Category Helicopter Performance, Stability, and Control

The longitudinal trim equations can be used to evaluate speed stability, or, as it is sometimes confusedly called, "longitudinal static stability.” The question here is whether with an inadvertent increase in speed with controls held fixed, the helicopter will pitch up and slow down, exhibiting speed stability, or pitch down and speed up in an unstable manner. The rotor flapping is always stabilizing, producing increased longitudinal flapping with increasing speed and a resulting nose-up moment, but the effect of the other components of the aircraft may be either stabilizing or destabilizing. The most important component in this regard is the horizontal stabilizer. If it is carrying a download at the initial trim condition, it should develop more of a download as speed is increased, thus producing a stabilizing nose-up pitching moment. Figure 8.26 summarizes the possibilities for several initial stabilizer loadings. In reality, the contribution of the horizontal stabilizer is modified by changes in angle of attack which accompany the changes in speed. For example, the helicopter’s rate of descent that will exist at the higher- than-trim speed with the collective fixed at its trim value will increase the stabilizer’s angle of attack. This is destabilizing. The draggier the helicopter, the more destabilizing is this effect, since a higher speed will require a higher rate of descent to achieve it. Another destabilizing change in angle of attack can be traced to the decrease in main rotor downwash at the horizontal stabilizer at the higher speed. This effect is especially significant on tandem rotor helicopters where the aft rotor is directly affected by what the front rotor is doing.

The magnitude of the stability—or the instability—in terms of the change in moment per unit change of speed can be found using the charts of Chapter 3 along with the longitudinal equilibrium equations. For this calculation, the equilibrium conditions are determined in level flight as has just been done and then recalculated at a slightly higher (or lower) speed assuming no change in collective pitch to represent an inadvertant speed change. A change, however, in tail rotor pitch is allowed to maintain zero sideslip as a pilot would do by instinct. For this recalculation, the seventh illustrative example, "Helicopter in Dive at Constant Collective Pitch” of Chapter 3 can be used to establish the new conditions including the angle of climb, yc (or dive, yD). The new trim condition for the example helicopter at 135 knots (p = 0.35) with the same collective pitch as at 115 knots (p = 0.30) are listed in Table 8.6.

Solving for the trim solutions as was done at 115 knots gives:

TM = 20,496 lb

From the chart of Chapter 3 for a tip speed ratio of 0.35 and the trim collective pitch and thrust coefficient:

й, + В, — 8.6

SM

Thus the longitudinal cyclic pitch is

= 8.6 + .7 = 9.3°

The comparable value at 115 knots was 8.9°. Thus the calculations have demonstrated that in this speed regime, the example helicopter has speed stability since the pilot must hold more forward cyclic stick to dive at 135 knots than to fly level at 115 knots. Had he not done so, the increase in speed would have resulted in a nose-up maneuver with a subsequent decrease in speed. Figure 8.27 shows the results of this analysis in terms of stick position. The rigging curves of Appendix A have been used to convert cyclic pitch to stick position.

The curvature of the collective-Fixed speed sweep line on Figure 8.27 can be traced to the two destabilizing effects of the change in angle of attack discussed earlier. For this example, the dynamic pressure increased by 37% in going from 115 to 135 knots, but the download on the horizontal stabilizer increased only by 10%. This type of curvature is often seen in flight test results.

To a pilot, speed stability is seen as the change in stick position required to maintain a new speed. For example, on an unstable helicopter, an increase in speed

will initially require a forward stick motion to accelerate; but when finally trimmed out at this new speed, the stick will be aft of its initial trim point. Such a helicopter is possible to fly, but will tend to wander off its trim point unless constantly corrected. This may not be much of a problem in normal flight, but it is generally considered unacceptable for instrument flight, where the cues are poor and the pilot has other things to worry about. As a general rule, pilots prefer a level of speed stability which is just slightly positive. Too much speed stability could result in running out of forward stick travel—or at least putting the stick into an uncomfortable far-forward position at high speeds.

