Nutation instability of spinning solid rocket motor spacecraft

Dn YANG,Yongling XIONG,Qin REN,Xinyong WANG

aSchool of Naval Architecture and Ocean Engineering,Huazhong University of Science and Technology,Wuhan 430074,China

bDepartment of Mechanics,Huazhong University of Science and Technology,Wuhan 430074,China

cHubei Key Laboratory of Engineering Structural Analysis and Safety Assessment,Luoyu Road 1037,Wuhan 430074,China

Nutation instability of spinning solid rocket motor spacecraft

Dan YANGa,Yongliang XIONGb,c,*,Qian RENb,c,Xianyong WANGb,c

aSchool of Naval Architecture and Ocean Engineering,Huazhong University of Science and Technology,Wuhan 430074,China

bDepartment of Mechanics,Huazhong University of Science and Technology,Wuhan 430074,China

cHubei Key Laboratory of Engineering Structural Analysis and Safety Assessment,Luoyu Road 1037,Wuhan 430074,China

The variation of mass,and moment of inertia of a spin-stabilized spacecraft leads to concern about the nutation instability.Here a careful analysis on the nutation instability is performed on a spacecraft propelled by solid rocket booster(SRB).The influences of specific solid propellant designs on transversal angular velocity are discussed.The results show that the typical SRB of End Burn suppresses the non-principal axial angular velocity.On the contrary,the frequently used SRB of Radial Burn could amplify the transversal angular velocity.The nutation instability caused by a design of Radial Burn could be remedied by the addition of End Burn at the same time based on the study of the combination design of both End Burn and Radial Burn.The analysis of the results proposes the design conception of how to control the nutation motion.The method is suitable to resolve the nutation instability of solid rocket motor with complex propellant patterns.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Smaller and cheaper satellites in short development period attract more and more attention.1,2Attitude motion is essential in achieving successful space missions of these satellites.Due to increasing demand on reliability and expensive lessons,attitude motion of spacecraft has been studied considerably.A series offruitful research works related to attitude motion are obtained from observation and stability theory to control method and robust devices.1–16Spin stabilization is considered as a simpler attitude control method than three-axis positioning system.Compared to non-spinning system,the spinstabilized system has the advantage of minimizing the effect of mechanical energy dissipation such as the tumbling of a prolate spacecraft.

The spin stabilized system for orbit-raising of payloads was firstly used in the late of nineteen sixties.When spin is adopted to stabilize a spacecraft,it is assumed that the lateral angular velocity components disappear due to gyroscopic effects.Nevertheless,a slight tip-offinduced by lateral wobble remains from the release upon motor ignition.In the previous theory of jet damping,the wobble can be eradicated shortly after ignition under the effect of jet damping.This was explained in detail by Davis as follow, ‘The existence of this torque was first called to our attention by Rosser,who pointed out that if a rocket rotates about a transversal axis during burning,the gas must be accelerated laterally as it flows down the motor tube.17The reaction on the motor tube tends to damp the rotation.”However,the propulsion driven nutation instability in spin stabilized spacecraft still occurs frequently.The nutation instability is often called the PAM-D coning anomaly,as it was first observed in the McDonnell Douglas Payload Assist Module(PAM-D)propelled by a SRB(Solid Rocket Booster)Star-48.18

As the increase of size and operation time of the motor,the spin stabilized solid rocket propelled space vehicles exhibit the unpredictable nutation instability.Small wobbling induced during separation of spacecraft from its spin platform in the space shuttle bay was enormously amplified during the second half combustion of the solid propellant.The amplitude growth ceased abruptly as the motor burned out,and the final cone angle as large as 17°were reached in some flights.18Consequently,many spacecrafts failed to perform their designated mission as the result of the nutation instability,as illustrated in Fig.1.

Stewartson(1959)studied the stability of a spinning cylindrical tank with liquid by theoretical and experimental methods.19He found that the free oscillation of the liquid inside the tank resulted in the instability while the rotational axis was deflected.Byrne and Raynor(1964)measured the oscillation angle of a rocket motor with a tail-fin in horizontal direction.20They indicated that the occurrence of the oscillation angle was induced both by the decreasing of inertia moment as the mass of the motor decreased,and by the increasing of restoring moment of the tail-fin as the speed of the motor increased.Breuer and Southerland(1965)studied the effect of the jet damping by both experimental and theoretical methods.21In their theory,the instability of the angular velocity only occurred when the decreasing of inertia moment is stronger than the effect of jet damping term.Similar conclusion was obtained from their experimental works.

