An articulated rotor uses flapping, lead-lag, and feathering hinges (or elastomeric bearings) in an appropriate sequence.

The flapping hinge is employed to relieve the blade-root bending moment arising from the lift and centrifugal forces.

The lag hinge to alleviate the in-plane moment at the blade root due to drag force

The feathering (or pitch) bearing to change the angle of attack of the blade

There are three hinge sequences:Flap-Lag-Pitch (F-L-P)

Flap-Pitch-Lag (F-P-L)Lag-Flap-Pitch (L-F-P)

Sikorsky designs tend to use an L-F-P hinge sequence with colocated hinges.

Elastomeric bearing rotor hub system

In this section, the position and velocity of an element of a rotor blade are calculated. To express these vector quantities, the following coordinate systems are used. Different coordinate systems are used for the three hinge sequences considered.

The blade is assumed to be rigid. The pitch bearing is outboard of the flapping hinge and the flapping hinge is outboard of the lag hinge.

: lag hinge offset (m): distance b/w flapping and lag

hinges (m): lag angle

L-F-P Hinge Sequence

The flapping hinge is assumed to be perpendicular to the rotor shaft. The lag hinge is assumed to be parallel to the shaft. The feathering axis is assumed to coincide with the blade span axis and the center of gravity and aerodynamic pressure center of the blade sections are assumed to be on the blade span axis. With these assumptions , the following coordinate systems are used.

X1,Y1,Z1 - With its origin on the flapping hinge F; Z1 parallel to the shaft and pointing downward; Y1 parallel to the blade segment between the two hinges (coinciding with the feathering axis when flap angle is equal to zero) and pointing outward; Z1, Y1, and X1 forming an orthogonal set.

X2, Y2, Z2 - With origin on the lag hinge L; Z2 parallel to Z1; Y2 parallel to the blade segment between the shaft and pointing outward; Z2, Y2 and X2 forming an orthogonal set.

X3, Y3, Z3 - With origin at the hub center, otherwise the same as X2, Y2, Z2 system. This coordinate system will be referred to as the rotating shaft system.

X,Y,Z - This axis system will be referred to as hub-body system with origin also at the hub center; Z parallel to Z3; X pointing forward to the nose of the aircraft; Y toward the right. This system is related to the rotating shaft system through the azimuth angle.

It is important to note that the sign convention regarding the flapping and lagging angles. Positive for Flapping up and for lagging back.

With this sign convention, the component of the position vector r with respect to the rotating-shaft and the hub-body systems are respectively

The hub-body system is now related to the aircraft body-axis system and an inertia system . Note that we consider a shaft tilt angle i, and a general offset of the hub center of the rotor system from the center of gravity of the aircraft

The center of gravity of the aircraft is translating (), accelerating linearly (), and rotating () with respect to the inertial system.

Vector quantities are now expressed in terms of the hub-body system

Note that when expressed in the hub-body system ( also in the aircraft body-axis system)

Eqn. (3)

components of the offset between the aircraft center of gravity and the hub center in hub-body system ( positive when the hub center is forward of aircraft c.g.; positive when the hub center is to the right of aircraft c.g.; positive when the hub center is below the aircraft c.g.)

p,q,r = components of aircraft angular velocity in hub-body system, deg/sec

= components of aircraft angular acceleration in hub-body system, deg/sec^2

R = Rotor blade radius measured from the center of rotation, (m)

X,Y,Z = Hub-body system = Body axis system = Inertia system

= Components of velocity at the aircraft center of gravity in hub-body system (m/sec)

= Components of velocity at blade element due to rotation, flapping, pitching and lagging motion (isolated rotor) in hub-body system, (m/sec)

owing to the rotation of the system with respect to the inertia frame

Thus,

Relationships among Inertia, Hub-body and body-axis systems

It should be noted that, due to the shaft tilt angle i, the components of the vector quantities expressed above differ from those which are expressed in the aircraft body-axis system by a transformation

The linear velocity and angular velocity of the aircraft c.g. when expressed in the aircraft body-axis system, then in the hub-body system.

If the shaft has a and

If the shaft has a sideward tilt with an angle j, in addition to the forward tilt angle i, then is to be replaced by , which is a result of the sequence of rotational transformations.

The aircraft body-axis system,(), is related to the inertia system, () through the usual sequence of rotation of the Euler's angles,(). The components of a vector expressed in the two frames are related by the following transformation

From the figure, it is clear that the sign convention for the components () is as follows: is positive when the hub center H is forward of aircraft c.g. (along x-axis); is positive when H is to the right of c.g. (along y-axis); and is positive when H is below aircraft c.g. (along z-axis)

with the above considerations we are now in a position to determine the total velocity of the blade element.

The position vector with respect to the aircraft c.g. is:

expressed in the hub-body system,

Where are given already

The total inertial velocity of the blade element is

The total inertial velocity of the blade element is

Note that

Because

When expressed in the hub-body system, ()() can be obtained by a simple differentiation of equation given below w.r.t time to yield

The other terms on the R.H.S are due to aircraft motion

In the subsequent development, the velocity components of the blade element (eq.(7b)), which are due to aircraft motions, will be denoted by Equations (5),(6), and (7) thus completely define the total velocity components, in the hub-body system, of the blade element for the L-F-P hinge sequence.

7(a)

7(b)

7(c)

In mathematics, and in particular linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the condition -A = AT

The vector cross product also can be expressed as the product of a skew-symmetric matrix and a vector: