# Flight Mechanics of a Rotorcraft

Before reading about the functional principles, control algorithms and testing results of the RAMA project, it is important to have at least some very basic understanding of the helicopter flight mechanics, its characteristics and properties. This chapter is only informative, but should provide the necessary insight, needed for proper understanding of the rest of this website.

## Definition of the Coordinate System

At the beginning, it is necessary to define the coordinate systems (Figure figure1 - this figure was taken from the Wikipedia, the free encyclopedia), which will be used throughout this website. Standard aerospace system was adopted for the RAMA project. Two reference frames are defined; The earth frame and the body frame. Both are orthogonal Cartesian frames. The earth frame xyz has the x axis pointing to the north, the y axis pointing to the east and the z axis pointing downwards. The body frame XYZ is oriented to have the X axis pointing forward along the fuselage, the Y axis pointing to the right and the Z axis pointing down, to form a right-hand Cartesian frame. The X, Y and Z axes are sometimes called the roll, pitch and yaw axes of the vehicle.

The attitude of the vehicle in space is expressed by the three Euler angles. zyx rotation is used to obtain the transformation from the earth frame to the body frame - meaning that the first rotation, by angle ψ, takes place around the z axis, the second rotation (by angle Φ) around the newly-defined y' axis and the last rotation (by angle θ) around the new x'' axis (the x'' axis already coincides with the X axis of the body frame XYZ). This set of rotations yields the XYZ frame from the xyz. The attitude angles ψ, Φ and θ are called the yaw, pitch and roll angles of the vehicle (sometimes denoted YPR angles for short). The conversion from one reference frame to the other might be expressed by the rotation matrix R:

where the matrices R1, R2 and R3 represent the respective rotations.

As is widely known, the Euler angles suffer from singularities, occurring for the 90° tilt angles. This singularities cause the presence of the gimbal lock phenomenon in the attitude representation. The gimbal lock is a condition when a degree of freedom is lost and two attitude angles become dependent. For example, for 90° pitch angle the roll and yaw angles become dependent - the resulting attitude of the vehicle is be the same if the values of roll and yaw are swapped. Consider following sequence of rotations: yaw by 45°, pitch by 90° and roll by -30°. The resulting attitude is the same as for the sequence: yaw -30°, pitch 90° and roll 45°. The presence of a gimbal lock is indicated by one of the rotation matrices R1, R2 or R3 in equation (1) becoming an identity matrix.

The gimbal lock is not a problem for the attitude representation though; the attitude could still be unambiguously determined, given the attitude angles. It only becomes a problem if Euler angles are used for the trajectory reconstruction. It is very informative to study the history of the NASA’s Apollo project and their reasoning and possible solutions to this phenomenon: Gimbal Angles, Gimbal Lock, and a Fourth Gimbal for Christmas, Apollo Guidance and Navigation - Considerations of Apollo IMU Gimbal Lock - MIT Instrumentation Laboratory Document E-1344.

## Basic Aerodynamics

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Fig. 3: Angle of attack definition

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It is a well-known fact that an airfoil profiled body (or any body for that matter), subjected to the air flow, generates the drag and the lift (Figure figure2 ). Both drag and lift are non-linearly dependent on the relative speed of the flow and the angle of attack (Figure figure3 ), which is a relative angle between the airflow vector and the airfoil axis. The higher the angle of attack is, the higher is the generated lifting force (and the drag force alike), up until the point of stall - this is a point where the laminar flow around the lifting body becomes turbulent (Figure figure4 ). The angle of attack, corresponding to the situation when stall just begins to occur (for a given airfoil), is called the critical angle of attack. When the stall condition happens, the lifting force drops suddenly and very sharply - there is almost a complete loss of lift in this situation. Similar rule applies also to the relative speed of the airflow - the generated lift and drag increase non-linearly with the increasing speed, up to the critical Mach number of the body. Weird things begin to happen at critical Mach numbers and for the sake of simplicity, we shall assume that the airflow speed is always well below that limit for the rest of this text. This condition is met for most of the UAVs (although the speed of the tips of the rotor blades or propellers might be actually closer to the critical Mach number than one might intuitively think).

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An infinitely long wing, subjected to a constant speed airflow along its whole length, would generate uniform lift per length unit. For the real-life wings the situation is somewhat more complicated, because they actually have to end somewhere. Naturally, there is a pressure difference between the top and the bottom side of a wing; that is how the lift is generated. At the wing tip, this difference produces a parasite airflow from the bottom side to the top (as can be seen at Figure figure6 ), creating a vortex at the wing tip. This vortex creates turbulece and decreases the lift generated by the outward portion of the wing, so the lift force generated along the length of a wing decreases towards the wing tip, as depicted in Figure figure6 .

