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.

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.

The Dynamics of a Deformable Body Experiencing Large Displacements

1988 ◽  
Vol 55 (3) ◽  
pp. 676-680 ◽  
Author(s):  
W. P. Koppens ◽  
A. A. H. J. Sauren ◽  
F. E. Veldpaus ◽  
D. H. van Campen
Keyword(s):  
Rigid Body ◽  
Mass Matrix ◽  
Deformable Body ◽  
Equations Of Motion ◽  
Body Motion ◽  
General Description ◽  
Large Displacements ◽  
Displacement Fields ◽  
Cpu Time ◽  
Time Savings

A general description of the dynamics of a deformable body experiencing large displacements is presented. These displacements are resolved into displacements due to deformation and displacements due to rigid body motion. The former are approximated with a linear combination of assumed displacement fields. D’Alembert’s principle is used to derive the equations of motion. For this purpose, the rigid body displacements and the displacements due to deformation have to be independent. Commonly employed conditions for achieving this are reviewed. It is shown that some conditions lead to considerably simpler equations of motion and a sparser mass matrix, resulting in CPU time savings when used in a multibody program. This is illustrated with a uniform beam and a crank-slider mechanism.

Small Vibrations Superimposed on Non-Linear Rigid Body Motion

1997 ◽  
Author(s):  
A. L. Schwab ◽  
J. P. Meijaard
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rigid Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Connecting Rod ◽  
Non Linear ◽  
Linear Differential

Abstract In the case of small elastic deformations in a flexible multi-body system, the periodic motion of the system can be modelled as a superposition of a small linear vibration and a non-linear rigid body motion. For the small deformations this analysis results in a set of linear differential equations with periodic coefficients. These equations give more insight in the vibration phenomena and are computationally more efficient than a direct non-linear analysis by numeric integration. The realization of the method in a program for flexible multibody systems is discussed which requires, besides the determination of the periodic rigid motion, the determination of the linearized equations of motion. The periodic solutions for the linear equations are determined with a harmonic balance method, while transient solutions are obtained by averaging. The stability of the periodic solution is considered. The method is applied to a pendulum with a circular motion of its support point and a slider-crank mechanism with flexible connecting rod. A comparison is made with previous non-linear results.

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.

scholarly journals An Efficient Modelling of Flexible Airships

2006 ◽  
Author(s):  
Selima Bennaceur ◽  
Naoufel Azouz ◽  
Djaber Boukraa
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
General System ◽  
Rigid Body Motion ◽  
The Body ◽  
Body Motion ◽  
Modal Synthesis ◽  
Stiffness Matrices ◽  
Updated Lagrangian ◽  
Convenient Technique

This paper presents an efficient modelling of airships with small deformations moving in an ideal fluid. The formalism is based on the Updated Lagrangian Method (U.L.M.). This formalism proposes to take into account the coupling between the rigid body motion and the deformation as well as the interaction with the surrounding fluid. The resolution of the equations of motion is incremental. The behaviour of the airship is defined relatively to a virtual non-deformed reference configuration moving with the body. The flexibility is represented by a deformation modes issued from a Finite Elements Method analysis. The increment of rigid body motion is represented similarly by rigid modes. A modal synthesis is used to solve the general system equations of motion. Time constant matrices appears (i.e. mass and structural stiffness matrices), and we show a convenient technique to actualise the time dependant matrices.

A Variationally Consistent Approach to Constrained Motion

2016 ◽  
Vol 83 (5) ◽  
Author(s):  
John T. Foster
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonholonomic Systems ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Constrained Motion ◽  
Consistent Approach ◽  
Holonomic And Nonholonomic Systems ◽  
D'alembert's Principle

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.

