Our heavyweight helicopter equal in the world does not have
In Rostov started production of the most load-lifting rotary-wing car The Russian holding «Helicopt[...]
Routh’s Discriminant
It is not necessary to solve for the roots of the characteristic equation to determine whether the system is stable or unstable. Instead, we have only to examine the combination of coefficients known as Routh’s discriminant (R. D.). For a cubic equation of the form:
As* + Bs2 + Cs + D = 0 Routh’s discriminant, R. D. (3), is:
R. D.(3) = BC — AD For a quartic equation of the form:
As4 + Bs* + Cs2 + Ds + E = 0 Routh’s discriminant, R. D. (4), is:
R. D.(4)=BCD – AD2-B2E
(D must be positive)
For a quintic equation of the form:
As5 + Bs4 + Cs* + Ds2 + Es + F = 0
Routh’s discriminant, R. D. (5), is:
R. D. (5) = D(BC – AD)(BE – AF) – B(BF – AF)2 – F(BC – AD)2
Several tests of the discriminant are used to produce clues about the types of time history that are characteristic of the system:
|
If Then
|
The equilibrium equations of Chapter 8 can be converted into equations of motion by accounting for forces and moments corresponding to inertia effects associated with accelerations, either linear or angular, and combinations of velocities.
G W
XM + XT + XH + XV + XF= G. W. sin0 + —^(x – yr + zq)
£
^ G. W.
YM + YT + Yv + YP = – G. W. sin Ф +——————– (y + xr-zp)
і
G. W.
ZM + ZT + ZH + Zv + Zp – —G. W. cos 0 +——————- (z — x q + уp)
S
Rm + TMhM + ZMyM + YThT f Yvhv + YFhF + RF = lxxp — qr{lyy — Izz)
MM — XMhM + ZJM + MT — XThT + ZTlT — XHhH + ZHlH — Xvhv + MF + Z pip — Xphp = lyyq — pr (I zz — Ixx)
NM — YMlM — YTlT — Yyly + Nf — Ypip = Izzr — pq(Ixx — Iyy)
Figure 9.5 shows the sign convention used.
Thee centrifugal forces and the gyroscopic moments in the foregoing equations fall directly out of the analysis of the dynamics of a rigid body that can
|
FIGURE 9.5 Parameters in Equations of Motion |
simultaneously translate and rotate along and about all three axes. Many of us, however, need some graphical help to accept these terms. Figure 9.6 is an attempt to provide this help.
From a rigorous standpoint, the set of six equations of motion should be augmented with three more equations representing the coning, longitudinal flapping, and lateral flapping of the rotor, which is not attached very rigidly to the airframe. As was shown in Chapter 7, however, the time constant for the flapping of conventional rotor blades corresponds to one-quarter to one-half of a rotor
|
FIGURE 9.6 Illustration of Sources of Centrifugal and Gyroscopic Terms |
revolution. This rapid response justifies the use of the quasi-static assumption, which eliminates blade motions as separate degrees of freedom and simulates replacing the rotor with a black box at the top of the mast, which essentially produces forces and moments instantaneously in response to changes in flight condition or control inputs.
There are some studies, however, in which the coning and cyclic flapping of the rotor on a short time basis cannot be ignored. These include the prediction of the immediate response to a gust or to a control input or the design of a high-gain
stability and control augmentation system (SCAS). For these situations, methods for writing the expanded set of equations are outlined in reference 9.1.
Accepting the quasi-static assumption, the set of six equations is a perfectly general representation of the helicopter in flight and can be used in sophisticated computer programs and flight simulators in which the instantaneous values of the aerodynamic terms are continually calculated as a function of the flight conditions. Even without a computer, the equations of motion can be used in a meaningful analysis by converting them into linear partial-differential small-perturbation equations and treating them as the equations of a multi-degree-of-freedom spring-damper – weight system.
The use of the linear differential equation method of analysis relies on the assumption that the change in the conditions of a dynamic system can be represented by superposition of the individual linear effects of changes in the independent variables. In such a complicated system as an aircraft in flight, the linearity is only rigorously true for very small changes or "small perturbations” from the trim values of the independent variables; but experience has shown that the assumption is valid enough to make the method very useful for the stability analysis of all types of aircraft.
Using the longitudinal force equilibrium equation as an example, the conversion goes like this:
1. Gather all terms on the left-hand side into a single aerodynamic term, X, and use small-angle assumptions on the right-hand side.
G. W.
—— (x-yr + zq)
Note that in this analysis the linear displacements along the body axes, X, Y, and Z are used as prime degrees of freedom. It is more common to see the corresponding velocities x, у, and z (or u, v, and w) used, since the forces and moments are functions of velocities and accelerations but not of the displacements of the aircraft in flight. Two special problems, however, have influenced the choice of displacements instead of velocities: flight simulation of manual or automatic hover over a spot, and the dynamic analysis of a helicopter mounted in a wind tunnel on moderately flexible supports. Although neither of these problems will be addressed here, the present choice of variables allows them to be studied when they arise while putting practically no unnecessary complication into more routine
studies.
