Alberto Trevisan – firstname.lastname@example.org – Master’s Degree in Mechanical Engineering
Updated in Jenuary 2017
In full-scale helicopters the primary stabilizing method, in most cases the only one, is the aero-elastic flapping of the rotor blades due to its high mass and slow response. In model helicopters the stabilization method used is the flapping of the flybar, which is an aerodynamically damped gyroscope device used to increase the stability of the vehicle in the pitch and roll axes and assists in its actuation. This device is well spread in robot helicopters because their small size causes an inherent instability and a fast time-domain response. In model helicopters this is also the only damping mechanism because their rotor hub is hingeless (rigid), so the blades can’t move upwards and they are too stiff to flap significantly. This hingeless system is adopted to decrease the control time and give a better sensation of control to the pilot, otherwise in full scale helicopters the blades are free to flap, or springs are mounted on the rotor to increase the stability of the system. This system increases the time that the helicopter needs to respond to the control inputs. The flybar system permits also to reduce the forces that the actuators must apply to control the rotor. The first design of this system came in the 50’s for full-scale helicopters and it used the gyroscopic effect of a bar with weights to regulate the tilt angle of the blades. To reduce the flapping motion of this mechanism was used a separate damper, this is the so called Bell stabilizing system. Lately an airfoil replaced the damper and weights to generate the Hiller system. The modern design is called Bell-Hiller flybar mixer because puts together some design aspects of both the configurations.
- Blades and flybar roll e pitch angle variation for collective pitch in static conditions
- Blades and flybar roll e pitch angle variation for cyclic pitch in static conditions
The test consists in increasing the translation of the swashplate along the rotor for the collective pitch of the blades and inclining the swashplate for the cyclic pitch to analyze how the angles of the blades and of the flybar vary for this type of robot-helicopter. Knowing the kinematic correlation between the translation of the swashplate and the collective pitch is of fundamental importance in order to impose the right motion to the servos to archive a lifting force. A kinematic verification is made also for the cyclic pitch to assure that the mechanism doesn’t lock up at some angles and the flybar roll angle doesn’t become too high, which can compromise the stability and integrity of the system.
Thismechanism also induces a delay in the response to the pilot inputs, which is a counter part to the fact that the machine is less affected by external disturbances and quick changes in state (cross-wind, ground effect, etc.). The presence of the aerodynamic forces also affects the forces that the actuators must exert on the swashplate. Those effects can be proven by the following test conducted with and without the airfoils:
- Change in cyclic input tilt (stepped function)
To assess the behaviour of the mechanism through the different rotational speeds an eigenvalues calculation is needed:
- Identify the natural frequencies of the system in the linearized configuration of the rotating frame.
Effect of flexible blades
To test the different behaviour of the system with the presence of blades compliance, which is a source of damping in the real scale helicopters, a complete simulation with both cyclic and collective pitch as input has been made.
The modelling problem
This type of rotor hub is hingeless and corresponds to a good compromise between stability and performance in terms of readiness to the pilot’s input.
It is composed of several parts that will be shortly described underneath:
Main shaft: transmits the torque to the blades and puts in rotation the whole system.
Swashplate: is the main mechanism which permits to control the rotor. It is made of three parts, the outer ring which does not rotate and is fixed to the helicopter body, an inner ring that rotates with the system and has four connections to four rods, a ball joint which slides over the shaft and permits the rotations of the inner and outer rings. This system, as a whole, is capable to tilt in all directions and to move vertically. The swashplate is connected to two pairs of rods, one of them controls directly the blade tilt and the other mixes this output with the flybar mechanism. The rods of the system acts only in pushing, so they are also called pushrods.
Slider: can glide over the main shaft and it is blocked in the rotation around it by two guides.
Washout control arm: transmits the motion from the swashplate to the washout rod through the washout radius arm.
Flybar control frame: can rotate along the seesaw main axis and controls the flybar pitch angle.
Flybar seesaw: controls the roll movement of the flybar and its fulcrum is a hole in the head block which enables it to rotate perpendicularly to the control frame rotation axis. Because of two openings in the other direction it is also permitted the previous pitch movement. These two swinging structures, the inner and the outer, are connected only by 2 bearings in the rotation points of the flybar control frame, two other bearings are put in the head block to permit the complementary rotation.
Upper arm: gathers the inputs from the swashplate rod and from the flybar mechanism through the upper arm rod, its action is to rotate the blade carrier and definitively change the angle of attack of the blades.
Blade grip: rotates around its main axis and is connected to the head block by bearings. One blade grip can rotate independently by the other.
The possible commands for this system are two, the first is the cyclic control in which the fixed disc of the swashplate (i.e. the outer ring) is inclined around the ball joint, this causes a resulting inclination of the rotating disk (i.e. the inner ring). The effect is a periodic change in the pitch angle of the blades and of the paddles. This mechanism is used to change direction of flight.
The arm called Bell input in the picture permits to directly change the angle of attack of the blades from the swashplate, the response to this input is fast but is inherently instable so, in the cyclic control, this input is mixed with the Hiller’s one. This changes the pitch of the flybar which is affected by the gyroscopic effect and by the centrifugal force. So the pitch of the main blades is affected by the pitch of the paddles which flaps, in other terms the cyclic pitch is controlled by the mix of the Bell and Hiller input. In this condition the effect is that the main rotor blades are forced to return in the position with zero angle and with a higher tendency to the stability.
Another effect is present, in these conditions the flybar acquires a roll angle with generation of lift. This lift tilts the flybar plate and imposes a secondary cyclic input to the rotor hub, acting as an actuator to the rotor blades. So the advantage is that the forces required by the actuators are reduced. So this system acts not only as a stabilizer but even as a force reducer.
The collective pitch of the rotor blades is controlled by the translation along the shaft of the swashplate (in solid with the ball joint), so raising the swashplate a positive angle of attack arises in the blades, otherwise lowering the swashplate from the neutral position is generated a negative angle of attack useful in the autorotation of the helicopter. In these conditions the washout arm prevents the collective pitch to be affected by the paddles that in this condition have a null pitch angle (beta) and null angle of incidence. The slider is moved, carrying with itself the washout arms, only with the collective control. Using the cyclic control the slider remains stationary in its original position.
