Dynamic Stabilization of an Invert Pendulum

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Anya Demko

May 2014

Approved for the Division(Physics)

Lucas J. Illing

Acknowledgements

If I were to actually acknowledge all the people that got me to this point, this wouldbe the longest section of my thesis; so I’ll try to keep it short: To Asa, for teachingme what love is and getting me to Reed, both figuratively and literally. To theAnimanimals, for showing me what it means to belong and teaching me that I cancare. To the Farmhaus, for teaching me what it means to have a family. To theInstitute, for teaching me what it means to have a home. To the Physics Department,for teaching me that I’m able to learn things I never thought I could. To Lucas, forputting up with me, and to Jay, for building the pendulum and making this thesispossible. To the Badass Sparkle Princesses, for teaching me what it means to be partof a team and for being the most dependable support I’ve ever had. To Phoebe, forteaching me how to love and for putting up with my all of my antics. To Erica, forteaching me what it means for others to care about me. To Pink$$$, for teaching mehow to care about myself and for getting me to this point; honestly, this would notbe a Physics thesis if it weren’t for you. And to every other person that has been apart of my life here at Reed, thank you for teaching me how to become who I am;you are the best teachers I could have hoped for.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Derivation Using Torques . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Lagrangian Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Non Dimensional Parameter Space . . . . . . . . . . . . . . . 101.3 The Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Chapter 2: Experimental Procedures . . . . . . . . . . . . . . . . . . . . 152.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 3: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 General Observations in Parameter Space . . . . . . . . . . . . . . . . 213.2 Specific Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Down Stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Up Stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 Down Periodic . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.4 Erratic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Increase of θcrit as ε Increases . . . . . . . . . . . . . . . . . . . . . . 25

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

List of Figures

1.1 Pendulum in an inertial and non-inertial frame: pseudo forces . . . . 41.2 Pendulum in an inertial and non-inertial frame: angles . . . . . . . . 51.3 Pendulum system for Lagrangian derivation . . . . . . . . . . . . . . 71.4 Parameter space with theoretical regions of stability . . . . . . . . . . 111.5 Graph of the effective potential and θcrit . . . . . . . . . . . . . . . . 11

2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Maximum amplitude versus frequency graph for wave driver . . . . . 172.3 Obtainable parameter space given experimental limitations . . . . . . 17

3.1 Experimental observations in parameter space . . . . . . . . . . . . . 223.2 Angle and position graphs for different pendulum behavior . . . . . . 233.3 Graph of experimental and theoretical values for θcrti . . . . . . . . . 26

Abstract

This thesis explores the equilibrium positions of a pendulum with a vertically oscil-lating pivot point, with special attention given to the inverted position. Counter tointuition, this position can be stable for this system. The equation of motion and atheoretical criteria for inverted stability are determined. A pendulum is constructedusing a mechanical wave driver to oscillate the pendulum’s pivot point. By vary-ing the pivot drive frequency and amplitude, the parameter space governing variouspendulum behavior is explored. Experimentally determined regions of stability agreewell with theoretically predicted boundary lines.

Dedication

To every woman who has ever questioned their ability to do science.

Introduction

Most people’s first instinct when asked to make something stable is to attempt toremove all motion. If you give a person a rigid rod and ask them to make it standupright, they’ll probably try to find a flat surface and work to balance it perfectly,moving it every so slightly, hoping they get it just right. If they do manage to balancethe rod, intuitively they know that the slightest push will probably knock it over. Givethe person a bit more time and more supplies and maybe they’ll glue the rod to thetable or bury part of it underground anchored in concrete. But it’s still the sameidea: attempt to remove all motion.

Though at first it may seem counterintuitive, sometimes stability can be foundthrough constant motion. This type of stability is generally deemed dynamic stability,and it will be the driving force behind this thesis. The system this thesis will studyis a pendulum with a vertically oscillating pivot point. A traditional pendulum witha stable pivot point has two equilibrium positions: a stable equilibrium point in thevertically down position and an unstable equilibrium point in the vertically up, orinverted, position. The pendulum will stay inverted only if it is balanced exactlyright, and any infinitesimal displacement to either side will cause the pendulum toturn downward. However, if the pivot point of a pendulum is forced to oscillatevertically with the right frequency and amplitude, a stable equilibrium point can beinduced in the inverted position. Once placed in the inverted position, a verticallyoscillating pendulum will stay there and can withstand minor perturbations.

The stable inverted pendulum is often called Kapitza’s pendulum after the RussianPjotr Kapitza who was the first to do an extensive experimental analysis of the systemin the 1950s [1]. Much work has been done solving the analytical solution to thesystem of a pendulum with vertically oscillating pivot since then [2–5], and othergroups have designed various apparatus to experimentally examine the system withmore rigor [6, 7]. This thesis will follow various theoretical and experimental methodsdeveloped in past papers.

The Theory section will discuss the pendulum with a vertically oscillating pivotusing three different methods, each giving different insight to the behavior of thesystem. The Experimental Procedures section will discuss the apparatus used toexperimentally verify the behavior of a pendulum at various drive amplitudes andfrequencies. The Results section will compare the experimentally observed boundariesfor various pendulum behavior with the theoretically predicted boundary lines.

Chapter 1

Theory

There are numerous methods to determine the motion of a pendulum with an os-cillating pivot, each way giving a slightly different view on the same problem. Inthis section we will explore three methods. The first will follow Butikov [3] and usetorque to intuitively explain why the inverted pendulum position is stable. The sec-ond method will determine the Lagrangian for the system and classically derive theequation of motion for the system. We will make use of work by Blackburn [2] toarrive at a parameter space with boundaries delineating stability regions for the pen-dulum. The final method will follow Landau and Lifschitz [8] and use the concept ofan effective potential to show how a stable region develops around the inverted angle.

