Forces on the
Blade Grips of your Helicopter
Everything you ever wanted to know about centripetal force but were too scared
to think about!
by
Nigel Fraser Ker
Introduction
Have you ever wondered how much force is placed on those little bolts that hold
the rotor blades onto your model helicopter? Well, now
you can find out...this little article gives you the maths to work out the
forces for yourself. If you want to download the spreadsheet that
accompanies this article, you will find the link at the bottom of this page.
Acceleration
As the blades on a helicopter rotate, they are
constantly accelerating towards the centre of the circle. Surely not, I
hear you cry...they stay the same distance from the centre of rotation so how
can they be accelerating towards it? Well, this is really because of the
way we define acceleration which is described in terms of velocity (which is a
vector quantity) rather than speed (which is a scalar quantity). Lost
already? Well, think of it like this...if you apply a force to something
it will start to accelerate. If you stop applying the force, it will keep
on going in the same direction as it was last going and at a constant speed
(assuming there are no other forces involved like gravity, air resistance,
etc). When a blade is rotating around the shaft of a helicopter, there is
a constant force keeping the blade from flying off at a tangent. This
force is therefore said to be accelerating the blade towards the centre of
rotation. Yes, it does sound odd but believe me, it's true.
Centripetal Force
To work out the centripetal force on the blade grip, we use the equation:
Force = mass x acceleration
The mass is easy to work out - just weigh a blade. Remember, everything must be measured in units which are compatible with one another and the best thing to use is SI units. So, find out the weight of a blade in kilos.
The acceleration is a little more complicated. The acceleration acting on a rotating object is given by the following equation:
Acceleration = Angular velocity2 x Radius
Angular Velocity
The angular velocity is what we call the 'head speed' of the helicopter except
that it is not given in RPM (Revs Per Minute) but rather in radians per
second. Without getting too deeply into the maths, a radian is about 57.2958
degrees and there are 2 PI (i.e. 6.2832) of them in a circle. Therefore,
if your blades were rotating at 1 revolution per second, this would be the same
as saying that the angular velocity was 6.2832 radians per second (wow, this is
starting to make sense!). If they were rotating at 100 revolutions per
second, the angular velocity would be 628.32 radians per second, and so
on. Get the idea? (Remember, we all use revs per minute as
the measure of headspeed but you will have to divide this figure by 60 to give
you the rotations per second before you multiply by 6.2832 to get the
radians per second.)
Radius of Gyration
Okay, that's angular velocity sorted out, now let's look at the radius.
This is a little more complicated than it appears. All the weight of the
blade is not sitting at the tip, nor is it all situated at the root...it's
spread out along the whole length of the blade and what we need to find out is
the point at which the weight appears to act. This is known as the
Radius of Gyration and is given by the equation:
Radius of Gyration = Square Root of (Total Radius2 / 3)
So, taking an easy example - let's say your helicopter blades are 55cm long (i.e. 0.55m from the bolt hole to the tip). To get the total radius we have to add the distance from the bolt in the blade grip to the centre of the shaft (what I call the Grip Radius)...let's say this is 7cm (i.e. 0.07m). This gives a total disk radius of 0.62m and placed into the above equation we get a radius of gyration of 0.358m.
The Equation
So, the equation for the force (in Newtons) on the blade grips is as follows:
Force = mass of one blade x (2 x
PI x (RPM / 60))2 x Square
Root of ((Blade Size + Grip Radius)2 / 3)
Remember, be careful to stick to SI units!
Can't be bothered to work all this out yourself? I don't blame you...here's a 'ready reckoner' to help you work it out using Microsoft Excel - have fun!
Download an example in Adobe Acrobat format here
Please note that this article is for interest only and should not be used as the basis for designing or modifying mechanical devices, e.g. model helicopters. You should consult a qualified engineer before undertaking any design/testing/modification work. Always follow the manufacturer's instructions and the safety guidelines and rules of your national governing body.