ANALYSIS OF THE TAUTOCHRONE, BRACHISTOCHRONE AND CYCLOIDAL PENDULUM

BRIAN COVELLO, NATHANIEL STAMBAUGH

PREVIEW Background Simple pendulum

• Small angle approximation • Elliptic Integrals • A novel pattern taylor expression

The Brachistochrone The Cycloid The Tautochrone Bridge? The Cycloidal Pendulum

BACKGROUND 1599 Galileo studied cycloids 1659 Hyugens showed cycloid is the solution to the tautochrone problem

• The same time path • Cycloidal pendulum à period not dependent on

amplitude 1697 Bernoulli showed cycloid is the solution to the brachistochrone problem

• The least time path • Coincidence?

THE SIMPLE PENDULUM

SIMPLE PENDULUM – EULER LAGRANGE

SMALL ANGLE APPROXIMATION

à à

What if we attempt to solve the nonlinear second order ODE through a taylor expression?

TAYLOR EXPRESSION

•  No obvious pattern •  Let’s start collecting terms

anyway…

CONTINUED… N=0,1

•  Begin with the first term…

•  Collect the linear terms n=1…Imagine the multiplicative possibilities that will generate a first degree linear term

CONTINUED… N=2

CONTINUED… N=3

CONTINUED… N=4

•  Fill in a0 as needed to the powers of “y” we are dealing with …

SEPARATE BASED ON PARTITIONS…

GENERALIZED PATTERN

TAYLOR EXPRESSION

•  For comparison…elliptic integrals

BRACHISTOCHRONE

BRACHISTOCHRONE – EULER LAGRANGE

BRACHISTOCHRONE – EULER LAGRANGE

TIME FROM TOP TO BOTTOM

TIME FROM SOME INITIAL Y

Same time from some initial y as from the top! •  This is known as the tautochrone (same time)

TAUTOCHRONE

EVOLUTE OF A CYCLOID Parameterization for evolute:

CYCLOIDAL PENDULUM