Dynamic Stabilization of an Invert Pendulum. A Thesis Presented to The Division of Mathematics and Natural Sciences Reed College

Transcription

1 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

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3 Approved for the Division (Physics) Lucas J. Illing

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5 Acknowledgements If I were to actually acknowledge all the people that got me to this point, this would be the longest section of my thesis; so I ll try to keep it short: To Asa, for teaching me what love is and getting me to Reed, both figuratively and literally. To the Animanimals, for showing me what it means to belong and teaching me that I can care. To the Farmhaus, for teaching me what it means to have a family. To the Institute, 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, for putting up with me, and to Jay, for building the pendulum and making this thesis possible. To the Badass Sparkle Princesses, for teaching me what it means to be part of a team and for being the most dependable support I ve ever had. To Phoebe, for teaching me how to love and for putting up with my all of my antics. To Erica, for teaching me what it means for others to care about me. To Pink$$$, for teaching me how to care about myself and for getting me to this point; honestly, this would not be a Physics thesis if it weren t for you. And to every other person that has been a part 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.

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7 Table of Contents Introduction Chapter 1: Theory Derivation Using Torques Lagrangian Derivation Non Dimensional Parameter Space The Effective Potential Chapter 2: Experimental Procedures Experimental Setup Procedure Chapter 3: Results General Observations in Parameter Space Specific Behaviors Down Stable Up Stable Down Periodic Erratic Increase of θ crit as ɛ Increases Conclusion References

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9 List of Figures 1.1 Pendulum in an inertial and non-inertial frame: pseudo forces Pendulum in an inertial and non-inertial frame: angles Pendulum system for Lagrangian derivation Parameter space with theoretical regions of stability Graph of the effective potential and θ crit Experimental set-up Maximum amplitude versus frequency graph for wave driver Obtainable parameter space given experimental limitations Experimental observations in parameter space Angle and position graphs for different pendulum behavior Graph of experimental and theoretical values for θ crti

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11 Abstract This thesis explores the equilibrium positions of a pendulum with a vertically oscillating pivot point, with special attention given to the inverted position. Counter to intuition, this position can be stable for this system. The equation of motion and a theoretical criteria for inverted stability are determined. A pendulum is constructed using a mechanical wave driver to oscillate the pendulum s pivot point. By varying the pivot drive frequency and amplitude, the parameter space governing various pendulum behavior is explored. Experimentally determined regions of stability agree well with theoretically predicted boundary lines.

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

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15 Introduction Most people s first instinct when asked to make something stable is to attempt to remove all motion. If you give a person a rigid rod and ask them to make it stand upright, 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 balance the rod, intuitively they know that the slightest push will probably knock it over. Give the person a bit more time and more supplies and maybe they ll glue the rod to the table or bury part of it underground anchored in concrete. But it s still the same idea: attempt to remove all motion. Though at first it may seem counterintuitive, sometimes stability can be found through 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 study is a pendulum with a vertically oscillating pivot point. A traditional pendulum with a stable pivot point has two equilibrium positions: a stable equilibrium point in the vertically down position and an unstable equilibrium point in the vertically up, or inverted, position. The pendulum will stay inverted only if it is balanced exactly right, and any infinitesimal displacement to either side will cause the pendulum to turn downward. However, if the pivot point of a pendulum is forced to oscillate vertically with the right frequency and amplitude, a stable equilibrium point can be induced in the inverted position. Once placed in the inverted position, a vertically oscillating pendulum will stay there and can withstand minor perturbations. The stable inverted pendulum is often called Kapitza s pendulum after the Russian Pjotr Kapitza who was the first to do an extensive experimental analysis of the system in the 1950s [1]. Much work has been done solving the analytical solution to the system of a pendulum with vertically oscillating pivot since then [2 5], and other groups have designed various apparatus to experimentally examine the system with more rigor [6, 7]. This thesis will follow various theoretical and experimental methods developed in past papers. The Theory section will discuss the pendulum with a vertically oscillating pivot using three different methods, each giving different insight to the behavior of the system. The Experimental Procedures section will discuss the apparatus used to experimentally verify the behavior of a pendulum at various drive amplitudes and frequencies. The Results section will compare the experimentally observed boundaries for various pendulum behavior with the theoretically predicted boundary lines.