It should be pointed out that the change in pitching moment with speed is unique to rotary-wing aicraft and does not normally exist for fixed-wings. For airplanes, any change in control position while changing speed is due only to the resulting change in angle of attack and not to the change in speed itself (except for high-speed airplanes, for which compressibility effects might be important). Airplanes also have inherent speed stability through another mechanism, however. For a given propeller pitch or jet power setting, the change in forward propulsive force is stabilizing; that is, if the airplane slows down, the propulsive force increases, thus accelerating the aircraft to its original trim speed—an automatic "cruise control.” A rotary wing aircraft that has auxiliary propulsion, such as an

autogiro or a compound helicopter, will also benefit from this inherent speed stability and need not be subjected to the analysis and tests that are necessary on a pure helicopter, which gets its propulsive force from tilt of the rotor.

The foregoing method is rigorous in accounting for the contributions of every aircraft component to the equations of equilibrium. A study of the numerical values in the example shows, however, that only a few terms dominate; the others have little effect on the final solutions. This observation leads to an approximate method involving solving the moment equation using only initial trim values to give the approximate value of the longitudinal flapping:

TABLE 8.5

Elements of the Longitudinal Equilibrium Equations for the Example Helicopter at 115 Knots

Physical Dimensions (See Appendix A)

Initial Trim Forces (From Chap 3)

Dimension

Main rotor disc area Main rotor shaft incidence Main rotor long, offset Main rotor vert, offset Tail rotor long, offset Tail rotor vert, offset Horiz. stab, area Horiz. stab, aspect ratio Horiz. stab, incidence Horiz. stab, angle of zero lift Horiz. stab. long, offset Horiz. stab. vert, offset Vert. stab. long, offset Vert. stab. vert, offset Fuselage long, offset Fuselage vert, offset

The numerical values called for in this equation were all given in Table 8.5. The result is:

ax = -.019 rad = — 1.Г

This is exactly the same as the —1.1° calculated by the more rigorous method. The pitch attitude, 0, can also be approximated since:

/ Д» ^ =0 + Ar + Af + Dy + ДЛ

aTpp = 0 + <*■+/* = – —————– =—– z——–

V G. W. – L„ — Lf )

For the example calculation:

%-ax = —.038 rad = —2.2*

and thus

© = -2.2 + 1.1 =-1.1° which compares to the more exact value of —0.9°.

The fuselage will produce side forces, yawing moments, and rolling moments as a function of sideslip angle. If the calculations are being done without the benefit of wind tunnel tests, the fuselage sideforce (in wind axes) and the yawing moment can be estimated using the same procedures outlined for fuselage lift and pitching moment above. The rolling moment of the fuselage due to sideslip is caused by its dihedral effect and can be either slightly positive or slightly negative. A method for estimating this moment using the physical parameters of the fuselage and wing is given in reference 8.1, but since flight test experience has shown that the dihedral effect is strongly, but mysteriously, influenced by the interference of the main rotor wake on the tail rotor and the vertical stabilizer, the estimation of fuselage dihedral effect has a very low priority, so the method will not be repeated here.

M/q = C/w (Fuse. Volume)

Source: Harris et at., “Helicopter Performance Methodology at Bell Helicopter Textron," AHS 35th Forum, 1979.

For an airplane—which has lateral symmetry—the six equations of equilibrium can be conveniently dealt with in two groups; the longitudinal equations consisting of X, Z, and M and the lateral-directional equations consisting of Y, R, and N. A helicopter is not quite as symmetrical as an airplane, and there is some cross­coupling as described in Chapter 7. Thus, from a rigorous point of view, the trim equations should be determined from a simultaneous solution of all six equations; but from a practical point of view, they can—and will—be treated as two independent sets.

The equilibrium equations in X, Z, and M can be used to find the longitudinal trim conditions of the helicopter. In Chapter 3 an approximate method was used based only on the equivalence of the X and Z equilibrium equations (in wind axes) which was satisfactory for performance calculations. That approximation ignored the pitching moments by assuming that the tip path plane was always perpendicular to the rotor mast. Now the moment equation will be used with the other two equations (in body axes) to find the magnitude of longitudinal flapping, along with the other parameters, which must exist if the aircraft is in trim.

