Edward J. Haug

A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems

ABSTRACT ABSTRACT A recursive formulation of the equations of motion of constrained mechanical systems with closed loops is derived, using tools of variational and vector calculus. Kinematic couplings between pairs of contiguous bodies presented in Part 1 of this paper are generalized. Lagrange multipliers are introduced to account for the effects of joints that are cut to define a tree structure. Constraint Jacobian terms are added to the reduced variational equations derived in Part I. Cut-joint constraint acceleration equations are derived, to complete the reduced equations of motion. Lagrange multipliers associated with each cut-joint are eliminated at the first junction body encountered that permits closing the loop that constraints in cut joint. The inductive algorithm developed in Part I is used to calculate accelerations for the system. A multi-loop compressor is analyzed to illustrate use of the method.

Methods of Design Sensitivity Analysis in Structural Optimization

Design sensitivity analysis plays a central role in structural optimization, since virtually all optimization methods require computation of derivatives of structural response quantities with respect to design variables. Three fundamentally different approaches to design sensitivity analysis are presented. These have been used extensively in the structural optimization literature. They are the virtual load method, the state space method, and the design space method. An analysis of these methods indicates that the state space and design space methods are more general than the virtual load method. Moreover, the virtual load method, when applicable, is a special case of the state space method. Any one of these procedures may be incorporated into an optimality criterion or a mathematical programming method for structural optimization.

Shape Design Sensitivity Analysis of Elastic Structures

ABSTRACT ABSTRACT Design problems in which the shape of two- or three-dimensional elastic bodies plays the role of design are studied. Five prototype problems are formulated in a unified variational form, with performance measures involving natural frequency, displacement, and stress in the structure. The material derivative of continuum mechanics and an adjoint variable method of design sensitivity analysis are used to develop an explicit formula for variation of performance measures in terms of normal movement of the boundary of the physical domain. Examples are presented for beams, membranes, shafts, and three-dimensional elastic solids.

Dynamics of mechanical systems with Coulomb friction, stiction, impact and constraint addition-deletion—I theory

Abstract This paper presents a unified formulation of the equations of motion for constrained multibody systems with Coulomb friction, stiction, impact and constraint addition-deletion. Cartesian generalized coordinates are employed, and basic variational principles of mechanics are used for derivation of the governing equations of each of the physical modes of behaviour considered. It is shown that the apparently different physical phenomena considered are in fact closely related mathematically. Computational algorithms to implement the formulations are presented in matrix form in Part I of the paper and are implemented for planar and spatial systems in Parts II and III.

Dynamic analysis of planar flexible mechanisms

The traditional approach to dynamic analysis of mechanisms and machines is based on the assumption that the systems are composed of rigid bodies. The assumption however, does not always hold. Members are elastic, so they deflect when they are subjected to external loads or inertial forces. Particularly at high speed, a linkage may undergo severe elastic deformation due to its own inertia, to an extent that it can no longer perform its intended function in actual operation. With the speed of machinery constantly increasing, an improved mathematical model of machine dynamics is needed to predict system response to transient inputs. Several authors [l]-[9] have recently introduced models of planar mechanical systems, treating flexible members rather than the traditional rigid links. Winfry [l] originally proposed an analysis method in which the effect of member flexibility is modeled by applying a structural dynamics stiffness technique, using the assumption of superposition (uncoupling) of gross rigid-body motion and a small elastic deformation. Later [2] he utilized a reduction of coordinates technique to determine a particular deflection in a mechanism, based on the same method and assumptions as in [l]. Erdman and co-workers [3] introduced the idea of kineto-elastodynamics, which is the study of the motion of mechanisms consisting of elements that deflect due to external loads or internal body forces. Their method is based on a flexibility matrix approach of structural analysis. They assume linear superposition of small elastic deformation due the inertia forces arising in rigid body motion of the system. They model a planar mechanism as combinations of cantilever beams, two force members, and simply supported beams with end moments. Subsequently, Imam and co-authors [4] presented a general method of deflection analysis for planar linkages, including multi-loop mechanisms, extending the method for kineto-elastodynamic analysis of mechanisms [3] and introducing the rate of change of eigenvalues. Sadler and Sandor [S] have employed lumped parameter models for simulating planar motion of mechanism components that are considered as simply supported beams subject to plane bending in kineto-elastodynamic analysis of mechanisms. Longitudinal deflections are neglected and the assumption of small transverse elastic deformations is made, in order that linear Euler-Bernoulli beam theory would apply. Finite difference formulas are applied to

Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces

Numerical algorithms for mapping boundaries of manipulator workspaces are developed and illustrated. Analytical criteria for boundaries of workspaces for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs are well known, but reliable numerical methods for mapping them have not been presented. In this paper, a numerical method is first developed for finding an initial point on the boundary. From this point, a continuation method that accounts for simple and multiple bifurcation of one-dimensional solution curves is developed. Second order Taylor expansions are derived for finding tangents to solution curves at simple bifurcation points of continuation equations and for characterizing barriers to control of manipulators. A recently developed method for tangent calculation at multiple bifurcation points is employed. A planar redundantly controlled serial manipulator is analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion control. Using these methods, more complex planar and spatial Stewart platform manipulators are analyzed in a companion paper.

Dynamics of Articulated Structures. Part I. Theory

ABSTRACT A finite element based method is developed for geometrically nonlinear dynamic analysis of spatial articulated structures; i.e., structures in which kinematic connections permit large relative displacement between components that undergo small elastic deformation. Vibration and static correction modes are used to account for linear elastic deformation of components. Kinematic constraints between components are used to define boundary conditions for vibration analysis and loads for static correction mode analysis. Constraint equations between flexible bodies are derived in a systematic way and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A lumped mass finite element structural analysis formulation is used to generate deformation modes. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled e...

Design sensitivity analysis of elastic mechanical systems

Abstract Adjoint variable methods are employed to develop efficient numerical methods of computing design derivatives of performance measures of elastic mechanical systems. Taking advantage of symmetry in finite element matrices and differential operators, a sensitivity analysis method is developed and demonstrated on a variety of static and dynamic structures and on vibration and impact absorbers. Where direct numerical comparisons are possible, the method is shown to be five to ten times faster than methods previously employed.

Design Sensitivity Analysis in Structural Mechanics.II. Eigenvalue Variations

Abstract The dependence of eigenvalues of boundary-value operators of structural mechanics on design variables that specify material properties and distribution is characterized. Prototype problems considered include vibration of strings, membranes, beams, plates, and plane elastic slabs and buckling of beams. Symmetry and positive definiteness properties of the elliptic differential operators that govern system response are used to show that the eigenvalues depend continuously on design. Further, it is shown that simple eigenvalues are Frechet differentiable with respect to design, but that repeated eigenvalues can only be expected to be Gateaux (directionally) differentiable with respect to design. The latter fact is shown to have substantial consequences in classes of optimal design problems in which the fundamental eigenvalue is known to be repeated at an optimum design. Explicit and computable formulas for derivatives (first variation) of both simple and repeated eigenvalues of each of the prototype ...

A recursive formulation for flexible multibody dynamics, Part II: closed loop systems

Abstract This paper presents a recursive formulation for dynamics of flexible multibody systems. Deformation modes are used to represent elastic deformation of each body, relative to a body reference frame that is permitted to undergo large displacement and rotation in space. Kinematic relations between contiguous bodies that are connected by articulated joints are defined by relative joint coordinates. A recursive variational vector calculus method is presented for efficient formulation and solution of the equations of motion on parallel processors. A manipulator and a rotating blade with geometric nonlinear effects are studied to illustrate computational efficiency of the recursive formulation.