STABILITY OF A ROTOR WITH ALFORD FORCES UNDER THE INFLUENCE OF NON-ISOTROPIC SUPPORT STIFFNESS AND GYROSCOPIC MOMENTS

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47th St., New York, N.Y GT-33 The Society shall not be responsible for statements or opinions advanced in papers or discussion at meetings of the Society or of its Divisions or Sections, or printed in its publications. Discussion is printed only if the paper is published in an ASME Journal, Papers are available from ASME for 15 months after the meeting. Printed in U.S.A. Copyright 1993 by ASME STABILITY OF A ROTOR WITH ALFORD FORCES UNDER THE INFLUENCE OF NON-ISOTROPIC SUPPORT STIFFNESS AND GYROSCOPIC MOMENTS BY VICENTE C. GALLARDO SR. ENGINEER - APPLIED MECHANICS GENERAL ELECTRIC AIRCRAFT ENGINES CINCINNATI, OHIO ABSTRACT Cross-axis coupled forces such as those due to Alford and Thomas can produce a whirl instability in a gas turbine rotor. This paper demonstrates the effect of non-isotropic support stiffness and gyroscopic loads on the severity of the instability. Non-isotropy in the support stiffness tends to suppress it; whereas the gyroscopic effect enhances the potential of the rotor for unstable motion at the same time influencing the effectiveness of the support nonisotropy. Simple criteria for a flexible rotor's stability for forward and backward whirls are obtained from the Routh-Hurwitz criteria and given in terms of the required Alford coefficient, which reduces to the required damping when gyroscopic effects are omitted. INTRODUCTION Rotor Dynamic instabilities induced by a cross-coupled stiffness have become a classic problem in turbomachines. The basic mechanism is a circular field of force that is proportional to the rotor displacement; hence, the name, which comes from analogy with a spring or stiffness force, which is proportional to displacement. One of the earliest mechanisms is hysteretic damping or damping in a rotating shaft (Kimball, 1923). It had been shown by Smith (1933) that the damping in the rotor produces cross-stiffness forces whose de-stabilizing effect may be suppressed by non-isotropy in the rotor support stiffness. More recently Thomas (1958) and Alford (1965), independently, have found a cross-stiffness-dominated rotor instability associated with a varying blade tip clearance (as the rotor orbited within the casing) of compressors and turbines and dependent on the power or torque level. This phenomenon is still being studied, in particular the mechanism that generates the cross-coupled aerodynamic forces at the blade tips. Some recent work in this have been reported: by Ulrichs (1977), Vance and Laudadio (1984) and in the "Handbook of Rotor Dynamics" edited by Erich (1992). Presented at the International Gas Turbine and Aeroengine Congress and Exposition Cincinnati, Ohio May 24-27, 1993

2 In these papers, analyses of the rotor instability have been made on a Jeffcott rotor where the rotor disk was assumed at the middle of a uniform shaft, and gyroscopic moments were neglected. This model has led to a startling conclusion that: a very small amount of non-isotropy in the support stiffness is sufficient to suppress the instability. Martinez-Sanchez (2/22/82) reached this conclusion from an examination of the stability criterion (via Routh-Hurwitz) that was developed by Alford (2/4/88) from the extension of his earlier work (Alford 1965) to a rotor with non-isotropic support stiffness; and from the concept of work done by the cross-coupled forces through the coherence (or lack of coherence) in the trajectories of the Alford forces and the motion of the rotor. (It should be noted that Alford's equations of motion are identical to Erich's (1989). The recent interest in the Alford instability and its not too infrequent occurrences contradict the theoretical predictions of its rarity or low practical probability. To quote Martinez-Sanchez (2/22/88): "...The other aspect... of these forces is the somewhat unsettling thought that we may have been chasing a ghost, if it turns out that these very real forces have great practical difficulty producing the instabilities that seem to follow naturally from them". The focus of this paper is to re-examine the Alford instability mechanism and to attempt to eliminate the dichotomy of experimental frequency and theoretical rarity. The root of the theoretical conclusion may lie in the Jeffcott model of the rotor, wherein there is no crosscoupling between horizontal and vertical degrees of freedom other than from the cross-coupled stiffness forces (Alford or hysteretic damping). For instance, see Alford (1988) and Erich (1989). Without the crosscoupled forces, the homogeneous equations of motion are uncoupled, and the eigenvectors or normal modes are simple planar motions. With planar motions, a field of cross-coupled stiffness forces would not be as efficient in storing energy in the system as with one where the natural motion is whirling or orbital. As a consequence, the instability would be more easily suppressed by damping or a small amount of nonisotropy in the support stiffness. However, if the natural modes of the rotor is whirling, i.e., crosscoupled, then the Alford forces would be much more de-stabilizing. The simplest mechanism to produce cross-coupled whirling normal modes of the rotor is the gyroscopic effect. The primary objective of this paper is to demonstrate the effect of gyroscopic forces and non-isotropic support stiffness on the Alford instability or others that depend on a cross-coupled stiffness. It had been brought to the author's attention of a more extensive investigation by Childs (1978) on the space shuttle engine using a modal approach. This investigation included a flexible rotor and a flexible casing, hysteretic damping, gyroscopic forces and Alford forces. However, because of its scope and complexity, the resulting stability analysis is a large complex eigenvalue problem which yielded numerical results. Conclusions about the effect of any parameter can only be reached inductively. However, Childs briefly discussed both gyroscopic effect and Alford forces on rotor stability in a degenerate two-degree of freedom case. 2