Large amplitude oscillation of those spinning spacecrafts used in business satellites was observed since 1980s,and later the stability of spinning solid rocket motor had been widely studied.Eke(1983)was among the first to give a detailed explanation of the nutation instability of a spinning solid rocket motor.22He derived the attitude dynamics equations of a variable mass spinning solid rocket motor.The attitude dynamics equations were used to analyze the attitude motion of the spacecraft propelled by Star-48.He assumed the system was axisymmetric,and ignored the deflection of jet.Then Coriolis force dominated the instability of the spacecraft.While the gas inside the chamber is assumed flowing along a single axis,the mass and inertia tensor are monotone decreasing as the function of time,then the lateral angular velocity weakens with time.Webster(1985)was among the first to suppress the nutation instability by active control,23however,the mechanism of the nutation instability was unclear.He installed a jet device on the side of the spacecraft.While the angle of nutation exceeded a threshold,the jet device started to work.Mingori and Yam(1986)simulated the attitude motion of an axisymmetric rigid tank with interior non-axial flow.24The internal mass moves under the effect of an elastic restoring force.While the sloshing of attitude motion locates in front of the spacecraft mass center,and the slags locate at the rear of the motor chamber.After combustion,these slags exhibit strong sloshing behavior.Mingori and Yam showed the nutation instability occurred while those moving mass of slags located at behind the center of mass of the vehicle,they also showed that the prediction was in agreement with some flight data.

Roach etal.(1990)simulated the unsteady three dimensional internal flow field during the combustion of Star-48 by computational fluid dynamics(CFD),they suggested a lateral force arisen by those asymmetrical vortex extending to the nozzle of the motor played an important role on the increase of lateral angular velocity.25On the other hand,Misterek et al.(1991)performed a two-dimensional simulation on the motor of Star-48.26They compared the force acted on the surface of the nozzle and jet damping.The numerical results showed that jet damping played a more important role than internal gas flow.On the contrary,they disagreed that the internal gas flows were responsible for the increase of nutation motion.Cochran and Kang(1991)pointed out that the accumulation of slags at the rear of the motor led to instability.They modelled the sloshing slag as a special pendulum,in the worst situation,the pendulum moved back and forth in a natural frequency.27When the frequency of the sloshing increases,the angle of nutation can be magnified.Or and Challonner(1992)performed similar simulation mentioned above and extended the work of Mingori and Yam,24and simplified the equation of motion.28,29He pointed out that his simple model predicted an approximate result,which was in agreement with the real nutation angular velocity by telemetry.Therefore,he thought that the root of the nutation instability was the resonance between the moving frequency of sloshing and the spacecraft.However,the parameters he adopted were not realistic as used by Cochran and Kang.27

Yam et al.(1997)provided an alternative derivation of the dissipative results based on the second method of Lyapunov.30These derivations were based on the perturbation analysis of the undamped eigenvalues and yielded some insights into the stability phenomena.Their results showed that a coning growth similar to that observed in space was possible by assuming certain time histories of spacecraft parameters,and provided insights on how the maximum axis spin rule for torque free spinning bodies was modified in the presence of thrust.Kang and Cochran(2004)studied the attitude instability of a spin-stabilized,thrusting upper stage spacecraft based on a two-body model consisting of a symmetric main body representing the spacecraft and a spherical pendulum representing the liquefied slag pool entrapped in the aft section of the rocket motor.31Their analysis showed that the pendulum was subject to a combination of parametric and externaltype excitation by the main body and the energy from the excited pendulum was fed back into the main body to develop the coning instability.When one of the resonance conditions and real flight data were used,their results matched well with the observed motion before and after burnout of the motor on a typical spin-stabilized upper stage.They presented some numerical examples to explain the mechanism of the cone angle growth and how disturbance moments were generated.

In 2002,NASA’s CONTOUR satellite was lost during the working ofits STAR-30 SRB.Before the SRB worked,CONTOUR was kept spin-stabilized at nominal spin rate for 6 weeks.However,a mishap occurred near the end of the SRB burn.In this accident,the influence of slag on nutation stability was absent,because STAR-30 did not has an embedded nozzle.The jet-damping torque was initially thought as one of the most likely potential causes,but it was finally confirmed to be irresponsible for this mishap by a careful investigation.6,16The significant changes in the mass properties caused by SRB led to concern about the stability of the spacecraft.