Dynamic Modeling and Simulation of Flexible Robots With Prismatic Joints

1990 ◽  
Vol 112 (3) ◽  
pp. 307-314 ◽  
Author(s):  
Ye-Chen Pan ◽  
R. A. Scott ◽  
A. Galip Ulsoy
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Numerical Procedure ◽  
Differential Algebraic Equations ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Integration Scheme ◽  
System Response ◽  
Constraint Equations ◽  
Prismatic Joints

A dynamic model for flexible manipulators with prismatic joints is presented in Part I of this study. Floating frames following a nominal rigid body motion are introduced to describe the kinematics of the flexible links. A Lagrangian approach is used in deriving the equations of motion. The work done by the rigid body axial force through the axial shortening of the link due to transverse deformations is included in the Lagrangian function. Kinematic constraint equations are used to describe the compatibility conditions associated with revolute joints and prismatic joints, and incorporated into the equations of motion by Lagrange multipliers. The small displacements due to the flexibility of the links are then discretized by a displacement based finite element method. Equations of motion are derived for the cases of prescribed rigid body motion as well as prescribed joint torques/forces through application of Lagrange’s equations. The equations of motion and the constraint equations result in a set of differential algebraic equations. A numerical procedure combining a constraint stabilization method and a Newmark direct integration scheme is then applied to obtain the system response. An example, previously treated in the literature, is presented to validate the modeling and solution methods used in this study.

Biaxial Vibrations of an Elasto-Plastic Beam With a Prescribed Rigid-Body Rotation

2003 ◽  
Author(s):  
M. Dibold ◽  
J. Gerstmayr ◽  
H. Irschik
Keyword(s):  
Rigid Body ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Body Motion ◽  
Plastic Behavior ◽  
Plastic Strains ◽  
Time Step ◽  
Implicit Midpoint Rule ◽  
Linear Elastic

In the present work, the development of plastic strains in a flexural beam is studied. The beam is modeled as a Bernoullieuler beam, where large rigid-body rotations and biaxial bending in the small strain regime are studied. The deformation is split into the spatial deformation of a hinged-hinged beam and the movement of the second support. Neglecting axial displacements of the beam, this support moves on a sphere. In the present paper, the latter motion is considered as prescribed. The beam thus is assumed to possess only flexural degrees-offreedom. Such a problem is frequently to be encountered in machine dynamics or robotics. We assume the stiffness of the beam to be considerably lowered due to catastrophic environmental influences, such that the deformations relative to the rigid-body motion, albeit small, reach the plastic regime. The equations of motion are derived by Hamilton’s principle. The potential energy follows from the internal energy due to the elastic part of the deformation and the potential due to gravity. Plastic strains are treated according to the theory of eigenstrains, which act as sources of self-stress upon the linear elastic beam. The biaxial deflections are discretized in space by means of Legendre polynomials. The plastic strains are discretized over length, height and width of the beam by small plastic cells. The plastic strains are computed in every time-step by a suitable iterative procedure. An implicit midpoint rule, which preserves the total energy of the system, is used for integration of the equations of motion. Linear elastic/perfectly plastic behavior is exemplarily treated in a numerical study.

Modelling lateral manoeuvres in fish

2012 ◽  
Vol 697 ◽  
pp. 1-34 ◽  
Author(s):  
K. Singh ◽  
T. J. Pedley
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Body Dynamics ◽  
The Body ◽  
Body Motion ◽  
Dynamic State ◽  
Parameter Study ◽  
Element Formulation ◽  
Turn Angle ◽  
Impulsive Input

AbstractWe propose a method to model manoeuvres in self-propelled flexible-bodied fish by modelling the hydrodynamics coupled to the body inertia. Flexible body motion is prescribed and the equations of motion are solved for the position of the centre of mass and rotation of the body. The governing equations are formulated by applying the conservation of linear and angular momentum. Two independent methods to model the fluid dynamics are pursued: Model 1 is an extension of elongated-body theory, modified for self-propulsion and flexible motion. Model 2 applies a numerical boundary-element formulation with the fish modelled as an infinitely thin rectangular body. The manoeuvring response to an impulsive input is first examined to understand the rigid-body characteristics of the fish. A flexible bend action is included to model C-bends of the type observed during escapes in fish. Models 1 and 2 are used to cross-verify the respective implementations as well as to develop physical insights into manoeuvring. A parameter study shows that fish of intermediate body depths are best adapted to rapid turns whereas the initial dynamic state of the fish is instrumental in affecting the sign as well as the magnitude of the turn angle, for a prescribed bend deflection. Computations for combined swimming and turning show that the initial rigid-body dynamics of the fish is much more effective than the induced effect of the prior shed wake in enhancing the turning response.

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