S TABILITY A NALYSIS OF P ENDULUM WITH V IBRATING B ASE Abstract: A simple pendulum can be unstable at the inverted position, however, it has long been known that adding a vibrating base can change the stability—making it stable at that particular position. Our analysis explores this unusual phenomenon by separating the “fast” and “slow” motion, introducing effective potential, and using the averaging technique. Mentor: Dr. Ildar Gabitov Thomas Bello Emily Huang Fabian Lopez Kellin Rumsey Tao Tao March 24, 2014
Transcript
STABILITY ANALYSIS OF PENDULUM WITH VIBRATING BASE
Abstract: A simple pendulum can be unstable at the inverted position, however, it has long been known that adding a vibrating base can change the stability—making it stable at that particular position. Our analysis explores this unusual phenomenon by separating the “fast” and “slow” motion, introducing effective potential, and using the averaging technique.
Mentor: Dr. Ildar Gabitov
Thomas Bello
Emily Huang
Fabian Lopez
Kellin Rumsey
Tao Tao
March 24, 2014
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STABILITY ANALYSIS OF PENDULUM WITH VIBRATING BASE
1. INTRODUCTION 1.1. Simple vs. Vibrating Pendulum
People maybe well acquainted with simple pendulum problems. It is stable downward vertically, and unstable at inverted position. However, when adding a vibrating base on the pivot of the simple pendulum, the system seems to be stable at the inverted position. Simple pendulum swings in a smooth motion (see figure 1a) and is often modeled as !!!!!!
+ 𝑔𝑙𝑠𝑖𝑛𝜃 = 0, where g is the gravitational acceleration, l is the length of the pendulum,
and 𝜃 is the angular displacement about downward vertical. However, adding a vibrating base would make the motion no longer smooth. It creates small but rapid oscillations on top of the swing, which adds difficulties in the modeling (See figure 1b).
Figure 1 Figure 2
1.2. Background
This subject, pendulum with vibrating base, is in fact well explored by scholars and scientists in the history. In 1908, a scientists call A. Stephenson first questioned that upper vertical position of the pendulum might be stable when the driving frequency is fast. No one could scientifically answer this unusual and counterintuitive phenomenon until almost fifty years later, a Russian physicist Pyotr Kapitza successfully analyzed the stability of this system by separating the motions into fast and slow, and introduced a new concept called Effective Potential.
2. OBJECTIVES For our purpose of midterm report, we are going to explore the stability of the inverted position by
1) Derive the Lagrangian for the vertical position 2) Find the Effective Potential using the Averaging technique 3) Analyze the stability at each stationary position
3. VARIABLES 𝑑!: Amplitude of base oscillations 𝜔: Frequency if base oscillations
𝑙: Length of the pendulum 𝜃: Counter clockwise angular displacement of pendulum
𝑔: Gravitational constant
𝑘: Kinetic energy 𝑈: Potential energy
4. EQUATION OF MOTION
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Referring to the figure 3 below, we separate the pendulum to x and y direction. The horizontal x-axis is defined as the positive to the right and the y-axis is defined as the positive to the downward. We also assume the gravitation is pointing downward and angles are zero in the y direction with positive in the counterclockwise direction. The following equations are derived according to the x and y directions:
x = 𝑙 sin 𝜃
y =𝑙 cos 𝜃 + 𝑑! sin(𝜔𝑡)
where sin(𝜔𝑡) is the motion of the vibrating base. Then, the velocity in each direction will be: Vx = ẋ = 𝜃 𝑙 cos 𝜃
Vy = ẏ= -𝜃 𝑙 sin 𝜃 + 𝑑!𝑤𝑠𝑖𝑛 (𝜔𝑡)
The velocity in each direction is the fundamental input for the later analysis in this project. 4.1. Kinetic and Potential Energy
In physics, the kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes.
K = !!𝑚 𝑉!! + 𝑉!!
Potential energy is energy stored in a system of forcefully interacting physical entities. The SL unit for measuring work and energy is the joule (J).