Body geometry and joints
The main geometry parameters of the system are presented below, note that for simplicity the rotor and the whole paddles have been modelled with plastic density (ABS) and the other parts in steel:
|Number of rotor blades||2|
|Rotor radius [m]||0.56|
|Half mass of rotor blades [kg]||0.1174|
|Stabilizing bar radius [m]||0.22|
|Inside radius of paddle [m]||0.144|
|Half mass of paddles [kg]||0.01326|
|Distance between swashplate and rotor blades [m]||0.0823|
|Average chord length of paddle [m]||0.043|
|Average chord length of blade [m]||0.0456|
|Total mass [kg]||0.6418|
Parts have been modelled with different densities:
ρ(ABS) ≅ 1.07⋅10^(-6) kg/(mm^3 ) for blades and paddles
ρ(Steel) ≅ 7.81⋅10^(-6) kg/(mm^3 ) for all the remaining parts
The geometry was imported from Cad software using the Parasolid extension and merging different parts in their right positions, these are the blades in the blade carrier and the paddles with the rod connecting them to the flybar control frame.
- Some parts have been merged using the command, Booleans: Merge two bodies. Even if the parts in reality are not made of the same material, for example the rods or the shaft, they have been modelled with the same material due to the characteristics of the command.
- The mass of the parts has been obtained by imposing the density of the material (plastic or steel) to the part and using the geometry from the Cad. Even if this is an approximation it is acceptable for our purpose and moreover, because the mass of the parts is very low.
- Some CM of the parts has been modified in order to obtain the most accurate equilibrium configuration. To do this the common CM of the symmetric parts has been manually adjusted using the tool Aggregate Mass. Even if the operation has been carried out with accuracy still some hundreds of millimetre differences are present, whcih leads to some little instabilities. The results of this inaccuracy will been explained.
The joins configuration is summarized in the table below:
|Body i||Body j||Type of Joint|
|Ball joint||Swashplate outer ring||Spherical|
|Ball joint||Swashplate inner ring||Spherical|
|Swashplate outer ring||Swashplate inner ring||Parallel|
|Swashplate inner ring||Swashplate rod||Spherical (-1)|
|Swashplate inner ring||Washout radius arm||Spherical|
|Washout control arm||Slider||Rotational|
|Washout control arm||Washout rod||Spherical (-1)|
|Washout rod||Flybar control frame||Spherical|
|Flybar control frame||Flybar seesaw||Rotational|
|Flybar seesaw||Upper arm rod||Spherical (-1)|
|Upper arm rod||Upper arm||Spherical|
|Upper arm||Blade grip||Rotational|
|Upper arm||Swashplate rod||Spherical|
|Paddle||Flybar control frame||Fixed|
|Mian blade||Blade grip||Fixed|
- This configuration is repeated symmetrically in the other part of the mechanism because the mechanism, in order to prevent dynamic instabilities, is made symmetric.
- The swashplate mechanism has 3 DOF, two rotations about perpendicular axis in its own plane (cyclic inputs) and one translation along the shaft (collective input). Because the outer and inner rings of the swashplate rotate around the ball joint, which is a portion of a sphere with 2 parallel bases included in two parallel secant planes, the spherical joint that connects those parts to the ball joint are overlapping on its centre of mass which is also the rotational centre. A configuration where the spherical joints are put on the centres of mass of their respective bodies will lead to a locked mechanism. These three DOFs of the swashplate are controlled by three servo-actuators fixed to the helicopter’s body; they also prevent the rotation of the outer ring which is in-built with the fixed frame. Another independent DOF is the rotation of the shaft. So the controllable DOFs are four, three from the swashplate and one from the shaft.
- It is also present an internal DOF, this additional motion is the flapping of the flybar, so the final DOFs are five. To properly count these DOFs, the Planar joint that was punt at a first stage has been substituted with a Parallel joint. The first joint constraints the rotations along the axes in plane and also the translation along the other direction (Z axis), the latter constraint doesn’t exist in the reality because the inner and outer rings are not fixed between each other and they slide one over the other simply because of gravity. The correct joint is so the parallel joint which constraints only the rotations on the plane (X/Y axes).
- The main blades and the paddles are linked with fixed joint to the respective carriers to permits the use of different materials for the parts (steel for the blade grips and the control frame and plastic for the blades and paddles).
- The internal rotation of the rods along their axes has been blocked with a motion which displacement is: 0 * time, in one of the two spherical joints at the extremities. To make this the Z axis of the joint must be aligned with the direction of the rod. An equivalent joint could be a Hook joint which preserve the same properties. This configuration doesn’t change anything from the kinematics point of view but erases the internal labilities.
- All the parts have been modelled as rigid bodies, this doesn’t take into account the flexibility of the main blades which is a source of damping and gives a further stabilizing effect to the system, even in hingeless rotors.
The Grubler count for the whole mechanism leads to this result:
DOF = 6*21 – 5*1(R) – 3*1(S)(-1) – 3*1(S) – 4*1(C) – 2*1(P) – 3*4(S)(-2) – 5*4(R) – 3*4(S)(-2) – 5*1(T) – 5*2(R) – 3*4(S)(-2) – 3*2(S) – 5*4(R) = 126 – 121 = 5
Kinematic, dynamics and eigenvalues simulations have been carried out. The GSTIFF SI2 solver has been used for the integration of equations.
Avoiding the rotation of the shaft, the DOFs of the mechanism are four, they can be seen in the video where can be noticed the decoupling of the movements. The simulation involves a time of 5s (End Time) and a Step Size of 0.01 has been used.
