ma x = -bv x + F rod.

Transcription

1 Notes on Dynaical Systes Dynaics is the study of change. The priary ingredients of a dynaical syste are its state and its rule of change (also soeties called the dynaic). Dynaical systes can be continuous in tie, with the rule being a differential equation, or discrete in tie, with the rule being a difference equation. The dynaical rule can be linear or nonlinear. The rule can be deterinistic or stochastic (eaning containing soe eleent of randoness). Physical dynaics refers to systes with rules of change that are derived fro the equations of otion (Newton s or otherwise). In physical dynaics, the state of the syste is usually a set of positions and velocities for each of the syste s particles. In physical dynaics, one is typically interested in predicting future states given all forces acting and soe starting inforation. 1. Exaples of Siple Physical Dynaical Systes Exaple #1: Consider a sall ass connected to a assless, rigid rod being pushed through a container of viscous fluid. For siplicity, suppose that the ass in the fluid is neutrally buoyant. That is, ignore gravity. Suppose that the rod exerts a force of constant agnitude on the ass. The situation is shown in the figure to the right. (The ends of the container are sealed and the rod fits snugly in a sall hole on the left face.) As the ass oves through the fluid it experiences a frictional force opposing its otion the agnitude of which is bv, where b is a coefficient describing the gooiness of the fluid and the crosssection of the ass and v is the ass s instantaneous speed. Suppose the ass oves only horizontally to the right and the rod pushes to the right. In a cartesian coordinate syste with x horizontal and positive to the right, Newton s nd Law for the ass is a x = -bv x + F rod. Of course, the acceleration is the second derivative of position with respect to tie. The custo in continuous tie dynaical systes is to express the dynaical rule as a first derivative of the state variables position and velocity, for physical dynaics. Thus, in this for, this proble is expressible as two first-order differential equations: v x = - b v x + F rod (1) Throughout these notes, we use the notation that a dot eans derivative with respect to tie. Note that the second equation can be solved directly for v x as a function of tie. The result can then be plugged into the first equation to get x(t). The differential equations (1) are linear in that x and v x and their derivatives appear raised to the 1 power only. Two solutions of a linear equation can be added to obtain a third solution. A general solution to the velocity differential equation can then be expressed as a su of two parts one that takes care of the forcing ter and a second that satisfies the hoogeneous equation what s left without forcing. The two parts are called particular and hoogeneous solutions, respectively. For the second equation in (1), the hoogeneous equation is v xh = - b v xh, which has a solution of the for v xh = C exp(-bt / ), where C is soe constant that depends on initial conditions. As tie goes on the Physics 4550, Fall 003 Dynaical Systes 1

2 hoogeneous part of the general solution vanishes, leaving only the particular (force dependent) part. The late-tie behavior of this equation always reduces to a particular for in which forcing and friction are in balance no atter what the initial state of otion is. In the language of dynaical systes, this latetie behavior is called an attractor of the dynaics; all initial states are attracted to it. A general solution to this proble is v x = v T + (v 0 - v T )exp(-bt / ) x = x 0 + v T t + ( b)(v 0 - v T )(1- exp(-bt / )) where v 0 is the ass s initial velocity, x 0 is its initial position, and v T, the particular (or attractor) solution, is v T = F 0 b. (We ve replaced F rod by the constant force F 0.) You can see fro these equations that as t Æ, v x Æ v T and x Æ v T t, irrespective of the ass s initial state of otion. The attracting value v T is also soeties called the terinal velocity: In the attracting state, the power input, F 0 v T, equals the power dissipated, bv T. Exaple #: Consider the sae proble as above, only now let F rod = F 0 cos(w D t). This is exactly the sae proble when w D is zero. The equations of otion for the ass on the rod becoe v x = - b v x + F 0 cos(w D t) It is natural to expect that the ass will eventually also oscillate horizontally with the sae frequency, w D, as that of the driving force. That is, because of friction, the otion of the ass will eventually attain an attracting state. The particular or attractor value of v x let s call it v xa to reind ourselves we re not talking about a general velocity cannot be written siply as v ax cos(w D t). This is because (1) the left hand side (LHS) of the velocity differential equation would not be zero (as it was in the previous exaple when the particular solution was substituted) it would be a sine, and () the right hand side (RHS) would contain only cosines. The LHS would then at ost equal the RHS only at special ties, not, as is required, at all ties. A correct particular solution can be obtained by assuing that v xa is a su of cosines and sines. This can be done copactly by setting v xa = v ax cos(w D t +f). The particular solution in the previous exaple depended on F 0 and b. To ensure that v xa is a solution, plug it into the velocity differential equation above and deterine what v ax and f ust be to ake the equation true. The algebra is a bit tedious but the results can be expressed as tan(f) = - w D b v ax F 0 = (w D ) + b () Physics 4550, Fall 003 Dynaical Systes