U = mgy
Based on the results in the Equation of Motion section:
𝐾 = !!𝑚(𝜃!𝑙! + 𝑑!
!𝜔! sin!(𝜔𝑡) − 2𝜃𝑙 (sin 𝜃)𝑑!𝜔 sin(𝜔𝑡) ) (1)
U = mg (𝑙 cos 𝜃 + 𝑑! sin(𝜔𝑡)) (2)
5. THE LAGRANGIAN AND THE EULER-LARANGE EQUATION
Now that we have derived Kinetic and Potential energy in terms of the variables for our system, we will use the Lagrangian to further analyze the motion of the Pendulum with a Vibrating Base. Where Kinetic Energy is K, and Potential Energy is U, the Lagrangian (L) is defined as follows:
𝐿 = 𝐾 − 𝑈 (3)
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5.1. Why Lagrangian
The Lagrangian is very useful for us because, in a sense, it contains everything that we need to know about the motion of our system. The Lagrangian also demonstrates a very interesting principle known in Physics as the Principle of Least Action. This principle basically states that in nature, a system will always act in a way such that the ‘Action’ of the system is minimized. It happens that the Action of the system is also defined as the area under the Lagrangian vs Time curve, and the Principle of Least Action states that the mechanics and motion of the system will be such that this area is at a minimum. The importance of this will become clearer when we introduce the Euler-Lagrange Equation.
5.2. Derivation of the Lagrangian
For now, we will derive the Lagrangian for the Pendulum with a Vibrating Base. Recall the expressions we derived for Kinetic (1) and Potential Energy (2). By inserting these into (3), we get the following expression for the Lagrangian.
This certainly is the Lagrangian, but we realize that it would be convenient if we had something a little bit simpler to work with. Fortunately for us, we can take advantage of two properties of the Lagrangian to greatly simplify our result. We can realize that for our purposes:
i) The Lagrangian does not depend on constants. L = aL
ii) The Lagrangian does not depend on functions of only time. L = L + f(t)
By pulling a length term and a mass term out of our Lagrangian (4), we are left with the following expression for the Lagrangian.
𝐿 = 𝑚𝑙(!!𝑙𝜃! + 𝑑!𝜔𝜃𝑠𝑖𝑛𝜃 𝑠𝑖𝑛 𝜔𝑡 − 𝑔𝑐𝑜𝑠𝜃 + !
!!𝜔!𝑑!! 𝑠𝑖𝑛! 𝜔𝑡 −
!!!!𝑐𝑜𝑠 𝜔𝑡 ) (5)
If we notice that mass and length are constants, and that the final two terms in the expression are functions of only time (They do not depend on the angle of the pendulum in any way), then we can use the two properties discussed above to simplify the Lagrangian. Thus (5) becomes:
𝐿 = !!𝑙𝜃! + 𝑑!𝜔𝜃𝑠𝑖𝑛𝜃 𝑠𝑖𝑛 𝜔𝑡 − 𝑔𝑐𝑜𝑠𝜃 + !
!!𝜔! (6)
This is our final version of the Lagrangian.
5.3. The Euler-Lagrange Equation
The next step is for us to use the Euler-Lagrange Equation. This equation was formulated in the 1750’s by Euler and Joseph Lagrange. Fascinatingly, the solutions to this differential equation will yield the functions for which a system is stationary. From here, stability of these stationary points can be analyzed. The Euler Lagrange Equation can be presented as follows:
𝑑𝑑𝑡𝜕𝐿𝜕𝜃
−𝜕𝐿𝜕𝜃
= 0
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or !!"
!"!!= !"
!" (5.1 & 5.2)
We can break this into three steps in order to show the mathematics behind this process. First, we will find the partial derivative of (6) with respect to θ.
!"!!= 𝑑!𝜔𝜃 sin 𝜔𝑡 𝑐𝑜𝑠𝜃 + 𝑔𝑠𝑖𝑛𝜃 (7)
Now we will take the partial derivative of (6) with respect to 𝜃.
!"!!= 𝑙𝜃 + 𝑑!𝜔 sin 𝜔𝑡 + 𝑠𝑖𝑛𝜃 (8)
And if we take the time derivative of (8) we get:
!!"