3 The present work is an attempt to provide a simple criterion as well as obtain a clearer insight on the physics of this instability. And as it will be shown later, there are four possible instability configurations. The first of which corresponds to Childs (1978). Finally, it should be noted that by virtue of the assumed modal displacement shapes and the sign convention (Figure 2), the angular displacement in the vertical plane has a negative sign, or equal to the negative of the bending slope. For the case where the vertical mode shape is not monotonically increasing, but first increases then decreases so that at the gyro station the slope is reversed, then the signs of the gyro forces in equation of motion are also reversed. Whirl Directions of the Alford Forces It is generally known that the Alford forces in a turbine stage are directed towards a forward whirl. The aerodynamic force accelerating the turbine in the direction of rotation is higher at the side of minimum radial clearance than at the position of maximum clearance. Thus the resultant aerodynamic forces produce a torque in the direction of rotor spin. Alford (1965) attributes this to greater efficiency at the minimum clearance side. This is illustrated in Figure la. In fans and compressors, the Alford forces are drag forces which act opposite the direction of rotation. However, there is a lack of concensus on which side (minimum clearance side or maximum clearance side) is more highly loaded tangentially. Thus the whirl direction of these forces depends on their calculated or measured values. One can use the theory of propellers to support the argument that the side of maximum clearance is more highly loaded (in drag) by virtue of a larger tip vortex. This would result in a forward whirl. But it is possible to argue also, that at the minimum clearance side, the air axial velocity is slowed down resulting in a larger angle of attack and a higher drag force; in this case the cross-coupled aerodynamic forces would be for a backward whirl. Thus, in the subsequent analyses, both forward whirl and backward whirl will be investigated individually, regardless of their origin. The forces are the net Alford forces obtained by summing them over all loading stages. In summary, the net modal whirl forces (see Figure 2) are given below FORWARD WHIRL Fx --qy Fy = qx BACKWARD WHIRL Fx = qy Fy = qx (1) (2) Though the same symbol is used for the Alford coefficients for the forward whirl and the backward whirl, these are not equal. 3

4 The Eouations Of Motion The equations of motion of a spinning rotor had been derived and were known for sometime. Basically, there are four physical degrees of freedom of a modified Jeffcott rotor spinning at constant speed and within a rigid cylindrical casing: 1. Vertical Translation 2. Horizontal Translation 3. Vertical Plane Angular Displacement 4. Horizontal Plane Angular Displacement However, these are reduced to two by using stationary rotor normal modes: one in the vertical plane and another in the horizontal plane. The angular modal displacement is the slope of the translational displacement in a given plane. Considering initially forward whirl Alford forces, the equations of motion of the spinning rotor is derived in terms of the non-spinning modes as was done by Gallardo et al (1981), we have: Lo 21[71 [Kg qt1= 0 (3) M 4110 C Kr where the modal displacement shapes in the vertical plane and horizontal plane are as those sketched in Figure 2a. These show nodal deflections that are monotonically increasing with increasing axial distance from the origin. Note especially the sign convention for the angular displacements. The variables in (3) are defined below: M = rotor modal mass, assumed to be the same in both planes Kx,Ky = non-spinning modal stiffnesses in the horizontal (x) and vertical (y) planes Cx,Cy = modal damping G - modal gyroscopic inertia, OYOZI I - rotor diametral moment of inertia 0 = constant rotor speed Ox,Oy = non-spinning rotor bending modal displacement shapes 4,01y - dtx, dix, modal bending slope dz dz or angular q = modal Alford force displacement shape coefficient, q (.100x0y q0= physical Alford force coefficient x(t) - horizontal plane modal generalized coordinate; x also denotes the horizontal plane y(t) = vertical plane modal generalized coordinate; y also denotes the vertical plane z = longitudinal coordinate - () = d dt distance t - time

