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.

Nonlinear Dynamic Analysis of a Structure Subjected to Multiple Support Motions

1981 ◽  
Vol 103 (1) ◽  
pp. 27-32 ◽  
Author(s):  
V. N. Shah ◽  
A. J. Hartmann
Keyword(s):  
Rigid Body ◽  
Dynamic Analysis ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Analysis ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Superposition Method ◽  
Modal Superposition ◽  
Modal Superposition Method ◽  
Multiple Support

A modal superposition method for the nonlinear dynamic analysis of a structure subjected to multiple support motions is presented. The nonlinearities are due to clearances between the components and their supports. The finite element method is used to derive the equations of motion with the nonlinearities represented by a pseudo force vector. The 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. The modal superposition method is used to analyze the dynamic response of one loop of the nuclear steam supply system. This loop has nonlinear supports and is subjected to nonuniform seismic excitations at the supports. It is shown that the computational cost of the modal superposition method is lower than that of the direct integration.

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.

Modal Superposition Method for Computationally Economical Nonlinear Structural Analysis

1979 ◽  
Vol 101 (2) ◽  
pp. 134-141 ◽  
Author(s):  
V. N. Shah ◽  
G. J. Bohm ◽  
A. N. Nahavandi
Keyword(s):  
Seismic Analysis ◽  
Equations Of Motion ◽  
Test Problems ◽  
Superposition Method ◽  
Structural Dynamic ◽  
Large Wave ◽  
Modal Superposition ◽  
Nonlinear Structural ◽  
Direct Integration Method ◽  
Modal Superposition Method

A modal superposition method for analyzing nonlinear structural dynamic problems involving impact between components is developed and evaluated. The finite-element method is used to express the equations of motion with nonlinearities represented by pseudo force vector. Three test problems are solved to verify this method. This has demonstrated the applicability of this method to seismic analysis of large, complex structural systems. It is concluded that the modal superposition method has a significant cost advantage over the direct integration method for problems with large wave fronts and the source of nonlinearities restricted to a limited portion of the structure.

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.

The Motion of a Compliantly Suspended Rigid Body Constrained by a Frictional Surface

2001 ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Rigid Body ◽  
Friction Force ◽  
Coulomb Friction ◽  
Unilateral Constraint ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Elastic Behavior ◽  
Frictional Surface ◽  
Elastic Suspension ◽  
Controlled Motion

Abstract In this paper, the quasistatic motion of an elastically suspended, unilaterally constrained rigid body is studied. The motion of the rigid body is determined, in part, by the position controlled motion of its support base and the behavior of the elastic suspension that couples the part to the support. The motion is also determined, in part, by contact with a frictional surface that both couples the rigid body to unilateral constraint surfaces and generates a friction force. The unknown friction force, however, is determined in part by the unknown direction of the rigid body motion. We derive a solvable set of equations that simultaneously determines both the friction force and the resulting rigid body motion. This set of equations requires that the friction and motion at the point of contact are oppositely directed. Solution involves the use of rigid body kinematics, the Coulomb friction coefficient, the commanded motion of the support, and the spatial elastic behavior of the coupling.

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.

Equations of Motion of an Aircraft Wing Modeled As a Composite Plate-Beam Model Including the Rigid-Body Motion

2001 ◽  
Author(s):  
Samir A. Emam ◽  
Ali H. Nayfeh ◽  
Scott L. Hendricks
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Composite Beam ◽  
Composite Plate ◽  
Equations Of Motion ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Beam Model ◽  
Aircraft Wing ◽  
Hamilton Principle

Abstract The equations of motion of an aircraft wing modeled as a composite beam are presented. The contribution of the rigid-body motion is taken into account; it affects the response of the wing, especially in maneuvers. The Hamilton principle is used to derive the equations of motion and the corresponding boundary conditions.

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.

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