The steering solution of rotors of the liftoplane is based on a four-gears pitch steering scheme. Cinematic scheme of such rotor is represented below and follows with explanation.
The scheme is performed in a manner, which indicates actual clearance of neighbored pieces and ensures missing of any collisions. The view should be understood as a transparent projection of depicted selected internal elements to faceplate of the rotor. Any intersections of selected elements on the scheme mean overlapping those elements in separated plans. The picture represents the rotor in the neutral articulation with PGS=(5;0;0) relative to base airflow and with indicated rotational direction, which is appropriate for the particular articulation. The rotor has a faceplate, on which some elements are mounted, which supports shafts of wings; those elements and shafts of wings aren't pictured on the scheme. The wing has a circular base, which is integral part of the wing. A bevel gear of big diameter is mounted on the circular base. A bevel pinion is meshed with the bevel gear and mounted on a linking shaft. Other end of the linking shaft has a miter gear, which is meshed with other miter gear, which is clustered with a pinion and fixed on respective shaft. The pinion is meshed with a pitch gear, fixed on other respective shaft. Also a steering pinion is meshed with the pitch gear and clustered with an entry gear. A grove follower is mounted on the last cluster and can move inside of a grove ring. The pitch gear with shaft, the steering gear, the entry gear and the grove follower are assembled as an earring inside of an earring’s shell, which can hold a number of supporting bearings. A central gear is meshed with the entry gears of all earring assemblies and clustered with the grove ring and with an internal gear. A pitch pinion is meshed with the internal gear. Central shaft is fixedly connected to the faceplate and used for powering the rotor.
The cluster of the central gear and the pitch pinion have ability to move in any radial direction up to some limit, changing the Gain and Skew of entire PGS-state. The earring’s shells have some "windows" for the pitch gears of neighbor earring assemblies, preventing collisions upon the steering. The pitch gear has its name, because it always synchronized in rotation with related wing. The pitch pinion has its name, because its rotation will change the Pitch of entire PGS-state. The steering pinion has its name, because it directly steers the pitch gear. The entry gear has its name, because it acts as entry interface for entire earring assembly. The pitch gear, the central gear, the entry gear and the steering pinion are base elements of the four-gears pitch steering scheme.
The chart above represents same cinematic scheme as before, but the rotor is in high negative gain articulation with PGS=(5;-40;0), which can be used upon gaining altitude. The scheme demonstrates how will be changed pitches of wings and positions of earring assemblies upon moving entire cluster with the central gear together with the pitch pinion for this articulation. In the high gain articulation still exists the enough clearance between earring assemblies, and the applied movement of the entire cluster is far from the maximal.
The chart above represents same cinematic scheme as before, but the rotor is in high positive gain articulation with PGS=(5;40;0), which can be used upon recuperative descent. The scheme demonstrates how will be changed pitches of wings and positions of earring assemblies upon moving entire cluster with the central gear together with the pitch pinion for this articulation. Remained clearance here is same as for previous scheme. Also indicated direction of rotation of entire rotor is opposite, since the articulation is related to recuperative descent.
The animation above represents a comparative view for changing of articulation over all three PGS-states presented before.
It will be very useful having an end use formula for obtaining a pitch variation of particular wings upon shifting of the central gear in the four-gears pitch steering scheme. The variation will be a function of instant distance between axis of the pitch gear and axis of the center gear. And the variation will be independent from orthogonal offset of the central gear from center of the rotor, when the distance is fixed. The last can be intuitive, but it isn't obvious. However it can be proved upon following thought analysis.
Let the central gear was moved orthogonal from some pitch gear, but their distance will be kept. This movement can be considered as rotation on some angle all four gears participated in the steering with a frozen meshing state. In such case the pitch gear will obtain additional variation, which is equal of the angle of the rotation of the system of those four gears. But in the case, the central gear also should obtain same additional variance as the pitch gear, because the meshing state is frozen. But actually the central gear is fixed from any rotation by the irrotational for this movement pitch pinion. And so the pitch pinion will imply a counteraction, which returns the central gear in its original angular position. The reversed rotation of the central gear will break the frozen meshing state of those four gears, and the pitch gear will be also returned to its original angular position, because all pitch gears are synchronized in their collective angular movement with the central gear by equality ratio.
The chart above explains a movement all gears relative to some pitch gear, upon changing its distance from the central gear. The chart pictures the pitch gear in horizontal position and with zero Skew articulation, but it is invariant for result will be obtained. At first, all participated gears have their correspondent radiuses based on their pitch diameters. The pitch gear has radius r1, the central gear has radius r2, the entry gear has radius r3 and the steering pinion has radius r4. At second, there exists a radius of circle where axes of all pitch gears are laid. It is referenced as R0. At third, there can be build a triangle with corners based on axes of the pitch gear, the central gear and the clustered entry gear and steering pinion. In the triangle, axis of the pitch gear is fixed against an offset, so it is referenced by O-letter. Axis of the central gear in the neutral position is referenced as A-letter, and axis of the cluster is referenced as B-letter for this case. In case of an offset Δr, last two points will be A1 and B1 respectively. Also for the triangle OAB can be assigned two corner angles for these O and A points as β0 and μ0 respectively. And for the case of shifting, they will be referenced as β1 and μ1 respectively. Additionally there can be considered two meshing points: between the central gear and the entry gear as C-letter, and between the pitch gear and the steering pinion as D-letter. Also for the case of shifting, they will be referenced as C1 and D1 respectively. And finally, the pitch variation of the pitch gear upon the shift of the center gear can be referenced as δ, and will be correspond to a reposition of the original meshing point D. The point D will be reposed in two instances. One instance will be laid on the pitch gear and it is referenced as H1, and other will be laid on the steering pinion and it is referenced as G1.
