Computational Analyses on the Dynamics of the Dipping Bird

Transcription

1 Computational Analyses on the Dynamics of the Dipping Bird Sean Murray and Glenn Moynihan Supervised by Prof. Stefan Hutzler

2 Abstract A simple differential equation was designed to display the motion of the Dipping Bird. Mathematica was used to solve the equations of motion. Short, medium and long time scales were observed in the motion of the bird, and results were compared to previous models and experiment. A model for two coupled dipping birds was developed, which accurately displayed in phase, anti-phase and period doubling motion. The coupled dipping birds were investigated as a chaotic system and some chaotic behavior was found, but attempts at creating a bifurcation table failed.

3 1 Introduction The dippy bird is a simple toy that originated in China in around It was patented as a toy in the US in 1946 and has been hanging around desktops, entertaining ever since. Much of the appeal of the toy comes from its apparent portrayal as a perpetual motion machine. If provided with a glass of water it seemingly oscillates and performs successive dips for an infinite amount of time. This however is not the case. The dippy bird is in fact quite a simple thermodynamic engine which will run for a few days if its constantly being wet by a glass of water, or a few hours if the head is only wet once. The physical apparatus of the dippy bird as shown in Fig1 consists of a a top bulb covered in felt the head, a bottom bulb the body, and a tube connecting the head to the body, which protrudes some way into the body the neck. The bird is filled with a highly volatile liquid like methelyne chloride and its gas in thermal equilibrium. There is also a bar, perpendicular to the neck about halfway up the neck which provides the axis for the bird to oscillate around. Figure 1: Diagrams of the physical dipping bird in vertical and horizontal (dipping position) positions The motion of the bird begins when the felt covering the head is wet with some kind of liquid (not necessarily water). If the air surrounding the head is not 100% saturated with the liquid then spontaneous evaporation will occur. The latent heat for the evaporation will be taken from the head, this causes the gas inside the head to cool down and condense. The pressure in the head will decrease and the pressure differential between the head and the body will cause the liquid in the body to be pushed up the neck of the bird. This causes the center of mass of the bird to rise, and the bird to start to tip over. At an angle close to 90 degrees from the vertical position, the neck of the bird rises out of the liquid in the body, the liquid in the neck and head flows back down to the body and the pressure difference between had and body equalises. The bird then tips back to its initial condition and the cycle starts over again. If the bird is re-wet after each cycle by letting the beak dip into a glass of the liquid being used, say, then the motion will continue on until it has evaporated all th liquid from the glass. The motion will be periodic and the time between successive dips (the dipping period) will remain unchanged throughout the entire process. 2

4 It is also possible to wet the head of the bird just once at the beginning. The bird will continue to dip until its evaporated all the liquid off its head. However the dipping period, will not remain constant. Since the rate of evaporation is proportional to the amount of liquid on the head, the rate of evaporation will decrease. Since the same work is required to move the liquid up the neck, and the power ( from rate of evaporation) decreases, the time taken to reach the critical angle increases. Hence the dipping period increases with time. 2 Aim of the Project The aim of the project was to design a simple differential equation that captured the quantitative motion of the dipping bird. This would be compared to two papers on the bird [1], [2].One, Guemez et al. (2003) with a involved computational model for a qualitative study of the dipping bird, and one, Lorenz (2006), which measured dipping bird motion directly through experiment. There are three different time scales we tried to capture in our model, firstly the short time scale of oscillations of the dipping bird. Secondly the time scale of successive dips of the bird ( the dipping period), and thirdly the long time scale of the increase in dipping period when the bird was not constantly re-wet after each dip. The final aim of the project was to try and introduce a force to our system to try ad induce chaotic motion in the dipping period of the bird. This was attempted by coupling two dipping birds together. 3 Computational Model of the Dipping Bird In the paper by Guemez et al. (2003) the dipping bird is analyzed as a pendulum with both its moment of inertia and torque described as functions of the height of liquid in the neck of the bird. The height of liquid in the neck was a function of time. When analyzing the motion of a physical dipping bird it seemed to act as a simple pendulum, whose angle of equilibrium about which it oscillated increased in time,until it dipped and reset that angle. We therefor constructed a model which was a damped compound pendulum with a simple linear driving force. Equating the torques on the bird of gravity, dampening and our driving force to the rate of change of angular momentum we derived the equations of motion: d 2 dt 2 (ml2 θ(t)) + mgl sin(θ(t)) + bl 2 θ lf (t, θ) = 0 (1) where m is the mass of the pendulum, l is the fixed length of the pendulum, θ is the angle between the pendulum and the vertical axis, b is a damping coefficient, and F is the driving force. Then to non dimensionalise our EOM we introduced a dimensionless time variable. then subbing this into equation (1) we get τ = t g l = w 0t d dt = dτ dt d dτ = g l d 2 dt = g d 2 2 l dτ 2 d dτ 3