1.1 Derivation Using Torques

To tackle the problem of the inverted pendulum using torques, we consider the systemin the non-inertial reference frame that follows the oscillating vertical axis, meaningthe frame is moving according to y = A sin(ωt). This is shown in Fig. 1.1. In thisframe, the pendulum will not be moving up and down and will only exhibit backand forth motion. Due to the nonzero acceleration of this frame, we must considerthe pseudo force of inertia. Under certain conditions, this pseudo force causes a netupward torque that works to bring the pendulum to the inverted position, as we willshow next.

The sign and magnitude of the inertial pseudo force depends on the position ofthe pivot point, y = A sin(ωt), by Fin = −my = mAω2 sin(ωt). We can see in Fig. 1.1that whether Fin is pointing up or down depends on the value of sin(ωt), meaning itdepends on whether the pivot is above or below its center position. Depending onthe magnitude of the pseudo force, the net force on the pendulum bob may actuallybe pointed upward.

But, averaged over a whole period of pivot oscillations, the inertial pseudo force is0, so this alone cannot be the reason for the pendulum bob turning upward. To fullyunderstand the action of the inertial pseudo force on the bob, we must think of theaverage torque it creates, which is not zero. We can qualitatively see from Fig. 1.2(a)that the lever arm for the period in which Fin is pointed up is larger than than when

4 Chapter 1. Theory

+y

+x

top

bottom

(a)

A sin(!t) > 0

A sin(!t) < 0

+y

F

Fin

in

+x

top

bottom

Fg

Fg

(b)

Figure 1.1: (a) The pendulum in an inertial frame with the pivot moving accordingto y = A sinωt. (b) The pendulum in a non inertial reference frame where the framemoves according to y = A sinωt. Here we include the pseudo force of inertia, Fin,which points either up or down depending on the sign of sin(ωt).

Fin is pointed down. This has to do with the position of the pivot and the variationin the angle of the pendulum bob that the pivot movement creates. We will nowdetermine this more rigorously.

We can think of the pendulum motion as broken up into two types: slow and fast.The slow motion is the large left-right rotations that a pendulum with a stationarypivot would exhibit, and the fast motion that is due to the rapidly oscillating pivot.To go along with these two types of motion we will have to consider two differenttime scales for the pendulum motion. Imagine a pendulum that is deflected fromthe vertical by an angle Θ on average. Instantaneously, this deflection angle can bebroken into the two parts, with each part created by one of the types of motion justdiscussed. There will be the slow deflection angle θ(t) caused by the left-right rotation,and the small, quickly changing angle δ(t) created by the fast pivot oscillations. SeeFig. 1.2(b 1&2). We can now write the instantaneous angle as Θ(t) = θ(t) + δ(t).Note that over one period of the fast oscillation, < δ(t) >= 0, and the slow motionremains essentially constant, so < θ(t) >= Θ. We can see from Fig. 1.2(b1) that wecan use the law of sines and the fact that δ is small to write δ(t) geometrically as:

y(t)

sin δ=

l

sin Θ=⇒ δ =

y(t)

lsin Θ =⇒ δ =

A

lsin(Θ) sin(ωt), (1.1)

where y(t) = A sin(ωt) is the motion of the pivot and l is the length to the center ofmass of the pendulum. We can then write the instantaneous deflection angle as:

Θ(t) = θ(t) + δ(t) = θ(t)− A

lsin(θ) sin(ωt). (1.2)

Since δ is small, we can say that

sin(Θ) = sin(θ + δ) = sin(θ) cos(δ) + sin(δ) cos(θ) = sin(θ) + δ cos(θ). (1.3)

1.1. Derivation Using Torques 5

+y

+x

+y

TOP

BOTTOM

FF

in

in

+x

top

top

bottom

bottom

Lever Arms

Fg

Fg

+y

+x

+y

+x

l

y(t)

(a1) (b1)

(t)

(t)

AveragePosition

Inertial(Laboratory)

Frame

(b2)(a2)

Non-Inertial(Co-Oscillating)

Frame

Figure 1.2: (a1) The pendulum in the inertial laboratory frame. In (a2) we haveswitched to the non-inertial frame that co-oscillates with the pivot oscillations. Qual-itatively we can see that the lever arm for when the pendulum is at the top of thepivot stroke, where Fin is pointing upwards, is longer than when the pendulum isat the bottom of the pivot stroke. This will work to create a net torque upward onthe pendulum. (b1) and (b2) show the angle θ(t) as it oscillates about some averageposition Θ by the small angle δ(t). We can use the law of sines to rewrite δ in termsof y(t), l, and sin(ωt).