There are two approaches for solving for the trim conditions from the equilibrium equations. The first is to write the three equations as Unear functions of three unknowns and then to solve them simultaneously. This method gives acceptable accuracy for most engineering purposes and will be illustrated in the following discussions. The second method is not constrained to Unear functions. It is done as an iterative procedure where values of the three unknowns are chosen initially based on estimates or guesses and then refined by going through the equations several times. This is well suited to high-speed computers and can readily be expanded to include aU six equiUbrium equations. As a matter of fact, in these types of computer programs, the trim conditions are simply faUouts of computing the performance using such methods as those presented in Chapter 3.

reduce the system to three linear equations in the three unknowns: 0, TM, and aX; In the equations of Table 8.4, the physical dimensions are, of course, known and the trim—or barred—terms for the aircraft components will already be known from previous performance calculations such as those done in Chapter 3- Table 8.4 also presents the numerical results for the example helicopter at 115 knots (|1 = .3). The elements that go into the equations for this example are tabulated in Table 8.5 which can serve as a check list for any helicopter.

Solving the three equations simultaneously yields the following solutions for the example helicopter:

Тд, — 20,5 86 lb 0 = -.0165 rad = -.9°

a. =-.019 rad =-1.1° –

SM

Besides the thrust, pitch attitude, and longitudinal flapping, the other trim value of interest is the longitudinal cyclic pitch. This can be determined from:

where the value of (Bx + ax ) is found from the performance charts of Chapter 3. For the example helicopter at 115 knots, this was determined during the illustration of the method for the "Entire Helicopter in Level Flight” to be:

(B, + %) = 7.8»

and now, knowing the trim value of longitudinal flapping calculated above:

Bx = 7.8- (-1.1) = 8.9°

Just as was done for the horizontal stabilizer, the aerodynamic characteristics of the vertical stabilizer of the example helicopter can be estimated from its physical parameters (Table 8.3).

Source: Von Mises, Theory of Flight, McGraw-Hill, 1944.

Fuselage

Conditions at the fuselage are shown in Figure 8.23. The various fuselage effects enter into the equations of equilibrium as:

X, = —Dp cos[0 – yc- e„f] + Lp sin[0 – Ус – гт]

YF = S. F.p cos p — DF sin p

ZF = —Lf cos[0 – yc – г„р] – Dp sin[0 – yc – £Mp]

MF = q{№/q)F, Nf = tf{N/q)F, RF =

Z

Rear View

FIGURE 8.23 Aerodynamic Conditions at the Fuselage

Lift, Drag, and Pitching Moments

Methods for estimating fuselage drag were outlined in Chapter 4. Methods for estimating fuselage lift and pitching moment prior to wind tunnel tests can be done either with the airplane methods of reference 8.1 or by direct comparison with wind tunnel results of previous helicopter fuselages. Figure 8.24 shows a compilation of the lift and pitching moment data as a function of angle of attack for several rather typical single-rotor helicopter fuselages in full-scale wind tunnel tests reported in references 8.18 and 8.19. The data has been nondimensionalized by dividing by the product of maximum fuselage width and length for lift data and the product of maximum fuselage width and fuselage length squared for moment data.

Another set of data reduced to coefficient form is given on Figure 8.25 for several fuselages of Bell helicopters. This data was taken from reference 8.2 and uses a different riondimensionalizing scheme than Figure 8.24.

The angle of attack of the fuselage is where

_ Vy TM

мр Vx 4qAM

The angle for zero lift can be assumed to be the same as for zero moment which in lieu of wind tunnel tests may be taken as the angle the aerodynamic cbrdline of the fuselage makes with the body axis.

In the equation for the drag of the vertical stabilizer, there are three new terms that must be evaluated; 6,; Cn, and АД. . The first two also occurred in the equation for the drag of the horizontal stabilizer and can be found by the same methods using Figures 8.17 and Figure 8.19. For the vertical stabilizer of the example helicopter, the required parameters have already been calculated for use in determining the slope of the lift curve. They are:

Xv= .21

A. R.^cff= 3.2

From Figure 8.17:

6, = .01

and from Figure 6.30 of Chapter 6, at a Reynolds number of 2 x 106 corresponding to a flight speed of 150 knots:

Cn = 0.0064

u0

The vertical stabilizer and tail rotor have a mutual interference sometimes called biplane effect, which results in the combination having a slightly higher induced drag than if they were isolated components producing the same side force and thrust. Using the methods derived for biplanes in reference 8.17, an equation for the additional interference drag can be written in helicopter terms:

where KlM is the interference factor, which is a function of the tail rotor radius, the span of the vertical stabilizer, and the separation distance between the two as shown in Figure 8.22. For the purposes of this estimate, the vertical stabilizer force, Yv, can be assumed to be the value calculated assuming no tail rotor interference effects; thus:

