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.
[1] The conventional Atwood's machine allows only "runaway" solutions (i.e. either the pendulum or counterweight eventually collides with its pulley), except for
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.
[3] The swinging Atwood's machine is a system with two degrees of freedom.
is the angle of the swinging mass relative to pointing straight downwards.
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
The above then becomes: Hamiltonian analysis may also be applied to determine four first order ODEs in terms of
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
However, due to energy conservation, the phase space is constrained to three dimensions.
, the Hamiltonian of the SAM is then:[4] Where Mt is the effective total mass of the system, This reduces to the version above when
[6] 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
These trajectories can be exactly solved algebraically, which is unusual for a system with a non-linear Hamiltonian.
[7] 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.
For certain conditions, system exhibits complex harmonic motion.
[1] The orbit is called nonsingular if the swinging mass does not touch the pulley.
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.
[1] 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
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): 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.
must instantaneously change direction, causing an infinite tension in the connecting string.
[1] For any initial position, it can be shown that the swinging mass is bounded by a curve that is a conic section.
The total energy of the system is therefore: However, notice that in the boundary case, the velocity of the swinging mass is zero.
can also be calculated for nonzero initial velocity, and the equation still holds in all cases.
gets arbitrarily large, the bounding curve approaches a circle.
[2] A new integrable case for the problem of three dimensional Swinging Atwood Machine (3D-SAM) was announced in 2016.
