A schematic of a single-cylinder engine mounted on springs and a viscous damper is shown in Fig. P3.66. The crank rotates about $O$ with a constant speed $\omega$. The connecting rod vertical plane. The center of gravity of the crank is at its axis of rotation.
(a) Derive the differential equation governing the absolute vertical displacement of the engine including the inertia forces of the crank and piston. but ignoring forces due to combustion. Use an exact expression for the inertia forces in terms of $m_r, m_p, \omega$, the crank length $r$, and the connecting rod length $l$. Write the differential equation in the form of $\mathrm{Eq}$, (3.1).
(b) Since $F(t)$ is periodic, a Fourier series representation can be used. Set up, but do not evaluate, the integrals required for a Fourier series expansion for $F(t)$.
(c) Assume $r / l \& 1$. Rearrange $F(t)$ and use a binomial expansion such that
$$
F(t)=\sum_{i=1}^{\infty} a_i\left(\frac{r}{l}\right)^i
$$
(d) Truncate the preceding series after $i-3$, Use trigonometric ideatities to approximate
$$
F(t) \approx b_1 \cos \omega t+b_2 \cos 2 \omega t+b_3 \cos 3 \omega t
$$
(e) Find an approximation to the steady-state form of $x(t)$.
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