Which Gearbox Ratio is Used to Maintain Speed Up an Incline?

[enc08]

Hi,

I don't get the right answer for this question. I don't see where I'm going wrong.

A vehicle has the following specifications
Drag coefficient = 0.33
Chassis dimensions 1.52m x 1.40m x 4.20m (w x l x h)
Rolling resistance = 200N
Vehicle's mass = 1200kg
Transmission efficiency at converting engine power to vehicle motion = 96%
Gearbox ratios: 3.25:1 (first gear), 1.9:1 (second gear), 1.25:1 (third gear), 0.94:1 (fourth gear)
Final drive ratio = 4:1
Rolling radius of wheels = 0.25m

The engine power is seen to be approximately constant at its maximum value Pmax = 59.4hp over the engine speed range 3000-6000rpm. With the engine at maximum power Pmax, the vehicle can maintain a speed of 40mph when climbing a hill at 20 degrees to the horizontal.

Q) Determine which of the four gearbox ratios can be used to maintain the speed of 40mph up the incline.

Take air density to be 1.2kg/m3, 1mph = 0.447m/s, 1hp = 0.7475kW

I use this equation to get the engine rpm for each gear ratio:

engine speed (rpm) = gear ratio X final drive ratio X vehicle speed X 60 / (2*pi*rolling radius)

So for first gear I get an engine rpm of 8880rpm
Second gear = 5190rpm
Third gear = 3414rpm
Fourth gear = 2500rpm

So my answer to is that gear 2 and gear 3 fulfill the conditions. However the answer says its gear 4. Any thoughts as to what I'm missing?

Thanks.

 

[Mech_Engineer]

The provided parameters give you enough information to calculate wind drag and rolling resistance; are you considering either of those to take into account in your calculations?

 

[billy_joule]

The problem seems poorly formed. No information is given about the power output at 2500 RPM so how could one say with any certainty that fourth gear will work at all?
Also, the question states P_max is required to maintain the given speed, so there is no way forth gear would work as P_2500RPM < P_max.

I agree with the OP's conclusion. There's no need to calc. power required to overcome wind drag, rolling drag etc, we are told it's 59.4 HP, gears 2 and 3 meet that requirement. There's an error in the question.

 

[Mech_Engineer]

The problem seems poorly formed. No information is given about the power output at 2500 RPM so how could one say with any certainty that fourth gear will work at all?

I disagree, the problem statement specifically states "The engine power is seen to be approximately constant at its maximum value Pmax = 59.4hp over the engine speed range 3000-6000rpm." Given a known power output and engine speed, net torque output (and therefore a force balance) can be developed for the vehicle.

Also, the question states P_max is required to maintain the given speed, so there is no way forth gear would work as P_2500RPM < P_max.

The problem is for the student to determine "which of the four gearbox ratios can be used to maintain the speed of 40mph up the incline." There are four gear ratios given, and it has to be determined which would be used based on a vehicle force balance.

I agree with the OP's conclusion. There's no need to calc. power required to overcome wind drag, rolling drag etc, we are told it's 59.4 HP, gears 2 and 3 meet that requirement. There's an error in the question.

The only "error" I see if there is one is that it doesn't specifically state that air drag is to be taken into account. But given the context of the question and the information presented (including coefficient of drag, chassis dimensions, and rolling drag) it seems obvious to me that a force balance must be considered which includes:

• Air Drag
• Rolling resistance (given as 200N)
• Torque required to lift car's weight up the incline (taking into account total mass and the tire diameter)

 

[billy_joule]

I disagree, the problem statement specifically states "The engine power is seen to be approximately constant at its maximum value Pmax = 59.4hp over the engine speed range 3000-6000rpm." Given a known power output and engine speed, net torque output (and therefore a force balance) can be developed for the vehicle.

Given it's to one decimal place I assumed it remained pretty close to that figure.

Either way, the power required for a 1200kg car to go 40mph up a 20deg incline is more than P_max, even before accounting for drag & rolling resistance. The car isn't even capable of doing what is claimed! Not surprising really, I've driven up the worlds steepest street (19 deg), there was no way the Suzuki Cultus I was driving (~100hp, ~1 ton) would've done 40 mph up it.
A pic from wiki:

 

[Mech_Engineer]

So for first gear I get an engine rpm of 8880rpm
Second gear = 5190rpm
Third gear = 3414rpm
Fourth gear = 2500rpm

O.P.:

You've calculated the engine speeds at each of the gear ratios which is a good start. Next you need to calculate the torque output of the engine given a known power output and RPM (see here: calculate engine power given torque and speed). Given those calculated torque output values, calculate the net force to the ground at the tires, and compare to the sum of the forces working against it (air drag, rolling drag, and incline drag).

To maintain speed, the net force to the ground from the engine needs to be greater than or equal to the sum of the drag forces.

 

[Mech_Engineer]

Either way, the power required for a 1200kg car to go 40mph up a 20deg incline is more than P_max, even before accounting for drag & rolling resistance.

Good catch, maybe the OP made a mistake transcribing the problem statement? It looks like a 59 hp car could only do 40mph up to an approx 12 deg. incline...

 

[SteamKing]

Hi,

I don't get the right answer for this question. I don't see where I'm going wrong.
I use this equation to get the engine rpm for each gear ratio:

engine speed (rpm) = gear ratio X final drive ratio X vehicle speed X 60 / (2*pi*rolling radius)

So for first gear I get an engine rpm of 8880rpm
Second gear = 5190rpm
Third gear = 3414rpm
Fourth gear = 2500rpm

So my answer to is that gear 2 and gear 3 fulfill the conditions. However the answer says its gear 4. Any thoughts as to what I'm missing?

Thanks.

I disagree with the book answer.

Fourth gear in this vehicle is essentially an overdrive gear, which is usually used for cruising on long stretches of roadway. Unless the vehicle is already at speed when encountering the incline, normal practice is to shift out of overdrive into a lower gear (higher overall gear ratio) to keep the engine in a speed range where it is making maximum power or close to it, during the climb. Otherwise, you are trying to lug the engine while going up the hill, which is not good for keeping the engine in good running condition. Once you get up the hill, then you can shift back into overdrive and resume cruising.

 

[Mech_Engineer]

I disagree with the book answer.

I suspect this isn't actually the answer from an answer book, but is in fact an incorrect answer given on another forum where this question has been cross-posted.

We will require several clarifications from the OP before moving forward; first and foremost, the stated horsepower of the vehicle is not enough to maintain 40mph up a 20 deg. incline.