In case of a rotorcraft, the situation is a little more complicated. The lifting force of the vehicle is generated by the main rotor. We may consider the rotor blades as “wings”, but those wings are not subjected to the uniform airflow. The angular velocity of the rotor might be considered constant, but the circumferential velocity is obviously not. It increases along the blade from the root to the tip, where it is at maximum. The lifting force distribution along the rotor blades is depicted in Figure figure5 - it increases (non-linearly, of course) along with the circumferential speed, up to the point where the tip vortex comes into play; then it starts to fall again to zero, reached at the blade tip.

The situation becomes even more complicated when the vehicle is not hovering, but flying at a constant velocity. Consider the case where a helicopter is flying forward, as shown in Figure figure9 . Assume that the rotor is turning clockwise. The air flow speed along a part of the blade is now given by the vector addition of the current circumferential velocity vector at the given point and the negative air speed of the vehicle (the vector of the airflow is logically directly opposite to the vehicle airspeed). It is obvious that the air flow speed (and consequently the generated lift) is not distributed uniformly along the rotor disc, but depends on the current angle of the blades relative to the velocity vector (uniform to the vehicle axis) - called the azimuth angle.

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The azimuth of a blade is the angle of the blade relative to the main axis of the vehicle, as shown in Figure figure9 . At 90°, a blade produces maximum lift, because the airflow generated by the movement of the vehicle blows directly opposite to the blade; at 270° azimuth the situation is reversed and the blade generates the lowest lift. The lift distribution in this case (one blade at 90° azimuth and the other at 270°) is depicted in Figure figure8 . It is obvious that a blade to the left of the fuselage (azimuth in the interval of 0°, 180°) is traveling against the “wind” caused by the vehicle movement, and is consequently generating more lift than the opposite blade to the right (in our case). The left side is therefore called the advancing side and the right side the retreating side. Advancing and retreating sides are always defined relative to the vehicle velocity vector. And in case the wind is blowing... Well, the point is surely clear enough by now.

The disbalance in the generated lift naturally produces a torque, acting at the vehicle. It might seem intuitive that in the case described above, this torque would cause the vehicle to roll to the right - but here comes the surprise. In fact, the torque in this case causes the vehicle to pitch backwards. This is because the gyroscopic effect of the rotor spin comes into play. This effect is causing a phase shift (a phase delay) between the lift force and the corresponding torque; this shift is typically by 90° (but could be less or more, depending on many factors, creating not only longitudinal, but also a lateral torque). Therefore, the maximum torque is acting with one blade at 180° azimuth and the other at 0°, causing the vehicle to pitch backwards. This effect works as a kind of natural negative feedback in the velocity control of the vehicle. In order to fly forward, the vehicle naturally has to pitch forward (to cause the portion of the rotor lifting force to accelerate the vehicle forwards as shown in Figure figure7 . With increasing speed, the vehicle tends to straighten again, decreasing the forward acting force in effect.

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Assume a helicopter in hover. There are two main forces acting on a rotor blade; the aerodynamic lift force, which was discussed in previous section, and the centrifugal force. Both forces do not share a common point of action; The acting point of the centrifugal force is situated in approx 58% of the length of the blade, while the acting point of the lift force is located a little more outwards (Figure figure10 ).

The vector sum of those two vectors determines the dihedral angle of the blade, measured between the plane perpendicular to the rotor axis and the blade. The rotor blades are usually hinged to allow for that movement, but the movement is restricted and damped by very stiff rubber dampers. So, in reality, the dihedral angle would be much less than the “natural” angle would be, because of the dampers restricting the movement. It is worth mentioning that the centrifugal force is normally about an order of magnitude greater than the lifting force, giving a relatively shallow dihedral angle even without the dampers. In practice (with the dampers involved), the dihedral angle of the blades is typically around 1-4°.

Let us now examine the effects of those two forces on the angle of attack of a blade, assuming that the blade is flexible and therefore can bend. The carbon fibre composite rotor blades, used on most UAVs, are usually very stiff and do not bend as much as the blades of their full-scale counterparts, so this aspect can be safely neglected on most of the UAVs; but it is still worth mentioning and plays a significant role on bigger rotorcrafts.

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It is very important to have the acting point of the centrifugal force in front of the acting point of the lifting force, as depicted at Figure figure11 . In this case, the torque, which is bending the blade, would tend to reduce the angle of attack, because the leading edge of the blade is bent downwards. This would in turn reduce the lifting force a little and an equilibrium is therefore quickly established, so the blade is stable (a negative feedback loop is present).