Rot X: STEP( time, 0 , 0 , 0.5 , 12d ) + STEP( time, 0.5 , 0 , 1 , -12d ) + STEP( time, 1 , 0 , 1.5 , -12d) + STEP( time , 1.5 , 0 , 2 , 12d )
Rot Y: STEP( time , 2 , 0 , 2.5 , 12d ) + STEP( time, 2.5 , 0 , 3 , -12d ) + STEP( time , 3 , 0 , 3.5 , -12d) + STEP( time, 3.5 , 0 , 4 , 12d )
Translation: STEP( time , 4 , 0 , 4.5 , 7 ) + STEP( time , 4.5 , 0 , 5 , -7 )
Instead of a scripted simulation, the choice was a sequence of step functions because all DOFs must stay activated for the entire period, with zero value (joint blocked) when other joints are moving, so the choice of stepped functions would have been equal to the scripted simulation in this case.
To analyze the correlation between the swashplate movements and the blades pitch, the paddles pitch and the paddles flap, some separate simulations have been carried out. This is important because the control of the mechanism, for example to give the wanted angle of attack to the blades in order to obtained a predetermined lift, is based on a known correlation between the imput and the output, preferably linear.
The first simulation is reported in Fig.4 and involves only the translation of the swashplate. In this configuration the inner plate, the outer plate and the ball joint translate as a unique body and (only here) the slider slides along the shaft and the washout control arm rotates in order to prevent a movement of the paddles. Fig.4 highlights that the rotation of the paddle is not involved. Although not depicted in Fig.4, also the flapping motion is not involved, because the seesaw is blocked by the symmetric actions of the two upper arms moved by the swashplate rod. As a result this input activates only the pitch of the blades (equal and opposite on the two blades, i.e. same angle of attack). Swashplate is moved upward in order to increase the lifting force. A linear correlation between the input and the output is obtained:
θ(blade rotation) [deg] = 1.2 * S(swashplate translation) [mm]
The second simulation is reported in Fig.5 and involves a rotation around an axis parallel to the axis of paddles. In this case the only part that is activated is the washout radius arm and the washout control arm. The swashplate rod does not move, and so only the paddles rotate gaining an angle of attack (equal rotation between the two paddles, i.e. opposite angle of attack) while the blades remain fixed. Once again, a linear correlation between input and output is obtained
θ(paddles pitch) [deg] = 1.6 * θ(swashplate rotation) [deg]
The third simulation is reported in Fig.6 and involves a rotation of the swashplate around an axis aligned with blade axis. In this configuration the washout radius arm does not move because it is exactly on the rotation axis, while only the swashplate rod moves causing a rotation of the blades. However, in this case the blades rotates in the same direction (opposite angle of attack) due to the inclination of the swashplate, on the contrary in the swashplate translation the blades rotate antimetrically (same angle of attack). Also in this case a linear relationship between input and output is obtained, while the paddles remain approximately unchanged
θ(blade rotation) [deg] ≅ 0.42 * θ(swashplate rotation) [deg]
In order to control the three movements discussed in Fig.4, 5 and 6 with a rotating main shaft, three actuators are usually employed, as shown in the sketch of Fig.7.
In the numerical model, three cylindrical bodies have been positioned under the three pins of the lower swashplate, a cylindrical joint between each of them and the ground was added. An inplane joint (translation along Z axis fixed) was also added between the cylindrical body and the swashplate pin’s marker. Every cylindrical joint was completed with a translational motion. The 0.5 in the function takes into account the position of the servos, that are spaced at 120° each around the swashplate, so when one is moving upwards, the other two must move downward, but since the radius is for those two lower and equal to cos(30°) = 0.5, this number is taken into account to prevent the mechanism to lock up.
This situation was considered to avoid the instabilities that are present in the mechanism when the shaft is rotating, carring with him the hole mechanism exept the lower swashplate ring. Those instabilities are due to the upper part and are transmitted to the swashplate because of the Parallel joint, which constraint the rotations in the plane between the two parts of the swashplate. Those instabilities causes the lower swashplate to rotate with the rest of the mechanism even if it has to be still. To permits the mechanism to rotate without imposing a rotational motion directly to the swashplate, witch would have led to the aformentioned problem, this actuators system was implemented.
|Body i||Body j||Type of Joint|
DOF = 6*3 – 4*3(C) – 1*3(Inplane) – 1*3(Motion) = 0
From the Grubler’s count, it can be seen that the number of DOFs in the mechanism has not been altered from this joints extension and there are no redundant constraint.
The 3s simulation in the video below shows the two movements discussed in Fig.5 and 6 together with the main shaft rotation, in order to highlight the cyclic variation of the blades and paddles angles when the swashplate is tilted. More precisely, in the simulation one actuator moves upward by 10mm in 0.5s while the other two move downward by 5mm in 0,5s; then the simulation continues for 2.5s. Summarizing, during half of the rotation the blades are tilted in one sense and during the opposite half in the other. The same movement is present in the paddle/flybar, so the change in blades and in paddles pitch has a phase equal to π, and the two movements are out of phase of π/2 angle. So, when the angle of attack of the paddles is at its maximum value, the blade’s one is at zero and vice versa.
Aerodynamic model for paddle/flybar
To perform the dynamic simulation the aerodynamic forces on paddle have been added to the model, because the focus is on the stabilizing mechanism that rely on paddle dynamics. It is assumed that the air speed is related to the rotation of the blade only. The shape of the paddles was approximated to the (symmetric) NACA 0012 airfoil to estimate its lift coefficient and centre of pressure.
To do so, the Reynolds coefficient was calculated first:
Re = (ρVd)/μ ≈ 64153
V = 22.044 m/s , rotational speed of Ω = 122 rad/s = 6990 deg/s = 1165 rpm
d = chord length = 0.043 m
ρ = 1.225 kg/m^3
μ = 1.81·E-5 Pa·s
Which is in the laminar region, so the stall region begins at low angles of attack. The value of the rotational speed was assumed as mentioned in the paper . The function of the lift coefficient was linearized taking into account the linear curve in the linear region, that does not change so much with different Reynolds numbers:
It can be seen that the value of angle of attack where the stall begins is very low, so it has been decided to exceed this number and model the response until the limit of:
θmax(paddles pitch) = 15°
Even if the response, to be coherent, should be lower than 10°. Should be stressed also that the linear slope doesn’t change so much changing the Reynolds number. The previous limitation is to be considered during the simulation to limit the input forces to have small angles in output, the whole procedure to have consistent results.