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17 Chapter 1 Theory There are numerous methods to determine the motion of a pendulum with an oscillating pivot, each way giving a slightly different view on the same problem. In this section we will explore three methods. The first will follow Butikov [3] and use torque to intuitively explain why the inverted pendulum position is stable. The second method will determine the Lagrangian for the system and classically derive the equation of motion for the system. We will make use of work by Blackburn [2] to arrive at a parameter space with boundaries delineating stability regions for the pendulum. The final method will follow Landau and Lifschitz [8] and use the concept of an 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 system in the non-inertial reference frame that follows the oscillating vertical axis, meaning the frame is moving according to y = A sin(ωt). This is shown in Fig In this frame, the pendulum will not be moving up and down and will only exhibit back and forth motion. Due to the nonzero acceleration of this frame, we must consider the pseudo force of inertia. Under certain conditions, this pseudo force causes a net upward torque that works to bring the pendulum to the inverted position, as we will show next. The sign and magnitude of the inertial pseudo force depends on the position of the pivot point, y = A sin(ωt), by F in = mÿ = maω 2 sin(ωt). We can see in Fig. 1.1 that whether F in is pointing up or down depends on the value of sin(ωt), meaning it depends on whether the pivot is above or below its center position. Depending on the magnitude of the pseudo force, the net force on the pendulum bob may actually be pointed upward. But, averaged over a whole period of pivot oscillations, the inertial pseudo force is 0, so this alone cannot be the reason for the pendulum bob turning upward. To fully understand the action of the inertial pseudo force on the bob, we must think of the average 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 F in is pointed up is larger than than when

18 4 Chapter 1. Theory (a) +y (b) +y bottom bottom top F in F g F in top A sin(!t) > 0 A sin(!t) < 0 +x F g +x Figure 1.1: (a) The pendulum in an inertial frame with the pivot moving according to y = A sin ωt. (b) The pendulum in a non inertial reference frame where the frame moves according to y = A sin ωt. Here we include the pseudo force of inertia, F in, which points either up or down depending on the sign of sin(ωt). F in is pointed down. This has to do with the position of the pivot and the variation in the angle of the pendulum bob that the pivot movement creates. We will now determine 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 stationary pivot 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 different time scales for the pendulum motion. Imagine a pendulum that is deflected from the vertical by an angle Θ on average. Instantaneously, this deflection angle can be broken into the two parts, with each part created by one of the types of motion just discussed. 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. See Fig. 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 motion remains essentially constant, so < θ(t) >= Θ. We can see from Fig. 1.2(b1) that we can use the law of sines and the fact that δ is small to write δ(t) geometrically as: y(t) sin δ = l y(t) = δ = sin Θ = δ = A sin(θ) sin(ωt), (1.1) sin Θ l l where y(t) = A sin(ωt) is the motion of the pivot and l is the length to the center of mass of the pendulum. We can then write the instantaneous deflection angle as: Since δ is small, we can say that Θ(t) = θ(t) + δ(t) = θ(t) A l sin(θ) sin(ωt). (1.2) sin(θ) = sin(θ + δ) = sin(θ) cos(δ) + sin(δ) cos(θ) = sin(θ) + δ cos(θ). (1.3)

19 1.1. Derivation Using Torques 5 (a1) +y (b1) +y Inertial (Laboratory) Frame bottom top +x y(t) (t) (t) l +x Average Position (a2) +y (b2) +y bottom Non-Inertial (Co-Oscillating) Frame F in F g top F in +x +x F g Lever Arms BOTTOM TOP Figure 1.2: (a1) The pendulum in the inertial laboratory frame. In (a2) we have switched to the non-inertial frame that co-oscillates with the pivot oscillations. Qualitatively we can see that the lever arm for when the pendulum is at the top of the pivot stroke, where F in is pointing upwards, is longer than when the pendulum is at the bottom of the pivot stroke. This will work to create a net torque upward on the pendulum. (b1) and (b2) show the angle θ(t) as it oscillates about some average position Θ by the small angle δ(t). We can use the law of sines to rewrite δ in terms of y(t), l, and sin(ωt).