5 The Alford force coefficient is added via the principle of virtual work. If one recalls Alford (1988) or Ehrich (1989), the above equation differ only in that it contains the gyroscopic terms ( 2 OG). Also, it is important to note that the equations of motion of a system with hysteretic damping (Smith, 1933) is formally identical to (3), if gyroscopic effects are not ignored. Characteristic Equation A direct stability criterion is derived rather than calculating the usual eigenvalues, so as to be able to assess more completely the effect of gyroscopic forces on the Alford instability. This is accomplished by applying the Routh-Hurwitz criteria on the characteristic equation of (3). First the canonical transformation: x(t),y(t) = (x,y)elt converts (3) to an algebraic equation. After dividing through by the modal mass and the square of w, (ref. frequency), it is simplified with the following definitions and substitutions: 0 = A/wx wx = X7R, horizontal stationary rotor frequency =X7,71 = pax, vertical stationary rotor frequency A = wy/wx a = q/m4 non-dimensional Alford coefficient = C/2Mwx critical damping ratio referred to the horizontal modal rotor frequency Cx = Cy = C9 assumed constant g =20/Mwoon-dimensional gyroscopic Thus: coefficient (32+.2m+1 (2( ) x- - z4-) 41A4 =0 (4) The condition for x and y to have non-trivial solutions is for the determinant of the coefficient matrix of (4) to equal zero. This yields the characteristic polynominal of (4) in P. 41,35 432A 2p[( ) + 23K] +(p24- c6 = 0 (5) Labeling the coefficients of fl() by A (), we have: f54+ PA, + At- pa cf A, = 0 Four Instability Configurations (6) Before applying the Routh-Hurwitz criteria to (5,6), let us re-examine them. It can be shown that the coefficient of 0 is not affected when the signs in front of the off-diagonal terms in (4) are reversed. However, it is altered when the sign within these off-diagonal brackets is changed! The change in the sign within the offdiagonal terms is effected by either of the following:

6 (1) The Alford forces are for backward whirl. (2) The vertical (or horizontal) modal displacement shape is positive but monotonically decreasing with distance from the origin at the gyro location. Thus the two kinds of whirl forces and the two gyroscopic possibilities yield four different configurations of instability! It will be shown that these four configurations of instability are governed by just two sets of instability criteria. These four configurations of Alford instability are illustrated in Table I and described as follows: The Routh-Hurwitz Stability Criteria Recall Equation (6), the characteristic polynomial: 3 + PIA 2 Pik 1 0 =0 (6) where the An coefficients are given in Equation 5. The Routh-Hurwitz (R-H) criteria for the polynomial (6) to have non-positive real (stable) roots are: A., A,,A,,A3> 0 (7a) A,AzAs> A, , (7b) All: Vertical and horizontal mode shapes both monotonically increasing - gyro; Forward whirl Al2: Same gyro as All but with backward whirl forces A21: Vertical (or horizontal but not both) mode displacement shape is positive but monotonically decreasing at the gyro location - gyro Forward whirl forces - This results in the same characteristic equation as Al2. A22: The same gyro as A21 but with backward whirl forces This results in the same characteristic equation as All. From these, the two essential instability configurations are: (1) All and A22 - less unstable and (2) Al2 and A21 - more unstable. Stability Criterion For The Less Unstable Configurations: All and A22 The. instability configurations All and A22 have formally identical characteristic polynomials and given in Equation (5). The difference is in the magnitudes of the forward whirl vs. backward whirl coefficients and the gyro coefficients. The first Routh-Hurwitz criterion is identically satisfied. The second Routh-Hurwitz criterion is evaluated by substituting the expressions for the An's from (5) into (7b) and expanding. The resulting inequality is a full quartic polynomial in the critical damping ratio C, the usual measure of stability. Unfortunately, this makes it impossible to obtain an explicit expression of the required critical damping ratio in terms of the of the system variables. 6