Cooper and Costello(2011)simulated the trajectory of a model projectile with different liquid payload configurations possessing stable behavior,32while one exhibited catastrophic flight instability.They computed the liquid payload moments by solving a linearized Navier–Stokes equation for a projectile undergoing coning motion.Their results showed how both stable and unstable liquid payload configurations influence flight trajectory parameters.Zhou et al.(2013)observed an unexpected and unstable coning motion occurred after the burnout of a spinning missile in a flight trial.33Their results demonstrated that the hinge moment might lead to the instability of spinning missiles under specific conditions.Ayoubi et al.(2011)extended the existing models in the literature,and derived the nonlinear equations of motion for a constant mass system with three momentum wheels,a spherical pendulum,and a dissipative moving mass or a nutation damper.34Their numerical simulations were in good agreement with the existing results.

To sum up,many mechanisms of nutation instability were put forward in the past decades.In general,the variation of the mass of the spacecraft,the pendulum motion of the sloshing slag inside the motor,and the internal asymmetrical gas flow are considered as the most important factors leading to the nutation instability of the spacecraft.The telemetry data also shows that the severity of the problem depends on spacecraft mass properties and other system parameters.Although many control methods are developed and implemented into the nutation control systems in the spacecraft,in which the conning instability appears,the controlling devices are often weight too much in addition to the cost and complexity.They cost a large percent of the effective payload of the vehicle.And this resolution is in indirect conflict with the philosophy of solid rocket space propulsion,which is based on its inherent simplicity and low cost.Furthermore,the control devices cannot always correct for the oscillations since the nutation instability cannot be predicted accurately.Therefore,it is crucial that the mechanism of the instability be understood completely in order that serious mission degradation can be avoided in the future orbit-raising operations.

Even though the variations of mass distribution have been widely linked to the attitude motion of satellites and comet,many of studies take the system as a whole because the mass distributions vary as functions of time in a complex or a poorly known manner.3,7,10,14In this paper,we consider the variations of mass and moment of inertia of SRB based on the typical designs of SRB.The equations of motion for the spinning spacecraft with the variation of mass and moment of inertia are derived in detail.Three classic solid propellant structure designs are presented to discuss the coning instability.The combination of classic solid propellant structure design is also studied,the results suggest a remedial design is available for suppressing the transversal angular velocity caused by Radial Burn.

2.Methodology

Rocket is a typical variable mass system.Here we separate the mass of the spacecraftBinto two parts,a solid rigid bodySand the solid fuelFafter the combustion process as shown in Fig.2.The mass of the system is composed by the solid rigid body and the fuel gas inside the spacecraft,whereR1is the radius of the outlet of the nozzle andleis the distance from the center of mass to the outlet of the nozzle,the superscript ‘*’represents the mass center of different parts,Ris the radius of the solid propellant,uis the velocity of gas.If we analyze the equation of motion in a reference coordinate system(x,y,z)of an absolute coordinate system(X,Y,Z)shown in Fig.3,where Rmis the position vector of mass center of the spacecraft,rmis the position vector of a given small element dmrelative to the mass center of the spacecraft in the reference coordinate system.We can derive the equation of attitude motion of the spacecraft and obtain the following equation,

Heretdenotes time,I is the moment vector of inertia of the spacecraft,α and ω denote angular acceleration and angular velocity of the spacecraft,respectively.M is the net moment offorce act on the spacecraft.ρ represents the density,and dVdenotes the volume of the small element.In the above equation,the third term on the left hand side is the moment of Coriolis force MC.The fourth term on the left hand side has the following expression by using Reynolds transport theorem,

whereAeis the exit area of the motor,n is the unit vector normal to the surface.The first term on the right hand side of Eq.(2)represents the rate of the relative angular moment of the system MH.While the second term on the right hand side of the above equation is the loss of the relative angular moment through the boundary of the system MT.It also represents the effective moment of the thrust vector relative to the center of mass of the spacecraft.

By using Reynolds transport theorem,the moment of Coriolis force has the following expression,

The first term of the above equation represents the change of moment arisen by the change of the moment of inertia.While the second term is thought as the moment of jet damping.To obtain the integral of this term,it is necessary to know the geometric configuration of the boundary of the system and velocity profile of the jet.In other word,the size of the combustion and the outlet of the nozzle have influence on the attitude of the spacecraft.

Substituting Eqs.(2)and(3)into Eq.(1),we obtain the equation of attitude motion of a spinning system as,

In the above equation,the terms containingare all caused by the relative motion of the internal gas inside the motor.However,it is very difficult to know the flow configuration for a motor with nutation instability.In this paper,we focus on the flight attitude due to the mass variation of the spacecraft.While the flow inside the motor is axisymmetric relative to the rigid body, so we haveWhen the flow is assumed to be axisymmetric inside the motor,we have MH=MT=0,i.e.the relative velocitycan be ignored except on the inlet and outlet of the system.Likewise,the last term of MCis negligible.

