Unilateral Impact and Contact of Elastic Structures Using Lagrangian Multipliers

2011 ◽  
Author(s):  
Sebastian Tatzko
Keyword(s):  
Equations Of Motion ◽  
Structural Equations ◽  
Contact Forces ◽  
Integration Scheme ◽  
Lagrangian Multiplier ◽  
Elastic Structures ◽  
Lagrangian Multipliers ◽  
Linear Elastic ◽  
Time Stepping ◽  
Numerical Effort

This paper deals with linear elastic structures exposed to impact and contact phenomena. Within a time stepping integration scheme contact forces are computed with a Lagrangian multiplier approach. The main focus is turned on a simplified solving method of the linear complementarity problem for the frictionless contact. Numerical effort is reduced by applying a Craig-Bampton transformation to the structural equations of motion.

Nonlinear Aeroelastic Behavior of Slender Wings Considering a Static Stall Model Based on Wagner Function

2013 ◽  
Vol 325-326 ◽  
pp. 172-179
Author(s):  
D. Badiei ◽  
M.H. Sadr ◽  
Sh. Shams
Keyword(s):  
Equations Of Motion ◽  
Linear Part ◽  
Beam Theory ◽  
Structural Equations ◽  
Aeroelastic Response ◽  
Linear Elastic ◽  
Model Based ◽  
Aerodynamic Model ◽  
Lift Curve ◽  
Nonlinear Aerodynamic

The aeroelastic behavior of high aspect ratio wings in an incompressible flow is investigated. The nonlinear nonplanar bending-bending-twisting motions of beam theory is used for the structural equations assuming large deformations with small strains, small Poisson effects, inextensional beam theory, and linear elastic material characteristics by neglecting warping and shear deformation. An Unsteady nonlinear aerodynamic static stall model based on the Wagner function is introduced and then is used for determination of aerodynamic loading of the wing. In this aerodynamic model, the static lift curve vs. angle of attack is approximated by a piece-wise curve and for each linear part of this curve a corrected Wagner theory is used. Combining these two types of formulation yields fully nonlinear integro-differentials aeroelastic equations of motion. The governing equations will be solved to predict the nonlinear aeroelastic response of a wing in the stall and post stall regions using Galerkin's method and a numerical method without the need of adding any aerodynamic state-space variables and their corresponding equations. The obtained equations are solved for some test cases and the obtained results are compared with the results given in the literature. Also a study is done to show effects of nonlinear aerodynamic static stall model on the limit cycle oscillations.

Nonlinear Aeroelastic Behavior of Slender Wings Considering a Static Stall Model Based on Wagner Function

2012 ◽  
Vol 186 ◽  
pp. 297-304 ◽  
Author(s):  
Davood Badiei ◽  
Mohammad Homayoon Sadr ◽  
Shahrokh Shams
Keyword(s):  
Equations Of Motion ◽  
Linear Part ◽  
Beam Theory ◽  
Structural Equations ◽  
Aeroelastic Response ◽  
Linear Elastic ◽  
Model Based ◽  
Aerodynamic Model ◽  
Lift Curve ◽  
Nonlinear Aerodynamic

The aeroelastic behavior of high aspect ratio wings in an incompressible flow is investigated. The nonlinear nonplanar bending-bending-twisting motions of beam theory is used for the structural equations assuming large deformations with small strains, small Poisson effects, inextensional beam theory, and linear elastic material characteristics by neglecting warping and shear deformation. An Unsteady nonlinear aerodynamic static stall model based on the Wagner function is introduced and then is used for determination of aerodynamic loading of the wing. In this aerodynamic model, the static lift curve vs. angle of attack is approximated by a piece-wise curve and for each linear part of this curve a corrected Wagner theory is used. Combining these two types of formulation yields fully nonlinear integro-differentials aeroelastic equations of motion. The governing equations will be solved to predict the nonlinear aeroelastic response of a wing in the stall and post stall regions using Galerkin's method and a numerical method without the need of adding any aerodynamic state-space variables and their corresponding equations. The obtained equations are solved for some test cases and the obtained results are compared with the results given in the literature. Also a study is done to show effects of nonlinear aerodynamic static stall model on the limit cycle oscillations.

