Nonlinear Modeling of Flexible Multibody Systems Dynamics Subjected to Variable Constraints

1989 ◽  
Vol 56 (2) ◽  
pp. 444-450 ◽  
Author(s):  
S. K. Ider ◽  
F. M. L. Amirouche
Keyword(s):  
Nonlinear Modeling ◽  
Equations Of Motion ◽  
Body Motion ◽  
Matrix Formulation ◽  
Closed Loops ◽  
Elastic Bodies ◽  
Minimum Dimension ◽  
Geometric Stiffening ◽  
Angular Velocities ◽  
Constrained Multibody Dynamics

This paper presents the geometric stiffening effects and the complete nonlinear interaction between elastic and rigid body motion in the study of constrained multibody dynamics. A recursive formulation (or direct path approach) of the equations of motion based on Kane’s equations, finite element method and modal analysis techniques is presented. An extended matrix formulation of the partial angular velocities and partial velocities for flexible (elastic) bodies is also developed and forms the basis for our analysis. Closed loops and kinematical constraints (specified motions) are allowed and their corresponding Jacobian matrices are fully developed. The constraint equations are appended onto the governing equations of motion by representing them in a minimum dimension form using an innovative method called the Pseudo-Uptriangular Decomposition method. Examples are presented to illustrate the method and procedures proposed.

On the Dynamics of Elastic Multibody Systems

1999 ◽  
Vol 52 (9) ◽  
pp. 275-303 ◽  
Author(s):  
Hartmut Bremer
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Complex Structure ◽  
Body Motion ◽  
Plate Vibrations ◽  
Elastic Bodies ◽  
Elastic Multibody Systems ◽  
Moving Beam ◽  
Dynamical Coupling ◽  
Left And Right

Due to the highly complex structure of the equations of motion, there exists a basic demand for procedures with minimum effort. This is achieved by the projection method applied to systems consisting of rigid and elastic bodies which undergo fast rigid body motions with superimposed small elastic deflections. The outlined method leads to different left and right Jacobians for the partial differential equations along with simple operators for determination of corresponding boundary conditions. When a Ritz series expansion is used for approximate solution, the left and right Jacobians become identical. The procedure is demonstrated for plate vibrations without rigid body motion and then applied to a single moving beam and finally augmented to multi beam systems. Special attention is hereby given to the effects of dynamical coupling which influence bending stiffness and connect bending with torsion. This review article has 55 references.

Nonlinear Equations of Motion for the Maneuvering Flexible Aircraft Wings

2006 ◽  
Author(s):  
S. Ahmad Fazelzadeh ◽  
Abbas Mazidi
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Body Motion ◽  
Dynamical Equations ◽  
Flexible Wings ◽  
Aircraft Wings ◽  
Angular Velocities ◽  
Nonlinear Equations Of Motion ◽  
Structural Motion

In this paper, the complete dynamical equations for the general maneuvering flexible wings with sweep and dihedral angles are formulated. These equations are valid for an isotropic non-uniform wing; include transverse shear and warping effects. The equations of motion and boundary conditions are derived using Hamilton’s variational principle. Interaction between rigid-body motion caused by the angular velocities of the general maneuver, and elastic deformations of the wing, results in nonlinear terms, form an important contribution to the final equations. For model validation, the simplified partial differential equations of pull-up maneuver are transformed into a set of differential equations through a Galerkin approach and finally the results of numerical simulation are presented. The combination of flexible structural motion and maneuver parameters are very effective on natural frequencies and instability boundaries.

Matrix Solution of Digitized Planar Human Body Dynamics for Biomechanics Laboratory Instruction

1992 ◽  
Author(s):  
David G. Alciatore ◽  
Lawrence D. Abraham ◽  
Ronald E. Barr
Keyword(s):  
Human Body ◽  
Body Position ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Iterative Solution ◽  
The Body ◽  
Body Motion ◽  
Laboratory Instruction ◽  
Matrix Formulation ◽  
Body Dynamics

Abstract The dynamics of planar human body motion, solved with a non-iterative matrix formulation, is presented. The approach is based on applying Newton-Euler equations of motion to an assumed 15 body segment model resulting in a system of 48 equations. The system of equations was carefully ordered to result in a banded system (bandwidth = 10) which is solved efficiently. The method is more favorable than a traditional iterative solution because it is more easily coded, reaction forces are more easily dealt with, and multiple solutions for a given body position can be readily obtained. The results described are limited to planar body motion but the method is easily extendible to general three-dimensional motion. A computer program was developed to process digitized body point coordinate data and calculate resultant joint forces and moments for each frame of data. This method of human body dynamics analysis was developed to support laboratory instruction for an Engineering Biomechanics course. Athletic activities are captured with a three-dimensional video digitizing system and the data is processed resulting in time histories of force and moment distributions throughout the body during the captured event. Computer software performs the analyses and provides real-time graphical illustrations of the kinematics and dynamics results. The dynamics results for the leg of a runner are presented here as an example of the application of the method.