20 6 Chapter 1. Theory We can now determine the average torques due to the force of gravity and the force of inertia. For both of these torques we will be averaging over the period of small oscillations. Over this period θ(t) is approximately constant. For the gravitational torque we have: < T g > = < F g l sin(θ + δ) > (1.4) = mgl (< sin(θ) > + < δ cos(θ) >) (1.5) = mgl (< sin(θ) > + < a ) l sin(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 seen that 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: < T in > = < F in l sin(θ + δ) > (1.9) = < maω 2 l sin(ωt) (sin(θ) + δ cos(θ)) > (1.10) = maω 2 l (< sin(ωt) sin(θ) > + < δ sin(ωt) cos(θ) >) (1.11) ( ) = maω 2 l 0 < sin(ωt) > < sin(θ) > + < δ sin(ωt) cos(θ) > (1.12) ( = maω 2 l < a ) l sin(θ) sin2 (ωt) cos(θ) > (1.13) = ma 2 ω 2 < sin(θ) >< sin 2 (ωt) >< cos(θ) > (1.14) = 1 2 ma2 ω 2 sin(θ) cos(θ). (1.15) In this instance, δ did not cancel since both it and F in depend on sin(ωt). We can see from Eq that when the pendulum rises above the horizontal, Θ > π, the torque 2 due to inertia is positive, or directed upward. To determine the magnitude of the torque needed to outweigh gravity, we can compare the two time averaged torques, Eq & 1.8: 1 2 ma2 ω 2 sin(θ) cos(θ) > mgl sin(θ) (1.16) A 2 ω 2 > 2gl (1.17) Aω > 2gl. (1.18) If Eq holds, then the inverted pendulum will be stable. This criteria has a physical interpretation. The LHS is the linear velocity of the pivot and the RHS is the maximum free fall velocity obtained by an object dropped from a height equal to the length of the pendulum. We can rewrite Eq another way in which we introduce two quantities, the non dimensional amplitude, ɛ = A and the non dimensional l

21 1.2. Lagrangian Derivation 7 +y m g y p (t) =A sin(!t) +x + l m Figure 1.3: Pendulum with a vertically oscillating pivot, length l, and bob of mass m. Note that θ = 0 is the y axis, or the vertically down position frequency, Ω = ω ω 0, where l is the length to the center of mass of the pendulum and ω o = g is the natural frequency of the un driven pendulum. Using ω l o, Ω and ɛ, Eq becomes: A ω > 2 (1.19) l ω o ɛ Ω > 2. (1.20) Eq gives a simple non dimensional criteria that makes it easy to identify which driving 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 as to why a stable inverted equilibrium point exists and gives the basic criteria for that equilibrium to be achieved. But this method does not give the equation of motion for a pendulum with oscillating pivot point. To get this we will use classical methods and 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 displaced by an angle θ from the downward position. See Fig The pendulum is restricted to movements in the X-Y plane. We will assume the rod is massless and the pendulum bob is a point mass, allowing us to assume the pendulum length is also the length to the center of mass of the pendulum.

22 8 Chapter 1. Theory We first determine the potential and kinetic energy of the pendulum. We know that the kinetic energy of a moving object is the sum of translational motion of the center of mass and rotational motion about that center of mass: T = 1 2 m ( ẋ 2 + ẏ 2) I cm θ 2, (1.21) where I cm is the rotational inertia of the pendulum about the center of mass. The potential 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 y p (t) = A sin(ωt). For these calculations we have set U = 0 at the x-axis. Using the constraint of the constant length of the pendulum, x 2 + (y y p ) 2 = l 2, we can write x and y in terms of l and θ. This allows us to determine the functions for 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) ẋ = l θ cos(θ) (1.25) ẏ = Aω cos(ωt) + l θ sin(θ) (1.26) Plugging Equations 1.25 and 1.26 into Equation 1.21 we can put the kinetic energy in terms of θ and θ: T = 1 (l 2 m 2 θ2 cos 2 (θ) + A 2 ω 2 cos 2 (ωt) + 2Aωl θ cos(ωt) sin(θ) (1.27) ) + l 2 θ2 sin 2 (θ) I θ o 2 T = 1 2 ( ml 2 + I o ) θ ma2 ω cos 2 (ωt) + maωl θ cos(ωt) sin(θ) (1.28) T = 1 2 I θ ma2 ω cos 2 (ωt) + maωl θ cos(ωt) sin(θ), (1.29) where we have defined I = ml 2 + I o, which by the parallel axis theorem is the rotational inertia about the pivot point of the pendulum. For the potential energy we plug Equation 1.24 into Equation 1.22: Using Equations 1.29 and 1.30, the Lagrangian is: U = mga sin(ωt) mgl cos(θ). (1.30) L = T U (1.31) = 1 2 I θ ma2 ω cos 2 (ωt) + maωl θ cos(ωt) sin(θ) (1.32) mga sin(ωt) + mgl cos(θ).

