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Propeller Tip Speed
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R. W. Stuart Tip Speed or Why did that prop rip up my finger so much? It is sometimes a bit difficult to realize the actual speed of the tip of an "active" model airplane propeller. Also, of the three recent verbal or published local (WRCC and LCRC) disclosures on this subject, two were wrong. One was either an error in reading a published article or an error in the article-- the other simply bad calculations. In order to insulate yourself from such bad info, I would like to show you how to figure your own tip speed, but with no "formulas" and little pain. Why no formula? The word itself is bad news because a formula represents some egghead's condensation and shorthand expression of his mental path to an answer. Unless you understand his terminology, dimensional preference (inches, feet, miles, meters, furlongs-- per second, minute, hour or fortnight) and his approach to the problem, you may well misuse the "formula". The tip of a moving propeller travels in a circle when the plane is static and in a slight spiral when the airplane is moving. We know the prop diameter, say 11 inches, which is the diameter of the circle described by the prop tip. The pitch of the prop is unimportant to this discussion. The distance traveled by the tip is one circumference (distance around the outside) of the circle for each revolution of the prop. The ratio of all circle circumferences to their diameters is a constant 3.14 to 1 ("pi") and being a ratio is dimensionless- inches circumference to inches diameter, miles circumference to miles diameter--all circles behave this way! So the distance traveled by the tip is 3.14 times the diameter of the prop for each revolution. If the diameter is 11 inches, the distance traveled by the tip in one revolution is 11 inches x 3.14 or 34.54 inches per revolution- call it 34.5 inches traveled per revolution. To continue we need to know the rpm (revolutions per minute) of the running system- not the exaggerated published rpm of the motor peddler, but an rpm meter reading of the system in question at the time in question. If you have five fliers at the field, you probably will have a glad meterman, but keep him and yourself behind the engine. So your little SUPER WHEEZE & BURP is turning up an actual 10,500 rpm. 10,500 rpm times 34.5 inches per revolution comes up 362250 inches that the tip travels per minute, but lets divide by 12 to change inches to feet with 30,187.5 feet tip travel per minute. 30,187.5 feet per minute divided by 5280 feet per mile becomes 5.72 miles tip travel per minute. Most people are more familiar with miles per hour so multiply 5.72 miles per minute by 60 minutes per hour to arrive at 343.2 miles per hour tip speed. This will dice your digits!! The flying airplane might have an air speed of 65 mph and a ground speed when flying in still air of 65 mph (if you are going to get hit by the flying airplane it is the ground speed which attracts your attention) and this speed is vectorially added to the rotational tip speed to determine the actual combined tip speed. The vectorial tip speed is 349.3 mph at a very slight forward angle to the rotational plane of the prop. If interested we could calculate the angle and spacing of the slashes across your hide. So how did my finger get so badly chopped up-- after all the tip speed was only about 350 mph. Looking backward from the 34.5 inches tip travel per revolution we can calculate that the 34.5 inches of motion requires only 0.00571 seconds, or a half inch finger laceration takes 0.000083 seconds. Try and dodge that one!!! If the prop was a feather, little or no damage to your finger would result from an incident-- the feather might suffer a bit. This is the effect of an idea called momentum. Momentum depends on the weight of a moving object multiplied by its speed. Think about turning or stopping a big ocean freighter. Prop speed is rotational and its momentum depends not only on its speed and weight, but how far from the center of rotation the effective center of weight exists. For a prop the effective center of weight is about 40% out from the center of the prop. This (and the added rotational weight of motor parts) makes a pretty effective little hammer whipping along at 350 mph. Also props by their very shape are knives. To check our method of problem solving and our use of when to multiply, add, or divide we can use an old engineering trick called dimensional calculation. For the above problem consider: inches diameter of prop times dimensionless pi gives us inches circumference per revolution of prop. Inches circumference per revolution times revolutions per minute gives inches traveled per minute divided by inches per foot gives feet per minute traveled by the prop tip. Feet per minute traveled divided by feet per mile results as miles per minute. Miles per minute times minutes per hour gives miles traveled per hour (in miles per minute times minutes per hour the minutes cancel out leaving miles per hour). Or: Pi in.dia./rev. x rev./min. x ft./ in. x mi/ft. x min./ hr. = mi./ hr. Be aware that /rev. & rev./, in./ & /in., ft./ & /ft. and /min. & min./ all cancel out leaving mi./hr. Dimensional calculation thus proves our method. Note that multiplying by ft./in. is the same as dividing by in./ft. If we combine all the numerically constant relationships such as inches per foot, feet per mile, "pi", and minutes per hour into one number which remains constant; and multiply this constant by the two variables, prop diameter and rpm, we then have a "formula" which will calculate all tip speeds. The rotating wing flyers can use the same system by keeping disc diameter in inches and using the rpm of the rotor. The momentum of the rotor is slightly more than for a prop since the increased diameter is offset by the decreased rpm if the motor hp is the same, but some momentum increase is due to the increased weight of the chopper blades. |
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