On the contrary, if the acting points of the forces would be swapped (as shown in Figure figure12 , the blade would be unstable. The torque would bend the leading edge of the blade upwards, increasing the lift, and therefore further increasing the torque (a positive feedback loop is established). This case might be actually stable too (given the structural parameters of the blade), but often leads to the effect called flutter. The angle of attack would increase, increasing the lift in turn, up to the point of stall (remember the previous section), when the critical angle of attack is reached. At this moment, the lifting force would rapidly decrease and the blade would snap back down because of its flexibility. This would suddenly reduce the angle of attack again and the cycle would repeat. This is obviously a very dangerous behavior, which often leads to the structural failure of the blade, suffering from this phenomenon. The mass of a blade must therefore be distributed so that the points of action of both forces are in stable configuration.

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Let us now consider the blade dynamics in the forward flight. Recall the disbalance of lift, generated by the rotor blades in that case, as described in previous section. The hinges, which allow for the camber movement of the blades to accommodate the dihedral angle, also serve as a partial mitigation of this problem.

Because the lifting force is increasing in the azimuth interval (270°, 90°), the blade would travel upwards (gradually increasing its dihedral angle) in the interval of (0°, 180°), reaching peak at 180°. Again, this is because the gyroscopic effect comes into play; the maxim dihedral angle is reached 90° after the peak force. Because the peak force is acting at 90°, peak dihedral angle is achieved at 180°. Analogically, the blade would travel down between the azimuth of (180°, 0°), with the lowest point occurring at 0°. This periodic vertical movement is known as the blade flapping, and it is one of the most important phenomenons of the helicopter flight. Let us investigate its consequences.

The blade flapping reduces the effective angle of attack of the blade traveling upwards and increases the effective angle of attack of the blade going downwards. This is because the blades no longer move directly against the airflow generated by the movement of the helicopter; their effective speed is now given by the vector sum of the velocity airflow vector, circumferential speed vector and the current flapping speed vector. This effect can be clearly seen in Figure figure13 . Because of this, in ideal case, the disbalance of lift, induced by the movement of the vehicle, would be negated by the effect of the blade flapping. In reality, it is not quite so, because the dampers in the camber hinges prevent the blades from flapping freely, so some of the disbalance still persists - but it is somewhat smaller than what would be the case without the hinges. Why are the camber dampers necessary will be explained in the next section.

In reality, there are many other effects affecting the blade flapping, but those are deliberately neglected in this text, for the sake of simplicity.

## Helicopter Control Fundamentals

In this section, the actuators and basic control principles of a helicopter shall be described. Basically, a helicopter has five inputs - the engine control (controlling the torque, which drives the main rotor), the collective control of the main rotor blades (controls the magnitude of the thrust the main rotor produces), the longitudinal and lateral cyclic control of the main rotor blades - controlling the “tilting” of the thrust vector, i.e. producing the rolling and pitching moments - and the tail rotor pitch (please note that the word “pitch” has a double meaning - it could mean the pitch of the helicopter body, or it could be used to describe the angle of attack of the blades; it is somewhat unfortunate, but it is a tradition). The tail rotor pitch controls the amount of thrust produced by the tail rotor, which is in turn used to control the yaw of the helicopter.

### Collective and Cyclic Controls

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The thrust of both the main and tail rotors is controlled by tilting (feathering) of their blades along their longest axis (changing their angle of attack). The blades are attached to the rotor head using the feathering shaft, to make the tilting possible. The speed of the main rotor is kept constant throughout the flight, as well as the speed of the tail rotor (which is usually driven by a drive shaft or a drive belt from the main rotor gear at a fixed gear ratio, so its speed is directly proportional to the main rotor speed). The engine is kept at constant speed all the time; it only has to react to the variable demand of torque (with is proportionally dependent on the produced thrust). A simple, independent P or PD controller could be used to control the engine speed; even a blunt feed-forward gain, derived from the collective control, is viable and often used in practice. It works well enough, because there is a lot of inertia in the whole system. The collective control tilts both blades of the main rotor together by the same angle, therefore controlling the amount of thrust produced by the rotor. The tail rotor control works in exactly the same manner. It is important to note that both rotors are able to produce thrust in both directions; i.e. the main rotor is able to thrust both upwards and downwards and the tail rotor clockwise as well as counterclockwise. For acrobatic flying, the amount of thrust produced by the main rotor is set to be symmetrical, meaning that the same thrust is available in both directions, allowing for inverted flight. The tail rotor is usually set so it can produce a little more thrust in the direction it has to compensate for the counter torque of the engine; i.e. counterclockwise for a clockwise spinning main rotor. This setting allows for the same available yawing speed in both directions.

