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

Applications

http://www.youtube.com/watch?v=Df6Rfsi6zSY

Future AnalysisAnalyzing Critical Values for the Horizontal Case

Analyzing Critical Values for Arbitrary Angles

Experimentation for comparison with theoretical findings.

Thank You

Questions?