Notes on the Periodically orced Harmonic Oscillator Warren Weckesser Math 38 - Differential Equations 1 The Periodically orced Harmonic Oscillator. By periodically forced harmonic oscillator, we mean the linear second order nonhomogeneous differential equation my + γy + ky = cos(ωt) (1) where m >, γ, and k >. We can solve this problem completely; the goal of these notes is to study the behavior of the solutions, and to point out some special cases. The parameter γ is the damping coefficient (also known as the coefficient of friction). We consider the cases γ = (undamped) and γ > (damped) separately. Undamped (γ = ). When γ =, we have the equation or convenience, define my + ky = cos(ωt). () ω = k m. This is the natural frequency of the undamped, unforced harmonic oscillator. To solve (), we must consider two cases: ω ω and ω = ω..1 ω ω By using the method of undetermined coefficients, we find the solution of () to be y(t) = c 1 cos(ω t) + c sin(ω t) + m(ω cos(ωt), (3) ω ) where, as usual, c 1 and c are arbitrary constants. Note that the amplitude of y p becomes larger as ω approaches ω. This suggests that something other than a purely sinusoidal function may result when ω = ω. 1
Beats. Consider the initial conditions y() = and y () =. Solving for c 1 and c gives c 1 = m(ω ω ) and c =, and thus y(t) = m(ω ω ) (cos(ωt) cos(ω t)). By using some trigonometric identities, we may rewrite this as ( ) ( ) y(t) = m(ω ω ) sin (ω ω)t (ω + ω)t sin. (4) Consider the two factors ( ) m(ω ω ) sin (ω ω)t and ( ) (ω + ω)t sin. Suppose ω ω ; then ω ω is small, and the second expression has a much higher frequency than the first. We see that the solution given in (4) is a high frequency oscillation, with an amplitude that is modulated by a low frequency oscillation. In igure 1, we consider an example where = 1, m = 1, and ω = 3. In the first three graphs, the solid lines are y(t) given by (4), and the dashed lines show the envelope or modulation of the amplitude of the solution. Note that the vertical scale is different in each graph: the amplitude increases as ω approaches ω. Remember that the solutions in igure 1 are for the special initial conditions y() = and y () =. Beats occur with other initial conditions, but the function that modulates the amplitude of the high frequency oscillation will not be a simple sine function. igure shows a solution to () with ω = 3.5, but with y() = 1 and y () =.. ω = ω : Resonance When ω = ω in (), the method of undetermined coefficients tells us that y p (t) = A cos(ω t) + B sin(ω t) can not be the particular solution, because each term also solves the homogeneous equation. In this case, we multiply the guess by t to obtain y p (t) = At cos(ω t) + Bt sin(ω t). This means that the particular solution is an oscillation whose amplitude grows linearly with t. This phenomenon is known as resonance. When we substitute y p into () and simplify, we obtain mω A sin(ω t) + mω B cos(ω t) = cos(ω t). (The terms containing a factor of t automatically cancel.) or this equation to hold for all t, we must have A = and B = /(mω ). Thus the general solution to () when ω = ω is y(t) = c 1 cos(ω t) + c sin(ω t) + mω t sin(ω t). Special Case. or the special initial conditions y() = and y () =, we obtain c 1 = c =, and so y(t) = t sin(ω t)/(ω ). This solution is the fourth graph shown in igure 1 (with = 1, m = 1 and ω = 3).