Qm у jv_

It V It

The zero lift angle of attack of the vertical stabilizer is affected by built-in camber. Some helicopters are designed with cambered fins to unload the tail rotor in forward flight in order to reduce loads in the blades and tail rotor control system. Despite some wishful thinking, this feature seldom reduces the total engine power required to fly at high speed, since the induced drag of the cambered vertical stabilizer will generally be higher than that of the tail rotor. If the camber is achieved by using a conventional airfoil, the angle of zero lift can be obtained from such collections of airfoil data as appear in references 8.1, 8.15, or 8.16. If the camber is achieved with a deflected trailing edge like a rudder, a simplified version of a method given in reference 8.1 can be used. The equation for the change in the angle of zero lift is:

where ct and (a5 /a5 ) are functions of the geometric parameters of the

Sidewash Angle

The sidewash angle at the vertical stabilizer is produced by lateral velocities induced by the main rotor, by the tail rotor, and by the sidewash caused by the

Source: Hoak, "USAF Stability and Control Datcom,” 1960.

fuselage in sideslip. There has been less attention paid to sidewash at the empennage than to downwash, so analysis will have to rely primarily on estimates. One set of available test data from reference 8.7 is given in Figure 8.21 for the Hughes AH-64 (without wings). The top figure shows the rather chaotic pattern existing at the empennage. (Note: These vectors are based on averages of some

Source: Logan, Prouty, & Clark, “Wind Tunnel Tests of Large – and Small-Scale Rotor Hubs and Pylons,” USAAVRADCOM TR-80-D-21, 1981.

even more chaotic instantaneous measurements.) The tendency of the flow to move to the right at the higher survey locations is due to the swirl in the main rotor wake. The lowest row of measurements is probably reflecting the effects of vortices generated by the nacelles.

The sidewash angle induced by the tail rotor may be assumed to be a function of the momentum value of induced velocity. Unless there is an appreciable separation distance, the effect may be assumed to be the same as that of the main rotor on the fuselage; that is:

where

In most cases, the sidewash induced by a sideslipping fuselage will be small enough to ignore. This is another way of saying that there is not yet much data on this effect. For those analyses that must at least give the appearance of completeness, it may be assumed that:

where this value is taken from Figure 8.15 for the downwash effect of the body alone. The final approximation is thus:

rf= 0.06|3

In calculating the slope of the lift curve, the effective aspect ratio of the surface must first be estimated. Since this can be influenced by the end-plating effects of the tail boom and a horizontal stabilizer, the estimating process must account for these components. The method of reference 8.1 makes use of the following equation:

where the three factors; (A. R.K/A. R.K+B), (A. R.V+B+H/A. K.V+B), and KH are given as functions of geometric parameters in Figure 8.19. As an illustration, the example helicopter parameters are:

Sv = 33 bv=U <>=4.25

2rx — 1.5 SH= IS x = 2.5 Ztf = o

The resultant values of the factors from Figure 8.19 are:

KH=M

and the resultant effective aspect ratio is:

A. R.Krff=3.2

Factor accounting for relative size of horizontal and vertical tails

Horizontal tail area Vertical tail area, measured from fuselage centerline Vertical tail span, measured from fuselage centerline Vertical distance between horizontal surface root chord and fuselage centerline, positive for surface below fuselage cpnterline

Source: Hoak, “USAF Stability and Control Datcom,” I960.

Once the effective aspect ratio has been determined, the corresponding slope of the lift curve, a„ can be found from Figure 8.6. For the example helicopter with a half-chord sweep of 27°, the slope is 3.0 per radian.

The forces on the vertical stabilizer play a primary role in the yawing moment equilibrium equation but also appear as small participants in some of the others. Figure 8.18 shows the geometric relationships that are used in evaluating these forces.

The equations for the forces on the vertical stabilizer are:

Xv = – Dv cos[p + iMv + r|T|/ + – Lv sin[|3 + iMy + Лі> + Ляк]

Yy = Ly costf + rMy + rTv + тц,] – D„sm[p + + Лтк + Лtr]

Zv = Xv sin[© – (e% + EFy + y,)]

where (3 is the sideslip angle and T| is the sidewash angle at the vertical stabilizer induced by the other components of the helicopter.