In order to tilt the helicopter (i.e. to make it roll or pitch), the cyclic control is used. There is the longitudinal cyclic for the pitch and the lateral cyclic for the roll tilting. The cyclic control is based on periodic changes of the angle of attack of the rotor blades with respect to their azimuth. It is best to explain its function using an example.

Imagine we want to induce a left roll. This is achieved by increasing the angle of attack of a blade in the azimuth interval of (0°, 180°) (with peak coming at 180°) and decreasing that angle in the opposite interval (which is (180°, 0°), as shown in Figure figure14 . Now recall the section Rotor Blade Dynamics and the blade flapping phenomenon. This action would cause the blades to reach the maximum dihedral angle at the azimuth of 270° and the minimum at 90°, tilting the imaginary rotor plane to the left in effect (Figure figure15 ). Also, this action produces a torque, tilting the helicopter body in the same direction.

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Now, remember the camber dampers, mentioned in previous section. As was already said, those dampers are in place to prevent unrestricted blade flapping. This is necessary in order to have some control authority over the vehicle. Imagine that there would be no dampers at all and the blades would be left to flap freely. In this case, the cyclic control would cause the blades to flap, but the flapping would have absolutely no effect on the helicopter body, because there would be no way how to translate the produced tilting torque from the blades to the body. So the dampers are necessary to transfer the torque from the blades to the fuselage. The stiffness of these dampers determines how much control authority the vertical movement of the rotor blades has over the helicopter body; the stiffer the dampers are, the more control authority is there. In turn, the more adverse effect would have the lift disbalance in the flight, as explained in section Rotor Blade Dynamics.

The longitudinal cyclic control works analogically. Let us now summarize the effects of the collective and the cyclic control of the main rotor blades. The collective control determines the basal angle of attack of both blades, controlling the overall amount of produced thrust; the cyclic control periodic blade tilting is superposed to that basal angle, inducing lift variations in the course of one revolution of the rotor, in turn producing the torque. The longitudinal and lateral cyclic could be naturally combined, so it is possible to induce both roll and pitch commands in one moment, causing the vehicle to tilt, for example, left forwards.

The problem with the cyclic control is that it usually has too much authority over the vehicle, causing too rapid response for a human pilot to handle; moreover, the reaction is strongly non-linear. As the bank angle of the vehicle increases, so is increasing the torque; it is obvious from the Figure figure15 . The gravity vector direction does not change, but the thrust vector acting point is shifting further away from it. This causes the spin of the vehicle to accelerate. At the bank angle of 90°, the situation reverses and the spin begins to decelerate again. So, when making a 360° roll (a very basic aerobatic figure) for example, the natural response of the vehicle is somewhat “herky-jerky”. This is very unpleasant behavior from the pilot’s perspective.

Therefore, a mechanical rate stabilizing system was devised right at the advent of the helicopter flight in the 1940s. This was the Bell’s system, later improved by Hiller, hence commonly known as the Bell-Hiller system. This system is obsolete nowadays, because it can be easily replaced by electronic gyro-stabilizers, but it is still very common in the UAV field. The Freya helicopter used in the RAMA project does have the Bell-Hiller system too, and therefore it in necessary to explain its function.

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The original Bell’s system consisted of a flybar (Figure figure16 ), which is a shaft, connecting two paddles, mounted over (or under) the main rotor. The flybar is usually mounted perpendicular to the rotor blades (seen from above). It can tilt freely, so the paddles can move up and down, and it can also be feathered, meaning that the angle of attack of the paddles is under control. The flybar is connected with the main blades feathering system by control rods, so that the tilt of the flybar causes tilting of the main rotor blades in the corresponding direction.

The flybar behaves like a small rotor; it would tilt (meaning the plane of its rotation would tilt) if the paddles are feathered. It also behaves like a gyroscope; it has the tendency to maintain its plane of rotation, so if the helicopter (along with the main rotor) would increase its bank angle (in any direction), the flybar would maintain its previous position (there are no camber dampers which would prevent it to do so). So, the plane of the flybar and the main rotor would no longer be parallel; and the flybar would therefore tilt the main rotor blades (via the connecting rods) and induce a countering cyclic control, up until the point when the planes of rotation are parallel again.

Similarly, if a command is applied onto the flybar, causing its plane of rotation to tilt, the flybar would induce a cyclic command to the main rotor, proportional to the relative angle of the flybar and main rotor disc; the grater the angle is, the greater is the command. As the helicopter begins to accelerate its rotation, this angle would decrease, reducing in turn the applied cyclic command.

