Coning Motion Stability of Wrap Around Fin Rockets

Science in China Series E: Technological Sciences

© 2007 Science in China Press

Springer-Verlag

Received June 14, 2006; accepted December 4, 2006 doi: 10.1007/s11431-007-0026-0 †Corresponding author (email: [email protected])

www.scichina.com www.springerlink.com Sci China Ser E-Tech Sci | June 2007 | vol. 50 | no. 3 | 343-350

Coning motion stability of wrap around fin rockets

MAO XueRui†, YANG ShuXing & XU Yong School of Mechatronic Engineering, Beijing Institute of Technology, Beijing 100081, China

Both the asymptotical stability criterion and the bounded stability criterion of the coning motion for wrap around fin (WAF) rockets are proposed through the analy-sis of coning motion equations, which can be easily used to determine the exis-tence of the coning motion during the rocket design. The correctness of the crite-rions is verified by mathematical simulation examples of a WAF rocket with differ-ent setting angles. It is also found that the setting angle of WAF has great effects on the rolling moment and side moment of the rocket.

coning motion, wrap around fins, setting angle, dynamic stability

1 Introduction

The wrap around fin is the stabilizing or control surface of a projectile, which has nearly the same curvature as the projectile body. It has been used in many kinds of projectiles for its advantages in tube-launching since 1950. Compared with flat fins, WAF has some special aerodynamic charac-teristics. For example, WAF configurations with zero setting angle may present roll reversals when transitioning through Mach 1, roll moments at zero angle of attack and significant side forces and moments when at an angle of attack due to asymmetry of the fins[1], which could in-duce the coning motion and compromise the dynamic stability of the projectiles.

This paper focuses on the WAF rockets stabilized with tail fins. Although the rockets are stati-cally stable, the dynamic stability is not promised, and the coning motion may appear due to small disturbance especially at high Mach numbers. Coning motion refers to the motion per-formed by a projectile flying with an angle (nutation angle) to the free stream velocity vector and undergoing a rotation about a line parallel to the free stream velocity vector and coincident with the projectile’s CG[2]. If the coning motion is unstable, the perturbation will grow with respect to time, and catastrophic flight failure will occur. If the coning motion is asymptotically stable, the perturbation will be eliminated and the projectile can fly stably. If the coning motion is bounded stable, the projectile will fly with a constant nutation angle and a constant roll angular velocity. In

344 MAO XueRui et al. Sci China Ser E-Tech Sci | June 2007 | vol. 50 | no. 3 | 343-350

this condition, the range and precision of the projectile will be significantly decreased by the coning motion.

To the statically stable rockets, coning motion stability is an important aspect of their dynamic stability characteristics. For example, in the US, unstable coning motion was detected 20 times out of 50 flight experiments for Nitehawk Rocket[3], and this motion also appeared in the sub-sonic and supersonic wind tunnel experiments for 2.75″ rockets[4]. In Spain, catastrophic coning motion made the velocity reduce 60% in just 1.5 seconds in the flight trials of a 140 mm artillery rocket[5]. In China, serious coning motion has been detected in the experiments for several kinds of unguided rockets[6].

Many researchers have worked on the coning motion and how to restrain it. Yi and Wang[7] proposed that the coning motion can be overcome by means of an internal moving mass. Morote and Liano[5] noticed the effects of cord and span of WAF on the dynamic stability of 140 mm rockets and concluded that clipping the span of WAF can mitigate the coning motion. Livshits[8] has studied the influence of aeroelasticity on the dynamic stability of the rocket and concluded that thrust misalignment and dynamic imbalance are important factors to the dynamic stability of the rocket. Lei and Wu analyzed the effects of spinning velocity and angle of attack on the coning motion stability and pointed out that the coning motion can be restrained by adopting nega-tive-fixed and negative-spin WAF rather than the commonly used positive-fixed and positive-spin WAF[9].

In the rocket design process, it is important to determine the existence of coning motion by stability analysis using the design parameters, and flight experiments can only be used to verify the design. As designers do now, flight experiments not only increase design time and cost, but also may fail to expose the problem due to insufficient experiments. But till now, the research on coning motion has been limited to experimental research and qualitative analysis, and there are no published theoretical results about the mechanism of coning motion and how to determine the stability characteristics of the coning motion, which restrict the design of WAF projectiles seri-ously.