23 1.2. Lagrangian Derivation 9 We would like to simplify this Lagrangian, and to do so we will use the fact that two Lagrangians that are identical up to a total time derivative give rise to the same equations of motion, i.e. they are equivalent: L(q, q, t) = L(q, q, t) + d F (q, t). (1.33) dt We can see that the second and fourth term in Equation 1.32 have no dependence on θ or θ and so will not contribute anything to the equation of motion. This means we should be able to find a function F such that d F (q, t) will cancel out these terms. dt These functions are clearly: t F 1 (t) = 0 F 2 (t) = 1 2 ma2 ω cos 2 (ωt) (1.34) t 0 mga sin(ωt). (1.35) We can therefore work with the simplified equivalent Lagrangian: L = 1 2 I θ 2 + maωl θ cos(ωt) sin(θ) + mgl cos(θ). (1.36) The second term in Eq has dependence on θ and so we will not be able to find a function to completely drop it from the Lagrangian. We do, however, want to find a function that will allow us to switch this center term for one that has no dependence on θ. This will make solving for the equation of motion simpler and will allow us to better see the varying force on the system that creates the effective potential (to be discussed in Sec. 1.3). The function that will do this is: F 3 (t) = maωl cos(ωt) cos θ. (1.37) Plugging Eq & 1.37 into Eq gives our final Lagrangian as: L = 1 2 I θ 2 maω 2 l 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 L L (1.39) dt θ θ 0 = I θ maω 2 l sin(ωt) sin(θ) + mgl sin(θ). (1.40) To more accurately mimic real life, we will add a velocity dependent friction term to Eq. 1.40, where b is the damping parameter. This leaves our final equation of motion as: I θ + b θ + [ mgl maω 2 l sin(ωt) ] sin(θ) = 0. (1.41)

24 10 Chapter 1. Theory Non Dimensional Parameter Space We would like to construct a parameter space that maps the stability boundaries of the pendulum. To do this, we must make the equation of motion, Eq. 1.41, non dimensional. First we convert to a normalized timescale using the drive frequency ω, ωt t. Noting that d ω d, and using the definition of the natural frequency of dt dt the pendulum, ωo 2 = mgl, we find: I Iω 2 dθ2 dθ + bω dt 2 dt + [mgl Amlω2 sin(t )] sin(θ) = 0 (1.42) dθ 2 dt + b dθ 2 Iω dt + [ω2 o ω Aω2 o 2 g sin(t )] sin(θ) = 0 (1.43) We will rename t as t and define the non dimensional quantities: Ω = ω ω o, Q = Iω o b, ɛ = Aω2 o g, (1.44) where Ω is the non dimensional frequency, Q is the quality factor and ɛ is the non dimensional amplitude. Substituting Eqs into Eq. 1.43, we get: θ + 1 ΩQ θ + [ 1 ɛ cos(t)] sin(θ) = 0 (1.45) Ω2 The behavior of the pendulum can now be determined by the three non dimensional factors in Eq Following Blackburn [2], we can argue that if our system is sufficiently underdamped (Q > 5), we can drop the damping term allowing us to create a 2-D parameter space. Using the small angle approximation on Eq with no damping, Blackburn shows that it reduces to the form of the Matheiu function. He goes on to show that there are three boundary lines separating four stability regions: ɛ = 2 Ω (1.46) ɛ = Ω 2 (1.47) ɛ = Ω, 2 (1.48) which are plotted in Fig Vertically down is stable between the Ω axis and Eq Vertically up is stable between Eqs & The Effective Potential The Langrangian derivation gives the equation of motion of the pendulum, which we then use to find what parameters make the inverted position stable. But by simply

25 1.3. The Effective Potential 11 Boundary Lines = p 2 1 = = Figure 1.4: Map of parameter space with theoretical boundary lines drawn. Different regions are numbered and arrows indicate the behavior present in that region. In region 1, only vertically down is stable. In region 2, both up and down are stable. In region 3, only up is stable. In region 4, the pendulum exhibits rotational motion. (a) crit (b) 80 q Critical HdegreesL = 5 0 p 4 p 2 3 p 4 p (radians) 5 p 4 3 p 2 7 p 4 2 p e Harb.L Figure 1.5: (a) Graph of the effective potential, U eff. Notice the well centered at π. Depending on the value of Ω and ɛ, the width and depth of this well will change. (b) Graph of how θ crit increases as ɛ increases. This corresponds to a widening of the potential well. In this graph, Ω = 5.

