A state-space dynamical representation for multibody mechanical systems part II: Systems with closed loops

Summary In Part I the state space equations for systems with tree configurations are developed. Part II turns to systems with closed loops, for which one also must consider the consistency equations on the motions of bodies forming closed loops. Introducing all the constraint equations on the relative motion of contiguous bodies in the loops into the consistency equations and generalizing the procedure used in Part I, a state-space representation of the dynamical equations is obtained.With 4 Figures     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

A data representation for collaborative mechanical design

Collaborative design projects place additional burdens on design documentation practices. The literature on group design has repeatedly documented the existence of problems in design decision-making due to the unavailability of design information. This paper describes a data representation developed for collaborative mechanical design information. The data representation is used to record the history of the design as a sequence of design decisions. The resulting data base records the final specifications, the alternatives which were considered during the design process, and the designers' rationale for choosing the final design parameters. It is currently implemented in a computerized data base system under development at Oregon State University (OSU).     

A natural partitioning scheme for parallel simulation of multibody systems

A parallel partitioning scheme based on physical-co-ordinate variables is presented to systematically eliminate system constraint forces and yield the equations of motion of multibody dynamics systems in terms of their independent co-ordinates. Key features of the present scheme include an explicit determination of the independent co-ordinates, a parallel construction of the null space matrix of the constraint Jacobian matrix, an easy incorporation of the previously developed two-stage staggered solution procedure and a Schur complement based parallel preconditioned conjugate gradient numerical algorithm.     

The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics

In the present work, rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. Concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail. Both internal and external constraints are dealt with on an equal footing. The present approach thus circumvents the use of rotational variables throughout the whole time discretization. After the discretization has been completed a size‐reduction of the discrete system is performed by eliminating the constraint forces. In the wake of the size‐reduction potential conditioning problems are eliminated. The newly proposed methodology facilitates the design of energy–momentum methods for multibody dynamics. The numerical examples deal with a gyro top, cylindrical and planar pairs and a six‐body linkage. Copyright © 2006 John Wiley & Sons, Ltd.     

Model reduction for dynamical systems with quadratic output

Finite element models for structures and vibrations often lead to second order dynamical systems with large sparse matrices. For large‐scale finite element models, the computation of the frequency response function and the structural response to dynamic loads may present a considerable computational cost. Padé via Krylov methods are widely used and are appreciated projection‐based model reduction techniques for linear dynamical systems with linear output. This paper extends the framework of the Krylov methods to systems with a quadratic output arising in linear quadratic optimal control or random vibration problems. Three different two‐sided model reduction approaches are formulated based on the Krylov methods. For all methods, the control (or right) Krylov space is the same. The difference between the approaches lies, thus, in the choice of the observation (or left) Krylov space. The algorithms and theory are developed for the particularly important case of structural damping. We also give numerical examples for large‐scale systems corresponding to the forced vibration of a simply supported plate and of an existing footbridge. In this case, a block form of the Padé via Krylov method is used. Copyright © 2012 John Wiley & Sons, Ltd.     

A dynamical formalism for unrooted systems of rigid bodies

Summary This paper presents a general formalism for the dynamics ofunrooted systems (=systems without kinematical coupling to a Galilean frame). Starting point are Lagrange's equations in ordered form. A set of points, themain points — introduced by O. Fischer hundred years ago- and a fictious body, theaugmented generalized body, permit to separate the translation of the mass center from the motion in the other coordinates. The basic idea is to break up the virtual velocity of a particle into component velocities that can be readily expressed in algebraic form. This allows, after some tensor-algebraic manipulations, an elegant representation ofall inertia coefficents as a linear function ofone kind of tensor, namely thebasic kinetic tensor. Newglobal inertia tensors (GITs), a generalization of those introduced by M. Fayet, are defined. They allow to separate geometry from affine geometry. GITs of order zero are shown to be ageneralization of the reduced mass. A recursive method is presented for efficent formulation of GITs of order one and two. It is also briefly indicated how rooted systems can be interpreted as a special case of rooted ones. A new formula to compute generalized forces due to a nonhomogeneous Newtonian force field is proposed. Results for rooted trees are reviewed as far as they are necessary for the purposes of this paper. The whole method translates conveniently into efficent computer codes.     

Design sensitivity analysis of non-linear structural systems part I: Theory

A unified approach is presented for design sensitivity analysis of non-linear structural systems that include truss, beam, plane elastic solid and plate components. Both geometric and material non-linearities are treated. Sizing design variables, such as thickness and cross-sectional areas of components of individual members and built-up structures, are considered. A distributed parameter structural design sensitivity analysis approach is used that retains the continuum elasticity formulation throughout the derivation of design sensitivity analysis results. Using this approach and an adjoint variable method, expressions for design sensitivity in terms of design variations are derived in the continuous setting which can be evaluated numerically using analysis results of finite element analysis. Both total Lagrangian and updated Lagrangian formulations in non-linear analysis of solid mechanics are used for design sensitivity analysis. Numerical implementation of design sensitivity analysis results using existing finite element code will be presented in Part II of this paper.     