It is worth noticed that, the region where the lift coefficient starts to decrease and assume a negative derivative has not been taken into account. This phenomenon will lead to dynamic instabilities and flutter which is not considered in this work.
The lift force was modelled through the formula:
V = Ω·r
according to the linearized model. This formulation neglects the flapping motion of the flybar that introduces another velocity to the flybar’s motion which changes the angle of attack. So two phenomena are present:
- Torsional motion (paddle pitch): increase of lift force due to the lift coefficient
- Flapping motion: arises another velocity component that varies the angle of attack:
which is negative if the paddle is moving upwards (lower attack angle) and positive if the paddle is moving downwards (higher attack angle).
So the final equation that takes into account these effects is:
Vx = Ω·r
A = 3111.04 mm^2
was also introduced the parameter B = 0.97 which takes into account the tip loss in a finite length airfoil due to tip vortex effect that reduces the effective length of the paddles.
The equation in Adams takes the following form:
Force paddle 1 = 0.5 * (0.1185 * ((.Flybar_project.oscillint_oscillext_RotZ) – (.Flybar_project.Paddle1_Zvelocity / .Flybar_project.Paddle1_Xvelocity))) * 1.225E-09 * 3111.04 * (.Flybar_project.Paddle1_Xvelocity)**2 * 0.97
The formula for the other blade changes for a minus in front, to take into account the fact that the two paddles are rigidly interconnected so when one has a positive lift the other presents a negative one.
The point of application of the forces was set in a static position even if this point changes with the angle of attack from a more centred position to an advanced position near the leading edge increasing the angle of attack.
The position was chosen supposing an equal distribution in time of the point through the various positions for the various angle of attach reached. This is probably not the best solution because the movement of the flybar involves accelerations and decelerations, so the time spend in the various position during the rotation is not equal but as a first approximation was assumed so. Definitively a medium of the position for the angles between 0° and 15° was made.
Chord length ≅ 43 mm
Medium position of the centre of pressure (cp) = 0.1625
Point position from the leading edge = 43 mm · 0.1625 = 7 mm
This position is a little advanced with respect to the classical 0.2 point but considering that the blade reaches easily angles of attack near the stall this assumption seems reasonable. The position through the width was chosen as the intermediate position by reason of simplicity (i.e. 33 mm from the inner edge).
- This calculation is set on the assumption that the paddle has a little width so the velocity is not varying so much and it can be approximate with the velocity at the midpoint.
- The effect of drag coefficient (Cd) is assumed negligible.
- βf ≤ 15°
- θpaddle ≤ 10°, this condition was relaxed in some cases extending its value to 15°, as mentioned previously.
The geometry used is the following:
Modeling of friction in joints
At some joints was insert some friction, both static and dynamic, and some torsional damping. The reasons for this choice can be summarize in:
- Friction: in order to avoid the instability at low speed that can be present in the early part of the simulation or near the stability. This is also a form of correction of the little unbalancies mentioned in the mechanism desctiption paragraph. These values of friction are adopted to model the ball bearings, so their values are quite low in order not to affect so much the simulation.
- Damping: to exclude some vibrations that are present in the system was added those terms that helps to analyze properly the system and add an equivalent effect to be more similar to the real one.
Friction was added between the following rotational joints:
- Blade grip – Main shaft
- Blade grip 2 – Main shaft
- Flybar seesaw – Main shaft
- Flybar seesaw – External Flybar control frame
Damping was added between the following rotational joints:
- Flybar seesaw – Main shaft (flapping motion)
- Flybar seesaw – External Flybar control frame (pitching motion)
An eigenvalues analysis of the flybar was performed in order to obtain the natural frequencies of flapping and pitching modes. This was performed for the linearized system at the equilibrium without external forces. The system was analyzed at different rotation speeds. Four simulations at 300 RPM (1800 deg/s), 600 RPM (3600 deg/s), 900 RPM (5400 deg/s) and 1165 RPM (6990 deg/s) were performed, linearizing the system at four different simulation times: 5s, 10s, 15s, 20s in order to verify the consistency of eigenvalues results. A rotating marker on the main shaft centre of mass was used as a reference for the linearization of the system.
Flybar flapping motion:
All the eigenvalues have negative real part and two non-null imaginary values so the vibration is damped and asymptotically stable. The results, at different times for the same rotational speed, presents a uniform response so can be said that already after five seconds the mechanism is in stable conditions. The behavior of the flapping motion, increasing the rotational speed, is presented in the table below and in Fig.14, where is visible the increase of the flapping natural frequency with the growth of the rotational speed. An almost linear correlation is present between the rotational velocity and the undamped natural frequency for the flapping motion of the system in analysis, in the speed range considered.
|ωn [Hz]||5 s||10 s||15 s||20 s|
Paddles’ pitching motion:
The same procedure aforementioned was used to calculate the natural frequencies for the pitching motion of the flybar. All the values present negative real parts so the pitching motion is asymptotically stable and damped. As previously, also in this case the system after five seconds is quite stable and the values for the undamped natural frequencies are plotted in the graph below. Can be seen a different trend of the values from the flapping motion, in this case the natural frequency trend is to decrease with the progressive growing of the rotational speed. A parabolic trend can catch the decrease of the values.
|ωn [Hz]||5 s||10 s||15 s||20 s|
Dynamic simulation with aerodynamic forces on flybar only
The whole system was considered as the one presented in the kinematic analysis section, but in this case (to avoid instabilities), three Translational Spring-Damper were added with very high parameters to simulate a fixed constraint. The reason for this choice was to maintain the system unchanged from the point of view of the Grubler count but with the need of a stiffer mechanism without the presence of the lower plate’s translation.
To have a more realistic simulation the weight of the flybar was increased, which will cause that the centrifugal and gyroscopic effect will affect more the system and will count more with respect to the aerodynamic force than the previous simulations. In the real mechanisms, a metallic insert of reduced dimensions is added to the flybar, for this reason in this simulation the mass of the paddle was doubled, increasing the density to:
ρ'(paddles) = 2.14·10^-06 kg/mm3
This choice of density was also proven with by an optimization analysis, not reported here, which proves that the choice made results good in terms of reducing the peak acceleration in the blades. This configuration of the paddle is also used in real applications where to the paddles, shaped as an airfoil, are added some metallic masses in order to find the most convenient configuration.