1 ω=3.5 Beats and Resonance 1 5 1 15 5 3 35 4 45 5 ω=3.3 5 1 15 5 3 35 4 45 5 5 ω=3.1 5 5 1 15 5 3 35 4 45 5 1 ω=3 (resonance) 1 5 1 15 5 3 35 4 45 5 t igure 1: Solutions to () for several values of ω. The other parameters are = 1, m = 1 and ω = 3. The initial conditions are y() = and y () =. The solid curves are the actual solutions, while the dashed lines show the envelope (or modulation) of the amplitude. Beats with Nonzero Initial Conditions 1 1 5 1 15 5 3 35 4 45 5 t igure : Solution to () for ω = 3.5, with the initial conditions y() = 1 and y () =. 3
3 Damped (γ > ). To solve (1) with the method of undetermined coefficients, we first guess Y p (t) = A cos(ωt) + B sin(ωt). We then check that neither term in Y p solves the homogeneous equation. When γ >, there are three possible forms for the homogeneous solution (underdamped, critically damped, and overdamped), but in all cases, the homogeneous solutions decay to zero as t increases, so neither term in Y p can be a solution to the homogeneous equation. So our first guess for Y p will work. Substituting this guess into (1) and choosing the coefficients to make Y p a solutions yields Y p (t) = (ω ω ) m m (ω ω ) + γ ω cos(ωt) + ωγ m (ω ω ) + γ sin(ωt) (5) ω Recall that an expression of the form A cos(ωt) + B sin(ωt) may be written as R cos(ωt δ), where R = A + B and tan δ = B/A. (R is the amplitude and δ is the phase angle.) With this, we have Y p (t) = m (ω cos(ωt δ), (6) ω ) + γ ω where The general solution to (1) is the usual ( δ = arctan ωγ m(ω ω ) y(t) = y h (t) + Y p (t). As mentioned earlier, when γ >, the homogeneous solution will decay to zero as t increases. or this reason, the homogeneous solution is sometimes called the transient solution, and Y p is called the steady state response. ). Example. Consider the initial value problem y + y + 1y = cos(t), y() = 1/, y () = 4. We have m = 1, γ =, k = 1, = 1 and ω =. We also find ω = k/m = 1. The homogeneous solution is y h (t) = c 1 e t cos(3t) + c e t sin(3t), and, using (5), we find the particular solution to be The solution to the initial value problem is Y p (t) = 3 6 cos(t) + 1 13 sin(t). y(t) = 8 13 e t cos(3t) + 14 13 e t sin(3t) + 3 6 cos(t) + 1 13 sin(t). igure 3 shows y h, Y p and y = y h + Y p. 4
.5 y h (t).5.5 1 3 4 5 6 7 8 9 1 y p (t).5.5 1 3 4 5 6 7 8 9 1 y(t).5 1 3 4 5 6 7 8 9 1 t igure 3: Graphs of y h, Y p and the full solution to (1) with parameter values m = 1, γ =, k = 1, = 1, and with initial conditions y() = 1/ and y () = 4. 3.1 Amplitude of the steady state. Let us consider the particular solution given by (6). The amplitude of the steady state is m (ω ω ) + γ ω. We see that the amplitude depends on all the parameters in the differential equation (recall that ω = k/m, so the amplitude depends on k through ω ). In particular, the amplitude of the steady state solution depends on the frequency ω of the forcing. Let s take = 1, m = 1, ω = 3, so we have 1 (ω 9) + γ ω. The amplitude as a function of ω for several values of γ is shown in igure 4. Note that as γ gets smaller, the peak in the response becomes larger, and the location of the peak approaches ω. The graph of the amplitude when γ =, for which the response blows up at ω, is included for comparison. 5
.4.35 Amplitude of the Steady State γ=1 γ=.3.5. γ=.15.1 γ=4.5 γ=8 1 3 4 5 6 7 8 9 ω igure 4: Amplitude of the steady state response as a function of ω for several values of γ. The curve associated with γ = has a vertical asymptote at ω = 3. 4 Brief Summary 1. The second order linear harmonic oscillator (damped or undamped) with sinusoidal forcing can be solved by using the method of undetermined coefficients.. In the undamped case, beats occur when the forcing frequency is close to (but not equal to) the natural frequency of the oscillator. (This is because the homogeneous solution and the particular solution are both sinusoidal functions, and their frequencies are close to each other. Whenever two sinusoidal functions with close frequencies are added, beats will occur.) 3. In the undamped case, resonance occurs when the forcing frequency is the same as the natural frequency of the oscillator. 4. In the damped case (γ > ), the homogeneous solution decays to zero as t increases, so the steady state behavior is determined by the particular solution. 5. In the damped case, the steady state behavior does not depend on the initial conditions. 6. The amplitude and phase of the steady state solution depend on all the parameters in the problem. Words to Know: harmonic oscillator, damped, undamped, resonance, beats, transient, steady state, amplitude, phase 6