Characterisation of barium titanate-silver composites, part I: Microstructure and mechanical properties

The paper presents an investigation of the influence of silver particles on the microstructure and mechanical properties of barium titanate. Barium titanate-silver composites have been prepared by ball milling precursor powder constituents; followed by drying, sieving and calcination prior to powder compaction. After sintering the green compacts, microstructural analysis was undertaken involving measurement of grain size, silver particle size, phase composition and phase content. Characterisation of mechanical strength, toughness, hardness and stiffness was also undertaken. Reaction product phases between silver and barium titanate could not be detected. The dispersed silver particles were shown to inhibit densification. Silver particles below 1 μm in size were intragranular and attached to domains. The size of the intergranular silver particles increased with silver content. An increase in silver content improved whereas strength, hardness and stiffness decreased, while toughness was unchanged.     

A numerical method for computing domains of attraction for dynamical systems

The backward mapping approach for computation of global domains of attraction of asymptotically stable non-critical equilibrium points of dynamical systems is presented. A basis for the proposed approach is an extension of Lyapunov's direct method due to LaSalle and Lefschetz. An iterative process that converges to the global domain of attraction of an asymptotically stable equilibrium point is formulated. The method applies to both continuous time and discrete time multidimensional systems. It is shown that the backward mapping approach proposed by C. S. Hsu for spiral equilibrium points of second order discrete time systems is a particular case of the algorithm presented here. The proposed method can be used for autonomous systems as well as for systems with periodic coefficients. When applied to discrete time formulation of dynamical systems, the method can be used to determine the regions of stability of periodic solutions. The paper concludes with a number of illustrative examples that demonstrate the usefulness of the proposed approach.     

A nonlinear dynamical systems framework for structural diagnosis and prognosis

This work aims to establish a nonlinear dynamics framework for diagnosis and prognosis in structural dynamic systems. The objective is to develop an analytically sound means for extracting features, which can be used to characterize damage, from modal-based input-output data in complex hybrid structures with heterogeneous materials and many components. Although systems like this are complex in nature, the premise of the work here is that damage initiates and evolves in the same phenomenological way regardless of the physical system according to nonlinear dynamic processes. That is, bifurcations occur in healthy systems as a result of damage. By projecting a priori the equations of motion of high-dimensional structural dynamic systems onto lower dimensional center, or so-called ‘damage’, manifolds, it is demonstrated that model reduction near bifurcations might be a useful way to identify certain features in the input-output data that are helpful in identifying damage. Normal forms describing local co-dimension one and two bifurcations (e.g. transcritical, subcritical pitchfork, and asymmetric pitchfork bifurcations) are assumed to govern the initiation and evolution of damage in a low-order model. Real-world complications in damage prognosis involving spatial bifurcations, global bifurcation phenomena, and the sensitivity of damage to small changes in initial conditions are also briefly discussed.     

Compromise design of stochastic dynamical systems: A reliability-based approach

This paper presents a procedure for obtaining compromise designs of structural systems under stochastic excitation. In particular, an effective strategy for determining specific Pareto optimal solutions is implemented. The design goals are defined in terms of deterministic performance functions and/or performance functions involving reliability measures. The associated reliability problems are characterized by means of a large number of uncertain parameters (hundreds or thousands). The designs are obtained by formulating a compromise programming problem which is solved by a first-order interior point algorithm. The sensitivity information required by the proposed solution strategy is estimated by an approach that combines an advanced simulation technique with local approximations of some of the quantities associated with structural performance. An efficient Pareto sensitivity analysis with respect to the design variables becomes possible with the proposed formulation. Such information is used for decision making and tradeoff analysis. Numerical validations show that only a moderate number of stochastic analyses (reliability estimations) has to be performed in order to find compromise designs. Two example problems are presented to illustrate the effectiveness of the proposed approach.     

A novel nonsmooth dynamics method for multibody systems with friction and impact based on the symplectic discrete format

As multibody systems often involve unilateral constraints, nonsmooth phenomena, such as impacts and friction, are common in engineering. Therefore, a valid nonsmooth dynamics method is highly important for multibody systems. An accuracy representation of multibody systems is an important performance indicator of numerical algorithms, and the energy balance can be used efficiently evaluate the performance of nonsmooth dynamics methods. In this article, differential algebraic equations (DAEs) of a multibody system are constructed using the D'Alembert's principle, and a novel nonsmooth dynamics method based on symplectic discrete format is proposed. The symplectic discrete format can maintain the energy conservation of a conservative system; this property is expected to extend to nonconservative systems with nonsmooth phenomena in this article. To evaluate the properties of the proposed method, several numerical examples are considered, and the results of the proposed method are compared with those of Moreau's midpoint rule. The results demonstrate that the solutions obtained using the proposed method, which is based on the symplectic discrete format, can realize a higher solution accuracy and lower numerical energy dissipation, even under a large time step.     

Application of stochastic averaging for nonlinear dynamical systems with high damping

The asymptotic behavior of coupled nonlinear dynamical systems in the presence of noise is studied using the method of stochastic averaging. It is shown that, for systems with rapidly oscillating and decaying components, the stochastic averaging technique yields a set of equations of considerably smaller dimension, and the resulting equations are simpler. General results of this method are applied to stochastically perturbed nonlinear nonconservative systems in R⁴. It is shown that in such systems the contribution of the stochastic components in the damped modes to the drift term of the critical mode may be beneficial in terms of stability in certain cases.     

Relative rotation number for stochastic systems: dynamical and topological applications

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.