Instantaneous Invariants and Curvature Analysis of a Planar Four-Link Mechanism

1994 ◽  
Vol 116 (4) ◽  
pp. 1173-1176 ◽  
Author(s):  
An Tzu Yang ◽  
G. R. Pennock ◽  
Lih-Min Hsia
Keyword(s):  
Rigid Body ◽  
Fourth Order ◽  
Arbitrary Point ◽  
Canonical System ◽  
Planar Mechanisms ◽  
Operating Cycle ◽  
Curvature Analysis ◽  
Crank Angle ◽  
Moving Plane ◽  
Derivatives Of

This paper shows that the canonical system and the instantaneous invariants for a moving plane, which is connected to the fixed plane by a revolute-revolute crank, are functions of the derivatives of the crank angle. Then closed-form expressions are derived for the curvature ratios of the path generated by an arbitrary point fixed in the moving plane, in terms of the coordinates of the point and the instantaneous invariants of the plane. For illustrative purposes, numerical results are presented for the instantaneous invariants (up to the fourth-order) of the coupler of a specified crank-rocker mechanism, as a function of the input angle. In addition, the paper shows the variation in the first and second curvature ratios of an arbitrary coupler curve during the complete operating cycle of the mechanism. The authors hope that, based on the results presented here, a variety of useful tools for the kinematic design of planar mechanisms, with a rotary input, will be developed for plane rigid body guidance as well as curve generation.

Instantaneous Invariants and Curvature Analysis for Planar Mechanisms

1994 ◽  
Author(s):  
An Tzu Yang ◽  
Gordon R. Pennock ◽  
Lih-Min Hsia
Keyword(s):  
Closed Form ◽  
Analytical Procedure ◽  
Rigid Motion ◽  
Continuous Functions ◽  
Gear Train ◽  
Planar Mechanisms ◽  
Operating Cycle ◽  
Curvature Analysis ◽  
Trajectory Matching ◽  
Epicyclic Gear

Abstract This paper establishes a systematic procedure to determine the instantaneous invariants for a member in a planar mechanism and the curvature ratios of the path of a point fixed in the member. Closed-form expressions are derived for the instantaneous invariants and the curvature ratios as continuous functions of the input variable of the mechanism. The methods that are proposed in this paper can be applied to the design of planar mechanisms in general. For a given mechanism, the configuration defined by the input variable at which the member achieves the optimal approximation of a prescribed rigid motion can be determined. Alternatively, a point fixed in a member can be selected such that it will generate a trajectory matching a given curve with high precision. For purposes of illustration, the paper details the analytical procedure for a simple epicyclic gear train and a four-bar mechanism. The instantaneous invariants and the curvature ratios are generated for the complete operating cycle of each mechanism.

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.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

1994 ◽  
Author(s):  
Pierre Larochelle ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Numerical Results ◽  
Image Space ◽  
Dimensional Synthesis ◽  
Invariant Metric ◽  
Planar Mechanisms ◽  
Position And Orientation ◽  
Position Synthesis

Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

scholarly journals The Solution of Nonlinear Fourth-Order Differential Equation with Integral Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yanli Fu ◽  
Huanmin Yao
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Reproducing Kernel ◽  
Order Differential Equation ◽  
Integral Boundary Conditions ◽  
Reproducing Kernel Space ◽  
Integral Boundary ◽  
Derivatives Of ◽  
Fourth Order Differential Equation

An iterative algorithm is proposed for solving the solution of a nonlinear fourth-order differential equation with integral boundary conditions. Its approximate solutionun(x)is represented in the reproducing kernel space. It is proved thatun(x)converges uniformly to the exact solutionu(x). Moreover, the derivatives ofun(x)are also convergent to the derivatives ofu(x). Numerical results show that the method employed in the paper is valid.

scholarly journals Volume derivatives of the Grüneisen parameter in the limit of infinite pressure