On this basis,the jet damping torque and the moment arisen by the change of the moment of inertia are playing the most important role on the attitude motion of the system.Therefore,the equation of motion for the spinning system can be simplified as,

Supposing the solid rigid body of the spacecraft is axisymmetric.We have the same moment of inertia in both transversal directions,i.e.Ix=Iy=It.Then we can express the moment of inertia as I=It(i+j)+Izk.Izis the moment of inertia in the axial direction.And i,j,k are the unit vectors in the three directions,as shown in Fig.2.In the same way,we can express the angular velocity as ω = ωxi+ωyj+ ωzk and the angular acceleration asSo the left three terms in Eq.(5)can be written as,

Because the Reynolds number is very high at the exit of the nozzle,it is reasonable to assume the velocity of jet on the outlet is constant,i.e.constant.So the last term of Eq.(5)can be written as,

Hence we can rewrite Eq.(5)as,

where˙m=represents the mass flux of the spacecraft.

If the initial angular velocity of spacecraft is ω0,and transversal angular velocity is ωT= ωx+jωy,then based on Eqs.(10)and(11),we have,

So we can solve the transversal angular velocity,

where ωT0is the initial transversal angular velocity.Following Eq.(12),we have,

Therefore,the angular velocity of the system changes exponentially as the change of the system parametersandfrom Eqs.(15)and(16).

In order to obtain the attitude motion of the system,it is necessary to know the variations of these parameters with time.For example,the transversal moment of inertia of the system can be written as,

whereIsis the transversal moment of inertia of the rigid body relative to the center of mass of the spacecraftS*,Ifrepresents the transversal moment of inertia of the solid propellant relative to its center of massF*,msandmfdenote the mass of the payload and the rigid body part,and the mass of solid propellant,respectively.aandbrepresent the distance from the center of mass of the rigid body to the center of mass of the systemB*,and the distance from the center of mass of the solid propellant to the center of mass of the systemB*,respectively.OnlyIsandmsare fixed in the above equation.Here we ignore the fixed part and study the variable mass part,which corresponds to the solid propellant change as shown in Fig.4.

For the variable mass solid propellant,we can simplify it to an axisymmetric cylinder with length 2h.After ignition,the solid propellant moves backward during the combustion so that the mass and geometric distribution change.Here we study several classical solid propellant configurations to analysis the influence on the transversal and axial angular velocity.It is noted that our method is also effective for other real spinning solid rocket motors.

3.Results and discussion

3.1.Case A:End Burn

Fig.5 shows a typical design of a solid rocket motor with End Burn.The solid propellant change into gas flow gradually from the right end.The solid part in the figure shows the unburned propellant.Hence,the mass and the moment of inertia of the propellant vary with time.Substituting the mass and the moment of inertia and their derivatives into the equations of attitude,

From Eq.(19),we know ωz=constant,i.e.the axial angular velocity is not changed during the flight.Based on Eqs.(17)and(18),one can obtain the relation of transversal angular velocity,

Then we obtain,

It is noted that ωTdecreases during the process of combustion.It suggests that the transversal angular velocity dissipates after ignition for End Burn design.However,asincreases,the dissipation of the transversal angular velocity slows down.It indicates that for those slender designs of the solid propellant with End Burn,they are more stable on nutation stability than those with podgy designs.Asthe transversal angular velocity keeps its initial disturbance,i.e.ωT=ωT(0).

3.2.Case B:Radial Burn

For a Radial Burn design of the solid propellant as shown in Fig.7,the interface between solid propellant and gas flow moves radially during the combustion.

From the equations of the attitude motion,we have,

For the transversal angular velocity,

We also plotted the variation of transversal angular velocitythe interface between solid propellant and gas in Fig.9.It is observed that the variation of angular velocity is a monotone decreasing function whenR/his less than a critical value 1.63.Otherwise,the transversal angular velocity decreases first,then it rises again.Therefore,to achieve a better nutation stability,a small value ofR/hin the design of a Radial Burn is recommended.

3.3.Case C:Centripetal Burn

Be different to the Radial Burn,a Centripetal Burn is the design which the interface between solid propellant and gas moves toward the central axis as shown in Fig.10.

