Does RPM vary linearly with the speed of an airplane?

Question

So I assumed if a plane was flying at max RPM it would be flying at max speed too. but saying if it flew at half the max speed it would be flying at half the max rpm sounded wrong. Does it vary linearly or as a square root due to drag varying with velocity squared? I just need an approximate model for my project so apologies if I trigger anyone with the lack of accuracy.

Answer 1

It depends how accurate you are trying to be.  Let’s assume the model only has to deal with a horizontal flight-path. (Which is very inaccurate model.)



The drag force (D) is approximately proportional to v².  So D = Av² where A is a constant (you can try different values and decide on a suitable value).



The thrust (T) from the prop’ will be a very non-linear function of rpm. The shape of the graph will depend on engine and propeller design.  And it will depend on airspeed.  So at 1000rpm the thrust at 100km/h is not the same as at 200km/h.



So a mathematical model is extremely complicated.



Your best bet is make a crude (extremely inaccurate) model.



E.g. T = kR where k is a constant (you can try different values and decide on a suitable value) and R is the RPM.



Then the resultant force (F) in the direction of motion is:

F = T - D = kR – Av².



Mass of plane = m.  

Acceleration a = F/m = (kR – Av²)/m (positive when speeding-up, negative when slowing down)



When at a steady speed, a=0 so kR – Av², giving v = √(kR/A)



If that helps.

Answer 2

If the air was entirely incompressible then the speed would be proportional to RPM.  As an approximation the DIFFERENCE between the no load RPM and the actual RPM is determined by drag.  Now here is the interesting thing.  That drag is proportional to the square of the speed of the plane ( approximately) but at the higher speed more air enters the propeller.  So that until the blade tip is supersonic the difference between the no load speed and the actual speed should also be linear.

Ie if the blade surface is capable of moving in the air at some speed v when there is no pressure on the blade at all, you would note that v is proportional to RPM.  ie v = k * Rpm

if some particular plane is connected so that the speed of the plane is z* v where z must be less than one.  we get u = z* v = z*k* Rpm

ie that v is in fact proportional to RPM for a propellor driven craft where the blade tip is moving subsonically and the friction of the plane is proportional to v^2.  That is my model for the device.

Answer 3

The faster you go the more air resistance. Most planes have a max RPM for take off but then once you are in the air you throttle back to the appropriate RPM for cruising speed which gets you the best efficiency for the engine, time flying and fuel use.

Answer 4

Soooo many variables to consider !

A propeller driven aircraft is not rigidly connected to the air. 1/2 RPM would probably produce about 1/2 max thrust which may not be enough to keep some small planes in the air but in more powerful planes might produce a speed well above half max.

A car at 1/2 max RPM in top gear would indeed be going 1/2 max speed. Even then it would most likely not be at 1/2 power.



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