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Forces on a Helicopter

Helicopters are fascinating creatures. One of their distinguishing features is their ability to hover in place. This is essentially accomplished through a torque and force balance. For the forces we must look at Newton’s second law of motion:

Newton's Second Law

Notice the arrows on top of the force and acceleration terms. These mean we’re dealing with force vectors, and an acceleration vector. These vectors have three components because helicopters live in three dimensions. For a helicopter to remain stationary the acceleration must equal zero, i.e., all three components of the acceleration vector must be zero. By setting the acceleration vector to zero in our equation, we see that the left side of the equation must be zero as well. Thus, the only way to achieve zero acceleration is if the sum of the forces add up to zero.

For the torques we turn to Euler’s (simplified) equation of motion:

Euler's Equation of Motion

It looks suspiciously similar to the first force equation above. In essence, torques are exactly the same as forces except that they deal with rotation instead of translation. If the sum of the torques aren’t zero it will cause the helicopter to spin around. Thus, for a helicopter to remain stationary both the sum of the forces, and the sum of the torques must be zero.

Gravity is the only force that acts on a helicopter when it’s hovering, so all we have to do is supply a force that counteracts gravity and we’re set. This is the job of the main blades. However, these spinning blades introduce a new torque on the helicopter, which throws off the torque balance and causes the helicopter to spin in the opposite direction of the main blade direction. This new torque is balanced with the tail rotor (on a conventional helicopter) that pushes the tail of the helicopter in the opposite direction of the spin. Thus the torques are balanced once again.  But we’re not finished yet. Similarly to how the main blades caused a torque as well as a force, by using a tail rotor to balance the torque, another force has been introduced that pushes the helicopter to one side or the other (depending on which way the tail rotor is facing). Add another propeller you say? Well it turns out that this force can be balanced simply by tilting the helicopter so that the force from the main blades acts slightly to one side. This balances the small force caused by the tail rotor. Finally all the forces and torques are balanced and we have a stationary helicopter hovering in the air. The next time you see a helicopter hovering stationary, see if you can see how it leans very slightly to one side to counteract the force from the tail rotor. This can be seen in the following video:

Image from R/C Airplane World.

Wheel Imbalance Can Easily Be Detected When Driving on the Highway

Have you ever driven a car that vibrated uncontrollably at certain driving speeds, especially on the highway? If you have, you probably noticed that these large vibrations occur when you drive at one specific speed, as shown on your speedometer, and the vibrations die out when you travel a little lower or higher than that specific speed. Personally, I’ve driven a car that undergoes unusually large vibrations when I’m driving it at around 70 miles per hour. This phenomenon arises from an imbalanced rotor.

Rotors are objects that rotate, like wheels on a car. An imbalanced rotor is one whose center of mass is not in line with its axis of rotation. For a wheel, the axis of rotation would be the axle of the car. Ideally, if a wheel is perfectly circular and uniform, meaning its center of mass is exactly in the center of the wheel, and the axle for the wheel goes through the center of the wheel, then the wheel would be “balanced” because its center of mass and axis of rotation are in line (both at the center of the wheel). However, in the real world, these ideal cases are few and far between. Imbalances on a wheel, if they are drastic, can cause undesired vibrations.

Now, why does an imbalanced rotor tend to have large vibrations at certain speeds? The answer comes from the principles of resonance and resonant frequency. Every object has a resonant frequency at which the object experiences large vibrations or large-amplitude oscillations (resonance). Consider a small wine glass. If you drive the wine glass at its resonance frequency long enough (perhaps by directing sound waves at it), it will shatter due to the resulting resonance. Similarly, for an imbalanced rotor (such as a wheel), there are certain speeds of rotation (called “critical speeds”) that cause large-amplitude vibrations in the rotor (up-and-down and side-to-side). If one changes the speed of rotation away from this critical speed (either higher or lower), then the resonance will die out and the rotor will rotate more smoothly. Interestingly, at very high speeds, an imbalanced rotor will tend to balance itself out and rotate smoothly as if it was balanced to begin with.

When driving a car with imbalanced wheels at its critical speed, you might notice that the resonant vibrations feel uncomfortable to you, especially since you can feel the vibrations  from the steering wheel, which you are probably holding onto if you are moving on the road. Similarly, resonant vibrations are uncomfortable for your car as well (think of the wine glass analogy again). Prolonged periods of driving a car with an imbalanced wheel at its critical speed will subject the car to increased stress and cause it to wear out faster (and potentially break down as a whole due to failure under stress…just like people sometimes do). So, if you ever experience unusually large vibrations from your car when you are driving at particular speeds, you will know what the problem most likely is, and the strategic play would be to get the wheel imbalance fixed as soon as possible to avoid excessively damaging your car.