So the equation of motion for the axial angular velocity has the following relationship,

For the transversal angular velocity,we have,

We can obtain the integral of the above equation read,cates that the transversal angular velocity increases at the beginning until it reaches a maximum value for the large value ofR/h.At the end of the combustion,it decreases as the interface between the solid propellant and the gas moves toward the central axis.It suggests that the nutation angle could be magnified for the design with large value ofR/h.

3.4.Case D:Combination of Radial Burn and End Burn

Sometimes,both the End Burn and the Radial Burn are adopted in the design of solid rocket motor as illustrated in Fig.13.In this situation,we have,

So the variation of the axial angular velocity depends on the combustion on the both directions.The integral of the above equation reads,

The variation of axial angular velocity versus 1is plotted in Fig.14.It suggests that for any given value ofR/h,the axial angular velocity increases.For a larger value ofR/h,the curve moves close to the axis of 1at this time,the axial angular velocity diverges more rapidly than those with smaller value ofR/h.

In the same way,one obtains the following equation based on the equation of transversal angular velocity,

Andle=2h-z.The variation of transversal angular velocity is plotted in the same manner as shown in Fig.15.The transversal angular velocity decreases with the time.The curves are closed to the axis offor the low value ofR/h.AsR/hincreases to 2,the curve locates in the middle of the two axes.For large value ofR/hthe decreases of transversal angular velocity takes longer time.Comparing to the case of End Burn,the transversal angular velocity dissipates rapidly.

3.5.Discussion

Based on the calculation above,the variable parameters of the system are in associated with the size ofzandr.It avoids to introduce the complex function of time for those parameters.However,in order to implement the present model to a real spacecraft,it is useful to take the parameters with the function of time into account.The variations ofzandrlead to the variation of mass,moment of inertia,and so on.For a complex design of a solid rocket motor,it is difficult to set an accurate function with time for those parameters.In this situation,it is necessary to simplify the complex design to a combination of several simple cases.For those simple cases,it is not difficult to associate the geometric parameters to time.

For instance,for the case of End Burn,while the burn rate of the solid propellant isu,i.e.Then we havez=The parameters of the system can be linked toz.So one obtained the mass function with time,m=m0+˙mtwith˙m=-ρπR2uandm0=2ρπR2h.

In the same way,the moment of inertia can be written as,

For the radius of gyrationk, we haveHence we know that the moment of inertia is a cubic function of time,k2is a quadratic function of time,mandIzare linear function of time for the End Burn design.In the same way,we know thatmis a linear function of time,andItis a quadratic function of time for the Radial Burn design.For the combination of End Burn and Radial Burn design,Itis a quantic function of time andk2is a quadratic function of time.For a real spacecraft,the mass and the moment of inertia of payload are known and constant.If only the time-dependent function of those parameters of the system are known,it is the samemway to solve the motion of attitude of the spacecraft.

4.Concluding remark

Spin-stabilized solid rocket motor is often adopted as the last stage rocket for the launch of small satellite.The mass of small satellite is so light about 50 kg that extra control devices would be very expensive.However,the nutation instability of the spacecraft is fatal for these small satellites.A well-accepted explanation to the nutation instability is not clearly understood until now.The variation of the parameters of the system,the instability of the gas dynamics in the combustion chamber and the instability of the interaction between the thrust vector and the internal motion of slags are considered as the mechanism of nutation instability.Based on the property description of the spacecraft nutation instability we already have,this unstable phenomenon is studied in this paper.Instead of taking the SRB as a whole,the variations of mass,moment of inertia of a spacecraft with SRB are concretely studied based on the typical designs of solid propellant.

Base on the dynamic equations of the whole system with the assumption for the internal flow in the rocket motor,the influence of the variable mass on the system attitude dynamics is examined by using three classical propellant designs.We analyzed the effect of the variation of the mass and the moment of inertia on the angular velocities.From the results,it can be seen that the variation of the attitude is very different with varied design of solid propellant and varied burning pattern.The mass dissipation of the propellant is advantageous to the spinning motion in End Burn,which makes the transversal angular velocity tend to be stable gradually,but it is opposite in Radial Burn that the transversal angular velocity grows exponentially,thus causes the unstableness of the flight.

We also analyzed the propellant design of the combination of End Burn and Radial Burn.The influence on the angular velocity can be thought as the combination of the influence of each case.The combination of both Burns could eliminate transversal angular velocity of a spacecraft with nutation.It illustrates the nutation instability aroused by Radial Brun could be remedied by the addition of End Burn at the same time.The present method supplies an effective way to analyze the nutation instability for a SRB with more complex propellant structure.

