每日彙整: 2018-04-29

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧二‧ V 】

『借例使例』誰說不是學習法耶！雖尚未談及

Kane’s Method in Physics/Mechanics

mechanics provides functionality for deriving equations of motion using Kane’s method [Kane1985]. This document will describe Kane’s method as used in this module, but not how the equations are actually derived.

Structure of Equations

In mechanics we are assuming there are 5 basic sets of equations needed to describe a system. They are: holonomic constraints, non-holonomic constraints, kinematic differential equations, dynamic equations, and differentiated non-holonomic equations.

\mathbf{f_h}(q, t) = 0
\mathbf{k_{nh}}(q, t) u + \mathbf{f_{nh}}(q, t) = 0
\mathbf{k_{k\dot{q}}}(q, t) \dot{q} + \mathbf{k_{ku}}(q, t) u + \mathbf{f_k}(q, t) = 0
\mathbf{k_d}(q, t) \dot{u} + \mathbf{f_d}(q, \dot{q}, u, t) = 0
\mathbf{k_{dnh}}(q, t) \dot{u} + \mathbf{f_{dnh}}(q, \dot{q}, u, t) = 0

In mechanics holonomic constraints are only used for the linearization process; it is assumed that they will be too complicated to solve for the dependent coordinate(s). If you are able to easily solve a holonomic constraint, you should consider redefining your problem in terms of a smaller set of coordinates. Alternatively, the time-differentiated holonomic constraints can be supplied.

Kane’s method forms two expressions, F_r and F_r^{*} , whose sum is zero. In this module, these expressions are rearranged into the following form:

\mathbf{M}(q, t) \dot{u} = \mathbf{f}(q, \dot{q}, u, t)

For a non-holonomic system with oo total speeds and mm motion constraints, we will get o – m equations. The mass-matrix/forcing equations are then augmented in the following fashion:

\mathbf{M}(q, t) = \begin{bmatrix} \mathbf{k_d}(q, t) \\ \mathbf{k_{dnh}}(q, t) \end{bmatrix}

\mathbf{_{(forcing)}}(q, \dot{q}, u, t) = \begin{bmatrix} - \mathbf{f_d}(q, \dot{q}, u, t) \\ - \mathbf{f_{dnh}}(q, \dot{q}, u, t) \end{bmatrix}

 

之與『拉格朗日方法』的差異之處，何妨先讀 SymPy mechanics 範例乎？

A rolling disc

The disc is assumed to be infinitely thin, in contact with the ground at only 1 point, and it is rolling without slip on the ground. See the image below.

../../../../_images/rollingdisc.svg

We model the rolling disc in three different ways, to show more of the functionality of this module.

 

※ 謹慎

 

…★

kanes_equations(bodies, loads=None)

Method to form Kane’s equations, Fr + Fr* = 0.

Returns (Fr, Fr*). In the case where auxiliary generalized speeds are present (say, s auxiliary speeds, o generalized speeds, and m motion constraints) the length of the returned vectors will be o – m + s in length. The first o – m equations will be the constrained Kane’s equations, then the s auxiliary Kane’s equations. These auxiliary equations can be accessed with the auxiliary_eqs().

Parameters:

bodies : iterable

An iterable of all RigidBody’s and Particle’s in the system. A system must have at least one body.

loads : iterable

Takes in an iterable of (Particle, Vector) or (ReferenceFrame, Vector) tuples which represent the force at a point or torque on a frame. Must be either a non-empty iterable of tuples or None which corresponds to a system with no constraints.

 

……☆

 

甚或還可更上一層樓呦◎

Steady Motion of a Rigid Disk of Finite Thickness on a Horizontal Plane

Milan Batista
(Submitted on 2 Sep 2005 (v1), last revised 5 Sep 2005 (this version, v2))

The article discusses the steady motion of a rigid disk of finite thickness rolling on its edge on a horizontal plane under the influence of gravity. The governing equations are presented and two cases allowing for a steady state solution are considered: rolling on consistently rough ground and rolling on perfectly smooth ground. The conditions of steady motion are derived for both kinds of ground and it is shown that the possible steady motion of a disk is either on a straight line in a circle. Also oscillations about steady state are discussed and conditions for stable motion are established.

Comments: 28 pages, 7 figures
Subjects: Classical Physics (physics.class-ph)
Journal reference: International Journal of Non-Linear Mechanics, Volume 41, Issue 4, May 2006, Pages 605-621
DOI: 10.1016/j.ijnonlinmec.2006.02.005
Cite as: arXiv:physics/0509021 [physics.class-ph]
  (or arXiv:physics/0509021v2 [physics.class-ph] for this version)

 

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An Analytical Solution of the Equations of a Rolling Disk of Finite Thickness on a Rough Plane

Milan Batista
(Submitted on 5 Sep 2005)

In this article an analytical solution of equations of motion of a rigid disk of finite thickness rolling on its edge on a perfectly rough horizontal plane under the action of gravity is given. The solution is given in terms of Gauss hypergeometrical functions.

Comments: 12 pages, 2 figures
Subjects: Classical Physics (physics.class-ph)
Journal reference: International Journal of Non-Linear Mechanics, Volume 41, Issues 6-7, July-September 2006, Pages 850-859
DOI: 10.1016/j.ijnonlinmec.2006.06.002
Cite as: arXiv:physics/0509034 [physics.class-ph]
  (or arXiv:physics/0509034v1 [physics.class-ph] for this version)