26 12 Chapter 1. Theory looking at the equation of motion, it is hard to see why inverted stability occurs. In Section 1.1 we discussed the reason why this happens in terms of the time averaged torque on the system. We will now discuss inverted stability in terms of a time averaged potential, which we will call the effective potential. A pendulum with an oscillating pivot can be thought of as a pendulum with a stationary pivot but subject to a rapidly oscillating potential field, given in Landau and Lifschitz [8] as: U eff = U bg + F 2 vary(t) 2mω 2 (1.49) where U bg is the background potential that does not vary with time and F vary (t) is a time varying force that can be identified from the Lagrangian for a system. This force must be varying rapidly enough that the underlying background potential can be treated as constant over a period of the rapid oscillations of F vary. This is a similar argument to that made in Section 1.1. There we broke the pendulum bob s motion down into a slow and a fast component and discussed how that motion effected the angle of the pendulum bob. This slow and the fast motion also give rise to the two pieces of the effective potential. In our system, the background potential is the potential due to gravity, U bg = U g = mgl cos(θ), and F vary (t) can be derived from Eq. 1.38, the Lagrangian for our system. We know that the Lagrangian is derived from the kinetic and potential energy for a system, with kinetic energy being dependent on the first time derivative of the spacial coordinates, and the potential being dependent on the spacial coordinates. We can see that in our Lagrangian, Eq. 1.38, the second and third terms are dependent on only θ, so they may be considered the potential energy terms. Only the second term varies with time, so this term must give rise to the time varying force we need in Eq Since we are in polar coordinates, we have that the force in the θ direction is F = 1 d U where l is the radius, which in this case is the length of the pendulum. l dθ The force associated with the second term in Eq is then: F vary = U vary (1.50) = 1 d [ maω 2 l sin(ωt) cos(θ) ] (1.51) l dθ = maω 2 sin(ωt) sin(θ). (1.52) For the time average of Eq over a period of pivot oscillations, τ = 2Π, we get: ω

27 1.3. The Effective Potential 13 fvary 2 = 1 τ 2π 0 = m2 A 2 ω 4 sin 2 (θ) 2π m 2 A 2 ω 4 sin 2 (ωt) sin 2 (θ)dt (1.53) τ 0 sin 2 (ωt)dt (1.54) = m2 A 2 ω 4 sin 2 (θ) π 2 π (1.55) = m2 A 2 ω 4 sin 2 (θ). 2 (1.56) Finally, plugging Eq into Eq we get for the effective potential: U eff = mgl cos(θ) + 1 m 2 A 2 ω 4 sin 2 (θ) 2mω 2 2 (1.57) = mgl cos(θ) + ma2 ω 2 sin 2 (θ) 4 (1.58) ] U eff = mgl [ cos(θ) + A2 ω 2 (1 cos(2θ)). 8gl (1.59) Fig. 1.5(a) shows a plot of the effective potential. The width of this well determines the 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 potential function and solving for the zeros: U eff = sin(θ) + A2 ω 2 4 sin(θ) cos(θ) = 0 (1.60) 8gl = sin(θ) [1 + ɛ2 Ω 2 ] 2 cos(θ) = 0 (1.61) ( = θ = 0, ± cos 1 2 ), (1.62) ɛ 2 Ω 2 where we have scaled mgl to equal 1, and have substituted in Ω and ɛ. Zero is the vertically down position, and ± cos ( ) 1 2 ɛ 2 Ω are the two sides of the potential well. 2 We will define θ crit as measured from the vertically up position (+y axis), meaning that as the pendulum becomes more stable, θ crit will increase. This is shown in Fig. 1.5(b).