Response to swashplate cyclic input (i.e. swashplate tilt)
The behavior of the flybar as a consequence of a swashplate cyclic input is shown in Fig.18, 19, 20 and in the video below. The simulation elapses 5s and the integration step is 0.001. In hovering flight mode (free rotation of the system at the constant rotation speed), the rotor was subjected to a swashplate rotation that induced a cyclic pitch and then the opposite motion passing through the zero value and returning to the initial position, that can be seen in the following figure. The actuator motions are reported the following (main actuator motion in Fig.17; other motion have the same shape, amplitude halved and opposite direction)
Main actuator translation: STEP( time , 1 , 0 , 1.2 , 3 ) + STEP( time , 1.2 , 0 , 1.4 , -3 ) + STEP( time , 1.4 , 0 , 1.6 , -3 ) + STEP( time , 1.6 , 0 , 1.8 , 3 )
Other actuators translation: STEP( time , 1 , 0 , 1.2 , 3*0.5 ) + STEP( time , 1.2 , 0 , 1.4 , -3*0.5 ) + STEP( time , 1.4 , 0 , 1.6 , -3*0.5 ) + STEP( time , 1.6 , 0 , 1.8 , 3*0.5 )
The motion of the three actuators result in the swashplate tilt of approx 6° [deg] (linear aerodynamic model) as shown by the dashed grey line in Fig.18.
It should be stressed that the blade pitch is both controlled by the collective pitch (i.e. translation of the swashplate, which is a pilot input) and by the dynamics of the flybar (internal flapping degree of freedom, which is therefore not a pilot input). Note also that, without the ‘resistance’ due to the flybar’s inertia during the rotation, the cyclic pitch input would not affect the blades pitch but will end in a motion of the upper arm and of the flybar seesaw only.
Can be seen that with the simple flybar, without the aerodynamic forces, the system can catch the movement induced in the system. As a result, the system acts correctly and in the system are present low vibrations. The blades angles reached in this configuration are compatible with the results of the kinematic investigation ( θblades rotation ≈ 0.42 θswashplate rotation ). Adding the paddles’ aerodynamic force to the system, it can no longer respond in a fast way as previously since the second peak is very low. The system results also with more vibrations induced by the flapping motion of the flybar itself, which is transmitted to the blades. An important result is also that the flybar acts as multiplier, because the angles reached from the blades are much higher then the previous, this one again can be attributed to the influence of the flapping motion.
From these results, can be seen the so-called time delay that the flybar induce into the system, which has been referred in the introduction and found in the papers, making the system slower in the time domain response in comparison to the simpler one.
From this graph, in Fig. 19, can be appreciated the enhanced presence of flapping in the configuration with the presence of the aerodynamic force, indeed induced by the torque generated in the flybar by the paddles aerodynamic force.
The pitching movement is now limited to the value induced by the swashplate’s rotation, which is consistent with the one find in the kinematic analysis ( θpaddles pitch ≈ 1.2 θswashplate rotation ), as can be seen in Fig. 20. Because this value is bounded below by the swashplate, the influence of the flybar in this case is not present.
To make more clear how the flapping motion affects the whole system, a direct visual comparison of the two systems is here shown. On the left there is the mechanism considered as a Bell stabilizer, without the presence of the aerodynamic forces, on the right the mechanism with the aerodynamic forces acting on the paddles. In the period between the two actuator movements, it is clear how one mechanism has less vibrations arounf the stable configuration, instead the other is still flapping. So the dynamic of the mechanism with the Bell-Hiller mechanism results slowed down due to the flybar flapping motion.
To respond to the question if the flybar reduces the force needed to move the actuator, a study of the forces exerted on one actuator was carried out and the results are shown in Fig. 21. For the system with the flybar but without the application of the aerodynamic force can be seen a force that remains always positive, so since the reference direction of the motion is pointed downwards (Fig. 22), the force always points in the same direction. Considering the translational movement of the actuator, since in the first part it has an opposite movement compared to the positive direction, this force exerts its action against the movement. In the second part the actuator movement is descending, so concordant to the positive direction and the force is pointing in the same direction. Analyzing the response to the same test but with the aerodynamic forces, can be seen that the actuator’s reaction force in the first movement becomes even negative, so concordant to the actuator direction of motion. In other words, the force is helping the actuator in its motion and in this case it has in some points not to push but to pull to balance the force. In the second part of the actuator’s motion, the flybar is concordant with the direction of the movement exerting an enhanced force with respect to the previous configuration.
The reference system of the mechanism, presented in Fig. 22, points downward as the positive directions for the force and the displacement in the Z coordinate.
Regarding the power which involves the actuators motion in the Z direction, the following convention can be set after having analyzed the combination between force and displacement varying in time:
- P+ = power to be supplied to the system
- P- = power supplied by the system
The power has been calculated as the derivative of the work, where both the force and the displacement vary in time domain ( i.e. F=F(t) and s=s(t) ), so the derivative for the product of functions has been used.
Comparing the instantaneous power response of the two actuators, can be seen that in the configuration without the aerodynamic forces (Fig. 23) the peaks reach quite the same values as the other one (Fig. 24) but they are mainly positive, so we are in an unbalanced configuration where the actuators must provide the energy to the system in order to move the mechanism.
Otherwise, in the configuration with the aerodynamic forces depicted in Fig. 24, the situation presented is more symmetrical so the power that is required to move the system is balanced by the component that the system provided by the system to the external environment. The final result is more balanced in terms of loading application because the system can be described as already destabilized and so less energy is required to move the actuators in order to control the blades. Must be said that this result is however paid with a loss of power in the main shaft due to the drag and turbulences generated by the flybar. This destabilizing effect caused by the flapping motion is induced by the paddles, which causes also aerodynamic losses. This effect is neglected in the present work but can be quantified in 15 ÷ 20% of engine power loss, which is a significant amount and is one of the causes that induced to substitute this system with the electronic stabilizers made of gyroscopic sensors.
