= Solution
For $\phi=-\Omega$, $p_1=m(\dot x_1-R\Omega\sin\Omega t)$ and $p_2=m(\dot y_1+R\Omega\cos\Omega t)$. The equations are $\ddot x_1+\omega^2x_1=R\Omega^2\cos\Omega t$, $\ddot y_1+\omega^2y_1=R\Omega^2\sin\Omega t$. For $\Omega\ne\omega$,
$$
\boxed{x_1=A\cos\omega t+B\sin\omega t+\frac{R\Omega^2}{\omega^2-\Omega^2}\cos\Omega t,\quad
y_1=C\cos\omega t+D\sin\omega t+\frac{R\Omega^2}{\omega^2-\Omega^2}\sin\Omega t.}
$$
At <resonance>[resonance] the forced terms are $R\omega t\sin\omega t/2$ and $-R\omega t\cos\omega t/2$. Constants depend on the marble's initial position and <velocity>. A general cup orientation is obtained by multiplying the vector solution by $\mathcal R(-\alpha t)$; the physical inertial solution is unchanged by relabeling the cup axes. Resonant growth eventually invalidates the small-horizontal-motion/neglected-vertical-speed approximation.
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