# What Routh’s Discriminant Tells Us

Even without solving for the roots, the characteristic equation can be made to yield useful information by using Routh’s discriminant (R. D.). For a cubic equation:

R. D.(3) = BC — AD

or in this case:

g dM

Iyy dx

Table 9.17 lists the tests of Routh’s discriminant.

TABLE 9.17

 Tests of Routh’s Discriminant Test Consequence Application to Hovering Helicopter 1. All coefficients are positive. No pure divergence For most helicopters, both dX/dx and dM/dq are negative, thus В is positive. dM is positive, thus D is positive dx 2. R. D. is positive. No unstable oscillation Since дМ/дх is generally 3. R. D. = 0 Neutrally stable positive, this test is failed. Requires that дМ/дх be zero 4. R. D. is negative. Unstable Generally true due to sign of дМ/дх 5. D = 0 6. One coefficient is Non-oscillatory neutral stability Only true if дМ/дх = 0 Pure divergence or unstable No negative coefficients for negative. oscillation conventional helicopters

Note the strong role that the sign of the speed stability term, дМ/дх, plays in these tests, with a positive value leading to instability. (This is in contrast to the forward flight situation, where a positive sign on the speed derivative is usually necessary for stability, as will be discussed in the next section.) The equation for the derivative shows that it is made up of both hub stiffness (through offset flapping hinges, for instance) and the tilt of the thrust vector with a moment arm proportional to mast height. For a conventional helicopter, both effects have the same sign thus leading to Test 4 predicting an unstable oscillation. If the rotor were mounted under the helicopter, the contribution of the tilt of the thrust vector would change sign. For a teetering rotor with no hub stiffness, the sign of the entire derivative would be reversed as shown in Figure 9.10. Routh’s discriminant would then satisfy Test 2, and no unstable oscillation could occur. Test 6, however, would indicate a pure divergence, represented by the tendency of this helicopter to drift off into translational flight.

Several helicopters have actually been built with the low rotor position. One was the De Lackner "stand on” configuration shown in Figure 9.11. This

FIGURE 9.10 Effect of Rotor Location on Sign of Speed Stability Derivative

helicopter, however, had hingeless blades with high inherent hub stiffness, and consequently its stability did not benefit significantly from its unique rotor location. [13]

thrust as required and that in this case longitudinal control is used to prevent any pitching motion. If the purpose of the analysis is to study cross-coupling problems, these assumptions will have to be dropped and, instead of simply using two or three equations, all six equations should be treated simultaneously. The method of solution is the same as for the more restricted situation. The mathematical manipulation to solve for the roots becomes very tedious if done by hand but very easy if turned over to a modern computer.