6 Chapter 1. Theory

We can now determine the average torques due to the force of gravity and the forceof inertia. For both of these torques we will be averaging over the period of smalloscillations. Over this period θ(t) is approximately constant. For the gravitationaltorque we have:

< Tg > = < Fgl sin(θ + δ) > (1.4)

= −mgl (< sin(θ) > + < δ cos(θ) >) (1.5)

= −mgl(< sin(θ) > + <

a

lsin(Θ) sin(ωt) cos(θ) >

)(1.6)

= −mgl(< sin(θ) > +

a

l< sin(Θ) >

:0

< sin(ωt) > < cos(θ) >

)(1.7)

= −mgl sin(Θ), (1.8)

where we have used the definition of δ from Eq. 1.1 and < sin θ >= sin Θ. It is seenthat Eq. 1.8 is the same torque a pendulum with a stationary pivot would experience.For the torque due to the inertial force we find:

< Tin > = < Finl sin(θ + δ) > (1.9)

= < mAω2l sin(ωt) (sin(θ) + δ cos(θ)) > (1.10)

= −mAω2l (< sin(ωt) sin(θ) > + < δ sin(ωt) cos(θ) >) (1.11)

= −mAω2l

(

:0

< sin(ωt) > < sin(θ) > + < δ sin(ωt) cos(θ) >

)(1.12)

= −mAω2l(<a

lsin(θ) sin2(ωt) cos(θ) >

)(1.13)

= −mA2ω2 < sin(θ) >< sin2(ωt) >< cos(θ) > (1.14)

= −1

2mA2ω2 sin(Θ) cos(Θ). (1.15)

In this instance, δ did not cancel since both it and Fin depend on sin(ωt). We can seefrom Eq. 1.15 that when the pendulum rises above the horizontal, Θ > π

2, the torque

due to inertia is positive, or directed upward. To determine the magnitude of thetorque needed to outweigh gravity, we can compare the two time averaged torques,Eq. 1.15 & 1.8:

−1

2mA2ω2 sin(Θ) cos(Θ) > −mgl sin(Θ) (1.16)

A2ω2 > 2gl (1.17)

Aω >√

2gl. (1.18)

If Eq. 1.18 holds, then the inverted pendulum will be stable. This criteria has aphysical interpretation. The LHS is the linear velocity of the pivot and the RHS isthe maximum free fall velocity obtained by an object dropped from a height equal tothe length of the pendulum. We can rewrite Eq. 1.18 another way in which we intro-duce two quantities, the non dimensional amplitude, ε = A

land the non dimensional

1.2. Lagrangian Derivation 7

m

+y

+x

g

+ l

m

yp(t) = A sin(!t)

Figure 1.3: Pendulum with a vertically oscillating pivot, length l, and bob of massm. Note that θ = 0 is the −y axis, or the vertically down position

frequency, Ω = ωω0

, where l is the length to the center of mass of the pendulum and

ωo =√

gl

is the natural frequency of the un driven pendulum. Using ωo, Ω and ε,Eq. 1.18 becomes:

A

l

ω

ωo>√

2 (1.19)

ε Ω >√

2. (1.20)

Eq. 1.20 gives a simple non dimensional criteria that makes it easy to identify whichdriving frequencies and amplitudes will lead to inverted stability for a given pendulum.

The above torque based explanation laid out by Butikov [5] gives good intuition asto why a stable inverted equilibrium point exists and gives the basic criteria for thatequilibrium to be achieved. But this method does not give the equation of motionfor a pendulum with oscillating pivot point. To get this we will use classical methodsand derive the Lagrangian for the system.

1.2 Lagrangian Derivation

For our derivation we consider a pendulum of length l with a bob of mass m displacedby an angle θ from the downward position. See Fig. 1.3. The pendulum is restrictedto movements in the X-Y plane. We will assume the rod is massless and the pendulumbob is a point mass, allowing us to assume the pendulum length is also the length tothe center of mass of the pendulum.

8 Chapter 1. Theory

We first determine the potential and kinetic energy of the pendulum. We knowthat the kinetic energy of a moving object is the sum of translational motion of thecenter of mass and rotational motion about that center of mass:

T =1

2m(x2 + y2

)+

1

2Icmθ

2, (1.21)

where Icm is the rotational inertia of the pendulum about the center of mass. Thepotential energy is the gravitational potential energy due to the vertical displacement:

U = mgy(t), (1.22)

where for our pendulum the pivot is oscillating according to yp(t) = A sin(ωt). Forthese calculations we have set U = 0 at the x-axis.

Using the constraint of the constant length of the pendulum, x2 + (y − yp)2 = l2,we can write x and y in terms of l and θ. This allows us to determine the functionsfor position and velocity of the pendulum bob in terms of θ and θ:

x = l sin(θ) (1.23)

y = A sin(ωt)− l cos(θ) (1.24)

x = lθ cos(θ) (1.25)

y = Aω cos(ωt) + lθ sin(θ) (1.26)

Plugging Equations 1.25 and 1.26 into Equation 1.21 we can put the kinetic energyin terms of θ and θ:

T =1

2m(l2θ2 cos2(θ) + A2ω2 cos2(ωt) + 2Aωlθ cos(ωt) sin(θ) (1.27)

+ l2θ2 sin2(θ))

+1

2Ioθ

2

T =1

2

(ml2 + Io

)θ2 +

1

2mA2ω cos2(ωt) +mAωlθ cos(ωt) sin(θ) (1.28)

T =1

2Iθ2 +

1

2mA2ω cos2(ωt) +mAωlθ cos(ωt) sin(θ), (1.29)

where we have defined I = ml2 + Io, which by the parallel axis theorem is therotational inertia about the pivot point of the pendulum. For the potential energywe plug Equation 1.24 into Equation 1.22:

U = mgA sin(ωt)−mgl cos(θ). (1.30)

Using Equations 1.29 and 1.30, the Lagrangian is:

L = T − U (1.31)

=1

2Iθ2 +

1

2mA2ω cos2(ωt) +mAωlθ cos(ωt) sin(θ) (1.32)

−mgA sin(ωt) +mgl cos(θ).