5 d 2 θ dτ + sin(θ) + b dθ F (τ, θ) 2 mω 0 dτ mg now we group our parameters into single control variables α = 1 mg ; β = b mω 0 = 0 (2) giving the final EOM as θ + sin(θ) + β θ f(τ, θ) = 0 (3) here f is our driving force f(τ, θ) = α (τ τ 0 ) ; θ < θ max = 0 ; θ θ max It is a sawtooth function,fig16, increasing linearly with time and resetting when a maximum angle has been reached. This describes a shifting of the equilibrium angle about which the pendulum oscillates with time, which replaces in our model, to some extent, the changes in moment of inertia and torques described by Guemez (2003). The parameter α can be used in two ways, firstly it can be kept as constant, where it controls the period of successive dips of the bird. Secondly for the situation where the bird is only wet once,α can slowly decrease with time. α = e pτ, p 1 This makes the rate of increase in force slowly decrease with time, and simulates the gradual evaporation of liquid from the birds head over time, Fig17. Using different values for our input variables (Appendix C), we ran simulations of our dipping bird and compared them with results from the Gumez and Lorenz papers. Our program solved equation (3) in Mathematica using the inbuilt numerical differential equation solver NDSolve and output values of θ and ω with time. 4 Coupled Dipping Bird Model To set up a system to investigate chaotic motion of the dipping birds we used two dipping birds, with different parameters and initial conditions coupled by a spring between the two body s, shown in Fig2. Figure 2: Two dipping bird models with different lengths, the body joined by a spring, and d is the distance between the two body s 4

6 This was designed using two coupled differential equations similar to equation (1), with an extra torque term added on the end to account for the torque applied by the spring between the two birds. d 2 dt 2 (l2 1θ 1 ) + m 1 gl 1 sin(θ 1 ) + b 1 θ 1 l 1 F 1 (t, θ 1 ) cl 1 d cos(θ 1 φ) = 0 (4) d 2 dt 2 (l2 2θ 2 ) + m 2 gl 2 sin(θ 2 ) + b 2 θ 2 l 2 F 2 (t, θ 2 ) + cl 2 d cos(φ θ 2 ) = 0 (5) Here the constants are the same as before, with c being the coupling constant, φ the angle between the spring and the horizontal, and d = d d o describes the extension of the spring where d is the distance between the two bird body s and d 0 is the distance between the two bird body s when θ 1, θ 2 = 0. we get Using the same methods as before to non dimensionalise and tidy up equations (4) and (5), θ 1 + sin(θ 1 ) + β 1 θ 1 α 1 γ 1 d cos(θ 1 φ) (6) θ 2 + sin(θ 2 ) + β 2 θ 2 α 2 γ 2 d cos(φ θ 2 ) (7) with all symbols having usual meaning and γ 1,2 = c gm 1,2 We use Mathematica to solve these simultaneous differential equations with NDSolve and analyze the results for different parameters and initial conditions. 5 Results 5.1 Results for the Single Dipping Bird Using our model, we compare results with that from papers of Guemez and Lorenz Evolution of Angle with time Using our model, we output and analyze the angular evolution of the dipping bird with respect to time. Fig 3 shows the simplest case, where the bird has an initial angle of the critical angle, and no initial angular velocity.this model also uses the case of the bird being constantly re-wet after each dip. In comparing this graph with fig..., we can observe that our model agrees quite well with the Guemez et al. (2003) model. Also by taking the angle with respect to the horizontal Fig14, we can compare our model to the experimental results of Lorenz(2006) Fig13. It is seen that the general shape of the graphs are quite similar. There are a few aspects of the graph that compare very well. The short time oscillation is visible, and has a rather unchanged period. Initially dampening occurs and the envelope surrounding the motion decreases from above and below the x axis 5

