Double pendulum

Consider a hamionic oscillator connected to another hamionic oscillator (Fig. 5-13). Write the sum of forces on each mass, mi and m2. This is a classic problem in mechanics, closely related to the double pendulum (one pendulum suspended from another pendulum). 

For simplicity, take the specific case where ki = k2 = k. Write the matrix of force constants analogous to matrix (5-29). Diagonalize this matrix. What are the roots Discuss the motion of the double pendulum in contrast to two coupled, tethered masses (Fig. 5-1). 

Returning to the main menu and double-elieking on the eentre box entitled Pendulum IF (mamdani) will bring up the FIS rule editor, as shown in Figure A1.7. Highlighted are the anteeedents and the eonsequent of Rule 1. 

For the above polyol blend viscosity (Brookfield, ASTM D-2196) = 1500 mPa-S at 23° C. For the reaction mixture working (pot) life 20 min Gardner circular dry times [72°F, 54% relative humidity (RH)] surface dry = 1.0 h, hard dry = 2.0 h, mar free = 3.5 h. For the finished coating gloss (ASTM D-523) = 90+ at 60° impact (ASTM D-2794) = 60 in.-lb direct, 10 in.-lb reverse Tabor abrasion (ASTMD-4060,1000 g load, 1000cycles, CS-17 wheel) = 95.6 mg pendulum hardness = 180 s MEK double rubs (ASTM D4752-95, 50 double rubs) = softened. 

Figure 9.11. Drum dryers for solutions and thin slurries (Buflovak Equip. Div., Blow Knox Co.), (a) Single drum dryer with dip feed and spreader, (b) Double drum dryer with splash feed, (c) Double drum dryer with top feed, vapor hood, knives and conveyor, (d) Double drum dryer with pendulum feed, enclosed for vacuum operation.

Fig. 2. Double-drum dryer (uttiiospheriei. Dryers of this type handle a variety of food products of widely varying densities and viscosities dilute solutions, heavy liquids, or pasty materials. A number of products can be dried successfully with this kind of configuration, inasmuch as exposure to temperature above the boiling point is restricted to just a few seconds. The movable dram permits effective control over product film thickness. Feed may be hy perforated lube mrngh. pendulum, or various special configurations. (Bufiovah Division. Blau -Knox hunt dc Chemical Equipment, Inc)...

Atmospheric double drum dryer with vapor hood and pendulum feed. 

It is universal for a large class of period doubling scenarios. Physical examples of this route to chaos include the driven pendulum (Baker and Gollub (1990)) and ion traps (Blumel (1995b)). 

The purpose of this section is to illustrate the methods of Lagrangian and Hamiltonian mechanics with the help of a simple mechanical system the double pendulum. It is shown that although the equations of motion for this system look very simple, the double pendulum is a chaotic system. 

The trajectory in Fig. 3.2 is obviously very compUcated. This demonstrates that the double pendulum, even in its simpUfied version studied here, is capable of exhibiting very complicated motion. 

We will now formulate the dynamics of the double pendulum within the Hamiltonian approach. The generaUzed momenta of the double pendulum are given by... 

Fig. 3.2. Projection of a phase-space trajectory of the double pendulum on the plane.

We construct the surface of section for the double pendulum in the following way. 

We are now ready to plot a Poincare surface of section for the double pendulum. We choose E = -2 and 19 different initial conditions defined... 

Fig. 3.3. Poincare section for the double pendulum, (a) E = -2, (b) E = 2. The full lines indicate the dynamically accessible regions.

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

The salient features of the dynamics of our model molecule are best exhibited with the help of Poincare sections that have already proved useful in the analysis of the double pendulum presented in Section 3.2. Fig. 4.7 shows the rpp projection of an x = 0 surface of section of a trajectory for a = 0.1, uq = 10 and E = 4 started at 0 = 0.957T, x = sin(0), y = 0, z = cos(0) and tj = 1.42. The resulting y-p Poincare section clearly shows chaotic features. This indicates that the classical dynamics of the skeleton of the model molecule is chaotic. But the most striking feature of the model molecule is its fully chaotic quantum dynamics. This is proved by Fig. 4.8, which shows the chaotic quantum fiow of the molecule on the southern hemisphere of the Bloch sphere. Fig. 4.8 was produced in the following way. First we defined the Poincar6 section by p = 0, dp/dt > 0. Then, we ran 40 trajectories in x,y,z,r],p) space for a = 0.1, Uo = 10 and E = Q starting at the 40 different initial conditions... 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom ... 

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