Physics 430: Lecture 12 2-D Oscillators and Damped
Physics 430: Lecture 12 2-D Oscillators and Damped Oscillations Dale E. Gary NJIT Physics Department 5.3 Two-Dimensional Oscillators It is trivial to extend our idea of oscillators to other dimensions. For example, the spring arrangement in the figure at right oscillates in two dimensions. In general, the springs in the x and y directions could have a different springkconstants x k y . Note that these springs may represent binding forces of an atom in a molecule or crystal. If the spring constants are the same, the oscillator is called isotropic, and there is a single frequency k of / moscillation . There are two equations of motion, one for each dimension, given 2 x x by . y 2 y Although the solutions are the same for x and y, the constants of integration, which depend theinitial x(t ) on Ax cos( t x ) conditions, and not the same (in general): . y (t ) Ay cos(t y ) October 12, 2010
Isotropic Oscillator If we redefine the origin of time to coincide with the time that, say, the x position is at its maximum, this becomes x(t ) Ax cos(t ) , y (t ) Ay cos(t ) where is the relative phase = ( y x). Consider a ball bearing in a bowl. It may oscillate in only one direction, i.e. in the x direction or the y direction. This motion would correspond to the above equations when the constant Ay = 0 or Ax = 0, respectively. The ball could go in a straight line at an angle to the x axis, i.e. in both x and y. That would correspond to Ax = Ay, and = 0. The ball could go in a circle about the bottom of the bowl, which would correspond to Ax = Ay,y and = /2, in one direction, y or = /2 in y the other direction. x x x Some other possibilities: October 12, 2010 Anisotropic Oscillator As noted before, in general, the springs in the x and y directions k x kconstants could have a different spring (How could we do y.
this in the bowl and marble case?) In that case, the oscillation frequencies would be different in the two directions x kx / m , y ky / m, and the oscillator is called anisotropic (differs depending on direction). We can easily write down the solution as: x(t ) Ax cos( x t ) . y (t ) Ay cos( y t ) You can play with a java applet to see the orbits for this case. Click here October 12, 2010 5.4 Damped Oscillations Recall when we were discussing the drag force, that we characterized it as either being proportional to v, or to v2. A drag force, or other resistive force in an oscillator leads to the oscillations dying out after awhile, a phenomenon we call damped oscillations. Lets investigate a damped oscillator whose damping is proportional x . to v, or For a damped spring, for example, mx bx our kx. equation of motion becomes resistive force spring force mx bx kx 0. Writing it to emphasize that it is homogeneous: b or k
x x x 0. For later convenience, we will substitute b m m 2 , m where is called the damping constant. Large => large damping. As usual, we will also write k / m . o October 12, 2010 Damped Oscillator Equation With these substitutions, our damped oscillator equation of motion becomes x 2 x o2 x 0. This is the starting point for our complete discussion, which will be based on the solutions to this equation in various limits. You may already know how to solve such an equation in the general case. The solution to such a linear equationrt is to assume a solution of the x(t ) e , form 2 rt rt 2 rt r e 2 re o e 0. which, when substituted into the equation, gives and after cancelling the common term, we have what is sometimes r 2 2 r o2 0.
called the auxiliary equation: This reduces the solution to that of solving a quadratic in r, which calls 2 for use of the quadratic two solutions rThe 2 o2 are: r equation. 2 2 1 o e r1t e r2t 2 The general solution isr1t foundr2tby a linear combination o2 t 2 of o2 t and x(t ) C1e C2 e e t C1e C2 e October 12, 2010 , i.e. Undamped and Weakly Damped To understand the physics captured in the general solution x(t ) e
t C e 1 2 o2 t C2 e 2 o2 t lets look at some limits. For no damping at all ( = 0), we recover the usual solution for simple harmonic motion: 2 t o2 t x(t ) C1e o C2e C1e iot C2e iot . Now consider the case of weak damping ( < o). This case is easiest to visualize if2 we where o2write i1 , 1 o2 2 . When the damping is small, we can think of 1 as a small correction to the undamped oscillation frequency o. The complete solution is x(t ) e t C1ei1t C2 e i1t . , Graphically, this looks like the plot at right. The oscillation damps with an envelope given by the leading term et. Thus, here acts as a decay parameter.
( o ) et note, oscillation frequency is slightly lower October 12, 2010 Strong Damping The general solution x(t ) e t C e 1 2 o2 t C2 e 2 o2 t , has a qualitatively different behavior in the limit of strong damping ( > o), sometimes called overdamping. In this case, the radical 2 o2 , is purely real, so we may as well leave the solution in its original form r1t r2t x(t ) C1e C2 e C1e 2 o2 t C2 e
2 o2 t . The lack of a complex exponential is a clue that there is no real oscillation involved. In fact, both terms decrease exponentially and the motion looks like: x(t) x(t) t long-term behavior decays as e initial conditions initial xo = 0, vo 0 conditions Decay parameter (slowest decay term) is x 0, v = 0 o 2 o2o. t October 12, 2010 2 o2 t Critical Damping The last limit we want to discuss is critical damping, when = o. In this case, there is a mathematical issue that arises. Now our two solutions r1 2 o2
r2 2 o2 become one solution, r1 = r2. Mathematically, we have a problem, since with only one solution, we have only one arbitrary constant, which is not sufficientit does not give a complete solution. Fortunately (and in general), when the auxiliary equation 2 2 2 we r can o2 r 2 another 2 r 2solution (r )(as 0you can easily gives a repeated rroot, find check) x(t ) te rt . The general solution is then a linear combination of our two solutions: x(t ) C1e rt C2te rt C1e t C2te t . ( o ) The graph of the solution qualitatively looks like the overdamped case, but the decay parameter ( = o) is larger (i.e. the decay is faster). In fact, in the critical damping case the decay is faster than in any other case. Obviously, if you want to keep something stable against oscillations you want to arrange for it to be critically damped. October 12, 2010 A Closer Look at Decay Parameters The decay parameters that govern the drop in amplitude at long times are:
damping decay parameter none =0 0 under < o critical = o over > o 2 o2 . This dependencedecay can be graphed as below: parameter o October 12, 2010
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