式及约束方程,最后给出了一个典型多体系统的数值算例。     

A new approximate solution technique for randomly excited non-linear oscillators—II

The method of weighted residuals is applied to the reduced Fokker-Planck equation associated with a non-linear oscillator, which is subjected to both additive and multiplicative Gaussian white noise excitations. A set of constraints are deduced for obtaining an approximate stationary probability density for the system response. One of the constraints coincides with the previously proposed criterion of dissipation energy balancing, and the others are useful for calculating the equivalent conservative force. It is shown that these constraints imply certain relationships among certain statistical moments; their imposition guarantees that such moments computed from the approximate probability density satisfy the corresponding exact equations derived from the original equation of motion. Moreover, the well-known procedure of stochastic linearization and its improved version of partial linearization are shown to be special cases of this scheme, and they are less accurate since the approximations are not chosen from the entire set of the solution pool of generalized stationary potential. Applications of the scheme are illustrated by examples, and its accuracy is substantiated by Monte Carlo simulation results.     

Variational Lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion

The nonisothermal finite strain dynamics of a porous solid containing a viscous fluid is developed on the basis of a new thermodynamics of open systems and irreversible processes. The same theory is applicable to the mechanics of a nonporous solid with thermomolecular diffusion of a substance in solution. New fundamental concepts of “thermobaric” and “convective” potentials are presented in the context of porous solids. Field equations and Lagrangian equations with generalized coordinates are derived directly from a variational principle of “virtual dissipation”. Inclusion of nonlinear viscoelasticity and plastic behavior is indicated. Partial saturation of pore fluid is discussed. The theory is applicable to the mechanics of a non porous solid with thermolecular diffusion of several molecular species in solution, and under certain conditions to the analogous case of a porous solid containg a fluid mixture. It is shown how the Lagrangian equations provide the foundation of finite element methods.     

Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.     

A circular inclusion with circumferentially inhomogeneous imperfect interface in harmonic materials

In the following analysis, we present a rigorous solution for the problem of a circular elastic inclusion surrounded by an infinite elastic matrix in finite plane elastostatics. The inclusion and matrix are separated by a circumferentially inhomogeneous imperfect interface characterized by the linear spring-type imperfect interface model where the interface is such that the same degree of imperfection is realized in both the normal and tangential directions. Through the use of analytic continuation, a set of first-order coupled ordinary differential equations with variable coefficients are developed for two analytic potential functions. The unknown coefficients of the potential functions are determined from their analyticity requirements and some additional problem-specific constraints. An example is then presented for a specific class of interface where the inclusion mean stress is contrasted between the homogeneous interface and inhomogeneous interface models. It is shown that, for circumstances where a homogeneously imperfect interface may not be warranted, the inhomogeneous model has a pronounced effect on the mean stress within the inclusion.     

Dynamic modelling of the double wishbone motor-vehicle suspension system

In this paper the dynamic analysis of the double wishbone motor-vehicle suspension system using the point-joint coordinates formulation is presented. The mechanical system is replaced by an equivalent constrained system of particles and then the laws of particle dynamics are used to derive the equations of motion. Due to the presence of large number of geometric and kinematic constraints the velocity transformation approach is used to eliminate some constraints. The equations of motion in terms of the Cartesian coordinates of the particles are transformed to a reduced set in terms of relative joint variables by defining differential-algebraic equations in terms of the joint variables are equal to the number of degrees of freedom of the whole system plus the number of cut-joint constraints corresponding to cut of kinematical closed loops. Use of both the Cartesian and relative joint variables produces an efficient set of equations without loss of generality. The chosen suspension includes open and closed loops with quarter-car model.     

A new approach to the vakonomic mechanics

The aim of this paper was to show that the Lagrange–d’Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d’Alembert–Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.     

Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.     

多体系统中的冗余约束

冗余约束主要起因于系统奇异构型以及切断铰约束方程的自动生成, 它的存在对多体系统建模和求解都提出了更高的要求. 为了使系统运动方程可解, 需要从系统约束中分离出一组独立约束, 不同的独立约束组往往造成数值分析结果的不同, 但本文严格证明了理论上它们是一致的. 冗余约束在很大程度上加大了系统奇异构型附近违约量控制的难度. 针对这一困难, 本文提出了一种将约束稳定化和违约修正相结合的方法, 数值算例证明了方法的有效性. 鉴于物体的受力分析是实际工程的迫切需要也是多体系统动力学核心任务之一, 本文研究了冗余约束对约束反力的影响, 给出了判别铰约束反力是否唯一的实用准则, 针对两个铰连接同一对物体而引起的冗余约束, 提出了铰合成原理及其求解各自铰内接触力的方法, 并通过数值算例说明了方法的可行性.      

Reliability Index and Failure Probability

Abstract

Numerical algorithms for the solution of nonlinear algebraic equation systems are discussed. Special application to the mechanism and multibody system kinematic analysis, as well as to the problems of constraint stabilization during dynamics simulation is regarded. Special attention is paid to the approaches of a separate solution of the differential equations and constraint stabilization. Numerical procedures that are effective additions to the well-known algorithms based on the Newton-Raphson method are presented. The problems of loss of precision and achievement of large unreal increments of the varying parameters are discussed. The traditional Newton-Raphson method is modified by applying a step reduction procedure that is developed numerically for the symbolic form of kinematic and dynamic equations. An optimization method for stabilization of constraints using the mass matrix of dynamic equations is suggested. According to the objective function defined the stabilization procedure provides minimal deviations of the parameters and their velocities with respect to the solution of the differential equations. No generalized coordinate partitioning is required either for solution of the dynamic equations or for stabilization of the constraints. Several examples of kinematic analysis of single and four contour plane mechanisms and constraint stabilization are solved, and the results are compared. The advantages of the algorithms developed are tested with a high-degree of initial deviation from the real solution. It is also shown that the step correction algorithm could provide admissible solution even when, in many cases, the classical approaches are not reliable. An example of the direct and inverse kinematic problem solutions of the four-degrees-of-freedom spatial platform is presented.