7 However, the same polynomial is also a quadratic in the non-dimensional Alford coefficient "a"! This permits a solution of the inequality for "a". Thus instead of damping, a stability criterion is given as an allowable non-dimensional coefficient, i.e., a threshold value of the Alford nondimensional coefficient that must be exceeded before instability can occur. After some algebraic manipulations: oc( 293' gzi-4<1 (1-1:9)22) ) where the positive sign is the only physically practicable alternative. The interpretation of the allowable Alford coefficient criterion is opposite to the usual notion of the required damping, i.e., more damping, less stability and vice versa. Thus: 1) The greater the allowable Alford coefficient, the greater the stability. 2) The smaller the allowable Alford coefficient, the less stability. The Routh-Hurwitz criteria give a single Alford instability criterion for these configurations. The stabilityinstability threshold, where the inequality becomes an equality, is plotted in Figure 3 for = 0.1 and a range of gyroscopic coefficients and for 04 From Figure 3, the well known stabilizing effect of support stiffness asymmetry is substantiated. The effect of gyroscopic forces are seen to reduce stability at typical or low values of the gyroscopic coefficient. However, at higher values of the gyroscopic coefficient, two things are observed: 1) the gyroscopic effect becomes more stabilizing and 2) the minimum stability position relative to the support stiffness ratio migrates from the condition of symmetry towards decreasing symmetry. It would appear that the increased gyroscopic stiffening effect dominates the total stiffness of the system so that in effect it becomes more stiff and renders the stabilizing influence of asymetry less pronounced. Stability Criterion For The More Unstable Configurations: Al2, A21 The more unstable configurations Al2 and A21 are derived from the equations of motion (3) or (4) and the characteristic polynomial (5) by reversing the sign between the gyro and Alford terms. As stated earlier, this is achieved either by a change in the sign of the bending slope in one plane or by the direction of whirl in the Alford forces. This results in a subtraction of the components of the off-diagonal terms in the matrix equation of motion (3) or (4) and in the As coefficient in (5). The latter becomes: A, =2[30+Pz) With these changes, the first Routh- Hurwitz criterion is identically satisfied by all the An coefficients except for Al which must satisfy: or (1 41) >0 (8a) oc<.(1+ P2) 2 1 (8b) This condition is shown later to be a necessary condition or an asymptotic limit of the result of the second Routh- Hurwitz criterion with increasing gyro. 7

8 The second Routh-Hurwitz criterion, as in the previous instability configurations, results in a similar polynomial in or DV. The roots are similar except for sign. Thus, for the more unstable configurations, the instability criterion is: c(<-29.t+t 0- z. +2(f2) 4(24- pl) (9) The result of the second Routh-Hurwitz criterion for the more unstable configurations is similar yet somewhat different from the less unstable configurations. In the more unstable configuration, Equation [9] has no ambiguity about the effect of the gyroscopic coefficient on the allowable Alford coefficient; the allowable Alford coefficient is reduced by increasing the gyroscopic coefficient. This is shown in Figure 4. Hence, the gyroscopic effect leads to greater instability. However, the effect of the gyroscopic forces on the stability trend with respect to the frequency ratio or support stiffness asymmetry is very similar for both principal instability configurations. Large Gyroscopic Effect And The Two Routh-Hurwitz Conditions For The More Unstable Configurations frequency ratio. Yet still for a constant Alford coefficient, the required stiffness or frequency ratio for stability decreases with increasing gyroscopic coefficient. Recall that this trend is obtained from the Alford stability criterion (9) developed from the second (7b) Routh-Hurwitz criterion. The first Routh-Hurwitz condition (8b) also indicates an increase of the allowable Alford coefficient with increasing A (from A>0 to above 1.0). This may mean that (8b) is a necessary condition whereas (9) is the necessary and sufficient condition. Evaluating (8b) and (9) for a range of the gyroscopic coefficient shows that the allowable Alford coefficient from (8b) is always larger than that from (9); and at arbitrarily high (impracticable) values of the gyroscopic coefficient, the two curves appear to coalesce, but never cross. To show this to be true, combine (9) and (8b) and assume that cagg (80 o r 25-51:s / P9z + 2.i.t2-)2 4(1212) <1(1+ PI) 2 Clearing the radical, we have: (1--P9z < (1 0) It may be of academic interest to note, that at very large values of the gyroscopic coefficient (too large for practical purposes), the allowable Alford coefficient is no longer a minimum at the isotropic or asymmetry (A.--1) condition. Instead, the minimum migrates towards A-0, and the allowable Alford coefficient, increases with increasing support stiffness ratio or In (11), it is obvious that for all values of g, C and g, the left hand side is less than the right, which corroborates the initial assumption that the allowable Alford coefficient from the second Routh-Hurwitz condition is less than from the first condition. Thus one can conclude that (8b) is the asymptotic limit of (9) with increasing g.

