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[[File:Swinging Atwoods Machine.svg|thumb|The Swinging Atwood's machine. The smaller mass, labelled m, is allowed to swing freely whereas the larger mass, M, can only move up and down. Assume the pivots to be points.]] | |||
The '''swinging Atwood's machine''' (SAM) is a mechanism that resembles a simple [[Atwood machine|Atwood's machine]] except that one of the masses is allowed to swing in a two-dimensional plane, producing a [[dynamical system]] that is [[chaos theory|chaotic]] for some system parameters and [[initial conditions]]. | |||
Specifically, it comprises two masses (the pendulum, mass <math>m</math> and counterweight, mass <math>M</math>) connected by an [[Kinematics#Inextensible cord|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.<ref name="Tufillaro1984"/> | |||
The conventional Atwood's machine allows only "runaway" solutions (''i.e.'' either the pendulum or counterweight eventually collides with its pulley), except for <math>M=m</math>. However, the swinging Atwood's machine with <math>M>m</math> 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<ref name="Tufillaro1984"/><ref name="bound"/> due to the pendulum's [[reactive centrifugal force]] counteracting the counterweight's weight.<ref name="Tufillaro1984"/> 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]].<ref name="smiles and teardrops"/> | |||
==Equations of motion== | |||
[[Image:Swinging-atwood's-machine-4.5.gif|thumb|Motion of Swinging Atwood's Machine for M/m = 4.5]] | |||
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 <math>m</math> and the non-swinging mass be <math>M</math>. The kinetic energy of the system, <math>T</math>, is: | |||
:<math> | |||
\begin{align} | |||
T &= \frac{1}{2} M v^2_M + \frac{1}{2} mv^2_m \\ | |||
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) | |||
\end{align} | |||
</math> | |||
where <math>r</math> is the distance of the swinging mass to its pivot, and <math>\theta</math> is the angle of the swinging mass relative to pointing straight downwards. The potential energy <math>U</math> is solely due to the [[standard gravity|acceleration due to gravity]]: | |||
:<math> | |||
\begin{align} | |||
U &= Mgr - mgr \cos{\theta} | |||
\end{align} | |||
</math> | |||
We may then write down the Lagrangian, <math>\mathcal{L}</math>, and the Hamiltonian, <math>\mathcal{H}</math> of the system: | |||
:<math> | |||
\begin{align} | |||
\mathcal{L} &= T-U\\ | |||
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) - Mgr + mgr \cos{\theta}\\ | |||
\mathcal{H} &= T+U\\ | |||
&= \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) + Mgr - mgr \cos{\theta} | |||
\end{align} | |||
</math> | |||
We can then express the Hamiltonian in terms of the canonical momenta, <math>p_r</math>, <math>p_\theta</math>: | |||
:<math> | |||
\begin{align} | |||
p_r &= \frac{\partial{\mathcal{L}}}{\partial \dot{r}} = \frac{\partial T}{\partial \dot{r}} = (M+m)\dot{r}\\ | |||
p_\theta &= \frac{\partial {\mathcal{L}}}{\partial \dot{\theta}} = \frac{\partial T}{\partial \dot{\theta}} = mr^2 \dot{\theta}\\ | |||
\therefore \mathcal{H} &= \frac{p_r^2}{2(M+m)} + \frac{p_\theta^2}{2mr^2} + Mgr - mgr \cos{\theta} | |||
\end{align} | |||
</math> | |||
Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in <math>r</math> and <math>\theta</math>. First, the <math>\theta</math> equation: | |||
:<math> | |||
\begin{align} | |||
\frac{\partial {\mathcal{L}}}{\partial \theta} &= \frac{d}{dt} \left(\frac{\partial {\mathcal{L}}}{\partial \dot{\theta}}\right)\\ | |||
-mgr \sin{\theta} &= 2mr \dot{r}\dot{\theta} + mr^2 \ddot{\theta}\\ | |||
r\ddot{\theta} + 2\dot{r}\dot{\theta} + g\sin{\theta} &= 0 | |||
\end{align} | |||
</math> | |||
And the <math>r</math> equation: | |||
:<math> | |||
\begin{align} | |||
\frac{\partial {\mathcal{L}}}{\partial r} &= \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}}\right)\\ | |||
mr\dot{\theta}^2 - Mg + mg\cos{\theta} &= (M+m) \ddot{r} | |||
\end{align} | |||
</math> | |||
We simplify the equations by defining the mass ratio <math>\mu = \frac{M}{m}</math>. The above then becomes: | |||