So, the Bell’s system behaves like a purely proportional controller; The flybar tilting determines the set point and the relative angle between the flybar and the main rotor discs is the control error. The cyclic actuation is applied solely to the flybar and only the flybar controls the cyclic feathering of the main blades. The p constant of this controller is given by the weight and size of the paddles. The more the paddles weight, the more gyroscopic - and stabilizing - effect they have, so making them heavy would stabilize the system, but increase the reaction time to the setpoint change; on the contrary, making them lighter makes the response of the system faster, for partial loss of stability.

This system somewhat linearized the angular rate response of the helicopter, but induced a lag in the cyclic control response. The reaction to a setpoint change was somewhat lazy and the pilots were still not completely happy. Therefore, the system was improved by Mr. Hiller to make the response faster. Hiller introduced a feed-forward branch into the controller; he added control rods, which feed a portion of the setpoint directly to the main blades (so, a little portion of the actuator movement is applied directly to the blades and a greater portion to the flybar). This system retains the stability of the Bell’s system, but makes the response to the setpoint variations faster.

It was determined that the Bell-Hiller stabilizer consumes a significant portion of the engine power; typically about 15-20%. It is therefore much better to get rid of this system completely and introduce the electronic rate stabilizers, as the RAMA system does.

### Tail Rotor Control

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The tail rotor serves to control the yaw of the helicopter and is actuated in exactly the same manner as the collective control of the mail rotor (as was mentioned earlier). In a steady state, the tail rotor must compensate for the torque reaction, induced by the engine. This torque reaction acts in opposite direction than the engine torque; therefore, if the rotor revs clockwise, the fuselage tends to spin counterclockwise. Hence, in this case, the tail rotor must thrust to the left, In order to produce a clockwise counter-torque. This thrust naturally induces not only the torque (with respect to the main rotor axis, where the tail boom serves as a lever), but also draws the helicopter to that direction (i.e. to the left). To compensate this draw, the helicopter must bank to the right slightly, so that the main rotor could produce a counter-thrust (otherwise, the helicopter would accelerate leftwards). So, in the hover, the helicopter is never completely straight; it has to bank slightly, in order to maintain the steady state. The bank angle is typically around 3-5° (Figure figure17 ).

This bank also tilts the tail rotor slightly upwards, so now it tends to pitch the helicopter slightly forwards. However negligible this portion of the tail rotor thrust might be, it would still bank the helicopter slightly forwards in the long term, causing it in turn to accelerate slightly forwards. So, this thrust must be compensated by a tiny backward pitch of the vehicle. However, the bank angle in this direction is almost negligible in reality.

## Effects of the Rotor Speed

As was said earlier, the rotor speed is kept constant throughout the flight. This is because the rotor speed has great influence on the control authority of the helicopter. With increasing rotor speed, the controls are becoming more responsive, because the control surfaces (i.e. the blades) produce greater amount of thrust at the same angle of attack. The relation between the rotor speed and the generated lift is strongly non-linear, so the speed variations would largely disrupt the control loops. Either the control gains would have to be set over-conservatively, reducing the effectiveness of the control loops, or some sort of adaptivity (usually the gain scheduling) would have to be incorporated. Either way, the controllability of the helicopter would be compromised nevertheless. Hence, it is very desirable to minimize the rotor speed variations. Fortunately, this is not so hard, as was mentioned in the previous section, because of the great amount of inertia accumulated in the rotor.

The variations of the air speed of the helicopter in flight naturally have the same effect as rotor speed variations, but in this case, the influence is somewhat lower, because the flight speed is typically seven or eight times lower than the circumferential speed of the rotor tips; hence, the rotor speed variations have much greater effect than air speed variations. Usually, the PID control loops (the lowest-layer, i.e. the angular rate control loops) are robust enough to cope with these variations under reasonable flight conditions (slow flight); however, it might beneficial to incorporate some sort of adaptivity (gain scheduling), if it is desirable to exploit the full flight envelope of the vehicle.

## Centre of Gravity

The Centre of Gravity (CG) position is naturally of vital importance for any flying vehicle, because it greatly influences the stability and controllability of the vehicle. For helicopters, the CG should be positioned right at the main rotor axis, as low as possible. The lower the CG is, the greater is the natural stability of the vehicle, because the lever between the acting point of the lift force and the CG would be greater (think of the pendulum effect).

If the CG position would be shifted away from the main rotor axis, the main rotor thrust force and the gravity force would form a permanent torque, which would tend to turn the vehicle (either in the roll or the pitch), which would have to be permanently compensated for by the cyclic control. This is naturally undesirable, so the helicopter should be always balanced so that the CG is located on the main rotor axis.