Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions.

(a) To investigate this, find the synchronous solution $A \cos \Omega t+B \sin \Omega t$ to the generic forced oscillator

equation

(7)

$$

m y^{\prime \prime}+b y^{\prime}+k y=\cos \Omega t

$$

(b) Sketch graphs of the coefficients $A$ and $B,$ as functions of $\Omega,$ for $m=1, b=0,1,$ and $k=25$

(c) Now set $b=0$ in your formulas for $A$ and $B$ and resketch the graphs in part (b), with $m=1,$ and $k=25 .$ What happens at $\Omega=5 ?$ Notice that the amplitudes of the synchronous solutions grow without bound as $\Omega$ approaches 5 .

(d) Show directly, by substituting the form $A \cos \Omega t+$ $B \sin \Omega t$ into equation $(7),$ that when $b=0$ there are no synchronous solutions if $\Omega=\sqrt{k / m}$

(e) Verify that $(2 m \Omega)^{-1} t \sin \Omega t$ solves equation (7) when $b=0$ and $\Omega=\sqrt{k / m}$. Notice that this nonsynchronous solution grows in time, without bound.

Clearly one cannot neglect damping in analyzing an oscillator forced at resonance, because otherwise the solutions, as shown in part (e), are nonphysical. This behavior will be studied later in this chapter.