The Global Motion of a Rigid Body Subject to Periodic Body-Fixed Torques

1995 ◽  
Author(s):  
X. Tong ◽  
B. Tabarrok
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
The Body ◽  
Body Motion ◽  
Heteroclinic Orbits ◽  
Coordinate Frame ◽  
Global Motion ◽  
Stable And Unstable Manifolds ◽  
Body Subject

Abstract In this paper the global motion of a rigid body subject to small periodic torques, which has a fixed direction in the body-fixed coordinate frame, is investigated by means of Melnikov’s method. Deprit’s variables are introduced to transform the equations of motion into a form describing a slowly varying oscillator. Then the Melnikov method developed for the slowly varying oscillator is used to predict the transversal intersections of stable and unstable manifolds for the perturbed rigid body motion. It is shown that there exist transversal intersections of heteroclinic orbits for certain ranges of parameter values.

Seismic Analysis of Structures With Coulomb Friction

1983 ◽  
Vol 105 (2) ◽  
pp. 171-178 ◽  
Author(s):  
V. N. Shah ◽  
C. B. Gilmore
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Seismic Event ◽  
Seismic Analysis ◽  
Equations Of Motion ◽  
Relative Displacement ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition Method

A modal superposition method for the dynamic analysis of a structure with Coulomb friction is presented. The finite element method is used to derive the equations of motion, and the nonlinearities due to friction are represented by pseudo-force vector. A structure standing freely on the ground may slide during a seismic event. The relative displacement response may be divided into two parts: elastic deformation and rigid body motion. The presence of rigid body motion necessitates the inclusion of the higher modes in the transient analysis. Three single degree-of-freedom problems are solved to verify this method. In a fourth problem, the dynamic response of a platform standing freely on the ground is analyzed during a seismic event.

Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part II — Application and Results

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Experimental Data ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Open Loop ◽  
Body Motion ◽  
Universal Joint ◽  
Open Loop System ◽  
Numerical Examples ◽  
Kinematic Properties ◽  
Good Agreement

Abstract The application of the systematic procedures in the derivation of the equations of motion proposed in Part I of this work is demonstrated and implemented in detail. The equations of motion for each subsystem are derived individually and are assembled under the concept of compatibility between the local kinematic properties of the elastic degrees of freedom of those connected elastic members. The specific structure under consideration is characterized as an open loop system with spherical unconstrained chains being capable of rotating about a Hooke’s or universal joint. The rigid body motion, due to two unknown rotations, and the elastic degrees of freedom are mutually coupled and influence each other. The traditional motion superposition approach is no longer applicable herein. Numerical examples for several cases are presented. These simulations are compared with the experimental data and good agreement is indicated.

CONSISTENT APPROXIMATION FOR THE STRAIN ENERGY OF A 3D ELASTIC BODY ADEQUATE FOR THE STRESS STIFFENING EFFECT

2004 ◽  
Vol 04 (02) ◽  
pp. 279-292 ◽  
Author(s):  
YU. VETYUKOV
Keyword(s):  
Elastic Body ◽  
Strain Energy ◽  
Conventional Approach ◽  
Deformation Model ◽  
Geometrically Nonlinear ◽  
Consistent Approximation ◽  
New Strategy ◽  
Geometric Stiffening ◽  
Angular Velocities ◽  
Stiffening Effect

Starting from the fully geometrically nonlinear deformation model of a 3D elastic body, a consistent approximation for the strain energy in the vicinity of a pre-deformed state is obtained. This allows for the stress (geometric) stiffening effect to be taken into account. Additional terms arise in the strain energy approximation in comparison to the conventional approach, in which stiffening is incorporated in the form of a so-called geometric stiffness matrix. Computational costs of the new model are of the same order as that of the conventional approach. When compared to the fully geometrically nonlinear theory, the numerical analysis shows the suggested model to describe the dynamics of an elastic rotating structure better than the conventional approach. A new strategy is suggested to treat the non-constant pre-deformation, which is important for the flexible multibody simulations when angular velocities and interaction forces vary in time.