The chart above represents magnified essential part of the previous chart with additionally details for deducing the target variation formula. At first, there is presented a reposition of the original meshing point C to two points E1 and F1 laid on the central gear and the entry gear respectively. Here E1 is simple result of an offset the point C to the Δr vector. The changing of angle β0 to β1 upon the offsetting corresponds with the changing related meshing position, and so angle of this changing is referenced as β. Also same kind angle μ is referenced for the other meshing position. And so the target angle δ can be considered as a sum of angular changing of the meshing position β and a remainder θ equal to an additional rotation, imposed by the steering pinion itself. The angle θ has a complemented angle φ on the steering pinion, related to it by the simple gear ratio. The angle can be decomposed as a sum of the common change of meshing position β and an angle γ from rotation of the entire cluster. There angle β is secondarily pictured as the arc between points N1 and P1 on the circle of the entry gear, where B1P1 is parallel to OB and N1 is a simple crossing of OB1 with the circle of the entry gear. Also the angle γ is pictured on the circle as the arc between points P1 and Q1, where the last is projection of the point G1. The angle γ also can be expressed as sum of an angle η from changing of the meshing position of the entry gear, and a remainder λ. The angle η is equal to μ and is pictured as the arc on the circle of the entry gear between the points S1 and C1, where B1S1 is parallel to AB. The remainder λ is related to μ by the simple gear ratio. So now all components exist for deducing the target formula.
The diagram above represents the entire process of deducing the target formula by grouping subjects of it. At first, there is one design constraint of equality of two gear ratios to some K value. At second, there are target definitions of the reposition of primary meshing point on the pitch gear with followed definition of the variation angle itself. At third, there are two constant definitions for base angles for the neutral case. At fourth, there are primary definitions and relations for particular position of the pitch gear, including: base angles, for the shifted case, reposition of the primary meshing point on the center gear and simple equations for variations of these base angles. At fifth, there are secondary definitions, including: the first remainder at its complementary, angular variation of the cluster, and the second remainder with related change of meshing position. All these last-level four subjects, fusing together, bring intermediate relations, which are resolved by simple algebra to the result relation. The result goes to simplifying, based on constant sum of angles in triangle, providing the final result, which states: the pitch variation is equal to product of inversion of variation of the summit angle on the sum of one and reciprocation of the common gear ratio, where the gear ratio is defined as ratio of radius of the center gear to radius of the entry gear.
The diagram above represents a data flow and definition for an end use application of the pitch variation formula for particular distance of the pitch gear from the central gear. At first, the application routine should be initialized by constant definition values. At second, this initialization should be continued by value from constant relation, which is based on the cosine theorem and provides a value of the summit angle in the neutral case. After that, the routine can acquire an input of particular angular position of a wing, which equal to the angular position of its pitch gear and calculate an instant distance, using its instant definition. After that, the routine should substitute values from all mentioned subjects to the chain of instant relations and calculate the final variation value for its output.
The plot below represents a particular result of using the pitch variation formula, as a pitch deviation distribution over entire wings positions of the rotor.
The result is represented for width set of radial offsets of central gear for positive and negative gains. Sign of the Δr value is used for referencing to some gain selected on such way so it is same as the sign of the gain itself. So its ratio to R0 is used there as a gain parameter, which can be referenced as a linear gain. The result corresponds to the four-gears placement pictured on the gears geometry chart, which I reference as a normal assembling. But the cinematic scheme, which was represented before, uses other variant of assembling, which I reference as an inverted assembling. Correspondent result for the pitch deviation distribution over entire wings positions of the rotor for case of the inverted assembling is represented on the plot below.
There sign of Δr was changed for be same with the gain itself. The result reflects some advantage of the variant of inverted assembly: main operation modes utilize negative gain and have highly loaded wings near main point, which near of phase 0.25, so those wings have lower pitch deviation for case of using of the inverted assembling, permitting to have more exact handling and steering of those wings.
The chart above represents features of both normal and inverted assembling respectively by comparative way. For the last, the entry gear and the steering pinion are placed in the upper elongation relative to the pitch gear and the center gear on side of the zero-Skew direction. Also positive direction of the linear gain is represented for both variants as the black arrow over Main<―>Opposite indicator for the zero-Skew articulation.
Special interest has behavior in change of pitch in the main and opposite positions upon changing of the linear gain in its entire range. Result of this kind calculation is represented on the plot below.
Also can be interesting changing of the angular gain itself upon changing of the linear gain in its entire range. Result of this kind calculation is represented on the plot below.
The plot also introduces a Normalized Linear Gain (NLG, Gn), which is equal to the ratio of the linear gain to some maximal linear gain value related to a maximal constructive limit in offset of the central gear, or simple equal to ratio of the current offset to its limit. The last definition is pictured on the plot below the alternative scale based on the normalized linear gain. This normalized variant of gain is very useable for gain indication in case of using a mechanical indicator, since mechanics much simpler and exact upon measuring of a linear displacement.
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