Simulation with aerodynamic forces on flybar and on the flexible blades
To assess the stress field that may occur in the blades for a possible sequence of movements and therefore for a load history, another simulation has been made considering also the collective pitch.
The collective pitch was neglected in all the previous dynamic simulations because it does not affect the flybar motion, which is the aim of this study. Nevertheless, the lift is generated only by the collective pitch, which sustains the helicopter in flight, on the other hand the cyclic pitch controls the flight direction. This introduces high stresses in the system, which must be verified.
In this simulation, the flybar influence over the complete system is counted with the aerodynamic forces. In this simulation was also used as material wood in substitution of ABS. This because in real helicopter, blades can be made in this material, and, as result, stiffer blades are given. Material properties are:
ρ(wood) = 0.75·10^-06 kg/mm3
E(wood) = 9000 MPa
To perform the numerical analysis a Flexible body must be created in Adams. A new material with the following properties has been created first:
From a preliminary investigation, the Number of modes after having neglected the rigid motions already present in Adams is 18. The blade has been meshed with Solid tetrahedral elements with Quadratic shape function (second order elements). The Edge shape has been set on Mixed which controls the elements shape at the boundary edges, the options are: Straight: the mid-nodes are positioned at the average coordinates of the end nodes, Curved: the mid-nodes on boundary geometry will be projected onto the geometry resulting in curved element edges, Mixed: edges will be curved except when such curving would result in an invalid element (that is the recommended option). Was also recommended to set the Curvature based scaling on, that provides that the mesher uses the geometry curvature to provide mesh grading and smoother transitions. The important parameter in this phase is the value of the option: Edge tolerance, this specifies the edge tolerance to collapse at zero the small edges, cannot be less than zero or ½ of element size.
After a convergence analysis not reported here, the two main blades of the mechanism have been meshed with brick elements with the following size:
- Element size: 5mm
- Minimum element: 2mm
- Collapse small edges, tolerance: 2mm
That gives an estimated number of 3D elements equal to 7833, for each blade, and 14610 nodes. The elements size have been chosen considering the thickness of the blade that is 6.5 mm at the maximum height so the minimum size was set to permits a stratification of the Brick elements.
The best results for the dynamic simulation was found using, as a constraint, the nodes included in the volume of a sphere of given radius (user input), in this case this value was set to 6mm. The centre of the radius was positioned at the centreline of the blade on the line of the blade carrier bolt. The radius is entirely contained in the blade’s volume and the number of nodes constrained is approximately 195.
Similar results can be obtained using a constraining sphere of 8 mm of radius or using the option closest nodes, where must be set the number of nodes to constraint, in this case using a number of nodes around 250, a spherical volume will be considered as previously.
Another important parameter to take into count is the Damping ratio. The default option was chosen, which leads to the following nonzero damping coefficients:
- 1% damping for all modes with frequency lower than 100.
- 10% damping for modes with frequency in the 100-1000 range.
- 100% critical damping for modes with frequency above 1000.
Also the Inertia modelling is another parameter that can be set to modify a Flexible body. The options are the following:
- Rigid body: the flexible body is considered a rigid body de facto.
- Constant: the deformation and the inertia properties are decoupled.
- Partial coupling: simplified formulation, with a low impact on the results but time-saving.
- Full coupling: used to achieve full accuracy.
The default option is the Partial coupling, which has great efficiency but needs to be tested the model to verify not to require the full formulation. Not noteworthy differences were found using the two options, partial coupling or full coupling. The following results however have been acquired using the Full coupling option.
Much better results were found changing the Solver settings for the dynamic simulation. The Error was modified from 1·10^-03 to 1·10^-04, that gives smoother results, especially for the stresses results obtained from the coupled Finite element analysis. This parameter, otherwise, does not affect so strongly the results of the only dynamic simulation.
The density of the flybar was set equal to the optimal one found in the Optimization chapter:
ρ(paddles) = 2.14·10^-06 kg/mm3
- As expected, the influence of the flapping motion of the blades is limited, considering this type of simulation, since the variations with respect to the results obtained with the full rigid body simulation can be neglected.
To model the effect of the aerodynamic forces over the dynamics of the whole system and to assess the differences between the previous simulation, the Blade Element Theory was used. To assess the aerodynamic forces acting as lift, a modified theory is used with a discrete model, instead of a continuous one with the use of the integration.
Dividing the blade in 7 pieces and counting 8 marks, we can write:
First of all, some markers must be put on the spots pointed out by the Blade element theory, at a discrete and regular length and on the static approximate centre of pressure. The markers distribution is presented in the table below:
|Point||Centre radius [mm]|
Using as spacing between the markers:
Δx = 64.83 mm
Also in this case the position of the force point was set in an average position considering the same NACA profile of the paddles, which is the NACA 0012. This is an assumption driven form the fact that this type of “upper-middle scale” RC helicopters adopts simple airfoil geometries and in this case a common NACA profile is used. This is not true in full scale helicopter and in others mini-RC helicopters. Considering that the blades pitching is normally limited to about 10°, the centre of pressure was chosen for a medium position around 5°, which corresponds to the typical value given for profiles.
Chord length ≅ 45.57 mm
Medium position of the centre of pressure (cp) = 0.20
Point position from the leading edge = 45.57 mm·0.20 = 9.11 mm
At this point, a unit force was put on each marker to represent the aerodynamic force acting on each element in which the blade was divided. This was made to permits the program to identify each marker as an attachment point, also called Master node. This procedure was necessary to apply the force not to a single node identified by a marker, but to a cluster of nodes, this to compute more correctly the modes of the blade. To make so, when the blade was made flexible, all the Master Nodes was mashed as attachments, choosing the options Closest nodes. Since the geometry is very flat, grouping the attachment nodes with a cylindrical or spherical volume would be complicated, so was chosen the option Closest nodes. The number of nodes grouped in the option closest nodes is 80 nodes.
|Blade mesh||Blade carrier- Blade attachment||Master nodes attachments for forces
Element size = 5mm
Min size = 2mm
Collapse edge = 2mm
|Geometrical feature: Sphere||Geometrical feature: closest nodes|
|Radius = 6mm||Nodes number = 80|
The graphical result of this operation is reported in Fig. 27, where are highlighted the attached nodes with a series of lines linking them to the master nodes.
