Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions: Part I—Element Level Equations

1990 ◽  
Vol 112 (2) ◽  
pp. 203-214 ◽  
Author(s):  
S. Nagarajan ◽  
David A. Turcic
Keyword(s):  
Rigid Body ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Realistic Model ◽  
Open Loop ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Elastic Links

Equations of motion are derived using Lagrange’s equation for elastic mechanism systems. The elastic links are modeled using the finite element method. Both rigid body degrees of freedom and the elastic degrees of freedom are considered as generalized coordinates in the derivation. Previous work in the area of analysis of general elastic mechanisms usually involve the assumption that the rigid body motion or the nominal motion of the system is unaffected by the elastic motion. The nonlinear differential equations of motion derived in this work do not make this assumption and thus allow for the rigid body motion and the elastic motion to influence each other. Also the equations obtained are in closed form for the entire mechanism system, in terms of a minimum number of variables, which are the rigid body and the elastic degrees of freedom. These equations represent a more realistic model of light-weight high-speed mechanisms, having closed and open loop multi degree of freedom chains, and geometrically complex elastic links.

scholarly journals On Equations of Motion of Elastic Linkages by FEM

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Michał Hać
Keyword(s):  
Rigid Body ◽  
Shape Function ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Open Loop ◽  
Rigid Body Motion ◽  
Body Motion ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Flexible Mechanisms

Discussion on equations of motion of planar flexible mechanisms is presented in this paper. The finite element method (FEM) is used for obtaining vibrational analysis of links. In derivation of dynamic equations it is commonly assumed that the shape function of elastic motion can represent rigid-body motion. In this paper, in contrast to this assumption, a model of the shape function specifically dedicated to the rigid-body motion is presented, and its influence on elastic motion is included in equations of motion; the inertia matrix related to the rigid-body acceleration vector depends on both shape functions of the elastic and rigid elements. The numerical calculations are conducted in order to determine the influence of the assumed shape function for rigid-body motion on the vibration of links in the case of closed-loop and open-loop mechanisms. The results of numerical simulation show that for transient analysis and for some specific conditions (e.g., starting range, open-loop mechanisms) the influence of assumed shape functions on vibration response can be quite significant.

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.

Using Dual Cam Tracks for High Speed Rigid Body Motion

1998 ◽  
Author(s):  
Clay Cooper ◽  
Stephen Derby
Keyword(s):  
Rigid Body ◽  
High Speed ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Transfer Rates ◽  
Track System ◽  
Selection Of

Abstract Rigid Body Motion has long been one of the standard problems for kinematicians. For high speed transfer rates, an industrial example of using a dual cam track system to achieve better performance is documented. The dual track establishes both a positional and orientational location of the followers. The selection of this mechanism type is discussed.

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.

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.

Dynamic Analysis of Flexible Supercavitating Vehicles Using Modal Based Elements

2003 ◽  
Author(s):  
Jou-Young Choi ◽  
Massimo Ruzzene ◽  
Olivier A. Bauchau
Keyword(s):  
Numerical Model ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Structural Characteristics ◽  
Rigid Body Motion ◽  
Operating Conditions ◽  
Flight Mechanics ◽  
Body Motion ◽  
Supercavitating Vehicles ◽  
Elastic Modes

This presents a numerical model for the simulation of the flight mechanics behavior of flexible supercavitating vehicles. Supercavitating vehicles exploit supercavitation as a means to reduce drag and increase the underwater speed. In the proposed formulation, the vehicle’s rigid body motion is described by 6 degrees of freedom, which define pitch, yaw and roll motion and the displacement of the center of gravity with respect to a fixed inertial reference system. The forces applied to the vehicle include the control actions at the nose and at the fins, propulsion, gravity and cavity/vehicle periodic interactions associated to typical operating conditions. The elastic displacements are superimposed to the rigid body motion through a modal superposition technique. The mode synthesis is performed using Herting’s Transformation, which provides maximum flexibility in the selection of the elastic modes to be used for the used for the superposition, and the possibility of easily handling free-free modes. The developed numerical model predicts the dynamic response of the considered class of supercavitating vehicles resulting from assigned maneuvers. The analysis is motivated by the need of accurately modeling the structural characteristics of supercavitating vehicles in order to estimate vibrations in the structure and to envision and design systems that improve their guidance and control efficiency.

Generalized Kirchhoff equations for a deformable body moving in a weakly non-uniform flow field

1994 ◽  
Vol 446 (1926) ◽  
pp. 169-193 ◽  
Keyword(s):  
Flow Field ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Surface Deformation ◽  
Nonlinear Differential Equations ◽  
Deformable Body ◽  
Equations Of Motion ◽  
First Integral ◽  
Uniform Flow ◽  
The Body

The classical Kirchhoff’s method provides an efficient way of calculating the hydrodynamical loads (forces and moments) acting on a rigid body moving with six-degrees of freedom in an otherwise quiescent ideal fluid in terms of the body’s added-mass tensor. In this paper we provide a versatile extension of such a formulation to account for both the presence of an imposed ambient non-uniform flow field and the effect of surface deformation of a non-rigid body. The flow inhomogeneity is assumed to be weak when compared against the size of the body. The corresponding expressions for the force and moment are given in a moving body-fixed coordinate system and are obtained using the Lagally theorem. The newly derived system of nonlinear differential equations of motion is shown to possess a first integral. This can be interpreted as an energy-type conservation law and is a consequence of an anti-symmetry property of the coefficient matrix reported here for the first time. A few applications of the proposed formulation are presented including comparison with some existing limiting cases.

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