!"!!= 𝑙Ӫ + 𝑑!𝜔𝜃𝑐𝑜𝑠𝜃 sin 𝜔𝑡 + 𝑑!𝜔!𝑠𝑖𝑛𝜃𝑐𝑜𝑠 𝜔𝑡 (9)
Now we can plug (7) and (9) into the Euler-Lagrange Equation (5.2).
𝑙Ӫ + 𝑑!𝜔𝜃𝑐𝑜𝑠𝜃 sin 𝜔𝑡 + 𝑑!𝜔!𝑠𝑖𝑛𝜃𝑐𝑜𝑠 𝜔𝑡 = 𝑑!𝜔𝜃 sin 𝜔𝑡 𝑐𝑜𝑠𝜃 + 𝑔𝑠𝑖𝑛𝜃 (10)
If we notice that the cosθ terms cancel each other out, we can easily simplify (10) to get the following differential equation.
Ӫ + !!!!
!𝑐𝑜𝑠 𝜔𝑡 − !
!𝑠𝑖𝑛𝜃 = 0 (11)
This is our final form of the Euler-Lagrange Differential Equation. By solving this Differential Equation, we can find stationary points for the Pendulum with a Vibrating Base, and we can analyze these stationary points for stability. Unfortunately, it is plain to see that this Equation can not be solved easily.
6. AVERAGING METHODS AND THE EFFECTIVE POTENTIAL
From physics, it is understood that systems always tend to move towards positions of minimum potential energy. In a normal pendulum, we use this knowledge by analyzing minimum values of potential energy to determine stability points of a pendulum.
Although, as previously mentioned, the inverted pendulum with a vibrating base has similar characteristics as a normal pendulum with smooth sinusoidal motion, the small oscillations caused by the vibrating base disrupt the smooth motion of the pendulum throughout each period and cause difficulty in analyzing stabilities. In order to deal with this complication we employ averaging techniques where we take an average over the period of rapid oscillation in order to treat motion as a single, smooth function. This helps us derive what is called an effective potential.
The effective potential is by definition, a mathematical expression that combines two, often opposing, forces into a single potential. In our case, it combines the normal smooth forces that are applied in a sinusoidal pendulum with the fast forces that are created by the oscillating base. What this allows us to do is to employ the same techniques we use in a normal pendulum to determine stability conditions.
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6.1. Calculation of Effective Potential In order to find the effective potential we begin with our differential equation found using the Euler Lagrange Equation and the Lagrangian
𝜃 + !!+ !!!!
!cos(𝜔𝑡) sin (𝜃) (12)
Since the motion of the pendulum consists of the “smooth” motion of the swinging pendulum and
the “rapid” motion of the vibrating base, we separate the variable 𝜃 into two variables
𝜃 𝑡 = 𝑋 𝑡 + 𝜉(𝑡) (13)
Where 𝑋 is the “smooth” motion and 𝜉 is the small, “rapid” motion. The differential equation can then be expanded into the first order approximations:
𝑋 + 𝜉 = − !!sin 𝑋 − 𝜉 !
!cos 𝑋 − !!!!
!cos 𝜔𝑡 sin 𝑋 − 𝜉 !!!
!
!cos 𝜔𝑡 cos(𝑋) (14)
From here, we can equate the corresponding terms on the left and right sides of this equation. For the rapid motion, this is
𝜉 = −𝜉 !!cos 𝑋 − !!!!
!cos 𝜔𝑡 sin 𝑋 − 𝜉 !!!
!
!cos 𝜔𝑡 cos(𝑋) (15)
However, since we have the assumption that 𝜉 is very small compared to 𝜔, the first and last terms
on the right side of this equation can be safely neglected due to the 𝜔! in the middle term, yielding
𝜉 = − !!!!
!cos 𝜔𝑡 sin 𝑋 (16)
We then solve for 𝜉 using integration, making use of the assumption that 𝑋, the “slow” motion, remains nearly constant over the period of the rapid motion, and that all constants of integration must be zero, because the rapid motion is sinusoidal centered at zero. Thus, we have
𝜉 = − !!!!
