The Physical Pendulum and the Onset of Chaos Consider the uniform - - PowerPoint PPT Presentation

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The Physical Pendulum and the Onset of Chaos Consider the uniform - - PowerPoint PPT Presentation

Sep 23, 2022 378 likes •855 views

The Physical Pendulum and the Onset of Chaos Consider the uniform rod rotating about an end point in the figure. Starting from the definition of r the torque = F , (1) derive the differential equation the angular posi- O tion

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SLIDE 1

The Physical Pendulum and the Onset of Chaos

Consider the uniform rod rotating about an end point in the figure. Starting from the definition of the torque τ = r × F, (1) derive the differential equation the angular posi- tion θ must satisfy. (2) Derive a new differential equation if the pendu- lum is damped by a friction force Ff = −b v where b is some constant describing the the pendulum. (3) Derive a final differential equation if the pen- dulum is now also driven by a force Fdrive = FD sin(Ωt)ˆ θ. (4) Generate an algorithm for the differential equa- tion from Part 3. (5) What does the phase space look like for each set of conditions if the initial conditions are θ0 = 25◦ and ω0 = 0 rad/s or θ0 = 24◦ and ω0 = 0 rad/s?

m mgsin θ g θ O C mgcos θ L

Chaos – p. 1/3

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SLIDE 2

Getting Started - The Harmonic Oscillator

Hooke’s Law states that Fs = −kx where Fs is the restoring force ex- erted by a spring and x is the dis- placement from equilibrium where there is no net force acting on the

  • mass. See example here.
  • 1. What differential equation does x satisfy?
  • 2. What is the solution?
  • 3. How would you test the solution?
  • 4. What is the physical meaning of the constants in the solution?

Chaos – p. 2/3

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SLIDE 3

The Harmonic Oscillator - The Solution

The solution for Hooke’s Law is

x(t) = A cos(ωt + φ)

where x(t) is the displacement from equilibrium.

t 1 2 3 4 5 6 7 8 9 10 x(t)

  • 1
  • 0.5

0.5 1 A Cosine Curve

Period Amplitude Phase

Chaos – p. 3/3

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SLIDE 4

The Simple Harmonic Oscillator - An Example

A harmonic oscillator consists of a block of mass m = 0.33 kg attached to a spring with spring constant k = 400 N/m. See the figure below. At time t = 0.0 s the block’s displacement from equi- librium and its velocity are y =

0.100 m and v = −13.6 m/s. (1)

Find the particular solution for this oscillator. (2) Use a centered derivative formula to generate an algorithm for solving the equation

  • f motion.
  • Chaos – p. 4/3
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SLIDE 5

The Pendulum - Stating the Problem

The simple pendulum is an example of an oscillatory system where the restoring force is provided by gravity. Consider the pendulum shown in the figure.

  • 1. What differential equation does θ

satisfy?

  • 2. What differential equation does θ

satisfy for small angles?

  • 3. What is the solution?
  • 4. How would you test the solution?
  • 5. What is the physical meaning of the

constants?

  • 6. Redo Part 1 using torques.

m mgsin mgcos θ θ θ C g L O

Chaos – p. 5/3

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SLIDE 6

The Simple Pendulum - The Solution

The solution for simple pendulum is

θ(t) = A cos(ωt + φ)

where θ(t) is the angular displacement from equilibrium.