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29 Chapter 2 Experimental Procedures This section will discuss the problems and considerations that went into the experimental design of the pendulum. Ideally, the device used to oscillate the base of the pendulum would have two main characteristics: (1) a wide range of finely tunable, accurate, stable frequencies, and (2) a wide range of finely tunable, accurate, stable amplitudes. Achieving both of these characteristics is difficult, and having one usually means sacrificing the other. The pendulum design in this experiment follows that of Smith and Blackburn [7] and uses a mechanical vibrator, which has a wide range of finely controllable frequencies but a limited amplitude range. First, I will discuss the experimental setup and give a detailed description of the vibrator that will oscillate the pendulum pivot as well as the device used to measure the amplitude of pivot oscillations. Next, I will discuss how the limitations of the vibrator will effect the pendulum dimensions and the regions in parameter space able to 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 the base, and a linear variable differential transformer (LVDT) to measure the amplitude of pivot oscillations. Also used were a signal generator and amplifier to drive the vibrator 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 pendulum is housed in a large plexiglass frame to help with stabilization. The pendulum is stabilized through the middle and top of the plexiglass frame to reduce unwanted side to side motion in the pivot so as to keep the pivot motion in one plane. A rod extends through the top of the plexiglass frame and connects to the movable metal rod that passes through the center of the LVDT. The frame was clamped to the lab table and weighted down to reduce unwanted vibrations in the housing. The pendulum was designed with a variable length spherical bob so the pendulum could have a range of natural frequencies. The length range of the pendulum was between.5cm and 2cm, corresponding to natural frequencies between 3.5 Hz and 1 Agilent 33210A Waveform Generator, Tektronix TDS2012C Digital Oscilloscope.

30 16 Chapter 2. Experimental Procedures (b) To LVDT To LVDT (c) 2 cm Vibrator 30 cm Vibrator 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 pendulum apparatus. 8 Hz. Exact natural frequencies at the various pendulum lengths were determined experimentally by video recording the oscillations of the un-driven pendulum and tracking the motion of the bob in the program Tracker2 to obtain the period of the pendulum s oscillation. The device used to oscillate the pivot of the pendulum is the Pasco Mechanical Wave Driver SF It has a maximum current load of 1 A which puts a limit on the driving 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 past 20 Hz. The amplitude reduction at high frequencies is due to the inherent limitations of the vibrator. The smaller amplitudes at lower frequencies (6 14 Hz) have to do with the longer period of the input sine waves. The current to the vibrator is largest at the peak of the sine wave and for waves with long periods, this high current is sustained for long enough that it may damage the vibrator. We therefore chose smaller amplitudes to limit the peak current to 1 A. At higher frequencies, the input signal oscillation period is short enough that a high current at the peak is only a momentary spike and peak currents in excess of 1 A can be momentarily tolerated by the vibrator as long as the RMS current stays below 1 A.. Fig. 2.3 shows the reachable parameter space for various pendulum lengths given the amplitude limitations of the vibrator. A pendulum with center of mass at 1.5 cm allows access to Ω is between 3.5 and 6 and is between.1 and.6, which is the region we are most interested in exploring because it contains the most boundary lines between different stability regions. To measure the amplitude of oscillations of the vibrator, we used a RoHS DC2 Tracker is an open source video analysis software specifically designed for kinematics. It can be found here: dbrown/tracker/

31 2.1. Experimental Setup Amplitude HcmL Frequency HHzL Figure 2.2: An image of the Pasco Mechanical Wave Driver used to oscillate the pivot point. It has a wide range of finely controllable frequencies, but a limited amplitude range. A graph of the maximum amplitude of oscillations versus frequency is also shown. 20 Pendulum Lengths = 1cm 15 W Harb.L = 1.5 cm = 2cm e Harb.L Figure 2.3: Graph of the reachable parameter space for various pendulum lengths given the amplitude limitations of the mechanical wave driver shown in Fig. 2.2.