Dynamics of the Planetary Roller Screw Mechanism

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Matthew H. Jones ◽  
Steven A. Velinsky ◽  
Ty A. Lasky
Keyword(s):  
Steady State ◽  
Slip Velocity ◽  
Equations Of Motion ◽  
Viscous Friction ◽  
Contact Angles ◽  
Dynamic Equations ◽  
Roller Screw ◽  
Angular Velocities ◽  
Screw Mechanism ◽  
Lagrange’S Method

This paper develops the dynamic equations of motion for the planetary roller screw mechanism (PRSM) accounting for the screw, rollers, and nut bodies. First, the linear and angular velocities and accelerations of the components are derived. Then, their angular momentums are presented. Next, the slip velocities at the contacts are derived in order to determine the direction of the forces of friction. The equations of motion are derived through the use of Lagrange's Method with viscous friction. The steady-state angular velocities and screw/roller slip velocities are also derived. An example demonstrates the magnitude of the slip velocity of the PRSM as a function of both the screw lead and the screw and nut contact angles. By allowing full dynamic simulation, the developed analysis can be used for much improved PRSM system design.

Flexible Multibody Dynamics Based on a Fully Cartesian System of Support Coordinates

1993 ◽  
Vol 115 (2) ◽  
pp. 294-299 ◽  
Author(s):  
N. Vukasovic ◽  
J. T. Celigu¨eta ◽  
J. Garci´a de Jalo´n ◽  
E. Bayo
Keyword(s):  
Multibody Dynamics ◽  
Reference Frames ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Flexible Multibody Dynamics ◽  
Body Motion ◽  
Cartesian System ◽  
Flexible Multibody ◽  
Local Reference ◽  
Rigid Multibody

In this paper we present an extension to flexible multibody systems of a system of fully cartesian coordinates previously used in rigid multibody dynamics. This method is fully compatible with the previous one, keeping most of its advantages in kinematics and dynamics. The deformation in each deformable body is expressed as a linear combination of Ritz vectors with respect to a local frame whose motion is defined by a series of points and vectors that move according to the rigid body motion. Joint constraint equations are formulated through the points and vectors that define each link. These are chosen so that a minimum use of local reference frames is done. The resulting equations of motion are integrated using the trapezoidal rule combined with fixed point iteration. An illustrative example that corresponds to a satellite deployment is presented.

Simulation of Flexible-Link Manipulators With Inertial and Geometric Nonlinearities

1995 ◽  
Vol 117 (1) ◽  
pp. 74-87 ◽  
Author(s):  
Chris Damaren ◽  
Inna Sharf
Keyword(s):  
Space Shuttle ◽  
Equations Of Motion ◽  
Theory Of Elasticity ◽  
Dynamics Simulation ◽  
The Finite Element Method ◽  
Flexible Link ◽  
Geometric Nonlinearities ◽  
Geometric Stiffening ◽  
Inertial Nonlinearities ◽  
Function Selection

Several important issues relevant to modeling of flexible-link robotic manipulators are addressed in this paper. First, we examine the question of which inertial nonlinearities should be included in the equations of motion for purposes of simulation. A complete model incorporating all inertial terms that couple rigid-body and elastic motions is presented along with a rational scheme for classifying them. Second, the issue of geometric nonlinearities is discussed. These are terms whose origin is the geometrically nonlinear theory of elasticity, as well as the terms arising from the interbody coupling due to the elastic deformation at the link tip. Accordingly, a general way of incorporating the well-known geometric stiffening effect is presented along with several schemes for treating the elastic kinematics at the joint interconnections. In addition, the question of basis function selection for spatial discretization of the elastic displacements is also addressed. The finite element method and an eigenfunction expansion techniques are presented and compared. All issues are examined numerically in the context of a simple beam example and the Space Shuttle Remote Manipulator System. Unlike a single-link system, the results for the latter show that all terms are required for accurate simulation of faster maneuvers. Hence, the conclusions of the paper are contrary to some of the previous findings on the validity of various models for dynamics simulation of flexible-body systems.

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