The basic equations for lift and drag of the vertical stabilizer are:

Ly = J^}AyC

W

Each of the new terms in these equations will be discussed separately.

The dynamic pressure at the horizontal stabilizer, qH, is usually less than the free – stream value because of the loss of momentum due to the air passing around the main rotor hub and fuselage. This loss, of course, is greatest for the inboard regions of the stabilizer and is less outboard. For preliminary design purposes, Figure 8.9 can be used to obtain an estimate of what the average dynamic pressure

Source: Hoak, “USAF Stability and Control Datcom,” 1960.

Source: Harris et al., “Helicopter Performance Methodology at Bell Helicopter Textron,” AHS 35th Forum, 1979.

Source: Hoerner & Borst, Fluid-Dynamic Lift, published by Mrs. Hoerner.

ratio might be for a given configuration. One set of flight test data from reference

8.4 and three sets obtained during the powered wind tunnel tests reported in references 8.5, 8.6, and 8.7 are presented. Note that the presence of the rotor wake has a significant effect on the dynamic pressure ratio, making the distribution unsymmetrical and producing some values above unity, especially behind the advancing side of the rotor disc. The effect of the rotor can be even more dramatic for those speed conditions where the horizontal stabilizer is immersed in the rotor wake. As an illustration of this effect, Figure 8.10 presents the results of wind tunnel and flight tests reported in reference 8.8. The fact that the increase in dynamic pressure can be higher than the disc loading is a result of the distortion of the wake in low-speed flight discussed in Chapter 3 under "Correlation of Flapping with Test Results.”

If a wind tunnel model of a new design is being tested, the dynamic pressure ratio can be determined indirectly. This is done by holding the model at a constant angle of attack and varying the incidence of the horizontal stabilizer (or, as is more common, cross-plotting data from several pitch runs with different incidence settings). The change in measured pitching moment can then be used to evaluate the product of dynamic pressure ratio and the slope of the lift curve for the horizontal stabilizer:

A M

A/’h

This product will be valid for the stabilizer geometry tested and can be used directly in this form in the analysis. For design studies in which alternate stabilizers are being investigated, the ratio, qH/q can be found by assuming that the value of aH found from Figure 8.6 is valid.

Downwash Angle

The downwash angle at the horizontal stabilizer is generated by both the main rotor and the fuselage—although the latter effect is usually small unless the helicopter has a wing or has widely spaced engine nacelles with a combined span greater than the span of the stabilizer. The downwash due to the rotor can be estimated from the results of downwash surveys made in a wind tunnel and reported in reference 8.9. Figure 8.11 presents the test measurements, at a tip speed ratio of 0.23, which may be taken as typical of forward flight, for several

Source: Blake & Alansky, “Stability and Controi of the YUH-6IA,” JAHS 22-1, 1977.

vertical locations and two longitudinal locations. It may be seen that the induced velocity ratio, exceeds 2—the maximum theoretical value—in some

locations. The downwash angle due to the main rotor at the horizontal stabilizer, can be estimated from the curve of Figure 8.11 that most closely matches the longitudinal and vertical position of the stabilizer with respect to the main rotor. Using a mean value of vH/vl that corresponds to the span of the stabilizer, the downwash angle is:

or

vH D. L.

H vx Ц

As an illustration, the horizontal stabilizer of the example helicopter is located at X’/R = —1.08 and Z’/R = +0.3, and the semispan/radius ratio is 0.2. From Figure

8.11, the effective value of vH/vx is 1.5. At 115 knots and design gross weight, the corresponding downwash angle is 0.06 radians or 3.5°.

V’/p — __ c

Advancing Side Retreating Side

Source: Herson & Katzoff, “Induced Velocities near a Lifting Rotor with Non-Uniform Disc Loading,” NACA TR 1319, 1957.

Figure 8.11 also shows an effect that was not recognized as significant at the time of the test but has been recognized since then. It is the higher downwash behind the advancing side than behind the retreating side for locations just behind the hub. When helicopters with high disc loading and large horizontal stabilizers are flown, it is found that they have a coupling of pitch with sideslip as the stabilizer moves either to a high downwash region behind the advancing side or to a low downwash region behind the retreating side. Discussions of this effect are contained in references 8.10 and 8.11.