t 1 2 3 4 5 6 7 8 9 10 (t) θ

  • 1
  • 0.5

0.5 1

Period Amplitude Phase

A Cosine Curve

Chaos – p. 6/3

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SLIDE 7

Torque - Rotational Equivalent of Force

  • F = m

a → τ = r F⊥

F

Chaos – p. 7/3

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SLIDE 8

Linear → Rotational Quantities

Linear Rotational Quantity Connection Quantity

s s = rθ θ = s

r

vT vT = rω ω = vT

r = dθ dt

aT aT = rα α = aT

r = dω dt

KE = 1

2mv2

KER = 1

2Iω2

  • F = m

a τ = rF⊥

  • τ = I

α

  • p = m

v

  • L =

r × p

  • L = I

ω

Chaos – p. 8/3

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SLIDE 9

The Physical Pendulum and the Onset of Chaos

Consider the uniform rod rotating about an end point in the figure. Starting from the definition of the torque τ = r × F, (1) derive the differential equation the angular posi- tion θ must satisfy. (2) Derive a new differential equation if the pendu- lum is damped by a friction force Ff = −b v where b is some constant describing the the pendulum. (3) Derive a final differential equation if the pen- dulum is now also driven by a force Fdrive = FD sin(Ωt)ˆ θ. (4) Generate an algorithm for the differential equa- tion from Part 3. (5) What does the phase space look like for each set of conditions if the initial conditions are θ0 = 25◦ and ω0 = 0 rad/s or θ0 = 24◦ and ω0 = 0 rad/s?

m mgsin θ g θ O C mgcos θ L

Chaos – p. 9/3

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SLIDE 10

Moments of Inertia

Chaos – p. 10/3

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SLIDE 11

Nonlinear, Physical Pendulum Phase Space and Time Series

0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 Θrad Ωrads Phase Space for Θ025o red, Θ024o black 5 10 15 20 25 30 0.4 0.2 0.0 0.2 0.4 ts Θrad Time Series for Θ025o blue, Θ024o gray

Chaos – p. 11/3

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SLIDE 12

Nonlinear, Physical Pendulum Phase Space and Time Series

0.4 0.2 0.0 0.2 0.4 0.4 0.2 0.0 0.2 0.4 Θrad Ωrads Phase Space for Θ025o red, Θ024o black 5 10 15 20 25 30 0.4 0.2 0.0 0.2 0.4 ts Θrad Time Series for Θ025o blue, Θ024o gray

Chaos – p. 12/3

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SLIDE 13

Nonlinear, Damped, Driven, Physical Pendulum Phase Space and Time Series

10 5 5 10 15 2 1 1 2 Θrad Ωrads Phase Space for Θ025o red, Θ024o black 20 40 60 80 10 5 5 10 15 ts Θrad Time Series for Θ025o blue, Θ024o gray

Chaos – p. 13/3

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SLIDE 14

Code for Nonlinear, Damped, Driven, Physical Pendulum

(* Initial conditions and parameters *) th0 = 25.0*Pi/180; (* initial position in meters *) w0 = 0.0; (* initial velocity in m/s *) t0 = 0.0; (* initial time in seconds *) grav = 9.8; (* acceleration of gravity *) length = 14.7; (* length of pendulum *) mass = 0.245; (* mass of pendulum *) (* driving force amplitude and friction force. See below for more *) qDrag = 0.6; (* drag coefficient *) DriveForce = 11.8; (* DriveForce = 11.8; cool plot value *) DriveFreq = 0.67; (* driving force angular frequency *) DrivePeriod = 2*Pi/DriveFreq; (* period of the driving force *) (* step size *) step = 0.10;

Chaos – p. 14/3

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SLIDE 15

Code for Nonlinear, Damped, Driven, Physical Pendulum

(* limits of the iterations. since we already have theta(t=0) and we have calculated theta(t=step) then the first value in the table will be for t=2*step. *) tmin = 2*step; tmax = 80.0; (* condense the constants into coefficients for the appropriate terms. *) f1 = 1 + (3*qDrag*step/(2*mass*length)); f2 = 3*DriveForce*(stepˆ2)/(2*length); f3 = -3*grav*(stepˆ2)/(2*length); f4 = -1 + (3*qDrag*step/(2*mass*length)); (* set up the first two points. *) t1 = t0 + step; th1 = th0 + w0*step; (* get rid of the previous results for the table and proceed *) Clear[pdispl] Clear[tdispl]

