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Pendulum With Vibrating BaseMATH 485 PROJECT TEAM
THOMAS BELLO, EMILY HUANG, FABIAN LOPEZ, KELLIN RUMSEY, TAO TAO
BackgroundRegular vs. Inverted Pendulum: http://www.youtube.com/watch?v=rwGAzy0noU0
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
Simple Vibrating
Tasks
Derive the Lagrangian for the vertical position
Find Effective Potential using the Averaging technique
Analyze the stability at each stationary position
Variables 𝑑0 = 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠
𝜔 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠
𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑝𝑒𝑛𝑑𝑢𝑙𝑢𝑚
𝜃 = 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑝𝑒𝑛𝑑𝑢𝑙𝑢𝑚
𝑔 = 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 9.81
𝐾 = 𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦
𝑈 = 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
Equation of MotionPosition and Velocity
The X and Y coordinates:
x = 𝑙 sin 𝜃
y =𝑙 cos 𝜃 + 𝑑0 sin(𝑤𝑡)
The Velocity:
Vx = ẋ = 𝜃 𝑙 cos 𝜃
Vy = ẏ= - 𝜃 𝑙 sin 𝜃 + 𝑑0𝑤𝑠𝑖𝑛 (𝑤𝑡)
Kinetic and Potential Energy Kinetic Energy
◦ K = 1
2𝑚 𝑉𝑥
2 + 𝑉𝑦2
=1
2𝑚( 𝜃2𝑙2 cos2 𝜃 + 𝜃2𝑙2 sin2 𝜃 + 𝑑0
2𝜔2 sin2(𝜔𝑡) − 2 𝜃𝑙 (sin 𝜃)𝑑0𝜔 sin(𝜔𝑡)
=1
2𝑚( 𝜃2𝑙2 + 𝑑0
2𝜔2 sin2(𝜔𝑡) − 2 𝜃𝑙 (sin 𝜃)𝑑0𝜔 sin(𝜔𝑡) )
Potential Energy
◦ U = mgy
= mg (𝑙 cos 𝜃 + 𝑑0 sin(𝜔𝑡))
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:
K =1
2𝑚 𝜃2𝑙2 + 𝑑0
2𝜔2 sin2 𝜔𝑡 − 2 𝜃 𝑙 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡
𝑈 = mg (𝑙 cos 𝜃 + 𝑑0 sin(𝜔𝑡) )
The Lagrangian• The Lagrangian (L) is defined as: L = K – U
• In the case of the Pendulum with a Vibrating Base:
K =1
2𝑚 𝜃2𝑙2 + 𝑑0
2𝜔2 sin2 𝜔𝑡 − 2 𝜃 𝑙 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡
𝑈 = mg (𝑙 cos 𝜃 + 𝑑0 sin(𝜔𝑡) )
𝐿 =1
2𝑚 𝜃2𝑙2 + 𝑑0
2𝜔2 sin2 𝜔𝑡 − 2 𝜃 𝑙 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡 − mg (𝑙 cos 𝜃 + 𝑑0 sin(𝜔𝑡) )
The Lagrangian• The Lagrangian (L) is defined as: L = K – U
• In the case of the Pendulum with a Vibrating Base:
𝐿 = 𝑚𝑙1
2 𝜃2𝑙 +
1
2𝑙𝑑0
2𝜔2 sin2 𝜔𝑡 + 𝜃 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡 − g cos 𝜃 −𝑔𝑑0𝑙
cos(𝜔𝑡)
• 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.
𝐿 =1
2 𝜃2𝑙 + 𝜃 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡 − g cos 𝜃
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:
𝐿 =1
2 𝜃2𝑙 + 𝜃 (sin 𝜃)𝑑0𝜔 sin 𝜔𝑡 − g cos 𝜃
d
dt
𝜕L
𝜕θ−𝜕L
𝜕θ= 0
𝜃 +𝑑0𝜔
2
𝑙cos 𝜔𝑡 −
𝑔
𝑙𝑠𝑖𝑛𝜃 = 0
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
𝜉(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
𝜃 = 𝑋 + 𝜉
Final differential equation can be written as a total derivative in position, which corresponds to the effective potential energy of the system
𝑋 = −𝑑
𝑑𝑋
−𝑔
𝑙cos 𝜃 +
1
4
𝑑02𝜔2
𝑙2sin2 𝜃
General equation of motion relates positions and potential energy
𝑥 = −𝑑𝑈(𝑥)
𝑑𝑥
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
𝑈𝑒𝑓𝑓 =−𝑔
𝑙cos 𝜃 +
1
4
𝑑02𝜔2
𝑙2sin2 𝜃
Effective potential looks like potential energy of slow motion and kinetic energy of rapid motion
𝑈𝑒𝑓𝑓 =−𝑔
𝑙cos 𝜃 +
1
2 𝜉2
Stability AnalysisStability occurs at points of minimum potential energy
Angles 0 and 𝜋 are always equilibrium, but 𝜋 is unstable
When frequency exceeds minimum value, two additional unstable equilibria appear, and 𝜋 becomes stable
𝜔 ≥2𝑔𝑙
𝑑02
𝜃𝑠 = 0, 𝜋, cos−12𝑔𝑙
𝑑02𝜔2 , 2𝜋 − cos−1
2𝑔𝑙
𝑑02𝜔2
Two angles correspond to range of stability: inside those angles, pendulum returns to 𝜋, outside, pendulum falls down to 0
Unstable Caseg = 9.8, 𝑙 = 1, d = 0.1, 𝜔 = 20
Stable Caseg = 9.8, 𝑙 = 1, d = 0.1, 𝜔 = 50
Stable Caseg = 9.8, 𝑙 = 1, d = 0.1, 𝜔 = 70
Future AnalysisAnalyzing Critical Values for the Horizontal Case
Analyzing Critical Values for Arbitrary Angles
Experimentation for comparison with theoretical findings.
Thank You
Questions?