1.2. Lagrangian Derivation 9

We would like to simplify this Lagrangian, and to do so we will use the fact thattwo Lagrangians that are identical up to a total time derivative give rise to the sameequations of motion, i.e. they are equivalent:

L(q, q, t) = L(q, q, t) +d

dtF (q, t). (1.33)

We can see that the second and fourth term in Equation 1.32 have no dependence onθ or θ and so will not contribute anything to the equation of motion. This means weshould be able to find a function F such that d

dtF (q, t) will cancel out these terms.

These functions are clearly:

F1(t) = −∫ t

0

1

2mA2ω cos2(ωt) (1.34)

F2(t) =

∫ t

0

mgA sin(ωt). (1.35)

We can therefore work with the simplified equivalent Lagrangian:

L =1

2Iθ2 +mAωlθ cos(ωt) sin(θ) +mgl cos(θ). (1.36)

The second term in Eq. 1.36 has dependence on θ and so we will not be able to find afunction to completely drop it from the Lagrangian. We do, however, want to find afunction that will allow us to switch this center term for one that has no dependenceon θ. This will make solving for the equation of motion simpler and will allow us tobetter see the varying force on the system that creates the effective potential (to bediscussed in Sec. 1.3). The function that will do this is:

F3(t) = mAωl cos(ωt) cos θ. (1.37)

Plugging Eq. 1.36 & 1.37 into Eq. 1.33 gives our final Lagrangian as:

L =1

2Iθ2 −mAω2l sin(ωt) cos(θ) +mgl cos(θ) . (1.38)

We can now use Eq.1.38 to find the equation of motion for θ:

0 =d

dt

∂L∂θ− ∂L∂θ

(1.39)

0 = Iθ −mAω2l sin(ωt) sin(θ) +mgl sin(θ). (1.40)

To more accurately mimic real life, we will add a velocity dependent friction term toEq. 1.40 , where b is the damping parameter. This leaves our final equation of motionas:

Iθ + bθ +[mgl −mAω2l sin(ωt)

]sin(θ) = 0 . (1.41)

10 Chapter 1. Theory

1.2.1 Non Dimensional Parameter Space

We would like to construct a parameter space that maps the stability boundariesof the pendulum. To do this, we must make the equation of motion, Eq. 1.41, nondimensional. First we convert to a normalized timescale using the drive frequency ω,ωt→ t∗. Noting that d

dt→ ω d

dt∗, and using the definition of the natural frequency of

the pendulum, ω2o = mgl

I, we find:

Iω2 dθ2

dt∗2+ bω

dθ

dt∗+ [mgl − Amlω2 sin(t∗)] sin(θ) = 0 (1.42)

dθ2

dt∗2+

b

Iω

dθ

dt+ [

ω2o

ω2− Aω2

o

gsin(t∗)] sin(θ) = 0 (1.43)

We will rename t∗ as t and define the non dimensional quantities:

Ω =ω

ωo, Q =

Iωob

, ε =Aω2

o

g, (1.44)

where Ω is the non dimensional frequency, Q is the quality factor and ε is the nondimensional amplitude. Substituting Eqs. 1.44 into Eq. 1.43, we get:

θ +1

ΩQθ + [

1

Ω2− ε cos(t)] sin(θ) = 0 (1.45)

The behavior of the pendulum can now be determined by the three non dimen-sional factors in Eq. 1.44. Following Blackburn [2], we can argue that if our systemis sufficiently underdamped (Q > 5), we can drop the damping term allowing us tocreate a 2-D parameter space. Using the small angle approximation on Eq. 1.45 withno damping, Blackburn shows that it reduces to the form of the Matheiu function. Hegoes on to show that there are three boundary lines separating four stability regions:

ε =

√2

Ω(1.46)

ε = .45− 1.799

Ω2(1.47)

ε = .45 +1.799

Ω2, (1.48)

which are plotted in Fig. 1.4. Vertically down is stable between the Ω axis andEq. 1.47. Vertically up is stable between Eqs. 1.46 & 1.48

1.3 The Effective Potential

The Langrangian derivation gives the equation of motion of the pendulum, which wethen use to find what parameters make the inverted position stable. But by simply

1.3. The Effective Potential 11

Boundary Lines

=p

2

= .45 1.7992

= .45 + 1.7992

2

1

34

Figure 1.4: Map of parameter space with theoretical boundary lines drawn. Differentregions are numbered and arrows indicate the behavior present in that region. Inregion 1, only vertically down is stable. In region 2, both up and down are stable. Inregion 3, only up is stable. In region 4, the pendulum exhibits rotational motion.

0 p4

p2

3 p4

p 5 p4

3 p2

7 p4

2 p

(a) (b)

0.3 0.4 0.5 0.6 0.7 0.80

20

40

60

80

e Harb.L

qCriticalHdeg

reesL

(radians)

= 5

crit

Figure 1.5: (a) Graph of the effective potential, Ueff . Notice the well centered at π.Depending on the value of Ω and ε, the width and depth of this well will change. (b)Graph of how θcrit increases as ε increases. This corresponds to a widening of thepotential well. In this graph, Ω = 5.

12 Chapter 1. Theory

looking at the equation of motion, it is hard to see why inverted stability occurs. InSection 1.1 we discussed the reason why this happens in terms of the time averagedtorque on the system. We will now discuss inverted stability in terms of a timeaveraged potential, which we will call the effective potential.