A preliminary simplified numerical investigation was made using the formulas previously shown, where the area used in this formulation is the flat area calculated as the blade was a flat squared plate. The graphical output of this operation is reported in Fig. 28.
|Point||R [mm]||A [mm2]||F [mN]||Point||Centre [mm]|
At this point, in the Adams simulation, for every marker the translational velocity in Y and Z are measured to use them in the definition of the aerodynamic forces as previously done for the paddles.
An example of the implemented expression for the aerodynamic force on one marker is:
0.5*(0.1185*(.Flybar_project.portapala2_alberoprinc_Pitchpale-(-(.Flybar_project.Blade2_F1_Zvel)/ .Flybar_project.Blade2_F1_Yvel)))* 1.225E-09 * 3200.33 * (.Flybar_project.Blade2_F1_Yvel)**2
- The tip loss is not considered directly but introduced by the lift distribution neglecting the contribution of the edge of the blade. To do so the tip of the blade has not been considered in the calculation of the area.
To find the value of the blades pitching angle to generate a total lift force sufficient to sustain the weight of the helicopter in flight, the Blade element theory again was used, neglecting the induced velocity vi and the climb velocity V. Neglecting the inflow velocity causes a small error because it has a little influence on the phenomenon.
From some data sheet of similar helicopters, the flight weight was assumed with the motor and the batteries, which approximately is:
Mtot(helicopter) = 2.5 kg → Weight = 24.525 N
Must be said that, the weight value is largely affected by the batteries weight so changing the type of these this value can vary a lot.
From integration, assuming a θpitch = 5° and neglecting the climb and induced velocities derives:
which is sufficient to sustain the helicopter in hovering flight.
Using the relation between swashplate translation and blades rotation obtained in the kinematic part, the input to give to the actuators can be evaluated as follows:
θblade [deg] = 1.2·S(swashplate translation) [mm] → S = 4.17 mm
Response to cyclic input (i.e. swashplate tilt) combined with collective pitch (i.e. swashplate translation)
A 3.5s simulation with time step 0.001s and the following actuators motions (main actuator motion in Fig.29; other motions have the same shape, amplitude halved and opposite direction):
Main actuator translation: STEP( time , 1 , 0 , 1.5 , -4.17 ) + STEP( time , 1 , 0 , 1.2 , -3 ) + STEP( time , 1.2 , 0 , 1.4 , 3 ) + STEP( time , 1.4 , 0 , 1.6 , 3 ) + STEP( time , 1.6 , 0 , 1.8 , -3 )
Other actuators translation: STEP( time , 1 , 0 , 1.5 , -4.17 ) + STEP( time , 1 , 0 , 1.2 , 3*0.5 ) + STEP( time , 1.2 , 0 , 1.4 , -3*0.5 ) + STEP( time , 1.4 , 0 , 1.6 , -3*0.5 ) + STEP( time , 1.6 , 0 , 1.8 , 3*0.5 )
The actuators motions result in the translation and tilt of the swashplate, in Fig.30.
In the next picture (Fig. 31) is visible the effect of the flybar flapping that affects in a great way the dynamics of the blades changing the response of the system. The response of the blades is no more linear since the mechanism has a slower dynamics and cannot respond fast to the actuators input. In this case, the presence of the forces on the blades adds some more destabilizing effect to the system. However, it is then quite fast damped.
Differently from the rigid bodies simulation without paddles lift forces, in the last complete case the induced vibrations in the final part of the movement (when the actuator are no more active and in the original position) are more persistent because of the influence of the aerodynamic forces on the blades which, as seen, has the same influence as the aerodynamic forces on the paddles in destabilizing the system.
In the comparison, presented in Fig. 31, is visible how much the presence of the aerodynamic forces on the paddles combined with the effect of the lift forces on the blades give an altered response. The pitching angle results larger when subjected to a cyclic input and the dynamic behaviour results slowed, so the response to the close change of pitching angle is not so sudden and perceptible as before.
The flapping motion of the flybar (Fig. 32) is complementary to the response on the paddles since the torque induced by the latter is the source of the altered and slower response of the blades. It is also visible that the collective pitch does not influence the response of the flybar, beacuse until the cyclic pitch is not active, the flybar is in stability.
A similar behaviour, as the one registered in the simulation without the lift forces, is seen, which has comparable max values but with a more persistent vibration in magnitude which are, on the other hand, rapidly damped.
The response seen in the blades pitching (Fig. 31) is induced by the flybar flapping because, form the comparison, is visible the effect of the torque generated in the flybar control frame by the aerodynamic forces that gives a mixed input to the blades with the actuators’ one. The last part of the vibration is the effect of the lift force on the blades that, as seen, has a similar effect to the one of the flybar.
The flybar’s pitching response, reported in Fig. 33, is not influenced by the collective pitch and does not presents significant variations as in the simulation using the rigid bodies and without the lift forces. So can be said that the presence of the lift forces with the flexible bodies has not influence over this movement and also the combined type of input, collective and cyclic, does not change it. The values and the shape of the output response function in this case are the same as the one with the rigid bodies and without the lift forces.
As said before this movement is not influenced by the presence of external destabilizing forces.
From the stress chart can be seen a different behaviour with respect to the previous simulations because in this case was used a material with a lower value of density so the static value of stress reached is equally lower than the previous ones. Can be seen that, as expected, the collective pitch induces a constant value of flexion on which the value of the cyclic pitch induces another stress, which is also in this case mainly flexion.
The maximum values obtained are higher than the max stress the material is capable to withstand, so there is the need to change material, switching this with another one more resistant, as composites, such as glass fibre-epoxy or carbon-epoxy. These materials are also stiffer, reducing so the potential level of damping induced by the flapping motion of the blades which is however so low that can be ignored in this hingless rotor helicopters, so the performance change is null from this point of view.