!sin 𝑋 cos 𝜔𝑡 𝑑𝑡! = !!
!sin 𝑋 cos 𝜔𝑡 (17)
Returning to the separated differential equation, we can now remove 𝜉 and its equivalent expression found above to yield
𝑋 = − !!sin 𝑋 − 𝜉 !
!cos 𝑋 − 𝜉 !!!
!
!cos 𝜔𝑡 cos(𝑋) (18)
Now we apply the averaging technique, in which we take an average of each term over the period of the rapid movement. We can treat the slow motion as constant over this period, but the rapid movement, being sinusoidal, will average out to a constant value in the end. For now, we denote this averaging using bar notation:
𝑋 = − !!sın 𝑋 − 𝜉 !
!cos 𝑋 − 𝜉 !!!
!
!cos 𝜔𝑡 cos(𝑋) (19)
We now substitute our solved expression for 𝜉 in the last term and take advantage of the fact that the middle term only contains a factor of 𝜉 and is therefore negligibly small. This yields
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𝑋 = − !!sin 𝑋 − !!!!!
!!cos! 𝜔𝑡 cos(𝑋)sın(𝑋) (20)
This can be factored into differential form
𝑋 = − !!sin 𝑋 − !
!!!!!!
!cos 𝜔𝑡 sın 𝑋 !
!"!!!!
!cos 𝜔𝑡 sın 𝑋 (21)
which can be further reduced to
𝑋 = − !!"
− !!cos 𝑋 + !
!!!!!
!!cos! 𝜔𝑡 sın! 𝑋 (22)
The averaging allows us to substitute cos! 𝜔𝑡 = !!, since this is the average value of the square of
cosine over its period. Disregarding the abuse of notation, for simplicity in interpreting the results, we also return to using 𝜃 instead of 𝑋, and remove the averaging bars:
𝜃 = − !!"
− !!cos 𝜃 + !
! !!!!!
!!sin! 𝜃 (23)
We have obtained an equation in the same form as the general equation of motion for physical systems, which is
𝑥 = − !"!"
(24)
Where 𝑥 is the coordinate of the system and 𝑈 is the potential energy of the system. Thus, the equation we have arrived at via the averaging technique can be viewed as a sort of potential energy of the system, and the expression
𝑈!"" = − !!cos 𝜃 + !
! !!!!!
!!sin! 𝜃 (25)
is called the “effective potential” of the system.
6.2. Significance of the Effective Potential
It is imperative to note that this is not the actual potential energy of the pendulum, which depends only on gravity. Rather, because of the separation of variable and the averaging technique we employed, this equation simplifies the motion of the pendulum into a single smooth motion that acts as if its potential energy were the effective potential given by the above expression. If we substitute our previously found expression for 𝜉, we notice that
𝑈!"" = − !!cos 𝜃 + !
!𝜉! (26)
which represents the effective potential as a combination of the actual potential energy of the slow motion and the kinetic energy of the rapid motion. This gives us another way of viewing the potential energy as simply the energy that the system will have at any given point, as the vibration of the base results in a constant input of energy, as shown above, which is what creates the stability in the vertical position (for sufficient inputs of energy).
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7. STABILITY ANALYSIS
Now that we have the effective potential, we can use this to analyze the motion and stability of the system. Systems always tend to move towards positions of minimum potential energy, so by finding the minimum values of the potential energy expression, we can find the stability points. Taking the derivative and setting it equal to zero yields:
!"!"= 0 = !
!sin 𝜃 1 + !
!!!!!!
!"cos 𝜃 (27)
From this expression and its graph, we see that the critical values are:
𝜃! = 0,𝜋, cos!! !!!"!!!!! , 2𝜋 − cos!! !!!"
!!!!! (28)
All of these points represent equilibrium. 𝜃 = 0, the straight downward position, is always a minimum and therefore always stable, as is expected. The last two critical values only exist when the argument is less than 1, or
𝜔 ≥ !!"!!
