brian covello: mathematics research utilizing differential equations for a pendulum
DESCRIPTION
Brian Covello's work with the isochronous pendulum. Taken from www.wikipedia.org A pendulum is a weight suspended from a pivot so that it can swing freely.[1] 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 the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum, and also to a slight degree on the amplitude, the width of the pendulum's swing. From its discovery in around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s.[2] Pendulums are used to regulate pendulum clocks, and are used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length. The word 'pendulum' is new Latin, from the Latin pendulus, meaning 'hanging'.[3] The simple gravity pendulum[4] is an idealized mathematical model of a pendulum.[5][6][7] This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines.TRANSCRIPT
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