Formulation of Multibody Dynamics as Complementarity Problems

2003 ◽  
Author(s):  
J. C. Trinkle
Keyword(s):  
Discrete Time ◽  
Complementarity Problem ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Contact Forces ◽  
Integration Scheme ◽  
Time Step ◽  
Stick Slip ◽  
Frictional Contacts ◽  
Time Formulation

Multibody systems with rigid bodies and unilateral contacts are difficult to simulate due to discontinuities associated with gaining and losing contacts and stick-slip transitions. Methods for simulating such systems fall into two categories: penalty methods and complementarity methods. The former calculate penetration depths of virtual rigid bodies at every time step and compute restoring forces to repair penetrations, while the latter assume that the bodies are truly rigid and compute contact forces that prevent penetration from occurring at all. In this paper, we are concerned with complementarity methods. We present an instantaneous formulation of the equations of motion of multi-rigid-body systems with frictional contacts as a complementarity problem. The unknowns in this formulation are accelerations and forces at the contacts. Since it is known that this model does not always admit a finite solution, it is problematic to use it directly in an integration scheme. This fact motivates the discrete-time formulation presented second. Although the discrete-time formulation also takes the form of a complementarity problem, it does not suffer from non-existence, and thus it is suitable for simulation. Numerical results are compared to the exact solution for a sphere initially sliding, then rolling, on a horizontal plane.

The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

2020 ◽  
Vol 1 (1) ◽  
pp. 93-102
Author(s):  
Carsten Strzalka ◽  
◽  
Manfred Zehn ◽  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
A Priori ◽  
High Stress ◽  
Von Mises Stress ◽  
Detailed Knowledge ◽  
Variable Loading ◽  
Numerical Effort ◽  
Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

The Anchor-Last Deployment Problem for Inextensible Mooring Lines

1975 ◽  
Vol 97 (3) ◽  
pp. 1046-1052 ◽  
Author(s):  
Robert C. Rupe ◽  
Robert W. Thresher
Keyword(s):  
Numerical Model ◽  
Equations Of Motion ◽  
Simultaneous Equations ◽  
Mooring Line ◽  
Integration Scheme ◽  
Lumped Mass ◽  
Mooring Lines ◽  
Concentrated Masses ◽  
Predictor Corrector ◽  
Lagrange’S Method

A lumped mass numerical model was developed which predicts the dynamic response of an inextensible mooring line during anchor-last deployment. The mooring line was modeled as a series of concentrated masses connected by massless inextensible links. A set of angles was used for displacement coordinates, and Lagrange’s Method was used to derive the equations of motion. The resulting formulation exhibited inertia coupling, which, for the predictor-corrector integration scheme used, required the solution of a set of linear simultaneous equations to determine the acceleration of each lumped mass. For the selected cases studied the results show that the maximum tension in the cable during deployment will not exceed twice the weight of the cable and anchor in water.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

Multirate Integration for Multibody Dynamic Analysis With Decomposed Subsystems

1999 ◽  
Author(s):  
Sung-Soo Kim ◽  
Jeffrey S. Freeman
Keyword(s):  
Dynamic Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Low Frequency ◽  
Inertia Force ◽  
Integration Scheme ◽  
Integration Algorithm ◽  
Multibody Dynamic ◽  
Higher Order Derivative ◽  
Derivative Information

Abstract This paper details a constant stepsize, multirate integration scheme which has been proposed for multibody dynamic analysis. An Adams-Bashforth Moulton integration algorithm has been implemented, using the Nordsieck form to store internal integrator information, for multirate integration. A multibody system has been decomposed into several subsystems, treating inertia coupling effects of subsystem equations of motion as the inertia forces. To each subsystem, different rate Nordsieck form of Adams integrator has been applied to solve subsystem equations of motion. Higher order derivative information from the integrator provides approximation of inertia force computation in the decomposed subsystem equations of motion. To show the effectiveness of the scheme, simulations of a vehicle multibody system that consists of high frequency suspension motion and low frequency chassis motion have been carried out with different tire excitation forces. Efficiency of the proposed scheme has been also investigated.

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