32 18 Chapter 2. Experimental Procedures EC 2000 series linear variable differential transformer. LVDTs are good at taking accurate one dimensional distance measurements with a quick response time. This model LVDT has a response time of 200 Hz, which is quick enough to capture the motion of the pendulum, which has natural frequencies under 10 Hz. The LVDT comes with the necessary electronics built in. The sensitivity of the device is 200 mv per millimeter with a maximum possible signal output error of 25 mv, meaning an accuracy within.125 mm. The LVDT consists of two parts, an outer metal tube and a solid metal rod meant to move up and down inside the tube. Different positions of the rod correspond to different voltage values. To take voltage readings, the output of the LVDT was connected to a digital oscilloscope. Since we were only concerned with the amplitude of oscillations and not absolute distances, there was no need to calibrate a zero voltage with an x=0 position. To get the amplitude measurement, the peak to peak voltage value 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 fluctuation of about 40 mv that was frequency independent. This corresponds to an error of about 14 mv, or.07 mm. Since this is within the possible error of the LVDT, it is not possible to determine if this fluctuation is due to a variation in the amplitude of the vibrator or simply a signal error. 2.2 Procedure First a pendulum length was set and the natural frequency at this length was experimentally determined through obtaining the period of the un driven pendulum from Tracker. The frequencies and amplitudes needed to reach different points in parameter space for this natural frequency were then calculated. Most data runs were taken with a pendulum length of slightly more than 1.5 cm, corresponding to a natural frequency 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 parameter space (slowly increasing ɛ). Observations at each point were taken qualitatively by observing the behavior of the pendulum. For one data run moving across space at Ω = 5, the critical angle needed for the pendulum to rise to the inverted stable position was estimated. For each value of ɛ, a rod was held stationary in the laboratory frame and the pendulum was allowed to rest 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 the rod to the inverted state was estimated. Videos were recorded of the behavior of the pendulum in each of the stability regions and then analyzed in Tracker. Position data for the bob and the pivot were determined and exported for further analysis in Mathematica. To obtain the angle in the co oscillating frame, the pivot position was subtracted from the bob position and the arctangent of this new position was determined, or:

33 2.2. Procedure 19 ( ) X(t)bob X(t) pivot θ(t) = arctan Y (t) bob Y (t) pivot (2.1)

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35 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, and erratic behavior. Details about each type of behavior will be discussed below. Each dot of Fig. 3.1 indicates which of these four behaviors was observed at that point in parameter space. It can be seen that the experimental observations agree well with the theoretically predicted boundaries. The most interesting behavior was found around the point where ɛ =.35 and Ω = 4, which is around the region where the three theoretically predicted behavior regions intersect. One set of observations on the boundary line passing from up and down being stable to only up being stable showed interesting behavior in the down position. When manually pushing the bob to the down position, it would either settle into period down behavior or shift into erratic behavior that often ended up in 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 at the intersection point of all three boundary lines. Here, up was not stable, but nor was down. In the down state, the pendulum would exhibit either periodic motion or erratic motion. This was the only point where no stable position was found for the pendulum. 3.2 Specific Behaviors Fig. 3.2 shows angle and position graphs for the down stable, up stable, and down periodic states. The pendulum used had a natural period of.31 s. The pendulum motion shown in these graphs was taken at the same driving frequency, corresponding to Ω = 5. The period of the driving frequency was.059 s. The driving amplitude, corresponding to the ɛ value, was different for each behavior shown. The left column shows 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

36 22 Chapter 3. Results 10 Combinations of Observed Behaviors 8 Ê Ê = Down Stable Ê Ê = Up/Down Stable W Harb.L 6 Ê Ê Ê Ê Ê Ê Ê ÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÊÊ Ï Ï = Up Stable & Down Periodic = Up/Down Stable & Down Periodic Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ÚÚ Ï Ï Ï Ï = Down Periodic & Erratic 4 Ê Ê Ê ÙÙÏ ÏÏ e Harb.L Figure 3.1: Observed dynamic behavior in dimensionless parameter space spanned by the driving frequency Ω and driving amplitude ɛ. The combination of observed behaviors create three distinct regions (only down stable, up and down stable, up stable 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 three theoretically predicted regions.

37 3.2. Specific Behaviors 23 (a) Down Stable = 5 = Angle HradL Y HcmL Time HsecL X HcmL (b) Up Stable = 5 = Angle HradL 100 Y HcmL Time HsecL X HcmL (c) Down Periodic = 5 =.38 Angle HradL Y HcmL Time HsecL X HcmL Figure 3.2: Angle and Position Graphs: (a) Down Stable. The angle graph shows initial oscillations of the bob decaying to approx. 90. The position graph shows the bob 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 to approx. 90. The position graph shows the bob settling about the inverted position, as shown by the denser lines in the center. (c) Down Periodic. Shows the bob oscillating without decay about the 90 position. The position shows how the combination of pendulum and pivot motion create an inverted V in laboratory space. Note the fairly stable line density.

