What Is the Maximum Velocity of a RWD Vehicle on a 6% Grade?

[eqm]

Homework Statement

Determine the maximum velocity attainable by a vehicle with the following conditions:

• RWD
• 6% grade
• Weight = 20 kN
• CG is 1.25 m behind front axle and 0.5 m above ground level.
• Wheel base is 2.8 m.
• Effective rolling radius of wheel = 30 cm
• Coefficient of aerodynamic drag = 0.45 with frontal area 2.3m2
• ρ = 1.25 kg/m3
  
• Engine develops peak torque at 45 kW and 4000 rpm
• The rotating inertia of the gearbox is and engine is 0.454 kgm2
• The rotating inertia of each wheel with driveline is 1.76 kgm2
• coefficient of friction between road and tire μ = 0.8

Homework Equations

Wr = (W l1cosθ + Rah + W h sinθ)/L

Max Tractive Effort = μ Wr

Ra = 1/2 ρ V2 A CD

The Attempt at a Solution

To solve this problem, I was going to first determine if the wheel torque is limited by the vehicle motor or by road adhesion, then find the maximum velocity using the limiting torque. I can get motor torque but am stuck at this point as I am not given a gear ratio to be able to convert motor torque to wheel torque.

TM = 30 P nrpm / π = 107.43 N/m

I know how to solve for everything else, its really just the the conversion to wheel torque that I'm stuck at (unless I'm approaching this problem entirely wrong).

 

[billy_joule]

To solve this problem, I was going to first determine if the wheel torque is limited by the vehicle motor or by road adhesion, then find the maximum velocity using the limiting torque. I can get motor torque but am stuck at this point as I am not given a gear ratio to be able to convert motor torque to wheel torque.

TM = 30 P nrpm / π = 107.43 N/m

I know how to solve for everything else, its really just the the conversion to wheel torque that I'm stuck at (unless I'm approaching this problem entirely wrong).

Is that the entire problem statement? It appears over-prescribed or incomplete.
I think you are expected to assume the GR will be whatever is required for the engine to be at 4k RPM at max velocity, otherwise, the problem can't be solved without choosing a random ratio and power curve.
The inertia of the rotating components doesn't matter at max speed, and neither does the coefficient of friction, COG, wheel size or wheelbase (for any reasonably realistic case).

You don't need to consider torques at all, just draw a free body, apply dynamic equilibrium and use P = Fv to find max velocity.
You could then show there is sufficient friction.