A New Method of Stabilization for Holonomic Constraints

1983 ◽  
Vol 50 (4a) ◽  
pp. 869-870 ◽  
Author(s):  
J. W. Baumgarte
Keyword(s):  
Essential Feature ◽  
Equations Of Motion ◽  
New Method ◽  
Lagrangian Multipliers ◽  
Holonomic Constraints ◽  
Asymptotic Stabilization

A new method for the asymptotic stabilization of holonomic constraints is presented. The essential feature of this approach is the introduction of stabilizing momenta of constraint. The advantage of the method is the fact that to obtain the nonclassical Lagrangian multipliers in the equations of motion, the holonomic constraints need to be differentiated only once with respect to time.

A Recursive Householder Transformation for Complex Dynamical Systems With Constraints

1988 ◽  
Vol 55 (3) ◽  
pp. 729-734 ◽  
Author(s):  
F. M. L. Amirouche ◽  
Tongyi Jia ◽  
Sitki K. Ider
Keyword(s):  
Dynamical Systems ◽  
General Solution ◽  
Equations Of Motion ◽  
New Method ◽  
Systems Dynamics ◽  
Complex Dynamical Systems ◽  
Governing Equations ◽  
Householder Transformation ◽  
Constraint Equations ◽  
Computer Automation

A new method is presented by which equations of motion of complex dynamical systems are reduced when subjected to some constraints. The method developed is used when the governing equations are derived using Kane’s equations with undetermined multipliers. The solution vectors of the constraint equations are determined utilizing the recursive Householder transformation to obtain a Pseudo-Uptriangular matrix. The most general solution in terms of new independent coordinates is then formulated. Methods previously used for handling such systems are discussed and the new method advantages are illustrated. The procedures developed are suitable for computer automation and especially in developing generic programs to study a large class of systems dynamics such as robotics, biosystems, and complex mechanisms.

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.

The Hamiltonian dynamics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 564-583 ◽  
Keyword(s):  
Nonholonomic System ◽  
Integrable Function ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Holonomic System ◽  
Principal Function ◽  
Holonomic Constraints ◽  
Second Derivatives ◽  
General Dynamical System ◽  
Hamilton Jacobi Equation

A formalism is developed which makes it possible to express the equations of motion of a nonholonomic system in Poisson bracket form. The main difficulty which has to be overcome arises from the fact that the Lagrangian co-ordinates and their corresponding momenta do not form a canonical set. However, at each instant of time, these variables can be expressed in a unique way as functions of a canonical set called the locally free co-ordinates and momenta. Poisson brackets can be formed with respect to the locally free variables, and it is shown that these lead to the correct equations of motion for a general dynamical system subject to a given set of non-holonomic constraints. Hamilton’s principle applies to a non-holonomic system , so a principal function can be formed, and its properties are studied in the second part of this paper. In addition to the usual Hamilton-Jacobi equation, the principal function satisfies a set of equations corresponding to the set of constraints. It is shown that these equations imply an indefinite or non-integrable principal function. A non-integrable function is one for which the order of double differentiation is not reversible. A precise method is given for defining the principal function for a non-holonomic system, and it is shown how this leads to indefiniteness in its second derivatives.

Transient Response of Inelastically Constrained Rigid-Body Systems

1974 ◽  
Vol 96 (3) ◽  
pp. 1041-1047 ◽  
Author(s):  
K. C. Park ◽  
K. J. Saczalski
Keyword(s):  
Structural Material ◽  
Rigid Body ◽  
Transient Response ◽  
Equations Of Motion ◽  
Coupling Mechanism ◽  
System Of Equations ◽  
Structural System ◽  
Lagrangian Multipliers ◽  
Nonlinear Geometric ◽  
Incremental Equations

An energy rate balance is employed to develop the incremental equations of motion for a shock loaded, inelastically constrained rigid-body structural system. Lagrangian multipliers provide the coupling mechanism necessary to reduce the overall system of equations to a set of modified rigid-body equations which include the nonlinear geometric and structural material effects. Kinematic material hardening and a modified yield criteria are used. Examples illustrate the technique and are compared with experimental results.

A New Method for the Solution of a Differential Equation with Two-Point Boundary Conditions Applied to the Compressible Boundary Layer on a Yawed Infinite Wing

1956 ◽  
Vol 60 (552) ◽  
pp. 808-809
Author(s):  
L. F. Crabtree ◽  
E.R. Woollett
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Exact Solution ◽  
Stagnation Point ◽  
Close Agreement ◽  
Equations Of Motion ◽  
New Method ◽  
Compressible Boundary Layer ◽  
Point Boundary ◽  
Direct Solution

The compressible laminar boundary layer on a yawed infinite wing is considered in Ref. 1, where it is shown that the problem may be solved by a direct solution of the linearised equations of motion under certain assumptions. As an example of this procedure the boundary layer near a stagnation point was calculated. Tinkler has published solutions of the exact equations for the general Falkner-Skan case (Ref. 1) obtained on the M.I.T. differential analyser for several values of the parameter involved. It has been found that the numerical results of Ref. 1 were in error and the corrected results obtained by a new method are tabulated below. Tinkler's exact solution of the stagnation point flow for ω = 0·10 is also given for comparison, and it will be seen that there is close agreement

Dynamic modeling of structurally-flexible planar parallel manipulator

Robotica ◽  
2002 ◽  
Vol 20 (3) ◽  
pp. 329-339 ◽  
Author(s):  
Bongsoo Kang ◽  
James K. Mills
Keyword(s):  
Dynamic Modeling ◽  
Parallel Manipulator ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Structural Vibration ◽  
Structural Flexibility ◽  
Lagrangian Multipliers ◽  
Pzt Actuator ◽  
Lagrangian Equations ◽  
Planar Parallel Manipulator

This paper presents a dynamic model of a planar parallel manipulator including structural flexibility of several linkages. The equations of motion are formulated using the Lagrangian equations of the first type and Lagrangian multipliers are introduced to represent the geometry of multiple closed loop chains. Then, an active damping approach using a PZT actuator is described to attenuate structural vibration of the linkages. Overall dynamic behavior of the manipulator, induced from structural flexibility of the linkage, is well illustrated through simulations. This analysis will be used to develop a prototype parallel manipulator.

The Maggi or Canonical Form of Lagrange’s Equations of Motion of Holonomic Mechanical Systems

1990 ◽  
Vol 57 (4) ◽  
pp. 1004-1010 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Linear Algebra ◽  
Nonlinear Equations ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Holonomic System ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
New Variables

This paper formulates the simplest possible, or canonical, form of the Lagrangean-type of equations of motion of holonomically constrained mechanical systems. This is achieved by introducing a new special set of n holonomic (system) coordinates in terms of which the m ( < n) holonomic constraints are expressed in their simplest, or uncoupled, form: the first m of these new coordinates vanish; the remaining (n-m) (nonvanishing) new coordinates of the (n-m) degree-of-freedom system are then independent. From the resulting equations of motion: (a) The last (n-m) are reactionless canonical equations (the holonomic counterpart of the linear or nonlinear equations, either of Maggi (in the old variables), or of Boltzmann/Hamel (in the new variables)) whose solution yields the motion, while (b) the first m supply the system reactions, in the old or new coordinates, once the motion is known. Special forms of these equations and a simple example are also given. The geometrical interpretation of the above, in modern vector/linear algebra language is summarized in the Appendix.

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