A pendulum with an oscillating pivot can be thought of as a pendulum with astationary pivot but subject to a rapidly oscillating potential field, given in Landauand Lifschitz [8] as:

Ueff = Ubg +F 2

vary(t)

2mω2(1.49)

where Ubg is the background potential that does not vary with time and Fvary(t) isa time varying force that can be identified from the Lagrangian for a system. Thisforce must be varying rapidly enough that the underlying background potential canbe treated as constant over a period of the rapid oscillations of Fvary. This is a similarargument to that made in Section 1.1. There we broke the pendulum bob’s motiondown into a slow and a fast component and discussed how that motion effected theangle of the pendulum bob. This slow and the fast motion also give rise to the twopieces of the effective potential.

In our system, the background potential is the potential due to gravity, Ubg =Ug = −mgl cos(θ), and Fvary(t) can be derived from Eq. 1.38, the Lagrangian for oursystem. We know that the Lagrangian is derived from the kinetic and potential energyfor a system, with kinetic energy being dependent on the first time derivative of thespacial coordinates, and the potential being dependent on the spacial coordinates. Wecan see that in our Lagrangian, Eq. 1.38, the second and third terms are dependenton only θ, so they may be considered the potential energy terms. Only the secondterm varies with time, so this term must give rise to the time varying force we need inEq. 1.49. Since we are in polar coordinates, we have that the force in the θ directionis F = −1

lddθU where l is the radius, which in this case is the length of the pendulum.

The force associated with the second term in Eq. 1.38 is then:

Fvary = −∇Uvary (1.50)

= −1

l

d

dθ

[−mAω2l sin(ωt) cos(θ)

](1.51)

= −mAω2 sin(ωt) sin(θ). (1.52)

For the time average of Eq. 1.52 over a period of pivot oscillations, τ = 2Πω

, we get:

1.3. The Effective Potential 13

f 2vary =

1

2π

∫ τ

0

m2A2ω4 sin2(ωt) sin2(θ)dt (1.53)

=m2A2ω4 sin2(θ)

2π

∫ τ

0

sin2(ωt)dt (1.54)

=m2A2ω4 sin2(θ)

2ππ (1.55)

=m2A2ω4 sin2(θ)

2. (1.56)

Finally, plugging Eq. 1.56 into Eq. 1.49 we get for the effective potential:

Ueff = −mgl cos(θ) +1

2mω2

m2A2ω4 sin2(θ)

2(1.57)

= −mgl cos(θ) +mA2ω2 sin2(θ)

4(1.58)

Ueff = mgl

[− cos(θ) +

A2ω2

8gl(1− cos(2θ))

]. (1.59)

Fig. 1.5(a) shows a plot of the effective potential. The width of this well determinesthe minimum angle the pendulum must be at before it rises to the inverted position.We can solve for this angle analytically by taking the derivative of the potentialfunction and solving for the zeros:

U ′eff = sin(θ) +A2ω2

8gl4 sin(θ) cos(θ) = 0 (1.60)

=⇒ sin(θ)[1 +

ε2Ω2

2cos(θ)

]= 0 (1.61)

=⇒ θ = 0 , ± cos−1(− 2

ε2Ω2

), (1.62)

where we have scaled mgl to equal 1, and have substituted in Ω and ε. Zero is thevertically down position, and ± cos−1

(− 2

ε2Ω2

)are the two sides of the potential well.

We will define θcrit as measured from the vertically up position (+y axis), meaningthat as the pendulum becomes more stable, θcrit will increase. This is shown inFig. 1.5(b).

Chapter 2

Experimental Procedures

This section will discuss the problems and considerations that went into the experi-mental design of the pendulum. Ideally, the device used to oscillate the base of thependulum would have two main characteristics: (1) a wide range of finely tunable,accurate, stable frequencies, and (2) a wide range of finely tunable, accurate, stableamplitudes. Achieving both of these characteristics is difficult, and having one usu-ally means sacrificing the other. The pendulum design in this experiment follows thatof Smith and Blackburn [7] and uses a mechanical vibrator, which has a wide rangeof finely controllable frequencies but a limited amplitude range.

First, I will discuss the experimental setup and give a detailed description of thevibrator that will oscillate the pendulum pivot as well as the device used to measurethe amplitude of pivot oscillations. Next, I will discuss how the limitations of thevibrator will effect the pendulum dimensions and the regions in parameter space ableto be explored. I will conclude with a description of the experimental procedure.

2.1 Experimental Setup

The final experimental setup consisted of the pendulum, a vibrator to oscillate thebase, and a linear variable differential transformer (LVDT) to measure the amplitudeof pivot oscillations. Also used were a signal generator and amplifier to drive thevibrator and a power source and digital oscilloscope to view the output of the LVDT.1

Fig. 2.1 shows three views of the complete pendulum apparatus. The pendulumis housed in a large plexiglass frame to help with stabilization. The pendulum isstabilized through the middle and top of the plexiglass frame to reduce unwantedside to side motion in the pivot so as to keep the pivot motion in one plane. A rodextends through the top of the plexiglass frame and connects to the movable metalrod that passes through the center of the LVDT. The frame was clamped to the labtable and weighted down to reduce unwanted vibrations in the housing.

The pendulum was designed with a variable length spherical bob so the pendulumcould have a range of natural frequencies. The length range of the pendulum wasbetween .5cm and 2cm, corresponding to natural frequencies between 3.5 Hz and

1Agilent 33210A Waveform Generator, Tektronix TDS2012C Digital Oscilloscope.

16 Chapter 2. Experimental Procedures

Vibrator

To LVDT(b)

To LVDT

Vibrator30 cm

2 cm

(c)

Figure 2.1: Experimental Set Up. (a) Photo of the actual pendulum apparatus. (b)Front view diagram of pendulum apparatus. (c) Side view diagram of pendulumapparatus.

