Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Recent developments in advanced aircraft aluminium alloys

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the “absolute nodal coordinate formulation” which was recently introduced for the large deformation analysis of flexible multibody systems.

Flexible Multibody Dynamics: Review of Past and Recent Developments

In probabilistic structural analysis, the performance or response functions usually are implicitly defined and must be solved by numerical analysis methods such as finite-elemen t methods. In such cases, the commonly used probabilistic analysis tool is the mean-based second-moment method, which provides only the first two statistical moments. This paper presents an advanced mean-based method, which is capable of establishing the full probability distributions to provide additional information for reliability design. The method requires slightly more computations than the mean-based second-moment method but is highly efficient relative to the other alternative methods. Several examples are presented to demonstrate the method. In particular, the examples show that the new mean-based method can be used to solve problems involving nonmonotonic functions that result in truncated distributions.

Advanced probabilistic structural analysis method for implicit performance functions

This paper summarises the keynote talk opening the session on ‘Critical Distance Theories of Fracture’; my aim is to describe the methodology used, to demonstrate its applicability in predicting experimental data on fracture and fatigue, to discuss the fundamental basis of the theory and, to suggest areas for future work in this field. The Theory of Critical Distances (TCD) is the name which I have given to a group of theories used for predicting the effects of notches and other stress-concentration features. These theories have a long history, but currently they are being investigated by only a small number of researchers worldwide. This is a pity, because the TCD is capable of accurate predictions in a wide variety of situations. The essential elements are an elastic stress analysis and a material-dependent critical distance, a parameter which we call L. The value of L can be written in terms of other material parameters as: 

$$ L = \frac{1} {\pi }\left( {\frac{{K_c }} {{\sigma _o }}} \right)^2 $$

(1)

The Theory of Critical Distances

On the use of multiaxial fatigue criteria for fretting fatigue life assessment

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

In this paper, the equations of motion of flexible multibody systems are derived using a nonlinear formulation which retains the second-order terms in the strain-displacement relationship. The strain energy function used in this investigation leads to the definition of three stiffness matrices and a vector of nonlinear elastic forces. The first matrix is the constant conventional stiffness matrix, the second one is the first-order geometric stiffness matrix ; and the third is a second-order stiffness matrix. It is demonstrated in this investigation that accurate representation of the axial displacement due to the foreshortening effect requires the use of large number or special axial shape functions if the nonlinear stiffness matrices are used. An alternative solution to this problem, however, is to write the equations of motion in terms of the axial coordinate along the deformed (instead of undeformed) axis. The use of this representation yields a constant stiffness matrix even if higher order terms are retained in the strain energy expression. The numerical results presented in this paper demonstrate that the proposed new approach is nearly as computationally efficient as the linear formulation. Furthermore, the proposed formulation takes into consideration the effect of all the geometric elastic nonlinearities on the bending displacement without the need to include high frequency axial modes of vibration.

Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

Numerical estimation of bone density and elastic constants distribution in a human mandible

Introducing internal damping in multibody system simulations is important as real-life systems usually exhibit this type of energy dissipation mechanism. When using an inertial coordinate method such as the absolute nodal coordinate formulation, damping forces must be carefully formulated in order not to damp rigid body motion, as both this and deformation are described by the same set of absolute nodal coordinates. This paper presents an internal damping model based on linear viscoelasticity for the absolute nodal coordinate formulation. A practical procedure for estimating the parameters that govern the dissipation of energy is proposed. The absence of energy dissipation under rigid body motion is demonstrated both analytically and numerically. Geometric nonlinearity is accounted for as deformations and deformation rates are evaluated by using the Green–Lagrange strain–displacement relationship. In addition, the resulting damping forces are functions of some constant matrices that can be calculated in advance, thereby avoiding the integration over the element volume each time the damping force vector is evaluated.

Jaime Domínguez

Papers

On the use of multiaxial fatigue criteria for fretting fatigue life assessment

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

Geometrically Nonlinear Formulations of Beams in Flexible Multibody Dynamics

Numerical estimation of bone density and elastic constants distribution in a human mandible

An Internal Damping Model for the Absolute Nodal Coordinate Formulation