:<math>(\mu+1)\ddot{r} - r\dot{\theta}^2 + g(\mu - \cos{\theta}) = 0</math> | |||
Hamiltonian analysis may also be applied to determine four first order ODEs in terms of <math>r</math>, <math>\theta</math> and their corresponding canonical momenta <math>p_r</math> and <math>p_\theta</math>: | |||
:<math> | |||
\begin{align} | |||
\dot{r}&=\frac {\partial{\mathcal{H}}} {\partial{p_r}} = \frac {p_r}{M+m} \\ | |||
\dot{p_r} &= - \frac {\partial{\mathcal{H}}} {\partial{r}} = \frac {p_\theta ^2} {mr^3} - Mg + mg\cos{\theta} \\ | |||
\dot{\theta}&=\frac {\partial{\mathcal{H}}} {\partial{p_\theta}} = \frac {p_\theta} {mr^2} \\ | |||
\dot{p_\theta} &= - \frac {\partial{\mathcal{H}}} {\partial{\theta}} = -mgr\sin{\theta} | |||
\end{align} | |||
</math> | |||
Notice that in both of these derivations, if one sets <math>\theta</math> and angular velocity <math>\dot{\theta}</math> to zero, the resulting special case is the regular non-swinging [[Atwood machine]]: | |||
:<math>\ddot{r} = g \frac{1-\mu}{1+\mu}=g\frac{m-M}{m+M}</math> | |||
The swinging Atwood's machine has a four-dimensional [[phase space]] defined by <math>r</math>, <math>\theta</math> and their corresponding canonical momenta <math>p_r</math> and <math>p_\theta</math>. However, due to energy conservation, the phase space is constrained to three dimensions. | |||
===System with massive pulleys=== | |||
If the pulleys in the system are taken to have [[moment of inertia]] <math>I</math> and radius <math>R</math>, the Hamiltonian of the SAM is then:<ref name="Pujol2010"/> | |||
:<math>\mathcal{H}\left(r, \theta, \dot{r}, \dot{\theta} \right) = | |||
\underbrace{ \frac{1}{2} M_t \left( R \dot{\theta} - \dot{r} \right) ^2 | |||
+ \frac{1}{2} m r^2 \dot{\theta}^2 }_{T} | |||
+ \underbrace{ gr \left(M - m \cos{\theta} \right) | |||
+ gR \left( m \sin{\theta} - M \theta \right)}_{U}, | |||
</math> | |||
Where {{var|M}}<sub>t</sub> is the effective total mass of the system, | |||
:<math>M_t = M + m + \frac{I}{R^2}</math> | |||
This reduces to the version above when <math>R</math> and <math>I</math> become zero. The equations of motion are now:<ref name="Pujol2010"/> | |||
:<math>\begin{align} | |||
\mu_t ( \ddot{r} - R \ddot{\theta}) & = r \dot{\theta}^2 + g (\cos {\theta} - \mu ) \\ | |||
r \ddot{\theta} & = - 2 \dot{r} \dot{\theta} + R \dot{\theta}^2 - g \sin {\theta} \\ | |||
\end{align} | |||
</math> | |||
where <math>\mu_t = M_t / m</math>. | |||
===Integrability=== | |||
[[Hamiltonian system]]s can be classified as [[integrable model|integrable]] and nonintegrable. SAM is integrable when the mass ratio <math>\mu = M/m = 3</math>.<ref name="Integrable"/> The system also looks pretty regular for <math>\mu = 4 n^2 - 1 = 3, 15, 35, ...</math>, but the <math>\mu = 3</math> case is the only integrable mass ratio found so far. For many other values of the mass ratio (and initial conditions) SAM displays [[chaos theory|chaotic motion]]. | |||
Numerical studies indicate that when the orbit is singular (initial conditions: <math>r=0, \dot{r}=v, \theta=\theta_0, \dot{\theta}=0</math>), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of <math>\theta_0</math>. When <math>\theta_0</math> is small (near vertical), the trajectory describes a "teardrop", when it is large, it describes a "heart". These trajectories can be exactly solved algebraically, which is unusual for a system with a non-linear Hamiltonian.<ref name="smiles and hearts"/> | |||
==Trajectories== | |||
The swinging mass of the swinging Atwood's machine undergoes interesting trajectories or orbits when subject to different initial conditions, and for different mass ratios. These include periodic orbits and collision orbits. | |||
===Nonsingular orbits=== | |||
For certain conditions, system exhibits [[complex harmonic motion]].<ref name="Tufillaro1984"/> The orbit is called nonsingular if the swinging mass does not touch the pulley. | |||
{| width=78% class="collapsible <!--{{{state|collapsed}}}-->" style="border:1px #AAAAAA solid;" | |||
! colspan=4 style="background-color:#F2F2F2;"|Selection of nonsingular orbits | |||
|- | |||
