$6\mathrm{.}4.$ The drive train for a front wheel drive car is shown in Fig. $6\mathrm{.}27.$ The wheels interact directly with the ground, and if no slip occurs provide a force to accelerate or decelerate the car mass.

Figure $6\mathrm{.}27$: An automotive front$-$wheel drive.

$($a$)$ Consider the pair of drive wheels as an ideal rotary$-$to$-$linear transducer. Construct the transducer linear graph and determine the relationships among the translational force and velocity and the wheel angular velocity and torque. Show that the wheels are a power$-$conserving transducer in the ideal case.

$($b$)$ Construct a linear graph relating the torque input through the wheel axle to the linear velocity of the car, including the effects of the car mass, and assuming that the car aerodynamic drag and rolling resistance may be represented as a single equivalent linear damper.

$($c$)$ The transmission in a car is used to match the engine output to the car speed. If the transmission is a lossless gear train with a step$-$down ratio $N,$ what element would you use to represent the transmission? If it is desired to operate the engine at a constant speed of approximately $600\frac{r}{s}($approximately $1800$ rpm $)$ what are the gear ratios required for $($i$)$ traveling in town at $15\frac{m}{s}$ and $($ii$)$ highway driving at $30\frac{m}{s}?$ Assume that the tire diameter is $0\mathrm{.}75$ m $.$

$($d$)$ Consider the engine as a power limited source, which at $=0\frac{r}{s}$ has a static torque of $1000N-m$ and at a top speed of $200\frac{r}{s}$ develops no torque. Construct a model for the engine.

$($e$)$ Construct the linear graph model for the complete vehicle drive system.

$($f$)$ Derive the system state equations.