9 From these observations, the large gyroscopic forces are always destabilizing in that they reduce the value of the allowable Alford coefficient. However, the trend with respect to the support stiffness non-isotropy or frequency ratio is altered, i.e., the least stable condition migrates toward A=0 from that of A=1 (when the gyroscopic coefficient is small: g<1/2). An explanation is needed for using the horizontal plane frequency wv. In the foregoing, damping in both 'planes are assumed to be the same and constant. The latter implies that the critical damping ratio ç varies inversely with the frequency wv. Thus with the assumption of constint damping rate, C, less confusion would prevail if w is held constant while varying wt, or the frequency ratio A. Similaely, the non-dimensional Alford coeff4cient is inversely proportional to w,'. Thus one can start the frequency ratio at zero and increase it by increasing w. If instead the vertical stiffness or frequency w is considered to be the w reference qdantity, then one can start x from 0.0 until it equals and exceeds w By redefining the variables in (4), (5), (6) and (8b), (9), (10), and (11) in terms of w, one obtains formally identical stabflity criteria in terms of the redefined parameters. Conclusions The effect of gyroscopic forces on the stability of a rotor under the action of the cross-coupled Alford forces has been demonstrated with a simple modified Jeffcott rotor. Gyroscopic forces though strictly neither stabilizing nor de-stabilizing, do alter the natural modes of the rotor from planar to whirling when it spins. And it is this whirling which allows the Alford forces greater facility in storing energy in the rotor. There are four instability configurations differentiated by the rotor bending slopes in the two planes and by the whirl direction of the Alford forces. The four can be reduced to two prinicpal configurations: 1) Less Stable Configurations a) Vertical and horizontal rotor bending deflections are both monotonically increasing (or decreasing) - gyro. The Alford forces are in the forward whirl direction. b) Vertical and horizontal rotor bending deflections have opposite trends - gyro. See Table I. Alford forces are in backward whirl. 2) More Unstable Configurations a) Similar to the gyro of (la) but with backward whirl forces. b) Similar to the gyro of (lb) but with forward whirl forces. In the case of the less unstable configurations, small gyro forces decreases the allowable Alford coefficient, hence, resulting in greater instability. However, at larger values, gyroscopic forces improve stability. The more unstable configurations show a simple decrease in the allowable Alford forces with increasing gyroscopic coefficient. Thus in these cases, gyroscopic forces always reduce stability. 9

