PENDULA:•SIMPLE HARMONIC;

•DRIVEN;

•COUPLED.

A SIMPLE PENDULUM IS JUST A WEIGHT ON THE END OF A STRING OR A STICK.

WEIGHT

STICK(LENGTH L)

PIVOT

THE PERIOD OF A PENDULUM IS GIVEN BY THE FOLLOWING EQUATION:

T = 2π L/g

Where: T is the time of one complete swing of the pendulum;

L is the length of the stick from pivot to the center of gravity of the weight;

g is the acceleration of gravity (9.8 m/s/s)

Please note that this equation is only an approximation which applies to small amplitude swings.

The larger the swing of a pendulum the longer a period it has. Thus, a pendulum forms a non-linear system. This has some implications for the accuracy of some pendulum clocks.

Please notice that the period of the swing of a pendulum is INDEPENDENT of the weight, W.

Einstein made big time with this fact. It was one of the building blocks of his GENERAL THEORY OF RELATIVITY.

An UNCONSTRAINED Pendulum can swing in any and all directions.

An unconstrained pendulum will swing in any and all directions due to a variety of forces on it. This includes the CORIOLIS force imposed by the earth’s rotation.

Foucault made a lot of this through his pendulum. See Umberto Eco.

A CONSTRAINED Pendulum is forced by its construction to swing in only one or more specific directions.

A Pendulum can be DRIVEN. This means it’s forced to move by some externally-applied force.There are THREE common manners of driving a pendulum, as follows.

Oddly enough, a pendulum clock

doesn’t do ANY of these.

An easy way to drive a pendulum into GENERALIZED activity is to drive it as follows.

D

L

S

Depending on the values of S (the speed of rotation of the hub) D (the length of the arm) and L (the length of the pendulum STRING) the pendulum will move in:

•Simple Harmonic Motion;•Periodic Complex Motion;•Completely Chaotic Motion.

C0UPLED PENDULA

THE SIMPLEST CASE OF COUPLED PENDULA IS TO HAVE TWO IDENTICAL PENDULA COUPLED TOGETHER WITH A SPRING. LIKE SO:

WHEN ONE PENDULUM IS MADE TO SWING IT PUTS A FORCE ONTO THE OTHER THROUGH THE SPRING.

THIS GOES ON UNTIL ALL THE SYSTEM ENERGY IS CONCENTRATED IN THE SECOND PENDULUM, AND THE FIRST PENDULUM STOPS.

SINCE THE TOTAL ENERGY OF THE SYSTEM IS A CONSTANT, AS THE SECOND PENDULUM SWINGS MORE, THE FIRST PENDULUM SWINGS LESS.

THIS FORCE CAUSES THE OTHER PENDULUM TO START TO SWING. IT GRADUALLY SWINGS MORE AND MORE.

THEN THE PROCESS REVERSES, AND THE ENERGY IS TRANSFERRED BACK TO THE FIRST PENDULUM.

THEN THE PROCESS REVERSES. ETC.

SO, THE COUPLED PENDULA SIMPLY TRADE ENERGY AND MOTION BACK AND FORTH.

UNTIL THE FRICTION IN THE SYSTEM EATS UP ALL THE ENERGY, AND ALL MOTION STOPS

In such a system, the pendula behave normally.

But the INTERESTING thing is that the rate at which the energy is transferred back and forth is controlled by the strength of the spring.

The STRONGER the spring the faster the motion is transferred back and forth between the pendula.

Outside of changing the period of oscillation of the pendula, the only influence one has on a coupled system is the change the characteristics of the coupling between pendula.

MULTIPLE COUPLED PENDULA are much more complex than a simple coupled pair.

P1 P2 P3 P4 P5 P6

k12 k23 k34 k45 k56L

This is because the coupling between each of the pedula can be different.

It is VERY difficult to predict the behavior of such a system. This is because whenever one pendulum moves, it transfers energy to the next, which moves and transfers energy back to the first, AND on to the next.

It is EVEN MORE DIFFICULT, and perhaps impossible to force such a system to behave in a specified manner.

HOWEVER, it can be observed that the motion of such a

system is fascinating.

Coupled pendule can be arranged side-by-side, in tandem, and vertically.

This allows for the possibility of making either two- or three-dimensional arrays of coupled pendula.

Two Dimensional Array of Pendula

Three Dimensional Array of Pendula

PRACTICAL DETAILS

•The pendula in these systems should be IDENTICAL.

•The pivot bearings should be as low friction as possible.

•The action of the couplings should be the same in both push and pull mode.

•Everything should be aligned PERFECTLY so as to assure that all forces are axial.

•The support structure should be solid and non-moving.

APPLICATIONS•A big set of sculpted coupled pendula could be an acceptable art piece. The pendula could be actuated by the observers, by wind, by actuators keyed by noise, position of passers-by, color, etc. It is also possible to use such actuations to not only actuate the pendula, but also to change the coupling between them to change the action.

•Coupled pendula could be incorporated as differently shaped moving elements of a sculpture or mobile.

•It is possible to incorporate an appropriate set of coupled pendula into clockworks.

FINAL NOTE BACK TO THE FINE PRINT

Friction is the ultimate enemy of continued action of anything without energy input. (this is a law of THERMODYNAMICS.)

By being clever about applying power to each pendulum at critical times, it’s possible to make a system which will continue to oscillate for as long as desired.