Coning of a Rotor in Hover

A blade—whether hinged or cantilevered from the hub—seeks an equilibrium coning angle that is a function of the lift, centrifugal forces, and blade weight. The magnitude of the coning, a0) may be found by setting the moments at the hinge— or at the effective hinge—to zero:

^hingc = Afjift + Mcf + Mw — 0

For most engineering calculations of coning, it is sufficiently accurate to assume that the hinge is at the center of rotation. In this case, the moment due to lift is:

UL, — rdr

dr or for a blade with ideal twist:

Mlift = f – CL2Rac(Qt – ф,)гУг = – —Cl2pacR4

l 2 ‘ 3 a

The moment due to centrifugal force is:

where m is the mass per running foot and Ih is the blade’s moment of inertia about its flapping hinge. (It can be assumed to be equal to the rotor’s moment of inertia divided by the number of blades.)

<*0

radians Ф1

Setting the sum of the three moments equal to zero and solving for a0 gives:

The combination of terms, pacR*/Ih is a nondimensional parameter, y, known as the Lock number after a pioneer British autogiro aerodynamicist, C. N. H. Lock. It represents the ratio of aerodynamic to centrifugal forces. Two blades that have the same airfoil and aspect ratio, and are constructed of material with the same density, will have the same Lock number no matter what their radius. The heavier the blade, the lower the Lock number. For operational blades, it varies from 10 for lightly built blades to 2 for blades of tip-driven helicopters with concentrated masses at the tips. The rotor parameters for the example helicopter are given in Appendix A. From these values:

The last term in the equation for coning is the contribution of the dead weight of the blade where:

= – gmrdr

If the blade has a uniform mass distribution, then:

X–W‘

and the coning equation may be written:

2 CT/a 2gR

As a general rule, the weight contribution is negligible except for very large rotors. For the example helicopter at its hover conditions; the calculated coning is 4.4° if the last term is ignored and 4.3° if it is included.

Although the derivation of the equation for coning was based on the assumption of a blade with a flapping hinge, it is also valid for flexible blades cantilevered from the shaft without hinges since the aerodynamic and centrifugal forces overpower whatever structural stiffness the blades might have.

Some past analyses have mistakenly led to the conclusion that there is an upper limit to how big a rotor can be built because of excessive coning. That this is an erroneous conclusion is evident from the foregoing equation provided the Lock number, y, and the blade loading coefficient, CT/a, are constants. As a matter of fact, the coning will actually decrease because of the weight term as radius is increased while holding the same tip speed.

The equation provides a simple method of estimating the maximum coning that a rotor can develop. Earlier it was shown that CT/o is approximately 1/6 of the average lift coefficient, ct. Assuming a maximum value of 1.2 for this parameter gives a maximum value of 0.2 for CT/o. Later discussions will show that this is a reasonable upper limit. For the example helicopter with a blade Lock number of 8.1, this results in a maximum coning of just over 10°.

Review of Assumptions

The discussion of hover performance up to this point has been based on a number of simplifying assumptions. For normal rotors in normal flight conditions, the use of these assumptions gives good results consistent with a first approximation method; but when more accuracy is required, the assumptions must be challenged and their effects evaluated. The most important of the assumptions are:

• Assumption: The lifting portion of the blade extends from the center of rotation to the extreme tip.

Challenge: The lifting portion of the blade actually starts sonle distance outboard from the center, and the tips are only partially effective.

• Assumption: Induced velocities are uniform over the disc.

Challenge: Induced velocities are nonuniform.

• Assumption: The blades have ideal twist.

Challenge: No actual blades have ideal twist.

• Assumption: The blades are rigid torsionally so that no structural twisting takes place.

Challenge: No blade is infinitely rigid, and a rotating blade is subjected to twisting moments from several sources.

• Assumption: Blades have constant chord.

Challenge: Some blades are tapered.

• Assumption: The wake does not rotate.

Challenge: The wake does rotate.

• Assumption: There is no effect of tip vortices on the angle of attack of a following blade.

Challenge: Tip vortex interference has been found to be significant.

• Assumption: The airfoil lift and drag characteristics are the same as the NACA 0012 characteristics of Figure 1.10.

Challenge: Many rotors use airfoils different than the NACA 0012.

• Assumption: The airfoil characteristics are not a function of local stall or compressibility effects.

Challenge: Stall and compressibility effects impose significant power penalties in some flight conditions.

• Assumption: There are no effects due to radial flow.

Challenge: There are some effects due to radial flow.

• Assumption: The rotor is far from the ground.

Challenge: In some cases the rotor is close to the ground.

Some of these assumptions result in errors in the calculation of the hover performance; others are good assumptions in that their consequences are negligible. In order to differentiate between the two types, the assumptions will be discussed one at a time.