Damping Ratio

Since the ratio of rotational frequency to natural frequency for a rotor with hinge offset is lower than unity, the phase angle will be less than 90°, as can be seen from Figure 7.3. How much less depends on the damping ratio, c/cak, where ccrit is critical damping. The concept of critical damping comes from the study of a single pendulum-spring-damper system that has the differential equation:

/p + cp + £p = 0

The general solution to this equation is:

If the term under the radical is negative, the motion will be oscillatory. If it is positive, the motion will have a pure convergence with no oscillations. Critical damping is the value of c that makes the radical term zero—that is, that lies on the boundary between oscillatory and nonoscillatory motion:

(caJ22 = k

or

cr’c ^undamped

The damping of the blade, c, may be evaluated from the aerodynamic hinge moment due to flapping velocity:

MA = j r’ — acr’fi (r’ + e)Cldr

(Reader, please note that italic c stands for chord and roman c for damping; and that italic e stands for hinge offset and roman e for the base of natural logarithms.)

1 e

c =

e

Y _

A useful nondimensional parameter relating the inertia and aerodynamic characteristics of a blade is the Lock number, y, which was already introduced in Chapter 1.

cpaR4

For most blades, у is between 6 and 10. Adding a large tip weight such as a jet engine may lower

The factor, y/16, is a universal parameter that will arise again when rotor damping as a whole is discussed.