Lagrangian Procedure Research Articles

Rigid and flexible joint modelling in multibody dynamics using finite elements

Abstract This paper presents a finite element formulation of mechanical joints for multibody dynamics. The augmented Lagrangian procedure is used to express constraints of both holonomic and non-holonomic types. The method is first demonstrated on the holonomic case of the hinge joint. It is then used to develop a wheel capable of radial deformation, slipping, breaking and coupled lateral/torsional deformations. Both joints are formulated with the objective of being integrated into a multiple purpose finite element software for flexible mechanism analysis. Two application examples are presented to demonstrate the effectiveness of the method: the dynamic deployment of a curved reflector in space and the simulation of a nose landing gear during touch-down.

Quasi-incompressible finite elasticity in principal stretches. continuum basis and numerical algorithms

The formulation and numerical analysis of constitutive equations for finite elasticity in terms of principal stretches is considered in detail. On the theoretical side, we develop closed-form expressions for the tangent moduli which appear not to have been previously recorded in full generality in the literature. Quasi-incompressible response is accounted for by means of a three-field variational principle which makes use of a multiplicative decomposition of the deformation gradient into volume preserving and dilatational parts. The incompressible limit is enforced by means of a simple and effective augmented Lagrangian procedure. Two possible implementations are discussed. Numerical examples are presented that illustrate the effectiveness of the proposed formulation.

Optimal control of batch distillation via nonlinear programming

Up to now, batch distillation units have been controlled by means of dynamic programming or variational calculus, neither of which is easily implementable, particularly when multicomponent mixtures are considered. A quite different approach based on an augmented Lagrangian nonlinear programming procedure is presented in this paper. For the maximum distillate problem, an optimal policy for linear time-dependent reflux is determined for each distillation period. From three examples involving binary and ternary mixtures, it appears that the Lagrangian procedure is an efficient tool for solving batch distillation problems.

A dynamical study of the proposed Space Station type configuration

The planar libration-vibration interaction dynamics of the proposed Space Station is studied using a simplified model where the keel and solar panels are represented as beams and the central pressurized modules as a rigid body. In the beginning, the governing nonlinear, nonautonomous and coupled equations of motion are derived using the classical Lagrangian procedure. This is followed by a relatively general finite element analysis of the structure to arrive at the first seven system modes and associated frequencies. Next, a parametric analysis is undertaken, through numerical integration of the equations of motion in conjunction with the modal discretization, to assess the librational and vibrational response of the system. Finally, a closed form-solution of the simplified nonlinear problem is attempted using the variation of parameters (Butenin) method. Its validity is assessed through comparison with the numerical results over a range of system parameters and initial conditions. Results provide information pertaining to the levels of librational and vibrational response and associated acceleration fields, which may prove helpful in appropriately locating experiments and monitoring instruments. It may also aid in the planning of the control system. The analytical approach provides surprisingly close correlation with the more elaborate numerical procedure, thus promising a better physical appreciation of the complex interactions as well as a considerable saving in the computational cost.

An elasto‐plastic analysis of hydrostatic bulging of a circular sheet

Abstract An incremental finite element method for large elasto‐plastic deformation of metal has been developed. The method takes account of mixed boundary data using the updated Lagrangian formulation variational procedure. The code developed is applicable to the process of hydrostatic bulging of a circular sheet clamped at its periphery. The computation algorithm is based on a convenient explicit form of the hydrostatic pressure load correction stiffness matrix for isoparametric elements. Computer programs are able to determine stress, strain, residual stress, surface profile and spring back. Most predictions agree favorably with experimental results.

Stiffness Matrix for Geometric Nonlinear Analysis

A new stiffness matrix for the analysis of thin walled beams is derived. Starting from the principle of virtual displacements, an updated Lagrangian procedure for nonlinear analysis is developed. Inclusion of nonuniform torsion is accomplished through adoption of the principle of sectorial areas for cross‐sectional displacements. This requires incorporation of a warping degree of freedom in addition to the conventional six degrees of freedom at each end of the element. Problems encountered in the use of this and similar matrices in three‐dimensional analysis are described.

The influence of satellite flexibility on orbital motion

The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in e=(l/a 0)2,l anda 0 being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.

On the motion of a nearly spherical bubble in a viscous liquid

The equations of motion of a nearly spherical bubble immersed in an incompressible and viscous liquid are deduced using the Lagrangian procedure with a dissipation function. The results appear valid for only slightly viscous liquids.

Massless spin-3/2theory and the unmixed spinor representations of the Lorentz group

The Rarita-Schwinger theory of a massless spin-3/2 field belonging to the mixed spinor representations (1,1/2) and (1/2,1) and the unmixed spinor representations (1/2,0) and (0,1/2) is explored. The motion is indeterminate to within a group of gauge transformations. It is found that the gauge invariants transform according to the unmixed spinor representations (3/2,0) and (0,3/2). The gauge invariants are quantized by a new non-canonical coordinate-covariant Lagrangian procedure, and the anticommutators are found to be positive. It is shown that the energy-momentum tensor is irremediably gauge variant, thus ruling out any possibility of gravitational interaction. It is found that electromagnetic interaction is also quite impossible to achieve.

SOLUTION OF THE STELLAR STRUCTURE EQUATIONS IN EULERIAN COORDINATES.

ABSTRACT The equations of hydrostatic and thermal equilibrium, assuming only radiative energy transport and spherical symmetry, are solved in Eulerian coordinates by a suitable modification of the Henyey method. An Eulerian approach may possibly be more suitably extended to more spatial dimensions than the usual Lagrangian procedure. The principle advantage of this method is that the equations of hydrostatic and thermal equilibrium and Poisson's equation may be solved simultaneously.

On the Quantum Theory for a Finite-Sized Electron. I

Section I discusses briefly different types of models for a finite-sized electron from the classical point of view with each model being characterized by a particular type of charge-current distribution. Each of these models can be quantized by the Feynman Lagrangian procedure and the new problems encountered are analyzed in Sec. II. These problems are illustrated in Sec. III which goes into the detailed calculations for the first-order perturbation calculation of a finite-sized electron interacting with an external field.