Swinging Atwood's machine
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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 In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
that is chaotic for some system parameters and
initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
. Specifically, it comprises two masses (the
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, mass and counterweight, mass ) connected by an inextensible, massless string suspended on two frictionless
pulleys A pulley is a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable or belt, or transfer of power between the shaft and cable or belt. In the case of a pulley supported by a frame or shell that d ...
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 The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for th ...
of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic,
quasiperiodic Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpred ...
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 Reed College is a private liberal arts college in Portland, Oregon. Founded in 1908, Reed is a residential college with a campus in the Eastmoreland neighborhood, with Tudor-Gothic style architecture, and a forested canyon nature preserve at ...
, 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 Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
or
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
. 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 \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, \mathcal, and the Hamiltonian, \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_\theta: : \begin p_r &= \frac = \frac = (M+m)\dot\\ p_\theta &= \frac = \frac = mr^2 \dot\\ \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 \theta. First, the \theta equation: : \begin \frac &= \frac \left(\frac\right)\\ -mgr \sin &= 2mr \dot\dot + mr^2 \ddot\\ r\ddot + 2\dot\dot + g\sin &= 0 \end And the r equation: : \begin \frac &= \frac \left( \frac\right)\\ mr\dot^2 - Mg + mg\cos &= (M+m) \ddot \end We simplify the equations by defining the mass ratio \mu = \frac. The above then becomes: :(\mu+1)\ddot - r\dot^2 + g(\mu - \cos) = 0 Hamiltonian analysis may also be applied to determine four first order ODEs in terms of r, \theta and their corresponding canonical momenta p_r and p_\theta: : \begin \dot&=\frac = \frac \\ \dot &= - \frac = \frac - Mg + mg\cos \\ \dot&=\frac = \frac \\ \dot &= - \frac = -mgr\sin \end Notice that in both of these derivations, if one sets \theta and angular velocity \dot to zero, the resulting special case is the regular non-swinging Atwood machine: :\ddot = g \frac=g\frac The swinging Atwood's machine has a four-dimensional phase space defined by r, \theta and their corresponding canonical momenta p_r and p_\theta. 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 I and radius R, the Hamiltonian of the SAM is then: :\mathcal\left(r, \theta, \dot, \dot \right) = \underbrace_ + \underbrace_, Where t is the effective total mass of the system, :M_t = M + m + \frac This reduces to the version above when R and I become zero. The equations of motion are now: :\begin \mu_t ( \ddot - R \ddot) & = r \dot^2 + g (\cos - \mu ) \\ r \ddot & = - 2 \dot \dot + R \dot^2 - g \sin \\ \end where \mu_t = M_t / m.


Integrability

Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s can be classified as
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
and nonintegrable. SAM is integrable when the mass ratio \mu = M/m = 3. The system also looks pretty regular for \mu = 4 n^2 - 1 = 3, 15, 35, ..., but the \mu = 3 case is the only known integrable mass ratio. It has been shown that the system is not integrable for \mu \in (0,1) \cup (3,\infty). 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: r=0, \dot=v, \theta=\theta_0, \dot=0), the pendulum executes a single symmetrical loop and returns to the origin, regardless of the value of \theta_0. When \theta_0 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.


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. The orbit is called nonsingular if the swinging mass does not touch the pulley.


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 A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, and various loops. In general a periodic orbit exists when the following is satisfied: :r(t+\tau) = r(t),\, \theta(t+\tau) = \theta(t) The simplest case of periodic orbits is the "smile" orbit, which Tufillaro termed Type A orbits in his 1984 paper.


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: :r(0) = 0 The region close to the pivot is singular, since r is close to zero and the equations of motion require dividing by r. As such, special techniques must be used to rigorously analyze these cases. The following are plots of arbitrarily selected singular orbits.


Collision orbits

Collision (or terminating singular) orbits are subset of singular orbits formed when the swinging mass is ejected from its pivot with an initial velocity, such that it returns to the pivot (i.e. it collides with the pivot): :r(\tau) = r(0) = 0, \, \tau > 0 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. They were also referred to as teardrop, heart, or rabbit-ear orbits because of their appearance. When the swinging mass returns to the origin, the counterweight mass, M must instantaneously change direction, causing an infinite tension in the connecting string. Thus we may consider the motion to terminate at this time.


Boundedness

For any initial position, it can be shown that the swinging mass is bounded by a curve that is a
conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
. The pivot is always a
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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 m is released from rest at r=r_0 and \theta=\theta_0. The total energy of the system is therefore: : E = \fracM \dot^2+\frac m \left(\dot^2+r^2\dot^2\right) + Mgr - mgr \cos = Mgr_0 - mgr_0 \cos However, notice that in the boundary case, the velocity of the swinging mass is zero. Hence we have: : Mgr - mgr \cos=Mgr_0 - mgr_0 \cos To see that it is the equation of a conic section, we isolate for r: : \begin r&=\frac\\ h&=r_0\left(1-\frac\right) \end 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 h can also be calculated for nonzero initial velocity, and the equation still holds in all cases. The
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of the conic section is \frac. For \mu>1, this is an ellipse, and the system is bounded and the swinging mass always stays within the ellipse. For \mu=1, it is a parabola and for \mu<1 it is a hyperbola; in either of these cases, it is not bounded. As \mu gets arbitrarily large, the bounding curve approaches a circle. The region enclosed by the curve is known as the Hill's region.


Recent three dimensional extension

A new integrable case for the problem of three dimensional Swinging Atwood Machine (3D-SAM) was announced in 2016. Like the 2D version, the problem is integrable when M = 3m.


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, Emmanuel Jan (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," ''
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'' Vol.35(1). *Casasayas, J., N. B. Tufillaro, and A. Nunes (1989) "Infinity manifold of a swinging Atwood's machine," ''
European Journal of Physics The ''European Journal of Physics'' is a peer-reviewed, scientific journal dedicated to maintaining and improving the standard of physics education in higher education. The journal, published since 1980, is now published by IOP Publishing on beha ...
'' Vol.10(10), p173. *Casasayas, J, A. Nunes, and N. B. Tufillaro (1990) "Swinging Atwood's machine: integrability and dynamics," ''
Journal de Physique The ''European Physical Journal'' (or ''EPJ'') is a joint publication of EDP Sciences, Springer Science+Business Media, and the Società Italiana di Fisica. It arose in 1998 as a merger and continuation of ''Acta Physica Hungarica'', '' Anales de ...
'' Vol.51, p1693. *Chowdhury, A. Roy and M. Debnath (1988) "Swinging Atwood Machine. Far- and near-resonance region", ''
International Journal of Theoretical Physics The ''International Journal of Theoretical Physics'' is a peer-reviewed scientific journal of physics published by Springer Science+Business Media since 1968. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of ...
'', Vol. 27(11), p1405-1410. *Griffiths D. J. and T. A. Abbott (1992) "Comment on ""A surprising mechanics demonstration,"" ''
American Journal of Physics The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current F ...
'' 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 The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current F ...
'' 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 The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current F ...
'', 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"Smiles and Teardrops" (1982)2010 Videos of an experimental Swinging Atwood's MachineUpdate on a Swinging Atwood's Machine at 2010 APS Meeting, 8:24 AM, Friday 19 March 2010, Portland, ORInteractive web application of the Swinging Atwood's MachineOpen source Java code for running the Swinging Atwood's Machine
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