2019 ◽  
Vol 97 (1) ◽  
pp. 114-116 ◽  
Author(s):  
A. Dwivedi
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Fundamental Theorem ◽  
High Pressures ◽  
Grüneisen Parameter ◽  
Third Order ◽  
Gruneisen Parameter ◽  
The Third ◽  
Properties Of Materials ◽  
Derivatives Of

Expressions have been obtained for the volume derivatives of the Grüneisen parameter, which is directly related to the thermal and elastic properties of materials at high temperatures and high pressures. The higher order Grüneisen parameters are expressed in terms of the volume derivatives, and evaluated in the limit of infinite pressure. The results, that at extreme compression the third-order Grüneisen parameter remains finite and the fourth-order Grüneisen parameter tends to zero, have been used to derive a fundamental theorem according to which the volume derivatives of the Grüneisen parameter of different orders, all become zero in the limit of infinite pressure. However, the ratios of these derivatives remain finite at extreme compression. The formula due to Al’tshuler and used by Dorogokupets and Oganov for interpolating the Grüneisen parameter at intermediate compressions has been found to satisfy the boundary condition at infinite pressure obtained in the present study.

Simultaneous Coordination of Two Rigid Body Motions Through Infinitesimally Separated Positions

1975 ◽  
Vol 97 (2) ◽  
pp. 527-531
Author(s):  
M. N. Siddhanty ◽  
A. H. Soni
Keyword(s):  
Rigid Body ◽  
Fourth Order ◽  
Rigid Bodies ◽  
Mathematical Approach ◽  
Link Mechanism ◽  
Rigid Body Motions ◽  
Design Solutions ◽  
Mathematical Relationships

A generalized mathematical approach is developed to guide two rigid bodies for simultaneous coordination of their infinitesimally separated positions. Mathematical relationships are developed to incorporate up to fourth-order derivatives while specifying infintesimally separated positions. The approach is demonstrated by considering an eight-link mechanism. It is shown that for a maximum of five precision positions of the two rigid bodies, a maximum of 1024 design solutions are possible.

Curvature Analysis of Conjugate Surfaces via a Tensor Approach

2012 ◽  
Vol 163 ◽  
pp. 251-255
Author(s):  
Peng Wang ◽  
Yi Du Zhang ◽  
Jie Deng
Keyword(s):  
Rigid Body ◽  
Helical Gear ◽  
Body Motion ◽  
Local Geometry ◽  
Line Contact ◽  
Rotation Tensor ◽  
Curvature Analysis ◽  
Conjugate Surfaces ◽  
Invariant Approach ◽  
Derivation Process

A novel invariant approach is proposed for derivation of curvature relationship between conjugate surfaces of line contact. Unlike other approaches found in the literature, tensor analysis is applied to the derivation process in which rigid-body motion is represented by rotation tensor and the local geometry of surface is expressed by curvature tensor, and the final result is given in a clear and compact form. An example of helical gear is provided to illustrate the application of the proposed approach.

scholarly journals GAUGED LIFSHITZ MODEL WITH CHERN–SIMONS TERM

2013 ◽  
Vol 28 (09) ◽  
pp. 1350025 ◽  
Author(s):  
GUSTAVO S. LOZANO ◽  
FIDEL A. SCHAPOSNIK ◽  
GIANNI TALLARITA
Keyword(s):  
Fourth Order ◽  
Order Term ◽  
Equations Of Motion ◽  
Nontrivial Solutions ◽  
Oscillatory Solutions ◽  
Modulated Phases ◽  
Fourth Order Term ◽  
Spatial Derivatives ◽  
Chern Simons ◽  
Derivatives Of

We present a gauged Lifshitz Lagrangian including second- and fourth-order spatial derivatives of the scalar field and a Chern–Simons term, and study nontrivial solutions of the classical equations of motion. While the coefficient β of the fourth-order term should be positive in order to guarantee positivity of the energy, the coefficient α of the quadratic one need not be. We investigate the parameter domains and find significant differences in the field behaviors. Apart from the usual vortex field behavior of the ordinary relativistic Chern–Simons–Higgs model, we find in certain parameter domains oscillatory solutions reminiscent of the modulated phases of Lifshitz systems.

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