|[[File:Sam mu 2.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=2</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 3.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=3</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 5.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=5</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 6.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=6</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|- | |||
|[[File:Sam mu 16.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=16</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 19.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=19</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 21.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=21</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 24.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=24</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|} | |||
====Periodic orbits==== | |||
[[File:Sam type A.svg|thumb|right|Type A orbits for <math>\theta_0</math> ranging from 0.1 to 3.1.]] | |||
When the different harmonic components in the system are in phase, the resulting trajectory is simple and periodic, such as the "smile" trajectory, which resembles that of an ordinary [[pendulum]], and various loops.<ref name="smiles and teardrops" /><ref name="motions"/> In general a periodic orbit exists when the following is satisfied:<ref name="Tufillaro1984"/> | |||
:<math>r(t+\tau) = r(t),\, \theta(t+\tau) = \theta(t)</math> | |||
The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed '''Type A''' orbits in his 1984 paper.<ref name="Tufillaro1984" /> | |||
{| width=78% class="collapsible <!--{{{state|collapsed}}}-->" style="border:1px #AAAAAA solid;" | |||
! colspan=4 style="background-color:#F2F2F2;"|Selection of periodic orbits | |||
|- | |||
|[[File:Sam mu 1.665000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|A "smile" orbit of the swinging Atwood's machine for <math>\mu=1.665</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 2.394000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=2.394</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 1.172700 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=1.1727</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 1.555000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=1.555</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|} | |||
===Singular orbits=== | |||
The motion is singular if at some point, the swinging mass passes through the origin. Since the system is [[invariant (physics)|invariant]] under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards:<ref name=Tufillaro1984/> | |||
:<math>r(0) = 0</math> | |||
The region close to the pivot is singular, since <math>r</math> is close to zero and the equations of motion require dividing by <math>r</math>. As such, special techniques must be used to rigorously analyze these cases.<ref name="collision"/> | |||
The following are plots of arbitrarily selected singular orbits. | |||
{| width=78% class="collapsible <!--{{{state|collapsed}}}-->" style="border:1px #AAAAAA solid;" | |||
! colspan=4 style="background-color:#F2F2F2;"|Selection of singular orbits | |||
|- | |||
|[[File:Sam mu 10.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=10</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|[[File:Sam mu 25.000000 r0 500.000000 t0 1.570796 dr0 0.000000 dt0 0.000000.svg|thumb|center|An orbit of the swinging Atwood's machine for <math>\mu=25</math>, <math>\theta_0=\frac{\pi}{2}</math>, and zero initial velocity.]] | |||
|} | |||
====Collision orbits==== | |||
[[File:Sam type B.svg|thumb|right|Type B orbits for <math>\theta_0</math> ranging from 0.1 to 3.1.]] | |||
Collision (or terminating singular) orbits are subset of singular orbits formed when the swinging mass is ejected from the its pivot with an initial velocity, such that it returns back to the pivot (i.e. it collides with the pivot): | |||
:<math>r(\tau) = r(0) = 0, \, \tau > 0</math> | |||
