Swinging Atwood's machine - AbsoluteAstronomy.com

The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine

Atwood machine

The Atwood machine was invented in 1784 by Rev. George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration...

except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

that is chaotic

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

for some system parameters and initial conditions.

Specifically, it comprises two masses (the pendulum, mass

and counterweight, mass

) connected by a 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

. However, the swinging Atwood's machine with

has a large parameter space

Parameter space

In science, a parameter space is the set of values of parameters encountered in a particular mathematical model. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function....

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

Reactive centrifugal force

In classical mechanics, reactive centrifugal force is the reaction paired with centripetal force. A mass undergoing circular motion constantly accelerates toward the axis of rotation. This centripetal acceleration is caused by a force exerted on the mass by some other object. In accordance with...

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

Reed College is a private, independent, liberal arts college located in southeast Portland, Oregon. Founded in 1908, Reed is a residential college with a campus located in Portland's Eastmoreland neighborhood, featuring architecture based on the Tudor-Gothic style, and a forested canyon wilderness...

, 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

Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

or Lagrangian mechanics

Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....

. 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

Standard gravity

Standard gravity, or standard acceleration due to free fall, usually denoted by g0 or gn, is the nominal acceleration of an object in a vacuum near the surface of the Earth. It is defined as precisely , or about...

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:

And 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

Atwood machine

The Atwood machine was invented in 1784 by Rev. George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration...

The swinging Atwood's machine has a four-dimensional phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the 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 massy pulleys

If the pulleys in the system are taken to have moment of inertia

Moment of inertia

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

and radius

, the Hamiltonian of the SAM is then:

Where _t is the effective total mass of the system,

This reduces to the version above when

and

become zero. The equations of motion are now:

where

Integrability

Hamiltonian system

In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

s can be classified as integrable and nonintegrable. SAM is integrable when the mass ratio

. 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

Chaos theory

.

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.

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.

Periodic orbits

For certain conditions, system exhibits complex harmonic motion

Complex harmonic motion

Complex harmonic motion occurs when a number of simple harmonic motions are combined.Chords in music are an example of this phenomenon.Any continuous periodic function can be represented as a complex harmonic motion using its fourier series....

. A consequence is that when the different harmonic components are in phase, the resulting trajectory is simple and periodic, such as the "smile" trajectory, which resembles that of an ordinary pendulum

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...

, and various loops. In general a periodic orbit exists when the following is satisfied:

The following are plots of arbitrarily selected periodic orbits.

Selection of periodic orbits

Singular orbits

The motion is singular if at some point, the swinging mass passes through the origin. Since the system is invariant

Invariant

Invariant and invariance may have several meanings, among which are:- Computer science :* Invariant , an Expression whose value doesn't change during program execution* A type in overriding that is neither covariant nor contravariant...

under time reversal and translation, it is equivalent to say that the pendulum starts at the origin and is fired outwards:

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.

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):

Under these conditions, 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.

Some such trajectories are "hearts", "rabbit ears" and "teardrops", described in Tufillaro's initial paper as well as later ones.

Boundedness

For any initial position, it can be shown that the swinging mass is bounded by a curve that is a conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

. The pivot is always a focus

Focus (geometry)

In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

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. 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. The eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

of the conic section is