Chaos – p. 15/3

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SLIDE 16

Code for Nonlinear, Damped, Driven, Physical Pendulum

(* A centered second derivative formula is used to generate an iterative solution for the mass on a spring. first load the starting poin. *) thmid = th0; (*starting value of theta *) thplus = th1; (* second value of theta *) tmid = t0; (* create a table of ordered (theta,w). for each component the next value is calculated and then the variables incremented for the next interation. pdispl = {{th0, w0}}; tdispl = {{t0, th0}}; Do[thminus = thmid; thmid = thplus; tmid = tmid + step; thplus = (f2*Sin[DriveFreq*t] + 2*thmid + f3*Sin[thmid] + f4*thminus)/f1; wmid = (thplus - thminus)/(2*step); pdispl = Append[pdispl, {thmid, wmid}] ; tdispl = Append[tdispl, {tmid, thmid}] , {t, tmin, tmax, step} ];

Chaos – p. 16/3

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SLIDE 17

Chaos Lab 1 Results

10 20 30 40 50 60 100 80 60 40 20 tim es Θrad Tim e Series of the Physical Pendulum Red Θ025o Blue Θ024o 10 20 30 40 50 60 150 100 50 tim es Θrad Tim e Series of the Physical Pendulum 100 80 60 40 20 10 5 5 10 Θrad Ω rads Phase Space of the Physical Pendulum

Chaos – p. 17/3

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SLIDE 18

Visualizing Chaos - The Phase Space Trajectory

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 18/3

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SLIDE 19

Visualizing Chaos - Stroboscopic Pictures

Chaos – p. 19/3

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SLIDE 20

Visualizing Chaos - Stroboscopic Pictures

3D Scatter Plot

20 40 60 ts 10 10 Θrad 2 1 1 2 Ωrads

Chaos – p. 20/3

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SLIDE 21

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 21/3

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SLIDE 22

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 22/3

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SLIDE 23

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 23/3

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SLIDE 24

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 24/3

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SLIDE 25

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 25/3

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SLIDE 26

Visualizing Chaos - The Poincare Section

3 2 1 1 2 3 2 1 1 2 Θrad Ωrads

θ0 = 10◦

Chaos – p. 26/3

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SLIDE 27

Visualizing Chaos - The Time Series

10000 20000 30000 40000 400 300 200 100 times Θrad Time Series of the Physical Pendulum

Chaos – p. 27/3

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SLIDE 28

Calculating Chaos - The Poincare Series - 1

(* initial conditions and parameters *) t0 = 0.0; x0 = 1.0; v0 = 0.2; step = 0.01; (* get the second and third points on the curve *) t1 = t0 + step; x1 = x0 + step*v0; x2 = 2*x1 - x0 - (step*step*x1); v1 = (x2 - x0)/(2*step); (* put the first point in the table *) MyTable = {{x0, v0}, {x1, v1}}; (* rename stuff for the first point of the algorithm *) xminus = x0; xmid = x1; xplus = x2; tmin = t1 + step; tmax = 50.0;

Chaos – p. 28/3

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SLIDE 29

Calculating Chaos - The Poincare Series - 2

(* Use a Do loop and store the points when t = n\[Pi]. A centered formula is used to approximate the second derivative. Set parameters needed to test when to store the data. *) TimeTest = Pi; PeriodCounter = 1; (* main loop. *) Do[xminus = xmid; xmid = xplus; xplus = 2*xmid - xminus - (step*step*xmid); vmid = (xplus - xminus)/(2*step); If[t > TimeTest, MyTable = Append[MyTable, {xmid, vmid}]; PeriodCounter = PeriodCounter + 1; TimeTest = PeriodCounter*2*Pi ], {t, tmin, tmax, step} ]

Chaos – p. 29/3

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SLIDE 30

Harmonic Oscillator With Coupled Equations - 1

(* Solving the mass on a spring problem. Initial conditions and parameters *) x0 = 0.0; (* initial position in meters *) v0 = 2.0; (* initial velocity in m/s *) t0 = 0.0; (* initial time in seconds *) (* set up the first two points. step size *) step = 0.1; t1 = t0 + step; x1 = x0 + v0*step; v1 = v0 - ( step*kspring*x0/mass); xminus = x0; (* initial value of x *) vminus = v0; (* initial value of v *) xmid = x1; vmid = v1; mass = 0.33; (* the mass in kg *) kspring = 0.5; (* spring constant in N/m *)