10 The stabilizing effect of support stiffness asymmetry is reinforced by the results of this analysis. However, it's efficacy is influenced by the gyroscopic forces. As gyroscopic forces increase, the position of minimum allowable Alford coefficient (bucket) migrates from symmetry at no gyro forces towards decreasing symmetry. The four instability configurations may give some insight in the assessment of rotor modes and whirl forces for Alford instability. They show how the magnitudes and directions of the gyroscopic and cross-coupled Alford parameters are affected by the location of gyroscopic stations and aerodynamically loaded stages as well as by mode shapes in translation and slope. With respect to this, the gyroscopic effect is proportional to the rotor moment of inertia and the modal slopes at its location so that a judicious choice of the inertia station can ameliorate its effects. Similarly, the modal Alford force is proportional to the modal translation at the rotor disk location, so that a similar judicious control of the rotor mode shapes and rotor casing clearance may also minimize the cross-coupled forces. Ignoring gyroscopic forces and support stiffness asymmetry in the assessment of cross-coupled Alford instabilities may lead to non-conservative results in the former and too conservative results in the latter. Finally a review of the dynamical equations of a rotor with hysteretic damping shows its formal identity with the equations of the rotor with Alford forces. Thus similar conclusions with respect to gyroscopic effect and support non-isotropy on the stability of a rotor with hysteretic damping can also be made. Acknowledgment The writer is indebted to J.S. Alford for sharing his communications to- and from- Prof. Martinez-Sanchez. Joe's encouragement in the writing of this paper is deeply appreciated. REFERENCES 1. Alford, J.S., 1965, "Protecting Turbomachinery From Self Excited Rotor Whirl," Journal of Engineering for Power, 87 (4), , February 4, 1988, "Turbine Blade-Tip Clearance Excitation Forces", (Letter to Professor M. Martinez-Sanchez, M.I.T., Cambridge, MA) GE Aircraft Engines, Cincinnati, Ohio. 3. Childs, D.W., (1978) The Space Shuttle Main Engine High-Pressure Fuel Turbopump Instability Problem, pp , Vol. 100, Trns. of ASME, Journal of Engineering For Power, Jan. 4. Ehrich, F.F., 1989, "The Role of Bearing Support Stiffness Anisotropy in Suppression of Rotor Dynamic Instability," D.E. - Vol. 18-1, ASME, New York. 5., 1992, "Handbook of Rotordynamics," McGraw-Hill, Inc., New York. 6. Gallardo, V.C., A.S. Storace, Gaffney, E.F., Bach, L.J., and Stallone, M.J., 1981, "Blade Loss Transient Dynamic Analysis, Task II - Theoretical and Analytical Development," Vol. II, NASA CR165353, NASA Lewis Research Center, Cleveland, Ohio. 10

11 7. Kimball, A.L., 1923, "Internal Friction Theory of Shaft Whirling," Phys. Rev. [2)21, Martinez-Sanchez, M. February 22, 1988, Letter to J.S. Alford (in reply to Ref. 2), M. I.T., Cambridge, MA. 9. Smith, D.M., 1933, "The Motion of a Rotor Carried By a Flexible Shaft in Flexible Bearings," Proc. Roy. Soc, A142,92-118, London. 11. Ulrichs, K., 1977, "Leakage Flow in Thermal Turbo-Machines as the Origin of Vibration-Exciting Lateral Forces," TT-17409, NASA, Washington, D.C., March. 12. Vance, J.M. and Laudadio, F.J., 1984, "Experimental Measurement of Alford's Force in Axial Flow Turbomachinery, "ASME Paper 84-GE-140, 29th International Gas Turbine Conference and Exhibit, The Netherlands. 10. Thomas, N.J., 1958, "Unstable Oscillations of Turbine Rotors Due to Steam Leakage in the Clearance of the Sealing Glands and the Buckets," Bulletin Scientifique, A.J.M. Vol. 71,

12 TABLE I FOUR ALFORD INSTABILITY CONFIGURATIONS MODAL GYROSCOPIC FORCES MODAL ALFORD FORCE: FWD WHIRL MODAL ALFORD FORCE: BCK WHIRL, z...-r1, r--- MI + a + KX 4 / M.. + C, + KY +2nIkl xy 11.1/4-1 Al 1 Al 2 40te -(10O;( -Ok 410* 4 A21 A22 MY + Ci + KX -2n1k1 illott -01Ay 7 dr- II MY + CV + KY +2n14# -qo ig IX +qo 4x#IX, 12

13 z r= x4i, N FIG. 2A ASSUMED STATIONARY MODE SHAPES AND SIGN CONVENTION; 4,0.r MONOTONICALLY INCREASING FIG. 1A ALFORD FORCES FOR FWD WHIRL FIG. 2B ASSUMED STATIONARY MODE SHAPES WHERE VERTICAL AND HORIZONTAL DISPLACEMENTS AND SLOPES HAVE DIFFERENT TRENDS FIG. 1B ALFORD FORCES FOR BCK. WHIRL 13

14 2.0 II' FIG. 3 INSTABILITY CONFIGURATION All: EFFECT OF GYRO COEFFICIENT AND FREQUENCY RATIO ON ALFORD INSTABILITY FOR I= 0.10 FWD WHIRL / FIG. 4 INSTABILITY CONFIGURATION Al2: EFFECT OF GYRO AND FREQUENCY RATIO ON ALFORD INSTABILITY FOR S= 0.10;FWD WHIRL 14