The simplest case of collision orbits are the ones with a mass ratio of 3, which will always return symmetrically to the origin after being ejected from the origin, and were termed '''Type B''' orbits in Tufillaro's initial paper.<ref name="Tufillaro1984" /> They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance.<ref name="smiles and teardrops" /><ref name="smiles and hearts"/><ref name="motions"/><ref name="collision"/> | |||
When the swinging mass returns to the origin, the counterweight mass, <math>M</math> must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time.<ref name="Tufillaro1984"/> | |||
==Boundedness== | |||
For any initial position, it can be shown that the swinging mass is bounded by a curve that is a [[conic section]].<ref name="bound"/> The pivot is always a [[Focus (geometry)|focus]] of this bounding curve. The equation for this curve can be derived by analyzing the energy of the system, and using conservation of energy. Let us suppose that <math>m</math> is released from rest at <math>r=r_0</math> and <math>\theta=\theta_0</math>. The total energy of the system is therefore: | |||
:<math> | |||
E = \frac{1}{2}M \dot{r}^2+\frac{1}{2} m \left(\dot{r}^2+r^2\dot{\theta}^2\right) + Mgr - mgr \cos{\theta} = Mgr_0 - mgr_0 \cos{\theta_0} | |||
</math> | |||
However, notice that in the boundary case, the velocity of the swinging mass is zero.<ref name="bound"/> Hence we have: | |||
:<math> | |||
Mgr - mgr \cos{\theta}=Mgr_0 - mgr_0 \cos{\theta_0} | |||
</math> | |||
To see that it is the equation of a conic section, we isolate for <math>r</math>: | |||
:<math> | |||
\begin{align} | |||
r&=\frac{h}{1-\frac{\cos{\theta}}{\mu}}\\ | |||
h&=r_0\left(1-\frac{\cos{\theta_0}}{\mu}\right) | |||
\end{align} | |||
</math> | |||
Note that the numerator is a constant dependent only on the initial position in this case, as we have assumed the initial condition to be at rest. However, the energy constant <math>h</math> can also be calculated for nonzero initial velocity, and the equation still holds in all cases.<ref name="bound"/> The [[eccentricity (mathematics)|eccentricity]] of the conic section is <math>\frac{1}{\mu}</math>. For <math>\mu>1</math>, this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For <math>\mu=1</math>, it is a parabola and for <math>\mu<1</math> it is a hyperbola; in either of these cases, it is not bounded. As <math>\mu</math> gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region.<ref name="bound"/> | |||
==References== | |||
<references> | |||
<ref name="Integrable"> | |||
{{cite journal | |||
| title = Integrable motion of a swinging Atwood's machine | |||
| year = 1986 | |||
| journal = [[American Journal of Physics]] | |||
| pages = 142 | |||
| volume = 54 | |||
| issue = 2 | |||
| doi = 10.1119/1.14710 | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
|bibcode = 1986AmJPh..54..142T }}</ref> | |||
<ref name="smiles and teardrops">{{cite thesis | |||
| title = Smiles and Teardrops | |||
| last = Tufillaro | |||
| first = Nicholas B. | |||
| year = 1982 | |||
| publisher = [[Reed College]] | |||
}}</ref> | |||
<ref name="smiles and hearts">{{cite journal | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
| title = Teardrop and heart orbits of a swinging Atwoods machine, | |||
| year = 1994 | |||
| journal = The American Journal of Physics | |||
| volume = 62 | |||
| issue = 3 | |||
| pages = 231–233 | |||
| doi = 10.1119/1.17602 | |||
|bibcode = 1994AmJPh..62..231T }}</ref> | |||
<ref name="Tufillaro1984">{{cite journal | |||
| title = Swinging Atwood's Machine | |||
| year = 1984 | |||
| journal = [[American Journal of Physics]] | |||
| volume = 52 | |||
| issue = 10 | |||
| pages = 895–903 | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
| last2 = Abbott | first2 = Tyler A. | |||
| last3 = Griffiths | first3 = David J. | |||
| accessdate = 2010-09-26 | |||
| doi = 10.1119/1.13791 | |||
|bibcode = 1984AmJPh..52..895T }}</ref> | |||
<ref name="Pujol2010">{{cite journal | |||
| title = Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study | |||
| year = 2010 | |||
| journal = [[Physica D]] | |||
| volume = 239 | |||
| issue = 12 | |||
| pages = 1067–1081 | |||
| last1 = Pujol | first1 = Olivier | |||
| last2 = Perez | first2 = J.P. | |||
| last3 = Simo | first3 = C. | |||
| last4 = Simon | first4 = S. | |||