7 As the bird evolves in time, the equilibrium angle about which it oscillates increases, and the envelope stops narrowing symmetrically about the x axis θ increases to a critical angle, then falls back down and the motion repeats itself periodically t Figure 3: θ versus t, showing the angular evolution of the dipping bird with time. Simple case, with constant α, modeling bird re-wet after each dip. To model the bird in the situation where it is only wet once, we change α from a constant value to a exponentially decreasing with p=0.002, we get graph shown in Fig 4. Here we can see the dipping period of the bird increases in time over the 3 dips t Figure 4: Angular evolution of bird in time, with alpha non constant, p=0.002, modeling case where bird is only wet once. Note increase in period in successive cycles Fig 4 shows the third, long time scale as successive dips have longer periods. Over successive dips we can plot this period versus time in,shown in Fig 5, and compare it to Fig 12. 6

8 e Figure 5: The evolution of dipping period with time for p= Dipping period seems to exponentially increase in time We can see the general evolution of the dipping period seems to be exponential. This agrees quite well with the experimental results, by which we take our model of α to be a good one Problems with our Model There in one major dissimilarity between our model and the physical dipping bird. To better observe the dissimilarity, we changed the angle θ of the bird to be to the horizontal and not the vertical Fig 14, this allows us to compare it to experimental results from the Lorenz paper Fig 13. In the model, when the short term oscillations dampen away, the bird quite linearly decreases in θ till it hits the critical angle and falls back down. In the physical bird, it holds a steady angle for a long time and then, in a short time, quickly decreases angle in a non-linear way until it hits minimum angle and then falls back down. This is due to our oversimplification of our driving force to be a linear sawtooth force. While this does give a difference between our model and the physical results we feel its not overly important and that the model still captures the major characteristics of the motion of the dipping bird. 6 Coupled Dipping Birds In order to try and get chaotic motion with the dippy bird, we introduced an external driving force, through coupling the dipping bird with a second bird by means of a spring between the two body s. We set up the system as described in section 4 and analyzed the motion for different parameters of the two birds. In all coupled dipping bird models, we have kept α constant unless stated. This means we are analyzing the case where the birds are re-wet after each successive dip. 6.1 In Phase Motion To test our model was working as expected, we set the two birds to have the same physical parameters α and β and the same initial conditions. The resulting motion was expected to be the same as in the section 5 for the single bird model, for both birds. Here we plot the results in phase space showing bird1 and bird2 in Fig 6. 7

9 Figure 6: Phase space diagram of the evolution of bird 1 and 2 overlayed on each other. Both diagrams are identical showing the in phase motion of the two birds Both birds follow the same path and when this is compared to the phase space graph of the single bird model in Fig6, it is seen to be the same motion as the single bird model, Fig Antiphase Motion Another check on the model is to set the birds in anti phase motion. This is done by keeping the physical parameters of the birds the same but reverse the initial angle and angular velocity of one of the birds. The resulting phase space diagram is shown in Fig Figure 7: Phase space diagrams the evolution of bird one and bird two, when set up with initial conditions for anti-phase motion. the two diagrams are symmetrical about reflection through the origin, showing the two birds to be in an anti-phase state It can be clearly seen that the two motions are symmetrical about the origin, as is the expected result. From these two checks of phase and anti phase motion we conclude our model is accurately computing our expected setup of two coupled dipping birds. 6.3 Period Doubling We then looked at setting up our model to try and find chaotic motion. We wanted to find chaos in the dipping period of bird1 or bird2 in our coupled model, so in the following graph, we will measure the dipping period versus cycle number. Before looking for chaotic motion, we tried to find some parameters for the system where it would settle into a period doubling or tripling motion for one of the birds, like in classical chaotic systems. For the parameters in Appendix C, we managed to get bird2 to settle into a period doubling motion, shown in Fig 8. 8

10 s Figure 8: The period of bird 2 against cycle. After a few cycles, the bird can be seen to settle down into a period doubling motion It is quite clear from the diagram that bird2 has settled into a motion where its period repeats only every second cycle. this is period doubling behavior and is characteristic of chaotic systems. 6.4 Chaotic Motion To try and find chaotic motion, we varied the parameter α2 in bird2 over a small range, in 50 steps, and looked at the period of the bird over a few cycles at that value of alpha. The results for the periods of bird1 and bird2 are shown below in Fig 9 and Fig 10. All values recorded for period, are over 30 cycles, split between bird1 and bird2, at the value α2 are included. Period alpha Figure 9: Plot of dipping period of bird1 versus α2. For each value of α2 many periods are found, this shows a break in the conserved dipping period of uncoupled birds Period alpha Figure 10: Plot of dipping period of bird2 versus α2. There are also many periods for each value of alpha. There is also a trend in average period which decreases with increase in α2 9