38 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 it would 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) can be seen. The density of lines indicates how much time the bob spent at that spot in space Down Stable Down stable was defined as the pendulum bob not oscillating when in the vertically down position. Once a stable vertically down position was reached, the bob was manually perturbed to see if it would return to the vertical down position. Angle and position graphs for the down stable state at Ω = 5 and ɛ =.24 are shown in Fig. 3.2(a). On the angle graph, a line is drawn at 90. It can be seen that the bob exhibits decaying oscillations about 93. This slight difference from 90 is thought 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 is slightly shorter than the driving period of.059 s. The position graph shows the bob settling to the vertically down position. The densest 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 than 90. The bob was given an unlimited amount of time to become stable, but it usually did so within 10 seconds. The decay rate increased as ɛ increased, meaning that the bob would stabilize quicker with larger amplitude pivot oscillations Up Stable To determine the stability of this state, the bob was started in the down position and manually lifted. If the bob eventually rose to the vertically up position and stayed there without oscillation, it was defined as up stable. Once a stable up position was reached, the bob was manually perturbed to see if it would return to that stable up position. Angle and position graphs for the up stable state at Ω = 5 and ɛ =.35 are shown in Fig. 3.2(b). On the angle graph, a line is drawn at 90. It can be seen that the bob exhibits decaying oscillations about 95. This slight deviation from 90 is again thought to be due to a slight bias in the setup. The period of large oscillations was.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 than the 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 the vertical. The is in agreement with the final angle of bob settling around slightly more than 90.

39 3.3. Increase of θ crit as ɛ Increases 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 ɛ =.38 are shown in Fig. 3.2(c). The bob oscillated without decay between 50 and 130. The position graph shows the combination of back and forth motion of the pendulum and the up and 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 opposite extreme of this V shape. The period of the motion as taken from the angle graph is.13 ±.03 s, which is roughly twice that the driving frequency Erratic Erratic behavior was defined as behavior that was not stable and not periodic. Generally, erratic behavior would end in the pendulum being in constant rotation. This behavior was very hard to record and so there is no data available. 3.3 Increase of θ crit as ɛ Increases Fig. 3.3 shows the increase of θ crit as ɛ increases. The upward trend of θ crit, which corresponds to an increasing range of initial angles that result in a stable inverted position, can be seen in the experimental results. In nearly all cases, the observed angle was greater than the predicted value. This can be attributed in part to the method of determining θ crit. A rod was held stationary in the laboratory frame and the pendulum was allowed to rest on this rod as it oscillated. The rod was then slowly raised, and when the pendulum rose off the rod to the vertical position, the angle was recorded. There are two problems with this method. Since the pendulum was moving up and down while the rod remained stationary, there was a lot of bumping between the rod and pendulum which may have given the pendulum a slight initial upwards velocity. Second, it was difficult to determine which angle should be recorded as θ crit since the pendulum would run through a a range of angles over the course of its oscillations. It was unclear whether θ crit should be the angle when the pendulum was at the top of the pivot position or the bottom. An improvement to this method would be to have the rod in the co oscillating frame.

40 26 Chapter 3. Results 60 Θ Critical degrees Figure 3.3: Graph of the theoretically predicted θ crit versus ɛ (solid line) and the experimentally measured values (points). The general upward trend can be seen in the experimental results.

41 Conclusion This thesis experimentally explored the equilibrium states of a pendulum with a vertically oscillating pivot point, with a focus on determining under what parameters the inverted position was stable. Three methods were used to theoretically determine the equation of motion for the system and the criteria of stability for the inverted position: one using time-averaged torque, one using Lagrange s equation, and one using the effective potential. A pendulum was constructed to experimentally verify the theoretically predicted boundary lines separating different regions of stability. Experimental results agreed well with the theoretical predictions. It was also experimentally verified that the maximum angle the pendulum can deviate from the vertical position and still return to the vertical position increases as stability parameters increase. Overall this thesis shows the possibility of finding stability through motion. It may serve as a reminder that, though often right, sometimes intuition can obscure our vision to the true range of possibilities for solving a problem.

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43 References [1] P. Kapitza, Dynamic stability of a pendulum when its point of suspension vibrates, Sov. Phys. JETP 21, 588 (1951). [2] J. Blackburn, H. J. T. Smith, and N. Gronbech-Jensen, Stability and Hopf bifurcations 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. Phys A: Math Theor. 44, (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: Clarendon, 1951). [10] J. Holyst and W. Wojciechowski, The effect of Kapitza pendulum and price equilibrium, Physica A 324, 388 (2003). [11] I. Loram and M. Lakie, Human balancing of an inverted pendulum:position control by small, ballistic-like, throw and catch movements, J. Physiol 540 (2002).

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