| last5 = Weil | first5 = J.A. | |||
| doi = 10.1016/j.physd.2010.02.017 | |||
|bibcode = 2010PhyD..239.1067P }}</ref> | |||
<ref name="bound">{{cite journal | |||
| title = Unbounded orbits of a swinging Atwood's machine | |||
| year = 1988 | |||
| journal = American Journal of Physics | |||
| pages = 1117 | |||
| volume = 56 | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
| last2 = Nunes | first2 = A. | |||
| last3 = Casasayas | first3 = J. | |||
| doi = 10.1119/1.15774 | |||
|bibcode = 1988AmJPh..56.1117T }}</ref> | |||
<ref name="motions">{{cite journal | |||
| title = Motions of a swinging Atwood's machine | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
| journal = Journal de Physique | |||
| volume = 46 | |||
| issue = 9 | |||
| year = 1985 | |||
| pages = 1495–1500 | |||
| doi = 10.1051/jphys:019850046090149500 | |||
}}</ref> | |||
<ref name="collision">{{cite journal | |||
| last1 = Tufillaro | first1 = Nicholas B. | |||
| year = 1985 | |||
| title = Collision orbits of a swinging Atwood's machine | |||
| journal = Journal de Physique | |||
| volume = 46 | |||
| pages = 2053–2056 | |||
| doi = 10.1051/jphys:0198500460120205300 | |||
}}</ref> | |||
</references> | |||
===Further reading=== | |||
*Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems", ''Brazilian Journal of Physics'' Vol.28 n.4 São Paulo Dec. | |||
*Barrera, Jan Emmanuel (2003) ''Dynamics of a Double-Swinging Atwood's machine'', B.S. Thesis, National Institute of Physics, University of the Philippines. | |||
*Babelon, O, M. Talon, MC Peyranere (2010), "Kowalevski's analysis of a swinging Atwood's machine," Journal of Physics A-Mathematical and Theoretical Vol. 43 (8). | |||
*Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators," ''Physica Scripta'' Vol.35(1). | |||
*Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," ''European Journal of Physics'' Vol.10(10), p173. | |||
*Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," ''Journal de Physique'' Vol.51, p1693. | |||
*Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", ''International Journal of Theoretical Physics'', Vol. 27(11), p1405-1410. | |||
*Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" ''American Journal of Physics'' Vol.60(10), p951-953. | |||
*Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the Swinging Atwood Machine", ''Journal of Physics'' II France 1, p711-715. | |||
*Nunes, A., J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine," ''American Journal of Physics'' Vol.63(2), p121-126. | |||
*Ouazzani-T.H., A. and Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine," ''Il Nuovo Cimento B'' Vol. 110 (9). | |||
*Olivier, Pujol, JP Perez, JP Ramis, C. Simo, S. Simon, JA Weil (2010), "Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study," Physica D 239, pp. 1067–1081. | |||
*Sears, R. (1995) "Comment on "A surprising mechanics demonstration," ''American Journal of Physics'', Vol. 63(9), p854-855. | |||
*Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine", ''Mechanics Research Communications'' Vol. 33 (5), p711–716. | |||
==External links== | |||
*[http://www.physics.purdue.edu/research/ugrad_rsch/haley/sam.html Example of use in undergraduate research: symplectic integrators ] | |||
*[http://metric.ma.ic.ac.uk/articles/samos98/Swinging.pdf#search=%22swinging%20atwood's%20machine%22 Imperial College Course] | |||
*[http://www.sc.ehu.es/sbweb/fisica/oscilaciones/atwood/atwood.htm Oscilaciones en la máquina de Atwood] | |||
*[http://www.drchaos.net/drchaos/_.._files/sam_0.pdf "Smiles and Teardrops" (1982)] | |||
*[http://www.imub.ub.es/samni07.html 2007 Workshop] | |||
*[http://www-loa.univ-lille1.fr/~pujol/ 2010 Videos of an experimental Swinging Atwood's Machine] | |||
*[http://meetings.aps.org/Meeting/MAR10/Event/123612 Update on a Swinging Atwood's Machine at 2010 APS Meeting, 8:24 AM, Friday 19 March 2010, Portland, OR] | |||
*[http://www.openprocessing.org/visuals/?visualID=27494 Interactive web application of the Swinging Atwood's Machine] | |||
*[http://www.compadre.org/osp/items/detail.cfm?ID=11247 Open source Java code for running the Swinging Atwood's Machine] | |||
{{chaos theory}} | |||