2 Stability analysis of the coning motion

2.1 Coning motion equations

The coning motion of the projectile at a certain Mach number can be written as[10]

1d

,d

xxJ M

tω

= (1)

2 1 2d

( ) ,d

yx z yJ J J M

tω

ω ω+ − = (2)

2 2 1d

( ) ,d

zx y zJ J J M

tω

ω ω+ − = (3)

d ,d ztδ ω= (4)

where 1J and 2J are the polar moment of inertia and equator moment of inertia respectively;

xω is the roll angular velocity of the rocket, zω and yω are the angular velocity components

in and out of the nutation angle plane which contains the body axis and velocity vector; δ is

MAO XueRui et al. Sci China Ser E-Tech Sci | June 2007 | vol. 50 | no. 3 | 343-350 345

nutation angle (see Figure 1). xM consists of roll moment due to asymmetry of the warp around

fin and roll damping moment, yM consists of side moment and damping moment, and zM

consists of pitching moment and damping moment.

Figure 1 Sketch of the coning motion.

Making weighted least square fitting for the aerodynamic moment coefficients. According to the relations between the coefficients and the nutation angle, roll moment coefficient, roll damp-ing moment coefficient, side damping moment coefficient and pitch damping moment coefficient can be written as 2

1 2c c δ+ , 23 4c c δ+ , 2

7 8c c δ+ and 211 12c c δ+ . Side moment coefficient

and pitching moment coefficient can be written as 35 6c cδ δ+ and 3

9 10c cδ δ+ [11]. Therefore

2 2 11 2 4 3 4 4( ) ( ) ,x

CLyM c c y C c c yV

= + + + (5)

3 25 4 6 4 7 8 4 2( ) ( ) ,y

CLM c y c y C c c y yV

= + + + (6)

3 29 4 10 4 11 12 4 3( ) ( ) .z

CLM c y c y C c c y yV

= + + + (7)

For convenience, denote 1 2 2( ) /k J J J= − , C = SLq, where S is the reference area, L is the reference length and q is dynamic pressure.

Because the state of zero nutation angle (zero angle of attack and zero side slip angle) is an important equilibrium state during the flight of the projectile, take ( , , ) (0,0,0)y zω ω δ = as the

emphasis of analysis, and make the weight factor kw descend from zero nutation angle to large

nutation angles. In most cases, 0 ( 1,2)ic i< ≠ . The coning motion equations can be written as

2 2 11 2 4 3 4 4

1 11

3 25 4 6 4 7 8 4 2 1 32

2 23

3 29 4 10 4 11 12 4 3 1 24

2 2

3

( ) ( )

( ) ( )

( ) ( )

CLyCc c y c c yJ VJ

yC CLc y c y c c y y ky yyJ VJ

yC CLc y c y c c y y ky yy J VJ

y

⎡ ⎤+ + +⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥ + + + −⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ + + + +⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

, (8)

where [ ] TT1 2 3 4, , , , , , .x y zy y y y ω ω ω δ⎡ ⎤= ⎣ ⎦


When the left side of eq. (8) are zero, the equilibrium state of the system can be calculated. De-

note the equilibrium state as [ ] [ ]T T1 2 3 4 10 20 30 40, , , , , ,y y y y y y y y= , which is the solution of

2 21 2 40 3 4 40 10( ) / 0c c y c c y Ly V+ + + = , (9)

4 240 8 10 40 7 10 8 9 6 2 10 40 7 9 5 2 10[ ( / ) / ] 0y c c y c c c c c J kVy CL y c c c J kVy CL+ + − + − = . (10)

2.2 The asymptotical stability criterion of the coning motion

Obviously [ ] [ ]T T10 20 30 40 1 3, , , / ,0,0,0y y y y c V c L= − is the solution of the equilibrium eqs. (9) and

(10), and it is also an important equilibrium state of the system. If this state is stable, the rocket can fly in zero nutation angle state stably, which means the coning motion is stable.