8 Hz. Exact natural frequencies at the various pendulum lengths were determinedexperimentally by video recording the oscillations of the un-driven pendulum andtracking the motion of the bob in the program Tracker2 to obtain the period of thependulum’s oscillation.

The device used to oscillate the pivot of the pendulum is the Pasco MechanicalWave Driver SF-9324. It has a maximum current load of 1 A which puts a limit on thedriving amplitude. Fig. 2.2 shows the maximum amplitude as a function of frequency.The range of amplitudes is between 1 and 9 millimeters, with a marked drop-off past20 Hz. The amplitude reduction at high frequencies is due to the inherent limitationsof the vibrator. The smaller amplitudes at lower frequencies (6–14 Hz) have to dowith the longer period of the input sine waves. The current to the vibrator is largestat the peak of the sine wave and for waves with long periods, this high current issustained for long enough that it may damage the vibrator. We therefore chosesmaller amplitudes to limit the peak current to 1 A. At higher frequencies, the inputsignal oscillation period is short enough that a high current at the peak is only amomentary spike and peak currents in excess of 1 A can be momentarily tolerated bythe vibrator as long as the RMS current stays below 1 A..

Fig. 2.3 shows the reachable parameter space for various pendulum lengths giventhe amplitude limitations of the vibrator. A pendulum with center of mass at 1.5 cmallows access to Ω is between 3.5 and 6 and ε is between .1 and .6, which is theregion we are most interested in exploring because it contains the most boundarylines between different stability regions.

To measure the amplitude of oscillations of the vibrator, we used a RoHS DC-

2Tracker is an open source video analysis software specifically designed for kinematics. It can befound here: https://www.cabrillo.edu/ dbrown/tracker/

2.1. Experimental Setup 17

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

Frequency HHzL

Am

plitu

deHc

mL

Figure 2.2: An image of the Pasco Mechanical Wave Driver used to oscillate the pivotpoint. It has a wide range of finely controllable frequencies, but a limited amplituderange. A graph of the maximum amplitude of oscillations versus frequency is alsoshown.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

e Harb.L

WHarb

.L

= 1cm

= 1.5 cm

= 2cm

Pendulum Lengths

Figure 2.3: Graph of the reachable parameter space for various pendulum lengthsgiven the amplitude limitations of the mechanical wave driver shown in Fig. 2.2.

18 Chapter 2. Experimental Procedures

EC 2000 series linear variable differential transformer. LVDTs are good at takingaccurate one dimensional distance measurements with a quick response time. Thismodel LVDT has a response time of 200 Hz, which is quick enough to capture themotion of the pendulum, which has natural frequencies under 10 Hz. The LVDTcomes with the necessary electronics built in. The sensitivity of the device is 200 mVper millimeter with a maximum possible signal output error of 25 mV, meaning anaccuracy within .125 mm .

The LVDT consists of two parts, an outer metal tube and a solid metal rod meantto move up and down inside the tube. Different positions of the rod correspondto different voltage values. To take voltage readings, the output of the LVDT wasconnected to a digital oscilloscope. Since we were only concerned with the amplitudeof oscillations and not absolute distances, there was no need to calibrate a zero voltagewith an x=0 position. To get the amplitude measurement, the peak to peak voltagevalue from a series of pivot oscillations was recorded and then converted to millimeters.The amplitude of oscillations was then one half this value.

The peak to peak voltage readings of our oscillations tended to have a fluctuationof about 40 mV that was frequency independent. This corresponds to an error ofabout 14 mV, or .07 mm. Since this is within the possible error of the LVDT, it isnot possible to determine if this fluctuation is due to a variation in the amplitude ofthe vibrator or simply a signal error.

2.2 Procedure

First a pendulum length was set and the natural frequency at this length was exper-imentally determined through obtaining the period of the un driven pendulum fromTracker. The frequencies and amplitudes needed to reach different points in parame-ter space for this natural frequency were then calculated. Most data runs were takenwith a pendulum length of slightly more than 1.5 cm, corresponding to a naturalfrequency of around 3.8 Hz. For each data run, a frequency was set (which sets Ω)and the amplitude was incrementally changed to move horizontally across parameterspace (slowly increasing ε). Observations at each point were taken qualitatively byobserving the behavior of the pendulum.

For one data run moving across space at Ω = 5, the critical angle needed for thependulum to rise to the inverted stable position was estimated. For each value of ε,a rod was held stationary in the laboratory frame and the pendulum was allowed torest on the rod as it oscillated. A video was then taken as the rod was slowly raised.The video was analyzed in tracker and the angle at which the pendulum rose off therod to the inverted state was estimated.

Videos were recorded of the behavior of the pendulum in each of the stabilityregions and then analyzed in Tracker. Position data for the bob and the pivot weredetermined and exported for further analysis in Mathematica. To obtain the anglein the co oscillating frame, the pivot position was subtracted from the bob positionand the arctangent of this new position was determined, or:

2.2. Procedure 19

θ(t) = arctan

(X(t)bob −X(t)pivotY (t)bob − Y (t)pivot

)(2.1)

Chapter 3

Results

3.1 General Observations in Parameter Space

Fig. 3.1 shows the results of a scan of the parameter space. In the scanned region,four types of behavior were observed: down stable, up stable, down periodic, anderratic behavior. Details about each type of behavior will be discussed below. Eachdot of Fig. 3.1 indicates which of these four behaviors was observed at that point inparameter space. It can be seen that the experimental observations agree well withthe theoretically predicted boundaries.