2. Rewrite the left-hand side in terms of stability derivatives multiplied by pertinent changes in the six degrees of freedom and the four control inputs. Since the aerodynamics are affected only by velocities (with an exception to be covered later), only the first derivatives with respect to time appear in the equations, and all these variables as well as control motions are understood to now be changes from the initial trim conditions.
Note that the terms associated with the control inputs—main rotor collective pitch, tail rotor collective pitch, and rotor cyclic pitch—have all been gathered on the right-hand side in order to separate free from forced motion. The aerodynamic terms may be seen to be functions only of velocities with the exceptions of dZ/dz and dM./dz, which will later be shown to be associated with the time required for a change in rotor downwash to reach the horizontal stabilizer.
There are three methods for studying the flying qualities of the helicopter using these equations. The first method, discussed in the previous section, involves substituting the general solution in the form:
x = x(j)e"
From this, the characteristic equation can be formed; evaluated by Routh’s discriminant; and/or solved for its roots, which can be related. to periods and damping. This classic method will be illustrated in the remainder of this chapter. A second method involves programming the equations on a so-called analog computer. This uses electronic components consisting of capacitors, resistors, and
inductances to represent spring, damper, and mass terms, respectively, in the equations as if they were written for mechanical systems. Voltages within the circuits represent various degrees of freedom and can be used to produce time histories in response to simulated gusts or control motions. This method was in wide use from about 1955 to 1975, but has now largely been supplanted by the third method, using digital computers, which can both do the classical analysis and produce time histories more accurately than can the analog computers.
In any case, it is necessary first to write the equations of motion in numerical form by evaluating all the derivatives.
The stability derivatives can be evaluated by several methods. The most accurate is, naturally, also the costliest unless it is to be used over and over. That method consists of programming a computer to do the type of numerical calculations illustrated in Chapter 8 to solve first for the trim conditions and then for the changes in aerodynamic forces and moments due to sequential unit variations in each degree of freedom and control input. The heart of this type of program is the subroutine that analyzes the main and tail rotors. Programs with various levels of sophistication are operational in all the major helicopter companies and research organizations. A description of one will be found in reference 9-2.
A noncomputer means of obtaining values for the stability derivatives of the main and tail rotors consists of differentiating equations for the aerodynamic coefficients and flapping angles such as those found in Chapter 3. This procedure, widely used before computer technology matured, still has value for quick analysis and for verifying the more complex computer programs as they are being developed. A description of this method, along with results in the form of charts, is given in reference 9-3.
In the following pages, a third method will be used. It makes use of the rotor performance charts in Chapters 1 and 3 to determine changes to the non – dimensional rotor coefficients as basic parameters are varied.
In many analyses, the derivatives are presented as the ratio of the change in forces divided by the mass of the aircraft and the change in moments about each axis divided by the appropriate moment of inertia. That approach has two advantages: it makes the equations of motion look more compact, and it provides derivatives that can be compared with respect to magnitude from one aircraft to another even if they vary widely in size. This latter reason seems to be of more significance for fixed-wing aircraft than for helicopters, for which many different fuselages an be supported by rotors that are essentially alike. One disadvantage with this system is that these derivatives change magnitude with gross weight and with loadings that change the moments of inertia. In this book, we will leave the derivatives in their dimensional form in order to avoid making them any more abstract than they already are.
Rotor Derivatives near Hover
In the case of hover, only the aerodynamics of the two rotors are important, and the required derivatives can be simply evaluated using the charts and equations already presented in Chapters 1, 2, 3, and 7. The significant rotor derivatives are listed in Tables 9.1 through 9.4 with numerical values corresponding to the example helicopter. Table 9.1 gives the basic rotor derivatives. Tables 9.2 and 9.3 are separate tables for the dimensional derivatives of the main and tail rotors. Finally Table 9.4 is a summary table for the entire helicopter. These tables can serve as checklists for the analysis of any other helicopter.
|
TABLE 9.5 Rotor Derivatives in Forward Flight from Charts
|
|
Partials with Respect to Oq; Constant A’ |
|
FIGURE 9.7 Illustration of Rotor Derivative Extraction from Performance Charts of Chapter 3 |
The basic partial derivatives for the example helicopter have been obtained from the performance charts of Chapter 3 in level flight at 115 knots (jl = .3) and are tabulated in Table 9.5 in the form:
d Dependent Variable d Independent Variable
The derivatives with respect to |i have been obtained by taking the difference between charts for p = .25 and p = .35. The others are from the charts for jl = .30. Since the rotor parameters are plotted against CT/<5, it is used as an intermediate variable, as illustrated by Figure 9.7.
Other simple derivatives needed in the analysis are listed in tables 9.6 and 9.7 along with values for the example helicopter at 115 knots.