Chaos – p. 30/3

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SLIDE 31

Harmonic Oscillator With Coupled Equations - 2

(* limits of the iterations. since we already have y(t=0) and we have calculated y(t=step), then the first value in the table will be for t=2*step. *) tmin = 2*step; tmax = 25.0; (* create a table of ordered (t,x). for each component the next value is calculated and then variables are incremented for the next interation. tpos = Table[ {t, vplus = vminus - (2*step*kspring/mass)*xmid; xplus = xminus + (2*step*vmid); vminus = vmid; vmid = vplus; xminus = xmid; xmid = xplus }, {t, tmin, tmax, step} ];

Chaos – p. 31/3

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SLIDE 32

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax1000. s

Chaos – p. 32/3

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SLIDE 33

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s

Chaos – p. 32/3

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SLIDE 34

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax15000. s

Chaos – p. 32/3

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SLIDE 35

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax15000. s 0.01 0.00 0.01 0.02 0.03 0.04 1.280 1.285 1.290 1.295 1.300 1.305 1.310 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s

Chaos – p. 32/3

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SLIDE 36

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0., tmax15000. s 0.01 0.00 0.01 0.02 0.03 0.04 1.280 1.285 1.290 1.295 1.300 1.305 1.310 Θrad Ωrads Θ0 25 deg, Shift0., tmax10000. s 0.01 0.00 0.01 0.02 0.03 0.04 1.280 1.285 1.290 1.295 1.300 1.305 1.310 Θrad Ωrads Θ0 55 deg, Shift0., tmax10000. s

Chaos – p. 32/3

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SLIDE 37

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s

Chaos – p. 33/3

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SLIDE 38

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 0.95 1.00 1.05 1.10 1.15 0.10 0.05 0.00 0.05 0.10 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s

Chaos – p. 33/3

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SLIDE 39

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 0.95 1.00 1.05 1.10 1.15 0.10 0.05 0.00 0.05 0.10 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 1.060 1.062 1.064 1.066 1.068 1.070 0.000 0.001 0.002 0.003 0.004 0.005 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s

Chaos – p. 33/3

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SLIDE 40

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 0.95 1.00 1.05 1.10 1.15 0.10 0.05 0.00 0.05 0.10 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 1.060 1.062 1.064 1.066 1.068 1.070 0.000 0.001 0.002 0.003 0.004 0.005 Θrad Ωrads Θ0 25 deg, Shift0.25, tmax10000. s 2000 4000 6000 8000 10 000 1500 1000 500 tim es Θrad Θ0 25 deg, Shift0.25, tmax10000. s

Chaos – p. 33/3

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SLIDE 41

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax1000. s

Chaos – p. 34/3

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SLIDE 42

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s

Chaos – p. 34/3

slide-43
SLIDE 43

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax15000. s

Chaos – p. 34/3

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SLIDE 44

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax15000. s 0.44 0.46 0.48 0.50 0.52 0.54 1.04 1.05 1.06 1.07 1.08 1.09 1.10 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s

Chaos – p. 34/3

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SLIDE 45

Chaos Lab 2 Results

3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax1000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s 3 2 1 1 2 3 20 15 10 5 5 10 15 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax15000. s 0.44 0.46 0.48 0.50 0.52 0.54 1.04 1.05 1.06 1.07 1.08 1.09 1.10 Θrad Ωrads Θ0 25 deg, Shift0.05, tmax10000. s 0.44 0.46 0.48 0.50 0.52 0.54 1.04 1.05 1.06 1.07 1.08 1.09 1.10 Θrad Ωrads Θ0 55 deg, Shift0.05, tmax10000. s

Chaos – p. 34/3