The most interesting behavior was found around the point where ε = .35 andΩ = 4, which is around the region where the three theoretically predicted behaviorregions intersect. One set of observations on the boundary line passing from up anddown being stable to only up being stable showed interesting behavior in the downposition. When manually pushing the bob to the down position, it would eithersettle into period down behavior or shift into erratic behavior that often ended upin constant rotation. Once in one of these states, it would not change to the other.This boundary line is a well know bifurcation point [2] for the inverted pendulum,so this behavior makes sense. Another set of interesting behavior was found right atthe intersection point of all three boundary lines. Here, up was not stable, but norwas down. In the down state, the pendulum would exhibit either periodic motion orerratic motion. This was the only point where no stable position was found for thependulum.

3.2 Specific Behaviors

Fig. 3.2 shows angle and position graphs for the down stable, up stable, and downperiodic states. The pendulum used had a natural period of .31 s. The pendulummotion shown in these graphs was taken at the same driving frequency, correspondingto Ω = 5. The period of the driving frequency was .059 s. The driving amplitude,corresponding to the ε value, was different for each behavior shown. The left columnshows graphs of the pendulum angle in the co-oscillating frame of the pivot point.For these graphs, −90 is vertically downward and 90 is vertically upward. Note

22 Chapter 3. Results

Ê Ê Ê

Ê Ê Ê Ê Ê

Ê Ê Ê Ê Ê

Ê Ê Ê Ê Ê ÊÊ

Ê Ê Ê Ê Ê Ê

Ê Ê Ê ÊÊ

Ê Ê Ê Ê

Ê Ê

Ê Ê

‡ ‡

‡ ‡ ‡ ‡

‡‡‡ ‡ ‡ ‡ ‡

‡ ‡ ‡ ‡ ‡

‡‡‡ ‡

‡

‡‡

Ï ÏÏ

Ï Ï

Ï Ï

Ï Ï

ÚÚ

ÙÙ

0.0 0.2 0.4 0.6 0.82

4

6

8

10

e Harb.L

WHarb

.L

= Down Stable

= Up/Down Stable

= Up Stable & Down Periodic

= Down Periodic& Erratic

= Up/Down Stable & Down Periodic

Combinationsof Observed Behaviors

Figure 3.1: Observed dynamic behavior in dimensionless parameter space spannedby the driving frequency Ω and driving amplitude ε. The combination of observedbehaviors create three distinct regions (only down stable, up and down stable, upstable and down not stable) that were well predicted by theoretical boundary lines.Two sets of interesting behavior were observed at the intersection point of the threetheoretically predicted regions.

3.2. Specific Behaviors 23

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-120

-100

-80

-60

-40

-20

0

Time HsecL

AngleHradL

-0.5 0.0 0.5 1.0

-2.6

-2.4

-2.2

-2.0

-1.8

X HcmL

YHcmL

(a)

(b)

0 1 2 3 4 5 60

50

100

150

Time HsecL

AngleHradL

-1.0 -0.5 0.0 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

X HcmL

YHcmL

0 1 2 3 4 5 6

-120

-100

-80

-60

-40

-20

0

Time HsecL

AngleHradL

(c)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-3.7

-3.6

-3.5

-3.4

-3.3

-3.2

-3.1

X HcmL

YHcmL

Down Stable

Up Stable

Down Periodic

= 5

= 5

= 5

= .24

= .35

= .38

Figure 3.2: Angle and Position Graphs: (a) Down Stable. The angle graph showsinitial oscillations of the bob decaying to approx. −90. The position graph shows thebob settling about the downward position, as shown by the denser lines in the center.(b) Up Stable. The angle graph shows initial oscillations of the bob decaying toapprox. 90. The position graph shows the bob settling about the inverted position, asshown by the denser lines in the center. (c) Down Periodic. Shows the bob oscillatingwithout decay about the −90 position. The position shows how the combination ofpendulum and pivot motion create an inverted “V” in laboratory space. Note thefairly stable line density.

24 Chapter 3. Results

the differing time scale on the angle graph for (a) from the angle graphs for (b) and(c). In general, the pendulum would become stable in the down state quicker than itwould become stable in the up state.

The position graphs show the pendulum bob as it moved in the laboratory frame.Both the pivot motion and the oscillating motion (shown in the angle graphs) canbe seen. The density of lines indicates how much time the bob spent at that spot inspace.

3.2.1 Down Stable

Down stable was defined as the pendulum bob not oscillating when in the verticallydown position. Once a stable vertically down position was reached, the bob wasmanually perturbed to see if it would return to the vertical down position. Angleand position graphs for the down stable state at Ω = 5 and ε = .24 are shown inFig. 3.2(a). On the angle graph, a line is drawn at −90. It can be seen that thebob exhibits decaying oscillations about −93. This slight difference from −90 isthought to be due to a slight bias in the setup. The period of large oscillations is.41 ± .2 s, which is longer than the natural period of the pendulum from this run,which was .31 s. The period of small oscillations was roughly .050± .053 s, which isslightly shorter than the driving period of .059 s.

The position graph shows the bob settling to the vertically down position. Thedensest region and final position of the bob is slightly offset to the left of the vertical.This is in agreement with the final angle of bob settling around slightly more than90.

The bob was given an unlimited amount of time to become stable, but it usuallydid so within 10 seconds. The decay rate increased as ε increased, meaning that thebob would stabilize quicker with larger amplitude pivot oscillations.