11 The resulting periods don t show a classical bifurcation diagram, there is no continuous area of single period, and no obvious bifurcations into period doubling, or tripling. However, the graphs do show that for different values of alpha, we don t get periodic behavior in the dipping periods for both the birds. While this is not what was sought in setting up the model, it does give a somewhat nice result, that for some values of alpha, the constant dipping period of the bird is broken. Also, for different values of alpha, these seemingly random periods don t repeat. However we don t think the resulting motion is chaotic, there seem to be many closely clustered periods for both birds at most values of α2. Though there is a general decrease in dipping period of bird2, we are still reasonably happy with the model, as the desired result was to find chaotic motion in one of the birds, treating the other bird as just an external driving force. 7 Conclusions 7.1 Single Bird Model The single dipping bird model was very successful. It compared very well with both the computational model of Guemez et al. and the experimental model of Lorenz.All 3 time scales were successfully captured by our model. The dissimilarities between our model and the physical dipping bird can be explained by some oversimplification of our driving force, but this is quite a minor difference. 7.2 Coupled Dipping Bird We modeled a pair of coupled dipping birds by modeling a spring attached between two birds body s. Our model was set up with initial conditions to observe in-phase and anti-phase motion. In both instances the model correctly displayed expected behavior. We set up the model in conditions which led to period doubling for one of the birds. This was done for a small time step 10 7 and found to hold up. However in previous results, when period doubling/tripling was observed for a timescale, decreasing the timescale sometimes extinguished this behavior, while this was generally at larger time scales, it is possible our model is still too discrete and the period doubling doesn t exist. We varied a parameter, α2 of one of the birds over a small range and measured the periods observed at those values of α2, hoping to observe something similar to the classical bifurcation diagram. This was not observed, as there were no obvious bifurcations. However we did get a level of chaotic behavior, in that there were multiple periods observed for each value of α2, and they differed widely with α Further Work There is still some scope left in the project for further work in our project in both models. In the single model, a non linear driving force could be investigated to try and rectify the dissimilarity between the model and observed experimental results (Lorenz 2006). In the case of the coupled dipping bird, the other parameters of our model β or γ, could have been varied and a bifurcation diagram with it as the parameter could have been investigated. 10

12 Also for when we varied α2, maybe varying it over an even shorter range, and less discretely might have led to results closer to a classical bifurcation diagram. Finally it would have been nice to build a physical coupled dipping bird system and try to observe chaotic, or at least non periodic dipping period motion. References [1] R. Lorenz, Finite-time thermodynamics of an instrumented drinking bird toy, Am. J.Phys. 74(8), (2006). [2] J. Guemez, R. Valiente, C. Fiolhais, and M. Fiolhais, Experiments with the drinking bird, Am. J. Phys.71(12), (2003). 8 Appendices A Results from Gumez et a.l and Lorenz Figure 11: Guemez et al. computational simulation of the angular evolution of the dipping bird Figure 12: Guemez et al. measured increase of dipping period with time for dipping bird enclosed in a box 11

13 Energy Figure 15: The energy for a single dipping period as in Fig 14 with time. It is seen that energy is not conserved, which is in accordance with the dipping bird as it does work. However our model doesnt show energy constantly decreasing which would be physical. This is another failing of the model. Figure 13: Lorenz measured angular evolution of the dipping bird. Angle taken with respect to the horizontal axis B Extraneous Plots These plots are either not needed for the main aims of this report, or are repeated graphs in a slightly different format t Figure 14: Simple single dipping bird model as in Fig 3 but with angle with respect to the horizontal, to better compare with Fig 13 12

14 Force Figure 16: A plot of the driving force with time for constant α Force Figure 17: A plot of the driving force with time for non constant α, with p = Figure 18: Single dipping bird evloution in phase space with constant α2 13

15 C Paramaters Table 1: Single Dipping Bird θ ω 0 0 α 0.01 β 0.12 p 0 step-size 0.01 Table 2: Coupled Dipping Bird in Phase Bird 1 Bird2 (X, Y ) - (2,0) θ ω α β γ p 0 0 step-size Table 3: Coupled Dipping Bird Anti-phase Bird 1 Bird2 (X, Y ) - (2,0) θ ω α β γ p 0 0 step-size

16 Table 4: Coupled Dipping Bird Period Doubling Bird 1 Bird2 (X, Y ) - (2,0) θ ω α β γ p 0 0 step-size Table 5: Chaotic motion Bird 1 Bird2 (X, Y ) - (2,0) θ ω α 1/ k β γ p 0 0 step-size