Synthesis of a Spherical Cycloidal Mechanism

1976 ◽  
Vol 98 (4) ◽  
pp. 1342-1346
Author(s):  
R. Mayo ◽  
A. H. Soni
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Unit Sphere ◽  
Rotation Matrix ◽  
Fourth Degree ◽  
Rotational Displacement ◽  
Synthesis Technique ◽  
Moving Points ◽  
Crank Mechanism

Using rotational displacement matrices to describe spherical motions, closed-form synthesis equations are derived for two rotation matrix mechanisms. These equations are used to calculate the linkage parameters of a spherical cycloidal crank mechanism so that it will guide a rigid body through five finitely separated positions lying on the surface of a unit sphere. It is shown that the synthesis equation for calculating the coordinates of the fixed and moving points lying on the unit sphere is of fourth degree. This indicates that a maximum of four unique solutions of the mechanism are possible when five finitely separated positions are specified. A synthesis example is presented illustrating the proposed synthesis technique using a spherical cycloidal crank mechanism to guide a rigid body.

scholarly journals Closed-form time derivatives of the equations of motion of rigid body systems

2021 ◽  
Author(s):  
Andreas Müller ◽  
Shivesh Kumar
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Formulation ◽  
Dynamics Of Rigid Body ◽  
Control Forces ◽  
And Control ◽  
Derivatives Of ◽  
Insight Into

AbstractDerivatives of equations of motion (EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.

A Closed-Form Formalism for Controlled and Hybrid Rigid/Elastic Multibody Systems: Part II — Symbolic Implementation and Applications

1991 ◽  
Author(s):  
Junghsen Lieh ◽  
Imitiaz Haque
Keyword(s):  
Closed Form ◽  
Equations Of Motion ◽  
Structural Flexibility ◽  
Symbolic Language ◽  
Structural Vibrations ◽  
Optimal Control Strategy ◽  
Active Suspensions ◽  
Elastic Multibody Systems ◽  
System Order ◽  
Crank Mechanism

Abstract A program is developed on a DECstation using the symbolic language MAPLE which generates the equations of motion in a closed form and reduces the system order symbolically. A procedure that can make symbolic simplification and linearization is provided. The integration of shape functions is performed symbolically. Both nonlinear and linearized equations of motion with control are established in FORTRAN format. Several models including an elastic vehicle with active suspensions, an elastic robotic manipulator and an elastic slider-crank mechanism with both joint and structural flexibility are generated. Numerical simulation for the active vehicle model using an optimal control strategy is presented. The effect of active suspensions on vehicle and structural vibrations is briefly discussed. A comparison between the nonlinear and linearized robot models is given. Simulation results of the slider-crank mechanism are also presented.

Closed-Form Stress Intensity Factor Solutions for Surface Cracks With Large Aspect Ratios in Plates

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Kisaburo Azuma ◽  
Yinsheng Li ◽  
Steven Xu
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Closed Form ◽  
Influence Coefficient ◽  
Maximum Point ◽  
Surface Cracks ◽  
Fourth Degree ◽  
Aspect Ratios ◽  
Influence Coefficients

Abstract Alloy 82/182/600, which is used in light-water reactors, is known to be susceptible to stress-corrosion cracking. The depth of some of these cracks may exceed the value of half-length on the surface. Although the stress intensity factor (SIF) for cracks plays an important role in predicting crack propagation and failure, Section XI of the ASME Boiler and Pressure Vessel Code does not provide SIF solutions for such deep cracks. In this study, closed-form SIF solutions for deep surface cracks in plates are discussed using an influence coefficient approach. The stress distribution at the crack location is represented by a fourth-degree-polynomial equation. Tables for influence coefficients obtained by finite element analysis in the previous studies are used for curve fitting. The closed-form solutions for the influence coefficients were developed at the surface point, the deepest point, and the maximum point of a crack with an aspect ratio a/c ranging from 1.0 to 8.0, where a is the crack depth and c is one-half of the crack length. The maximum point of a crack refers to the location on the crack front where the SIF reaches a maximum value.

scholarly journals Baker–Campbell–Hausdorff–Dynkin Formula for the Lie Algebra of Rigid Body Displacements

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1185
Author(s):  
Daniel Condurache ◽  
Ioan-Adrian Ciureanu
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Lie Algebra ◽  
Lie Group ◽  
First Time

The paper proposes, for the first time, a closed form of the Baker–Campbell–Hausdorff–Dynkin (BCHD) formula in the particular case of the Lie algebra of rigid body displacements. For this purpose, the structure of the Lie group of the rigid body displacements S E ( 3 ) and the properties of its Lie algebra s e ( 3 ) are used. In addition, a new solution to this problem in dual Lie algebra of dual vectors is delivered using the isomorphism between the Lie group S E ( 3 ) and the Lie group of the orthogonal dual tensors.

On Saint-Venant's Problem for Helicoidal Beams

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Fem Analysis ◽  
Original System ◽  
Form Solution ◽  
Hamiltonian Matrix ◽  
Jordan Structure ◽  
Rigid Body Motions

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

On the closed form of Wigner rotation matrix elements

1996 ◽  
Vol 19 (2) ◽  
pp. 131-145 ◽  
Author(s):  
Shan-Tao Lai ◽  
P. Palting ◽  
Ying-Nan Chiu
Keyword(s):  
Closed Form ◽  
Rotation Matrix ◽  
Matrix Elements ◽  
Wigner Rotation ◽  
Rotation Matrix Elements

Use of Resultants and the Bernshtein Formula in the Dimensional Synthesis of Linkage Components

1994 ◽  
Vol 116 (4) ◽  
pp. 1171-1172 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Fourth Degree ◽  
Finite Precision ◽  
Closed Form Solutions ◽  
System Of Polynomial Equations ◽  
Resultant Theory

The applications of resultants and the Bernshtein formula for the dimensional synthesis of linkage components for finite precision positions are discussed. The closed-form solutions, which are derived from systems of polynomials in multiple unknowns by applying resultant theory, are in forms of polynomial equations of a single unknown. For the case of two compatibility equations, the closed form solution is a sixth degree solution polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree solution polynomial. For each case, the Bernshtein formula is applied to calculate the number of solutions of the system of polynomial equations. The calculated numbers of solutions match the degrees of the solution polynomials for both cases.

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