3 Note that when w D is set to zero (reeber, this condition is the sae as the previous exaple), v ax = F 0 b = v T (as we expect). In addition, tan(f) is zero, and so therefore is f. Thus, as we expect, when the driving force is constant (as in the previous exaple) the velocity and the driving force are exactly in phase. They aren t, however, if w D is not zero. In fact, the larger w D, the ore negative is tan(f) and the closer f is to -p /. Thus, at high driving frequencies the velocity lags behind the driving force by 90. Since the rate at which energy input fro the rod is dissipated into heat by friction is proportional to v ax, dissipation depends on driving frequency. In general, a spectru is a classification of behavior as a function of constituent frequencies. A power spectru is power dissipated as a function of the input driving frequency. For the proble considered here, (log-log) plots of dissipated power as a function of w D are shown to the right. The upper graph corresponds to a relatively sall ass, while the lower one is for a higher value. As long as (w D ) << b which is true for larger values of w D if is saller the dissipated power is approxiately constant, which eans that irrespective of drive frequency, the (velocity) response of the syste is log(driving frequency) roughly the sae. This liit is like the case of zero driving frequency. Because the input force and the velocity are in phase in this liit, the input force is always pushing the ass in the direction of its displaceent. Thus, the work done by the input force is always positive. On the other hand, when (w D ) >> b, the (velocity) response becoes uch saller. The syste response drops off sharply at high frequencies like 1/frequency. (The straight lines in the figure have slope = -.) In this liit the input force and velocity are 90 out of phase. Thus averaged over a coplete drive cycle, the input force does essentially zero work. This syste is an exaple of what is called a low band pass filter. A siple electrical analog of this syste is an AC driven LR circuit. Exaple #3: Now, iagine that the rod is replaced by an ideal, Hooke s Law spring with force constant k. In a suitable coordinate syste, the equations of otion becoe v x = -kx - b v x + F 0 cos(w D t) Soething new enters, naely the velocity differential equation now depends explicitly on the position of the ass. We can try the attractor velocity v xa = v ax cos(w D t +f) as in the previous exaple, but then we also have to use the attractor position x A = - v ax sin(w D t +f). Substitution of both into the velocity differential equation produces w D log(power) Physics 4550, Fall 003 Dynaical Systes 3

4 tan(f) = - (w D -w 0 ) bw D v ax F = 0 w D ([w D -w 0 ]) + (bw D ) (3) where w 0 = k is the so-called natural frequency of the undaped, undriven oscillator. Interestingly, fro the first equation of (3), when the driving frequency is uch less than the natural frequency tan(f) is positive. Consequently, the velocity leads the input force and thus the work done by it over a drive cycle is sall (as is the dissipation). At frequencies uch higher than the natural frequency, the velocity lags the input force and so its work (and the associated dissipation) is again sall. When the driving frequency is close to the natural frequency, however, the velocity is close to being in phase with the driving force and the work done log(driving frequency) (and the dissipation) are large. This phenoenon is called resonance. The situation is shown to the right. The upper graph corresponds to sall ass, the lower to large. A sall ass does not cause uch flexure in the spring, so is like the rigid rod exaple above. log(power) Exaple #4: The physical pendulu is a ass at the end of a rigid, assless rod. (A rod can push or pull, whereas a string can only pull. You ll see why I want that to happen in a bit.) The figure to the right shows a typical pendulu, free to swing under the influence of gravity with no friction. We assue that the pendulu oves in the (x, y) plane only. The cartesian coponents (x, y) of the ass s position are related through the constraint equation x + y = l. In the absence of friction and external driving, the forces on the ass are due to gravity (pointing down and with agnitude g) and the rod (pointing along the rod and with agnitude T). While g is a constant, T varies as the ass oves. It turns out that the variation in T is not iportant for our purposes. The dynaical rules for this proble are: x l q y T g x = v x v x = -T x y = v y v y = -g + T y The quantities T x and T y are, respectively, the x- and y-coponents of the rod force. It s easier to analyze this proble in polar coordinates (r,q) (as in the figure). The constraint equation iplies that r = l. We then have Physics 4550, Fall 003 Dynaical Systes 4

5 a tangential = lw = -g sin(q) {a centripetal = lw = -g cos(q)+ T } The first two equations are sufficient to deterine the angular velocity. In principle, the latter could then be inserted into the third equation to deterine T, if desired. As long as q reains sall, sin(q) ª q, and the two iportant dynaical equations are w = - gq l These are equivalent to a siple haronic oscillator with natural frequency given by w 0 = g / l. This is just Galileo s siple pendulu perfectly periodic with a frequency that is independent of the aplitude of the otion. When the angle is not sall, the dynaics becoes nonlinear because q appears in the sine function of the second differential equation. If we add friction and periodic driving, we get w = -w 0 sin(q)- (b )w + (F 0 l)cos(w D t) Because of the nonlinearity in (6), there are no known closed-for (e.g., sines, cosines) solutions. The best we can do is to integrate the equations nuerically. Nuerical integration often requires picking sall step sizes. But, what is eant by sall? Is 1 year sall? (It is copared with a century, but not with a second.) 1 second? (Ditto: 1 yr vs. 1 ns.) 1 nanosecond? (Ditto: 1 s vs. 1 ps.) To resolve such questions, one akes the equations to be integrated diensionless. This requires defining units of fundaental easures and is often a bit of an art. For the proble we are considering, one way of proceeding is to easure frequency in units of w 0 and tie in units of 1 w 0. We re going to reeber we are doing this without writing frequency and tie as different sybols! We are also going to introduce the diensionless quantities b = b w 0 and g = F 0 g. Then, w = -sin(q)- bw +g cos(w D t) (4) Differential equations whose solutions wiggle (or abruptly change slope) require extra care when integrating nuerically. The ethod of choice is called Fourth Order Runge-Kutta with Variable Step Size. (Don t worry about its details. You always have to look the up when you need the. The Excel sheet pendulu.xls integrates the above equations using the RK4 algorith.) Physics 4550, Fall 003 Dynaical Systes 5