(29)
If this is the case, then those two values are absolute maxima, and 𝜃 = 𝜋 becomes a local minimum of the effective potential, signifying that the straight up position is stable. Thus, the stability condition is given above, and the range of stability is between the two angles listed above.
To make things more clear, we can look at some graphs using some specific values. We can assign the variables the following values.
𝑔 = 9.8𝑚𝑠!
𝑑! = .1 𝑚
𝑙 = 1 𝑚
By plugging these values into the inequality derived above (29), we get that the frequency (ω) must be greater than or equal to 44.27 s-1 in order for a stable point to exist at θ = π. We can now graph the Effective Potential using the above values, and see how the graph changes as frequency (ω) changes.
7.1. Low Frequency
To begin with, we can look at an example where the frequency (ω) is below the critical value, and therefore the upright position will be unstable. For now: ω = 20
Graph 1.
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It is sometimes convenient to think about stability of a system, as a ball rolling down a hill. Obviously, the system will be stable at θ = 0 + 2πk, since this is the downright position of the pendulum. We are much more interested in what is happening at the upright position of our Pendulum with a Vibrating Base. In this case, the Pendulum is not vibrating fast enough, and the Pendulum is unstable. The two interior critical points, see (28), do not exist in this case. At θ = π, the system is stationary, but not stable. Let us observe however, what happens if we increase the frequency to 50 s-1.
7.2. Medium Frequency
Graph 2.
Here, the frequency is just barely past the critical value. In the next example, we will see that the system can be far ‘more stable’. This time however, the stability analysis (28) yields five critical points. The system is still arbitrarily stable at 0 and 2π, but this time the upright position (θ = π) is stable, where the range of stability can be calculated using equation (28). As long as the inverted pendulum remains within this range, it will return to the upright position. If the pendulum is pushed outside of the range, then it will return to the vertical position at the bottom (θ= 0).
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7.3. High Frequency
If we increase the frequency further, to ω = 70 for example, we can see that the range of stability increases slightly, and also the critical point at π becomes ‘more stable’.
Graph 3.
As the frequency continues to increase, the pendulum becomes ‘more stable’ at the upright position. This means that it would take a larger push in order to knock the pendulum from its upright position. This also means that, inside its range of stability, it will return to the stable position (θ = π) in less time.
8. APPLICATIONS Inverted pendulums are an integral part of many technological features we enjoy today. Although not the only example of an inverted pendulum with a vibrating base, a very common example is the Segway. Segway’s, introduced first in 2001, are a mechanically self-balanced vehicle used for transportation designed to stay upright regardless of operator. The way a Segway works is that it internally contains gyroscopic and tilt sensors that feed information to an electric motor. Similar to how a carpenter’s level works, the tilt sensor uses liquid to determine the stability of the machine and uses the sensors to send information to the electric motor about the direction the machine is being unbalanced. The electric motor then processes the information and spins the wheels at a frequency proportional to the amount the machine is being unbalanced and in the same direction of the unbalance to keep the Segway in the upright form. Segway use has expanded from personal use to commercial use for police officers and as a more modern form of transportation for physically disabled people.
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9. FUTURE ANALYSIS
We have seen, at least mechanically, that stability conditions exists for both a horizontal and an arbitrary pendulum similar to how we saw the conditions exist for the inverted pendulum. Through these observations we can safely hypothesize that regardless of where we originally assign our pendulum, there is a minimum frequency at which stability is achieved and a certain range for which this stability is maintained given the original position. Our task is to now use the same techniques as for the inverted pendulum to analyze the stability conditions for the horizontal and arbitrary pendulums and quantitatively back the hypothesis we generated from our mechanical observations. Once we have theoretical calculations for each of the three types of pendulums, we will then experimentally determine the accuracy of our calculations and explore potential sources of error. Intuitively, we expect error to occur prior to making any experimental calculation because we have made some assumptions that although are almost negligible aren’t necessarily completely true, such as the assumption that our rod is a point mass and not uniformly distributed mass. Through the experimentation we will be able to conclude just how much error actually occurred during our calculations.
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