翻訳と辞書 Words near each other ・ Swingin' New Big Band ・ Swingin' on a Rainbow ・ Swingin' on Broadway ・ Swingin' on the Moon ・ Swingin' Pretty ・ Swingin' School ・ Swingin' Stampede ・ Swingin' the '20s ・ Swingin' Time ・ Swingin' Utters ・ Swingin' Utters (album) ・ Swingin' with Bud ・ Swingin' with Raymond ・ Swinging (sexual practice) ・ Swinging at the Castle ・ Swinging Atwood's machine ・ Swinging Brass with the Oscar Peterson Trio ・ Swinging Bridge ・ Swinging Bridge (disambiguation) ・ Swinging Doors ・ Swinging Doors (song) ・ Swinging gate (American football) ・ Swinging Hannover ・ Swinging Hollywood ・ Swinging London ・ Swinging London Town ・ Swinging on a Star ・ Swinging on a Star (musical) ・ Swinging Out Live ・ Swinging Popsicle Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Swinging Atwood's machine ： ウィキペディア英語版 Swinging Atwood's machine The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions. Specifically, it comprises two masses (the pendulum, mass $m$ and counterweight, mass $M$) connected by an inextensible, massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight.〔 The conventional Atwood's machine allows only "runaway" solutions (''i.e.'' either the pendulum or counterweight eventually collides with its pulley), except for $M=m$. However, the swinging Atwood's machine with $M>m$ has a large parameter space of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic, quasiperiodic or chaotic, bounded or unbounded, singular or non-singular〔〔 due to the pendulum's reactive centrifugal force counteracting the counterweight's weight.〔 Research on the SAM started as part of a 1982 senior thesis entitled ''Smiles and Teardrops'' (referring to the shape of some trajectories of the system) by Nicholas Tufillaro at Reed College, directed by David J. Griffiths.〔 ==Equations of motion== The swinging Atwood's machine is a system with two degrees of freedom. We may derive its equations of motion using either Hamiltonian mechanics or Lagrangian mechanics. Let the swinging mass be $m$ and the non-swinging mass be $M$. The kinetic energy of the system, $T$, is: :$$ \begin T &= \frac M v^2_M + \frac mv^2_m \\ &= \fracM \dot^2+\frac m \left(\dot^2+r^2\dot^2\right) \end where $r$ is the distance of the swinging mass to its pivot, and $\backslash theta$ is the angle of the swinging mass relative to pointing straight downwards. The potential energy $U$ is solely due to the acceleration due to gravity: :$$ \begin U &= Mgr - mgr \cos \end We may then write down the Lagrangian, $\backslash mathcal$, and the Hamiltonian, $\backslash mathcal$ of the system: :$$ \begin \mathcal &= T-U\\ &= \fracM \dot^2+\frac m \left(\dot^2+r^2\dot^2\right) - Mgr + mgr \cos\\ \mathcal &= T+U\\ &= \fracM \dot^2+\frac m \left(\dot^2+r^2\dot^2\right) + Mgr - mgr \cos \end We can then express the Hamiltonian in terms of the canonical momenta, $p\_r$, $p\_\backslash theta$: :$$ \begin p_r &= \frac\\ p_\theta &= \frac\\ \therefore \mathcal &= \frac + \frac + Mgr - mgr \cos \end Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in $r$ and $\backslash theta$. First, the $\backslash theta$ equation: :$$ \begin \frac &= \frac \left(\frac &= 2mr \dot\dot + mr^2 \ddot\\ r\ddot + 2\dot\dot + g\sin &= 0 \end And the $r$ equation: :$$ \begin \frac &= \frac \left( \frac^2 - Mg + mg\cos &= (M+m) \ddot \end We simplify the equations by defining the mass ratio $\backslash mu\; =\; \backslash frac$. The above then becomes: :$(\backslash mu+1)\backslash ddot\; -\; r\backslash dot^2\; +\; g(\backslash mu\; -\; \backslash cos)\; =\; 0$ Hamiltonian analysis may also be applied to determine four first order ODEs in terms of $r$, $\backslash theta$ and their corresponding canonical momenta $p\_r$ and $p\_\backslash theta$: :$$ \begin \dot&=\frac \\ \dot &= - \frac - Mg + mg\cos \\ \dot&=\frac \\ \dot &= - \frac \end Notice that in both of these derivations, if one sets $\backslash theta$ and angular velocity $\backslash dot$ to zero, the resulting special case is the regular non-swinging Atwood machine: :$\backslash ddot\; =\; g\; \backslash frac=g\backslash frac$ The swinging Atwood's machine has a four-dimensional phase space defined by $r$, $\backslash theta$ and their corresponding canonical momenta $p\_r$ and $p\_\backslash theta$. However, due to energy conservation, the phase space is constrained to three dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Swinging Atwood's machine」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.