Denote the characteristic matrix of the system as ( )ijA a= , where 0

iij

j

ya

y⎛ ⎞∂

= ⎜ ⎟⎜ ⎟∂⎝ ⎠ and the sub-

script 0 denotes taking values of zero nutation angle state. The characteristic equation of the sys-tem is 0A Iλ− = , so

2 23 23 7 7 1111

2 21 2 2 2

22 22 9 7 9 1 51

2 2 22 3 23 2

0.

c CL c CL c c C LCLcJ V J V J V J V

Cc c c C L c c VCc Vk kJ c LJc L J V

λ λ λ

λ

⎡ ⎛⎛ ⎞ ⎛ ⎞− + − − +⎢ ⎜⎜ ⎟ ⎜ ⎟ ⎜⎢⎝ ⎠ ⎝ ⎠ ⎝⎣

⎤⎞+ − + + =⎥⎟⎟ ⎥⎠ ⎦

(11)

It can be judged by Louts criterion that if and only if the roll angular velocity of the rocket

1 1 3 7 9 5 2/ /y c V c L c c CL c J Vk= − < , all the solutions of eq. (11) have negative real parts, and ac-cording to Lyapunov First Theory[12], the system is stable. This is to say, the sufficient and neces-sary condition for the coning motion to be asymptotically stable is 1 1 3/y c V c L= − <

7 9 5 2/c c CL c J Vk , or

3

1 5

3 7 9 2 1.

2( )c c SL

c c c J Jρ

<−

(12)

Theorem 1 If and only if 3

1 5

3 7 9 2 12( )c c SL

c c c J Jρ

<−

, the coning motion of the rocket is asymp-

totically stable, which means the perturbed nutation angle can converge to zero. Because 5c is the side moment coefficient and 9c is the derivative of static steady moment

to nutation angle when the nutation angle is zero, it can be concluded that the coning motion sta-bility will be improved with the increasing of static steady moment and decreasing of the side moment.

2.3 The bounded stability criterion of the coning motion

If 1 3 7 9 5 2/ / ,c V c L c c CL c J Vk− > (13)

[ ] [ ]T T10 20 30 40 1 3, , , / ,0,0,0y y y y c V c L= − is still the solution of eqs. (9) and (10), but now, the

characteristic equation of the system has positive real part eigenvalues, which means the rocket cannot fly in zero nutation angle state stably.


From eqs. (9) and (10),

6 2 2 44 8 10 40 3 8 10 4 7 10 4 8 9 6 2 2 40

2 2 2 2 27 10 3 8 9 3 7 9 4 6 1 2 2 5 2 40

2 25 1 2 7 9 3

( / )

( / / )

/ 0.

c c c y c c c c c c c c c c c J kV CL y

c c c c c c c c c c c J kV CL c c J kV CL y

c c J kV CL c c c

+ + + +

+ + + + +

+ + =

(14)

Because 1 3 7 9 5 2/ /c V c L c c CL c J Vk− > , 1 0c > . Therefore when

2 25 1 2 7 9 3/ 0,c c J kV CL c c c+ > (15)

the equilibrium eq. (14) has physical solutions. According to Lyapunov First Theory, when

2 2 2 2

3 4 41 7 8 41 9 10 41 2 11 3 4 41 52 2

6 41 8 41 21 21 2 41 4 11 41 7 8 41

( )( )( 3 ) / ( )( /

3 / 2 ) (2 / 2 )( ),

C c c y c c y c c y J ky c c y c V L

Vc y L c y y ky c y V L c y y c c y

+ + + < +

+ + + + +

(16)

the solution corresponds to a stable state of the system, which means the nutation angle and roll angular velocity are both constant, called bounded stability state.

Theorem 2 When eqs. (13), (15) and (16) are satisfied, the coning motion of the rocket is bounded, which means the disturbed nutation angle can converge to a nonzero value.

3 Examples

Take the WAF rocket with different setting angles as the example to verify the correctness of the stability criterion deduced above.

From the discussion above, it is clear that roll moment and side moment of the rocket have critical effect on the stability of the coning motion and the stability of the rocket can be improved by decreasing these moments, which can be achieved by changing the setting angle of WAF[9]. Define the setting angle as positive when the concave side of the fin is windward relative to the free stream and negative when the convex side is windward (see Figure 2). In the numerical simulation, the side slip angle keeps zero, so the nutation angle is equal to the angle of attack.

Figure 2 Front views of the WAF rockets with different setting angles. (a) Positive; (b) zero; (c) negative.

