The sprung parts of a vehicle weigh $9\mathrm{.}79$ kN$,$ its center of gravity is $106\mathrm{.}7$ cm behind the front axle, and the wheelbase is $228\mathrm{.}6$ cm$.$ The combined stiffness of the springs of the front suspension is $24\mathrm{.}52$ kN$/$m and that of the rear suspension is $26\mathrm{.}27$ kN$/$m $.$ The radius of gyration of the sprung parts about a horizontal transverse axis through the center of gravity is $102\mathrm{.}6$ cm $(40\mathrm{.}4$ in$.).$

$1.$Calculate the natural frequencies of the pitch and bounce motions of the vehicle body.

$2.$Determine the locations of the oscillation centers.

$3.$Determine the parameters of the front and rear quarter car models.

$4.$Calculate body bounce and wheel hop frequencies for the quarter car models.