[[Category:Hamiltonian mechanics]] | |||
Revision as of 04:37, 8 November 2013
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 and counterweight, mass ) 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.[1]
The conventional Atwood's machine allows only "runaway" solutions (i.e. either the pendulum or counterweight eventually collides with its pulley), except for . However, the swinging Atwood's machine with 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[1][2] due to the pendulum's reactive centrifugal force counteracting the counterweight's weight.[1] 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.[3]
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 and the non-swinging mass be . The kinetic energy of the system, , is:
where is the distance of the swinging mass to its pivot, and is the angle of the swinging mass relative to pointing straight downwards. The potential energy is solely due to the acceleration due to gravity:
We may then write down the Lagrangian, , and the Hamiltonian, of the system:
We can then express the Hamiltonian in terms of the canonical momenta, , :
Lagrange analysis can be applied to obtain two second-order coupled ordinary differential equations in and . First, the equation:
We simplify the equations by defining the mass ratio . The above then becomes:
Hamiltonian analysis may also be applied to determine four first order ODEs in terms of , and their corresponding canonical momenta and :
Notice that in both of these derivations, if one sets and angular velocity to zero, the resulting special case is the regular non-swinging Atwood machine:
The swinging Atwood's machine has a four-dimensional phase space defined by , and their corresponding canonical momenta and . However, due to energy conservation, the phase space is constrained to three dimensions.
System with massive pulleys
If the pulleys in the system are taken to have moment of inertia and radius , the Hamiltonian of the SAM is then:[4]
Where Template:Vart is the effective total mass of the system,
This reduces to the version above when and become zero. The equations of motion are now:[4]
Integrability
Hamiltonian systems can be classified as integrable and nonintegrable. SAM is integrable when the mass ratio .[5] The system also looks pretty regular for , but the case is the only integrable mass ratio found so far. For many other values of the mass ratio (and initial conditions) SAM displays chaotic motion.
Numerical studies indicate that when the orbit is singular (initial conditions: ), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of . When is small (near vertical), the trajectory describes a "teardrop", when it is large, it describes a "heart". These trajectories can be exactly solved algebraically, which is unusual for a system with a non-linear Hamiltonian.[6]
Trajectories
The swinging mass of the swinging Atwood's machine undergoes interesting trajectories or orbits when subject to different initial conditions, and for different mass ratios. These include periodic orbits and collision orbits.
Nonsingular orbits
For certain conditions, system exhibits complex harmonic motion.[1] The orbit is called nonsingular if the swinging mass does not touch the pulley.
| Selection of nonsingular orbits | |||
|---|---|---|---|
Periodic orbits

When the different harmonic components in the system are in phase, the resulting trajectory is simple and periodic, such as the "smile" trajectory, which resembles that of an ordinary pendulum, and various loops.[3][7] In general a periodic orbit exists when the following is satisfied:[1]
The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed Type A orbits in his 1984 paper.[1]
Singular orbits
The motion is singular if at some point, the swinging mass passes through the origin. Since the system is invariant under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards:[1]
The region close to the pivot is singular, since is close to zero and the equations of motion require dividing by . As such, special techniques must be used to rigorously analyze these cases.[8]
The following are plots of arbitrarily selected singular orbits.