Flow fields of WAF rockets with different setting angles were numerically simulated and the correlation between the computed values and experimental values is considered good. The results indicate that as the setting angle increases, the roll moment coefficient and side moment coeffi-cient decrease while the static steady moment stays almost constant. The coefficients versus the angle of attack are presented in Figures 3―5. Both the results of numerical simulation and ex-periments indicate that the plot of moment coefficients versus angle of attack has similar trends when Mach numbers vary from 2 to 4.

Analyzing these cases of different setting angles with the method discussed in part 2 and sub-stituting the parameters into eqs. (12), (13), (15) and (16), it can be found that the case of positive


Figure 3 Static steady moment coefficient versus angle of attack (Ma=3).

Figure 4 Roll moment coefficient versus angle of attack (Ma=3).

setting angle does not satisfy the asymptotical stability criterion but satisfy the bounded stability criterion while the cases of zero and negative setting angles satisfy the asymptotical stability cri-terion. Calculate eqs. (1)―(4) numerically for these cases and the results have good agreement with analysis results (see Figures 6―8). Therefore the coning motion of the WAF rockets with negative or zero setting angle is always stable. But in these two cases, the roll angular velocity of the rocket is very small, which cannot vanish the thrust offset, eccentric mass and aerodynamic offset of the rocket to improve shooting precision and denseness. So a reasonable setting angle should produce a roll angular velocity large enough to eliminate the thrust offset, eccentric mass and aerodynamic offset and small enough to avoid coning motion instability.


Figure 5 Side moment coefficient versus angle of attack (Ma=3).

Figure 6 Nutation angle versus time (the case with positive setting angle).

Figure 7 Nutation angle versus time (the case with zero setting angle).

Figure 8 Nutation angle versus time (the case with negative setting angle).


4 Conclusion

In this paper, the coning motion of wrap around fin rockets is researched by analyzing the coning motion equations using Lyapunov First Theory. And it is found that the coning motion can be either asymptotically stable, bounded stable or unstable. Then both the asymptotical stability criterion and the bounded stability criterion are proposed which can be easily used to determine the exis-tence of the coning motion during the rocket design. Finally, the correctness of the criterions is verified by mathematical simulation examples of a WAF rocket with different setting angles. It is also found that the setting angle of WAF has great effects on the rolling moment and side moment of the rocket while has little influence on the pitching moment.

1 Wu J S, Ju X M, Miao R S. Advances in the research for aerodynamic characteristics of wrap-around fins. Adv Mech (in

Chinese), 1995, 25(1): 102―113 2 Winach P. Prediction of projectile performance stability and free-flight motion using CFD. AIAA-97-0421, 1997 3 Curry H, Reed J F. Measurement of magnus effects on a sounding rocket model in a supersonic wind tunnel. AIAA No.

66-754, 1966 4 Nicolaides J D, Ingram C W, Clare T A. An investigation of the non-linear flight dynamics of ordnance weapons. AIAA

No. 69-135, 1969 5 Morote J, Liano G. Stability analysis and flight trials of a clipped wrap around fin configuration. AIAA 2004-5055, 2005 6 Zhang C, Yang S X. A method to get the attitude of a rolling airframe missile. Trans Beijing Inst Tech (in Chinese), 2004,

24(6): 481―485 7 Yi Y, Wang Z H. Research on overcoming the coning motion of rotary spacecraft. Flight Dyn (in Chinese), 2001, 19(1):

81―84 8 Livshits D S. Dynamic stability of free flight rockets. AIAA-96-1344-CP, 1996 9 Lei J M, Wu J S. Coning motion and restrain of large fineness ratio unguided spinning rocket stabilized with tail fin. Acta

Aerodyn Sin (in Chinese), 2005, 23(4): 455―457 10 Mao X R, Yang S X, Xu Y. Research on the coning motion of wrap around fin projectiles. Can Aeron Space J, 2006, 52(3):

119―125 11 Hong J S. Analytical study of limit cycle phenomenon for reentry vehicle. Acta Aerodyn Sin (in Chinese), 2005, 23(2):

204―209 12 Zhang H X, Yuan X X, Ye Y D, et al. Research on the dynamic stability of an orbital reentry vehicle in pitching. Acta

Aerodyn Sin (in Chinese), 2002, 20(3): 247―259

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Coning Motion Stability of Wrap Around Fin Rockets