# EQUATIONS OF FLAPPING MOTION

Development of the equations of flapping motion begins by considering a rotor system with a single hinge mounted a distance eR from the axis of rotation. The shaft rotates with a constant angular velocity, ), and the blade flaps with angular velocity, |3. If axes are fixed in the blade (see Fig. 4.3) then the dynamic situation can be described using [4.1]:

L = Ixx m і + Ixx m2 m3 I

M = Iyym2 – Iyym3 m1 – mbXgRaz V (4.9)

N = (Ixx + Iyy)m3 – (Ixx – Iyy)m1 m2 + mbXgRay J Fig. 4.3 Rotor axes system.

The axes are set to be parallel to the principal axes with their origin at the hinge, such that the i-axis lies along the blade span, the j-axis is perpendicular to the span and the k-axis completes the right-hand set. Although the blades are in reality highly flexible, it will be assumed, for simplicity, that they are rigid. If Ixx is the moment of inertia about i and Iyy is the moment of inertia about j then Izz, the moment about k, is given by Izz = Ixx + Iyy. Note that m1, m2 and m3 are the angular rates about the principal axes and ax, ay and az represent the acceleration of the hinge.

4.4.1 Pure flap motion

Analysis begins by assuming that the blade is constrained so that only flap motion is possible, see Fig. 4.4. From Fig. 4.4 it can be seen that the angular components of velocity, m1, m2 and m3 are given by:

m1 =) sin p ‘j

m2 = -p і (4.10)

m3 = )cos p J

Also the absolute acceleration of the axis system, a0, is given by:

a0 = ax i + ay j + az k

a0 = – Q.2eR cos pi + )2eR sin p k

Hence:

ax = — Q2eR cos P ‘j

ay = 0 і (4.11)

az = )2eR sin P J

Now substituting Equations (4.10) and (4.11) into Equation (4.9) and assuming small P:

L = 0

— M = 1yyP + )2(iyy + mbXgeR)P > (4.12)

— N = 2 )1yy psin P J

In other words if a rotating blade flaps upwards then in order to maintain it within the flapping plane no feathering moment is required but a rearwards moment of 2)1yypsin p is necessary. This moment is equal and opposite to the Coriolis moment acting on the blade; it is usually so large that it is often relieved by a drag hinge. Therefore if the blade is hinged so that it is able to move in an in-plane sense then as the blade flaps up it will move forward or lead.