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 k of moscillation isotropic and there is a single frequency 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 the initial 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 y or y would correspond to Ax Ay y and 2 in one direction 2 in 4 2 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 Let s investigate a damped oscillator whose damping is proportional x to v or For a damped spring for example m x bx our kx equation of motion becomes resistive force spring force m x bx kx 0 Writing it to emphasize that it is homogeneous or b k For later convenience we willb substitute x x x 0 m m 2 m where is called the damping constant Large large damping As usual we will also write o k m 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 equation x t e rt is to assume a solution of the form r 2e rt 2 re rt 2 e rt 0 o which when substituted into the equation gives and after cancelling the common r 2 2 rterm o2 we 0 have what is sometimes called the auxiliary equation This reduces the solution to that of solving a quadratic in r which 2 2 2 calls for use ofr1the are r2 The two solutions quadratic 2 oequation o e r1t e r2t 2 2 o2 t 2 of The general solution C e combination o t x t C1eisr1t found C2 e r2t by ea tlinear C e and 1 2 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 let s 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 i ot C2e i ot Now consider the case of weak damping o This case is o2write i 1 easiest to visualize if2 we where 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 C1ei 1t C2 e i 1t Graphically this looks like the plot at right The oscillation damps with an envelope given by the leading term e t Thus here acts as a decay parameter o e t 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 sufficient it does not give a complete solution Fortunately and in general when the auxiliary equation 2 2 2 2 2 2 we r can o r another 2 r solution r as you 0 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 …