Pendulum With Vibrating Base MATH 485 PROJECT TEAM THOMAS BELLO, EMILY HUANG, FABIAN LOPEZ, KELLIN RUMSEY, TAO TAO Background Regular vs. Inverted Pendulum: http://www.youtube.com/watch?v=rwGAzy0noU0

Simple Vibrating History: In 1908, A. Stephenson found that the upper vertical position of the pendulum might be stable when the driving frequency is fast In 1951, a Russian scientist Pyotr Kapitza successfully analyzed this unusual and counterintuitive phenomenon by splitting the motion into 1. “fast” and “slow” variables 2. introducing the effective potentials Tasks

Derive the Lagrangian for the vertical position

Find Effective Potential using the Averaging technique

Analyze the stability at each stationary position Variables

푑0 = 푎푚푝푙𝑖푡푢푑푒 표푓 푏푎푠푒 표푠푐𝑖푙푙푎푡𝑖표푛푠 휔 = 푓푟푒푞푢푒푛푐푦 표푓 푏푎푠푒 표푠푐𝑖푙푙푎푡𝑖표푛푠 푙 = 푙푒푛푔푡ℎ 표푓 푝푒푛푑푢푙푢푚 휃 = 푐표푢푛푡푒푟푐푙표푐푘푤𝑖푠푒 푎푛푔푢푙푎푟 푑𝑖푠푝푙푎푐푒푚푒푛푡 표푓 푝푒푛푑푢푙푢푚 푔 = 푔푟푎푣𝑖푡푎푡𝑖표푛푎푙 푐표푛푠푡푎푛푡 9.81 퐾 = 푘𝑖푛푒푡𝑖푐 푒푛푒푟푔푦 푈 = 푝표푡푒푛푡𝑖푎푙 푒푛푒푟푔푦 Equation of Motion Position and Velocity The X and Y coordinates: x = 푙 sin 휃

y =푙 cos 휃 + 푑0 sin(푤푡)

The Velocity:

Vx = ẋ = 휃 푙 cos 휃

Vy = ẏ= -휃 푙 sin 휃 + 푑0푤푠𝑖푛 (푤푡) Kinetic and Potential Energy Kinetic Energy 1 ◦ K = 푚 푉2 + 푉2 2 푥 푦 1 = 푚(휃 2푙2 cos2 휃 + 휃 2푙2 sin2 휃 + 푑 2휔2 sin2(휔푡) − 2휃푙 (sin 휃)푑 휔 sin(휔푡) 2 0 0 1 = 푚(휃 2푙2 + 푑 2휔2 sin2(휔푡) − 2휃푙 (sin 휃)푑 휔 sin(휔푡) ) 2 0 0 Potential Energy ◦ U = mgy

= mg (푙 cos 휃 + 푑0 sin(휔푡)) The Lagrangian • The Lagrangian (L) is defined as: L = K – U • Take the simple case of a ball being thrown straight up.

Kinetic (K) Potential (U) The Lagrangian • The Lagrangian (L) is defined as: L = K – U • Take the simple case of a ball being thrown straight up.

Kinetic (K) Potential (U) Lagrangian The Lagrangian • The Lagrangian (L) is defined as: L = K – U • Take the simple case of a ball being thrown straight up. • The area under the Lagrangian vs. Time curve is known as the action of the system

Lagrangian The Lagrangian • The Lagrangian (L) is defined as: L = K – U • In the case of the Pendulum with a Vibrating Base:

1 K = 푚 휃 2푙2 + 푑 2휔2 sin2 휔푡 − 2휃 푙 (sin 휃)푑 휔 sin 휔푡 2 0 0

푈 = mg (푙 cos 휃 + 푑0 sin(휔푡) ) The Lagrangian • The Lagrangian (L) is defined as: L = K – U • In the case of the Pendulum with a Vibrating Base:

1 K = 푚 휃 2푙2 + 푑 2휔2 sin2 휔푡 − 2휃 푙 (sin 휃)푑 휔 sin 휔푡 2 0 0

푈 = mg (푙 cos 휃 + 푑0 sin(휔푡) )