The same trend of the blades pitching is visible on the Von Mises stress. This because the stresses induced in the blades, mainly by flexion in the blades main axis (i.e. lenght) and only for a very little part in shear, is generated by the pich angle in the blades which is strongly influenced by the flapping motion of the flybar.
σ(VM peak with aero forces) = 49.838 MPa ( t = 1.73 s )
σ(VM peak without aero forces) = 43.156 MPa ( t = 2.11 s )
From this graph can be seen that the main component of stress is the stress parallel to the blades that gives a flexion due to the blades own weight and the lift forces.
As seen in the previous graphs, the values of the stress in the blades exceed the yielding strength (similar to the ultimate strength) of the material used in this simulation, which is wood. So the need is to choose another material with better properties. Common Producers uses glass-epoxy or carbon-epoxy composites for these applications.
From the first kinematic analysis the relation between the input, seen as the translation and rotation of the swashplate, and the output of the blades rotation and paddles pitch was found for this particular mechanism. This result is important to assess the differences between the static value that a quantity has and its value influenced by the dynamics of the mechanism, mainly by the flybar flapping effect.
The flybar acts as a stabilizer considering the flybar pitch itself, lowering the output peak and damping the system when an impulsive external torque subjects the cyclic input, representing an external disturbance. This response generates a flapping motion in the flybar, which is the main characteristic of the flybar, that also affects the blades pitch since they are connected to the flybar seesaw. Considering an external disturbance, for example a lateral wind gust that runs over the helicopter, the flybar reacts with the gyroscopic effect, maintaining aligned the blades plane with the flybar’s one. Therefore, the output, as a variation of blades pitch, is much smaller than the one without the flybar and this effect is not influenced by the presence of the Hiller input with the aerodynamic forces on the paddles, so the same response is present both in the Bell mechanism and in Bell-Hiller configuration.
The Bell-Hiller stabilizer acts slowing down the response of the system subjected to a fast change in the cyclic input (since it is not affected by the collective pitch) and altering the response between input and output. This is due to the flapping motion that the aerodynamic forces induce on the flybar, that alter the response of the system. The flybar pitch is not altered by the presence or not of the aerodynamic forces. The aerodynamic forces also acts varying the force that the actuators must exert on the system to make it change configuration, in some cases changing the sign of it from pulling to pushing and making a more balanced system from the point of view of the instantaneous power that must be exerted.
The paddles configuration used by the producers, where some additional masses are added to the plastic paddles finds its reason in the fact that the optimum configuration, in terms of reduced initial acceleration peak of the blades compared to the paddles mass, is the one with a mass double than the one with the whole paddle in plastic (ABS). Therefore, considering paddles mass and blades peak angular acceleration, the better configuration is the one before mentioned.
Using a flexible bodies simulation was found that the flapping and torsional compliance does not give an additional damping in this type of helicopters since the blades are much stiffer than the ones in real helicopters, so the blades flapping motion has a negligible influence on the mechanism.
Adding the aerodynamic forces to the blades and using the flexible bodies simulation, we have come to the result that the lifting forces on the blades has a destabilizing influence as the flapping motion of the flybar but with a lower magnitude.
To make an overall check of the stresses on the blades and the mechanism responses, a complete simulation was performed considering the presence of the aerodynamic forces on the paddles and the combined effect of collective and cyclic pitch. The effect of the flybar, seen in the rigid body simulation, and the destabilizing effect caused by the blades aerodynamic forces are both present. As a result, the system presents a similar response as in the rigid body simulation but obviously a little more destabilized because of the combined presence of the aerodynamic forces in both paddles and blades. The main result of this simulation was the fact that the material chosen for the blades is not suitable for this application in terms of strength. Therefore, a carbon-epoxy blades or at least glass-epoxy must be adopted as done by the producers.
- Use of aluminum as material for the metallic parts instead of steel
Asses the stress field in the shell laminate with the dynamic loads (ply by ply)
 Tadashi Aoki, Shigeru Sunada, Hiroshi Tokutake and Yukio Otsuka; Analysis of Bell-Hiller Stabilizer Bar; Osaka Prefecture University, Sakai, Japan; in The Japan Society for Aeronautical and Space Sciences Presented at The 46th Aircraft Symposium 2008 on October 22, 2008.
 S. K. Kim and D. M. Tilbury; Mathematical Modeling and Experimental Identification of an Unmanned Helicopter Robot with Flybar Dynamics; University of Michigan; in Journal of Robotic Systems 21(3), 95–116, 2004.
 Mohammad Reza Sabaapour and Hassan Zohoor; Analysis of a Swashplate Mechanism of the Hingeless Rotor Hub with the Flybar in a Model Helicopter, Part I: Kinematics; Sharif University of Technology, Tehran, Iran; in Journal of System Design and Dynamics Vol. 4, No. 4, 2010.
 Bing Zhu and Zongyu Zuo; Approximate analysis for main rotor flapping dynamics of a model-scaled helicopter with Bell–Hiller stabilizing bar in hovering and vertical flights; Nanyang Technological University, Singapore, Beihang University, Beijing; 13 May 2015
 Douvi C. Eleni, Tsavalos I. Athanasios and Margaris P. Dionissios; Evaluation of the turbulence models for the simulation of the flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil; in Journal of Mechanical Engineering Research Vol. 4(3), pp. 100-111, March 2012.
 Vittore Cossalter; Meccanica applicata alle macchine; Edizioni Progetto; 2006
 MSC Software; Adams Tutorial Kit for Mechanical Engineering Courses
 MSC Software; Adams – Online Help
 Department of Control Engineering, Faculty of Electrical Engineering of the Czech Technical University in Prague; RAMA UAV Control System, Flight Mechanics of a Rotorcraft (https://rtime.felk.cvut.cz/helicopter/)
 Martin Barczyk and Alan F. Lynch; Control-Oriented Modeling of a Helicopter UAV with a Bell-Hiller Stabilizer Mechanism; University of Alberta, Edmonton, Canada; in American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
 Martin Barczyk; Nonlinear State Estimation and Modeling of a Helicopter UAV; University of Alberta, Edmonton, Canada; Ph.D. Thesis