4.7: Resonance
( \newcommand{\kernel}{\mathrm{null}\,}\)
Resonance occurs when the frequency of the inhomogeneous term matches the frequency of the homogeneous solution. To illustrate resonance in its simplest embodiment, we consider the second-order linear inhomogeneous ode ..x+ω20x=fcosωt,x(0)=x0,.x(0)=u0.
Our main goal is to determine what happens to the solution in the limit ω→ω0.
The homogeneous equation has characteristic equation r2+ω20=0, so that r±=±iω0. Therefore, xh(t)=c1cosω0t+c2sinω0t.
To find a particular solution, we note the absence of a first-derivative term, and simply try x(t)=Acosωt.
Upon substitution into the ode, we obtain −ω2A+ω20A=f, or A=fω20−ω2.
Therefore, xp(t)=fω20−ω2cosωt.
Our general solution is thus x(t)=c1cosω0t+c2sinω0t+fω20−ω2cosωt, with derivative .x(t)=ω0(c2cosω0t−c1sinω0t)−fωω20−ω2sinωt.
Initial conditions are satisfied when x0=c1+fω20−ω2,u0=c2ω0, so that c1=x0−fω20−ω2,c2=u0ω0.
Therefore, the solution to the ode that satisfies the initial conditions is x(t)=(x0−fω20−ω2)cosω0t+u0ω0sinω0t+fω20−ω2cosωt=x0cosω0t+u0ω0sinω0t+f(cosωt−cosω0t)ω20−ω2, where we have grouped together terms proportional to the forcing amplitude f.
Resonance occurs in the limit ω→ω0; that is, the frequency of the inhomogeneous term (the external force) matches the frequency of the homogeneous solution (the free oscillation). By L’Hospital’s rule, the limit of the term proportional to f is found by differentiating with respect to ω:
lim
At resonance, the term proportional to the amplitude f of the inhomogeneous term increases linearly with t, resulting in larger-and-larger amplitudes of oscillation for x(t). In general, if the inhomogeneous term in the differential equation is a solution of the corresponding homogeneous differential equation, then the correct ansatz for the particular solution is a constant times the inhomogeneous term times t.
To illustrate this same example further, we return to the original ode, now assumed to be exactly at resonance, \overset{..}{x}+\omega_0^2x=f\cos\omega_0t,\nonumber and find a particular solution directly. The particular solution is the real part of the particular solution of \overset{..}{z}+\omega_0^2z=fe^{i\omega_0t}.\nonumber
If we try z_p = Ce^{i\omega_0t}, we obtain 0 = f, showing that the particular solution is not of this form. Because the inhomogeneous term is a solution of the homogeneous equation, we should take as our ansatz z_p=Ate^{i\omega_0t}.\nonumber
We have \overset{.}{z}_p=Ae^{i\omega_0 t}(1+i\omega_0t),\quad\overset{..}{z}_p=Ae^{i\omega_0 t}\left(2i\omega_0-\omega_0^2t\right);\nonumber and upon substitution into the ode \begin{aligned}\overset{..}{z}_p+\omega_0^2z_p&=Ae^{i\omega_0t}\left(2i\omega_0-\omega_0^2t\right)+\omega_0^2Ate^{i\omega_0t} \\ &=2i\omega_0Ae^{i\omega_0 t} \\ &=fe^{i\omega_0t}.\end{aligned}
Therefore, A=\frac{f}{2i\omega_0},\nonumber and \begin{aligned}x_p&=\text{Re}\left\{\frac{ft}{2i\omega_0}e^{i\omega_0t}\right\} \\ &=\frac{ft\sin\omega_0t}{2\omega_0},\end{aligned} the same result as \eqref{eq:3}.
Find a particular solution of \overset{..}{x}-3\overset{.}{x}-4x=5e^{-t}.
Solution
If we naively try the ansatz x=Ae^{-t},\nonumber and substitute this into the inhomogeneous differential equation, we obtain A+3A-4A=5,\nonumber or 0 = 5, which is clearly nonsense. Our ansatz therefore fails to find a solution. The cause of this failure is that the corresponding homogeneous equation has solution x_h=c_1e^{4t}+c_2e^{-t},\nonumber so that the inhomogeneous term 5e^{−t} is one of the solutions of the homogeneous equation. To find a particular solution, we should therefore take as our ansatz x=Ate^{-t},\nonumber with first- and second-derivatives given by \overset{.}{x}=Ae^{-t}(1-t),\quad\overset{..}{x}=Ae^{-t}(-2+t).\nonumber
Substitution into the differential equation yields Ae^{-t}(-2+t)-3Ae^{-t}(1-t)-4Ate^{-t}=5e^{-t}.\nonumber
The terms containing t cancel out of this equation, resulting in −5A = 5, or A = −1. Therefore, the particular solution is x_p=-te^{-t}.\nonumber