1 퐿 = 푚 휃 2푙2 + 푑 2휔2 sin2 휔푡 − 2휃 푙 (sin 휃)푑 휔 sin 휔푡 − mg (푙 cos 휃 + 푑 sin(휔푡) ) 2 0 0 0 The Lagrangian • The Lagrangian (L) is defined as: L = K – U • In the case of the Pendulum with a Vibrating Base:

1 1 푔푑 퐿 = 푚푙 휃 2푙 + 푑 2휔2 sin2 휔푡 + 휃 (sin 휃)푑 휔 sin 휔푡 − g cos 휃 − 0 cos(휔푡) 2 2푙 0 0 푙

• We can take advantage of the following two properties to simplify our Lagrangian i) The Lagrangian does not depend on constants ii) The Lagrangian does not depend on functions of only time.

1 퐿 = 휃 2푙 + 휃 (sin 휃)푑 휔 sin 휔푡 − g cos 휃 2 0 The Euler - Lagrange Equation • Formulated in the 1750’s by Leonhard Euler and Joseph Lagrange. • Yields a Differential Equation whose solutions are the functions for which a functional is stationary • The Equation:

1 퐿 = 휃 2푙 + 휃 (sin 휃)푑 휔 sin 휔푡 − g cos 휃 2 0 d 휕L 휕L − = 0 dt 휕θ 휕θ

푑 휔2 푔 휃 + 0 cos 휔푡 − 푠𝑖푛휃 = 0 푙 푙 Averaging & Effective Potential .From our Euler-Lagrange Equation we want to derive Effective Potential Energy

.Effective Potential (Ueff) : is a mathematical expression combining multiple (perhaps opposing) effects into a single potential . Separate fast and slow components from Euler-Lagrangian into: . Ẍ(t) : “slow” motion: Smooth Motion . 휉 (t): “fast” motion: Rapid Oscillation .Averaging Technique: take an average over the period of the rapid oscillation in order to treat motion as single, smooth function .Assumptions . Fast components have MUCH higher frequency than slow components and a relatively low amplitude . Slow motion is treated as constant with respect to rapid motion period Finding Effective Potential….. Begin by separating variables into rapid oscillations due to vibrating base and slow motion of pendulum 휃 = 푋 + 휉 Final differential equation can be written as a total derivative in position, which corresponds to the effective potential energy of the system

푑 −푔 1 푑 2휔2 푋 = − cos 휃 + 0 sin2 휃 푑푋 푙 4 푙2 General equation of motion relates positions and potential energy

푑푈(푥) 푥 = − 푑푥 Effective Potential Continued….. Effective potential treats entire motion as single smooth motion and can be used for analysis as if it were the actual potential energy

−푔 1 푑 2휔2 푈 = cos 휃 + 0 sin2 휃 푒푓푓 푙 4 푙2

Effective potential looks like potential energy of slow motion and kinetic energy of rapid motion −푔 1 푈 = cos 휃 + 휉 2 푒푓푓 푙 2 Stability Analysis Stability occurs at points of minimum potential energy Angles 0 and 휋 are always equilibrium, but 휋 is unstable When frequency exceeds minimum value, two additional unstable equilibria appear, and 휋 becomes stable

2푔푙 휔 ≥ 2 푑0

−1 2푔푙 −1 2푔푙 휃푠 = 0, 휋, cos 2 2 , 2휋 − cos 2 2 푑0 휔 푑0 휔 Two angles correspond to range of stability: inside those angles, pendulum returns to 휋, outside, pendulum falls down to 0 Unstable Case

g = 9.8, 푙 = 1, d = 0.1, 휔 = 20 Stable Case

g = 9.8, 푙 = 1, d = 0.1, 휔 = 50 Stable Case

g = 9.8, 푙 = 1, d = 0.1, 휔 = 70 Applications

http://www.youtube.com/watch?v=Df6Rfsi6zSY Future Analysis .Analyzing Critical Values for the Horizontal Case . Analyzing Critical Values for Arbitrary Angles . Experimentation for comparison with theoretical findings. Thank You Questions?