Add Simplification

dM ж*-1»*+

Nearly the same results are obtained if the helicopter is assumed to be constrained vertically so that the Z-Force equation can be eliminated. Now the equations are:

It is often useful to see a more graphical representation of the system. One such is the block diagram of Figure 9.8. An even more graphic illustration is the mechanical analog of Figure 9.9, in which viscous dampers and screw jack actuated beam-rider weights are used to generate appropriate forces and moments.

The characteristic equation is:

, / 1 дХ 1 дм 2 g dM

si ——————– +———- s2 + л——– = 0

G. W./£ dx I„ dq } I„ dx

or, for the example helicopter:

i3 + .724j2 + .115 = 0

whose roots are:

sx = —.87, i2 } = .075 ± .355/

FIGURE 9.8 Block Diagram for Two-Degree-of-Freedom System Representing a Hovering Helicopter

FIGURE 9.9 Mechanical Analog of Hovering Helicopter

Note that these roots are almost the same as those obtained when the Z-Force equation was included. The only root missing represents the damped plunge mode.

The calculated period of the oscillation is 17.7 seconds, and the time to double amplitude is 9.2 seconds. Comparison with the same parameters obtained from the more complete set of equations demonstrates that some simplifying assumptions can often be very useful in reducing the complexity of the analysis of dynamic systems, especially when the modes of motion are only weakly coupled.

The procedure can be carried even a step further if only the period of the oscillation is required. This was first pointed out in reference 9-4 by Hohenemser, who noted that since the helicopter is oscillating about a point far above, the moment of inertia about its own center of gravity can be neglected. (This is analogous to a child on a swing.) If this suggestion is followed, the characteristic equation reduces to:

This equation has the form of that of a single-degree-of-freedom system consisting of a mass on a spring:

ms2 + k = 0

for which the natural frequency is:

or in this case:

For the example helicopter this simple approach gives a period of 15.7 seconds for the longitudinal oscillation in hover. Note that under the assumption of no fuselage moment of inertia, this value applies just as well to the lateral oscillation.

A generalization can be stated about the period of oscillation—it is essentially proportional to the square root of the rotor radius. The demonstration makes use of the following line of reasoning:

In hover

д я. q у

-r—^ = — 0O + 20j — 2

Jp J ‘ 1 ClR

and from Chapter 3, with the tip speed ratio, |i, set to zero:

Thus with only a small white lie concerning the coefficient of the induced velocity ratio:

. 16CT/g

dp a

The damping derivative with zero hinge offset is:

^ai, 16

dq у R

When these are substituted into the equation for the period, we have:

If it is assumed that most modern helicopters have nearly the same values of CT/o and Lock number, y, as the example helicopter, the period can be roughly approximated by:

P = 3-2 V/? sec

(it is interesting to note that the period of a pendulum is approximately V/73 seconds and so, by analogy, the helicopter is swinging from a support 33 radii above it.)

A human can learn to control an unstable vehicle like a hovering helicopter as long as the period is significantly longer than the total time delay in perceiving an error, processing the information in the brain, and moving the appropriate control to correct the error. The period of very small one-man helicopters

approaches the lower limit and of radio-controlled models generally goes below unless some means is used to change their characteristics. For this reason, most models use a version of the Hiller servorotor system that increases the damping by an order of magnitude and hence increases the period enough to make it compatible with the capabilities of the ground-based pilot.