3.2.2 Up Stable

To determine the stability of this state, the bob was started in the down position andmanually lifted. If the bob eventually rose to the vertically up position and stayedthere without oscillation, it was defined as up stable. Once a stable up position wasreached, the bob was manually perturbed to see if it would return to that stable upposition. Angle and position graphs for the up stable state at Ω = 5 and ε = .35 areshown in Fig. 3.2(b). On the angle graph, a line is drawn at 90. It can be seen thatthe bob exhibits decaying oscillations about 95. This slight deviation from 90 isagain thought to be due to a slight bias in the setup. The period of large oscillationswas .39 ± .2 s, which is longer than the natural period of the pendulum, which was.31 s. The period of small oscillations was roughly .070± .003 s, which is longer thanthe driving period of .059 s.

The position graph shows the bob settling to the vertically downward position.The densest region and final position of the bob is slightly offset to the left of thevertical. The is in agreement with the final angle of bob settling around slightly morethan 90.

3.3. Increase of θcrit as ε Increases 25

3.2.3 Down Periodic

Down periodic was defined as some repeated motion about the downward vertical.Angle and position graphs of this motion taken at Ω = 5 and ε = .38 are shown inFig. 3.2(c). The bob oscillated without decay between −50 and −130. The positiongraph shows the combination of back and forth motion of the pendulum and the upand down motion of the pivot, which together caused the bob to make an inverted“V” shape in space. Each period of pivot oscillations, brought the bob to the oppositeextreme of this “V” shape. The period of the motion as taken from the angle graphis .13± .03 s, which is roughly twice that the driving frequency.

3.2.4 Erratic

Erratic behavior was defined as behavior that was not stable and not periodic. Gen-erally, erratic behavior would end in the pendulum being in constant rotation. Thisbehavior was very hard to record and so there is no data available.

3.3 Increase of θcrit as ε Increases

Fig. 3.3 shows the increase of θcrit as ε increases. The upward trend of θcrit, whichcorresponds to an increasing range of initial angles that result in a stable invertedposition, can be seen in the experimental results. In nearly all cases, the observedangle was greater than the predicted value. This can be attributed in part to themethod of determining θcrit. A rod was held stationary in the laboratory frame andthe pendulum was allowed to rest on this rod as it oscillated. The rod was then slowlyraised, and when the pendulum rose off the rod to the vertical position, the anglewas recorded. There are two problems with this method. Since the pendulum wasmoving up and down while the rod remained stationary, there was a lot of bumpingbetween the rod and pendulum which may have given the pendulum a slight initialupwards velocity. Second, it was difficult to determine which angle should be recordedas θcrit since the pendulum would run through a a range of angles over the course ofits oscillations. It was unclear whether θcrit should be the angle when the pendulumwas at the top of the pivot position or the bottom. An improvement to this methodwould be to have the rod in the co oscillating frame.

26 Chapter 3. Results

æ

æ

æ

æ

æ

æ

æ

0.30 0.35 0.40 0.45 0.50 0.55 0.600

20

40

60

Ε Harb.L

ΘC

ritic

alHd

egre

esL

Figure 3.3: Graph of the theoretically predicted θcrit versus ε (solid line) and theexperimentally measured values (points). The general upward trend can be seen inthe experimental results.

Conclusion

This thesis experimentally explored the equilibrium states of a pendulum with a ver-tically oscillating pivot point, with a focus on determining under what parametersthe inverted position was stable. Three methods were used to theoretically determinethe equation of motion for the system and the criteria of stability for the invertedposition: one using time-averaged torque, one using Lagrange’s equation, and one us-ing the effective potential. A pendulum was constructed to experimentally verify thetheoretically predicted boundary lines separating different regions of stability. Exper-imental results agreed well with the theoretical predictions. It was also experimentallyverified that the maximum angle the pendulum can deviate from the vertical positionand still return to the vertical position increases as stability parameters increase.

Overall this thesis shows the possibility of finding stability through motion. Itmay serve as a reminder that, though often right, sometimes intuition can obscureour vision to the true range of possibilities for solving a problem.

References

[1] P. Kapitza, “Dynamic stability of a pendulum when its point of suspensionvibrates,” Sov. Phys. JETP 21, 588 (1951).

[2] J. Blackburn, H. J. T. Smith, and N. Gronbech-Jensen, “Stability and Hopfbifurcations in an inverted pendulum,” Am. J. Phys. 60, 903 (1992).

[3] E. I. Butikov, “On the dynamic stabilization of an inverted pendulum,” Am. J.Phys. 69, 755 (2001).

[4] E. Butikov, “Subharmonic resonances of the parametrically driven pendulum,”J. Phys A: Math Gen. 35, 6209 (2002).

[5] E. I. Butikov, “An improved criterion for Kapitza’s pendulum stability,” J. PhysA: Math Theor. 44, 295202 (2011).

[6] M. Michaelis, “Stroboscopic study of the inverted pendulum,” Am. J. Phys. 53,1079 (1985).

[7] H. Smith and J. Blackburn, “Experimental study of an inverted pendulum,” Am.J. Phys. 60, 909 (1992).

[8] L. Landau and E. Lifshitz, Mechanics (Oxford; New York: Pegamon Press,1976), 3rd ed.

[9] N. McLachlan, Theory and application of the Mathieu functions (Oxford: Claren-don, 1951).

[10] J. Holyst and W. Wojciechowski, “The effect of Kapitza pendulum and priceequilibrium,” Physica A 324, 388 (2003).

[11] I. Loram and M. Lakie, “Human balancing of an inverted pendulum:positioncontrol by small, ballistic-like, throw and catch movements,” J. Physiol 540(2002).