On the Determination of Joint Reactions in Multibody Mechanisms

2004 ◽  
Vol 126 (2) ◽  
pp. 341-350 ◽  
Author(s):  
Wojciech Blajer
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Algebraic Equation ◽  
Matrix Inversion ◽  
Differential Algebraic Equation ◽  
Joint Coordinates ◽  
Reduced Dimension ◽  
Absolute Coordinates ◽  
Coordinate Partitioning

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

scholarly journals Numerical simulation of finite dimensional multibody nonsmooth mechanical systems

2002 ◽  
Vol 55 (2) ◽  
pp. 107-150 ◽  
Author(s):  
B Brogliato ◽  
AA ten Dam ◽  
L Paoli ◽  
F Ge´not ◽  
M Abadie
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Ordinary Differential Equation ◽  
Mechanical Systems ◽  
Difficult Problem ◽  
Review Article ◽  
Differential Algebraic Equation ◽  
Complementarity Conditions ◽  
Finite Dimensional ◽  
Impact Phenomena

This review article focuses on the problems related to numerical simulation of finite dimensional nonsmooth multibody mechanical systems. The rigid body dynamical case is examined here. This class of systems involves complementarity conditions and impact phenomena, which make its study and numerical analysis a difficult problem that cannot be solved by relying on known Ordinary Differential Equation (ODE) or Differential Algebraic Equation (DAE) integrators only. The main techniques, mathematical tools, and existing algorithms are reviewed. The article utilizes 233 references.

The mixing of ground water and sea water in permeable subsoils

1958 ◽  
Vol 4 (5) ◽  
pp. 479-488 ◽  
Author(s):  
G. F. Carrier
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Ground Water ◽  
Sea Water ◽  
Original Model ◽  
Linear Ordinary Differential Equation ◽  
Mixing Process ◽  
Salinity Distribution ◽  
Permeable Media

The subterranean mixing in permeable media of sea water and ground water is studied. The model for this mixing process which was suggested by C. K. Wentworth is adopted, but is soon discarded in favour of a more tractable formulation whose equivalence to the original model is established. The analysis is carried to the point where the determination of the salinity distribution of the ground water in a given subsoil requires only the solution of an elementary linear ordinary differential equation.

scholarly journals Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Saima Akram ◽  
Allah Nawaz ◽  
Humaira Kalsoom ◽  
Muhammad Idrees ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
First Order ◽  
Multiple Periodic Solutions ◽  
Fine Focus ◽  
Nonautonomous Equation ◽  
Maximum Multiplicity

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

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