Another wind tunnel test in which the downwash angle at the horizontal stabilizer was measured directly is reported in reference 8.12. In these tests, a free-

0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10

Cjlo

FIGURE 8.12 Measured Downwash at Horizontal Stabilizer

Source: Bain & Landgrebe, “Investigation of Compound Helicopter Interference Effects," USAAVLABS TR 67-44, 1967.

floating stabilizer was used as a flow vane. Figure 8.12 shows the measured downwash angle for various combinations of fuselage, wing, and rotor at constant fuselage angle of attack as the rotor thrust was varied with collective pitch. The downwash angles due to rotor with and without the wing have been converted into induced-velocity ratios on the bottom portion of the figure by using the equations:

It may be seen that for this configuration, the rotor-induced velocity ratio is essentially the fully developed value of 2.0, which would also be read from Figure

8.11 for this stabilizer position, and that the presence of the wing has essentially no effect. Figure 8.13 shows yet another set of downwash measurements from the wind tunnel tests of the powered model of reference 8.5. Again the asymmetry of the lateral distribution is evident.

A more recent wind tunnel test produced the data on Figure 8.14 for the flow conditions at the stabilator position of the Hughes AH-64. The powered model used a rotor from a previous test that was somewhat small for the size of the fuselage. Thus the flow survey was made at two positions: one close to the rotor at low speeds, and one further back at high speed, where the influence of the fuselage and wings was more significant. Results taken from reference 8.7 for both survey locations are shown in Figure 8.14. Note that at low’ test speeds, the rotor wake increased the local dynamic pressure significantly above that of the wind tunnel.

The equation for the downwash at the stabilizer due to the fuselage with or without a wing can be written:

Since wings on helicopters are relatively small compared to the fuselage, the charts prepared by the airplane people are generally not directly applicable. Figure

8.15 (page 500) gives some test results for the downwash as measured by the floating stabilizer of the model of reference 8.12 for the fuselage alone and for three different-sized wings. These can be used as a rough guide for estimating the effect during preliminary design. Configurations with external engine nacelles, such as the example helicopter, can be assumed to have almost the same downwash characteristics as the small wing of Figure 8.15. In another wind tunnel test, this time on the Sikorsky S-76, reported in reference 8.13, the value of dtF /daf was measured as 0.15.

If a wind tunnel model with adjustable horizontal stabilizer incidence is being tested without a rotor, the fuselage-induced downwash can be determined by the following procedure: •

FIGURE 8.14 Measured Downwash Angles

Source: Logan, Prouty, & Clark, “Wind Tunnel Tests of Large and Small Scale Rotor Hubs and Pylons,” USAVRADCOM TR-80-D-21, 1981.

Source: Bain & Landgrebe, “Investigation of Compound Helicopter Aerodynamic Interference Effects,” USAAVLABS TR 67-44, 1967.

methods used for wings are directly applicable unless the configuration is such that the stabilizer can be considered to be operating in clean air. However, if values are needed before wind tunnel tests are done, the wing method is the only method readily available. The theoretical span-efficiency factor, 5, as a function of aspect ratio and taper ratio is given in Figure 8.17, which was taken from reference 8.14.

The zero lift profile drag coefficient, CDq, of the horizontal stabilizer can be estimated from Figure 6.30 of Chapter 6. For this case, the Reynolds number is based on the chord.

Horizontal Stabilizer Characteristics of the Example Helicopter The preceding methods can be used to make an estimate of the aerodynamic characteristics of a horizontal stabilizer from its physical parameters. For the example helicopter, these parameters and characteristics are shown in Table 8.2.

Evaluation of the slope of the lift curve of the horizontal stabilizer, aH, as a function of aspect ratio and sweep of the mid-chord line can be made from Figure 8.6, which has been adapted from reference 8.1. Test data for several stabilizers and wings of Bell helicopters are shown on Figure 8.7, taken from reference 8.2.

Many modern helicopters have end plates on their horizontal stabilizers in the form of small vertical surfaces. End plates tend to block the flow around the tip from bottom to top and thus reduce the tip vortex. The result is that the effective span—or the aspect ratio—is increased, with a corresponding increase in the slope of the lift curve. The increase in effective aspect ratio as a function of the – height-to-span ratio is given in Figure 8.8, which was taken from reference 8.3.