| Selection of singular orbits | |||
|---|---|---|---|
Collision orbits
Collision (or terminating singular) orbits are subset of singular orbits formed when the swinging mass is ejected from the its pivot with an initial velocity, such that it returns back to the pivot (i.e. it collides with the pivot):
The simplest case of collision orbits are the ones with a mass ratio of 3, which will always return symmetrically to the origin after being ejected from the origin, and were termed Type B orbits in Tufillaro's initial paper.[1] They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance.[3][6][7][8]
When the swinging mass returns to the origin, the counterweight mass, must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time.[1]
Boundedness
For any initial position, it can be shown that the swinging mass is bounded by a curve that is a conic section.[2] The pivot is always a focus of this bounding curve. The equation for this curve can be derived by analyzing the energy of the system, and using conservation of energy. Let us suppose that is released from rest at and . The total energy of the system is therefore:
However, notice that in the boundary case, the velocity of the swinging mass is zero.[2] Hence we have:
To see that it is the equation of a conic section, we isolate for :
Note that the numerator is a constant dependent only on the initial position in this case, as we have assumed the initial condition to be at rest. However, the energy constant can also be calculated for nonzero initial velocity, and the equation still holds in all cases.[2] The eccentricity of the conic section is . For , this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For , it is a parabola and for it is a hyperbola; in either of these cases, it is not bounded. As gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region.[2]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 2.0 2.1 2.2 2.3 2.4 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 3.0 3.1 3.2 Template:Cite thesis
- ↑ 4.0 4.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 6.0 6.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 7.0 7.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 8.0 8.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
Further reading
- Almeida, M.A., Moreira, I.C. and Santos, F.C. (1998) "On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems", Brazilian Journal of Physics Vol.28 n.4 São Paulo Dec.
- Barrera, Jan Emmanuel (2003) Dynamics of a Double-Swinging Atwood's machine, B.S. Thesis, National Institute of Physics, University of the Philippines.
- Babelon, O, M. Talon, MC Peyranere (2010), "Kowalevski's analysis of a swinging Atwood's machine," Journal of Physics A-Mathematical and Theoretical Vol. 43 (8).
- Bruhn, B. (1987) "Chaos and order in weakly coupled systems of nonlinear oscillators," Physica Scripta Vol.35(1).
- Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," European Journal of Physics Vol.10(10), p173.
- Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," Journal de Physique Vol.51, p1693.
- Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", International Journal of Theoretical Physics, Vol. 27(11), p1405-1410.
- Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" American Journal of Physics Vol.60(10), p951-953.
- Moreira, I.C. and M.A. Almeida (1991) "Noether symmetries and the Swinging Atwood Machine", Journal of Physics II France 1, p711-715.
- Nunes, A., J. Casasayas, and N. B. Tufillaro (1995) "Periodic orbits of the integrable swinging Atwood's machine," American Journal of Physics Vol.63(2), p121-126.
- Ouazzani-T.H., A. and Ouzzani-Jamil, M., (1995) "Bifurcations of Liouville tori of an integrable case of swinging Atwood's machine," Il Nuovo Cimento B Vol. 110 (9).
- Olivier, Pujol, JP Perez, JP Ramis, C. Simo, S. Simon, JA Weil (2010), "Swinging Atwood's Machine: Experimental and numerical results, and a theoretical study," Physica D 239, pp. 1067–1081.
- Sears, R. (1995) "Comment on "A surprising mechanics demonstration," American Journal of Physics, Vol. 63(9), p854-855.
- Yehia, H.M., (2006) "On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine", Mechanics Research Communications Vol. 33 (5), p711–716.
External links
- Example of use in undergraduate research: symplectic integrators
- Imperial College Course
- Oscilaciones en la máquina de Atwood
- "Smiles and Teardrops" (1982)
- 2007 Workshop
- 2010 Videos of an experimental Swinging Atwood's Machine
- Update on a Swinging Atwood's Machine at 2010 APS Meeting, 8:24 AM, Friday 19 March 2010, Portland, OR
- Interactive web application of the Swinging Atwood's Machine
- Open source Java code for running the Swinging Atwood's Machine