Acknowledgement

This work was supported by the National Natural Science Foundation of China(Nos.11502086 and 11502087).

Reference

1.Liu XK,Zhao H,Yao Y,He FH.Modeling and analysis of microspacecraft attitude sensing with gyrowheel.Sensors2016;16(8):1321–40.

2.Ran DC,Sheng T,Cao L,Chen XQ,Zhao Y.Attitude control system design and on-orbit performance analysis of nano-satellite-‘Tian Tuo 1’.Chin J Aeronaut2014;27(3):593–601.

3.Olson C,Russell RP,Bhaskaran S.Spin state estimation of tumbling small bodies.J Astronauti Sci2016;63(2):124–57.

4.Morozov VM,Mikhailov DD,Kalenova VI.On the stability of stationary motions of a system of coaxial bodies.Cosmic Res2016;54(2):164–9.

5.Cheng Y,Wang RX,Xu MQ,Li YQ.Simultaneous state and actuator fault estimation for satellite attitude control systems.Chin J Aeronaut2016;29(3):714–37.

6.Van der Ha JC.Lessons learned from the dynamical behaviour of orbiting satellites.Acta Astronaut2015;115(10–11):121–37.

7.Martin KM,Longuski JM.Velocity pointing error reduction for spinning,thrusting spacecraft via heuristic thrust profiles.J Spacecraft Rockets2015;52(4):1268–72.

8.Lin YY,Wang CT.Detumbling of a rigid spacecraft via torque wheel assisted gyroscopic motion.Acta Astronaut2014;93(1):1–12.

9.Lai YW,Liu JH,Ding YH,Gu DF,Yi DY.Precession-nutation correction for star tracker attitude measurement of STECE satellite.Chin J Aeronaut2014;27(1):117–23.

10.Janssens FL,Van der Ha JC.Stability of spinning satellite under axial thrust and internal mass motion.Acta Atronaut2014;94(1):502–14.

11.Doroshin AV,Krikunov MM.Attitude dynamics of a spacecraft with variable structure at presence of harmonic perturbations.Appl Math Model2014;38(7–8):2073–89.

12.Nazari M,Butcher EA.Analysis of stability and Hopf bifurcation of delayed feedback spin stabilization of a rigid spacecraft.Nonlinear Dyn2013;74(3):801–17.

13.Gui HC,Jin L,Xu SJ.Local controllability and stabilization of spacecraft attitude by two single-gimbal control moment gyros.Chin J Aeronaut2013;26(5):1218–26.

14.Jing WX,Xia XW,Gao CS,Wei WS.Attitude control for sSpacecraft with swinging large-scale payload.Chin J Aeronaut2011;24(3):309–17.

15.Janssens FL,Van der Ha JC.On the stability of spinning satellites.Acta Astronaut2011;68(7–8):778–89.

16.Van der Ha JC,Janssens F.Jet-damping and misalignment effects during solid-rocket-motor burn.J Guid Control Dynam2005;28(3):412–20.

17.Davis JL,Follin JW,Blitzer L.Exterior ballistics of rocket.Princeton:Van Nostrand;1958.p.33–45.

18.Flandro GA,Leloudis M,Roach R.Flow induced nutation instability in spinning solid propellant rockets;1989.Report No.:AL-TR-89-084.

19.Stewartson K.On the stability of a spinning top containing liquid.J Fluid Mech1959;5(4):577–92.

20.Byrne WJJ,Raynor S.An analysis of the yawing motion of a rocket with a varying mass.AIAA J1964;2(5):962–3.

21.Breuer DW,Southerland WR.Jet-damping effects:theory and experiment.J Spacecr Rockets1965;2(4):638–9.

22.Eke FO.Dynamics of variable mass systems with applications to the Star 48 solid rocket motor.Adv Astronaut Sci1983;54:671–84.

23.Webster EA.Active nutation control for spinning solid motor upper stage.AIAA,SAE,ASME,and ASEE,joint propulsion conference,1985 July 8–10.Monterey,CA;Reston(VA):AIAA;1985.p.1-7.

24.Mingori DL,Yam Y.Nutational instability of a spinning spacecraft with internal mass motion and axial thrust.AAS/AIAA astrodynamics conference;1986 August 18–20;Williamsburg,USA.Reston(VA):AIAA;1986.p.367–75.

25.Roach R,Burnley R,Flandro G,Burnette J.Internal ballistics interactions in spinning rocket nutation instability.AIAA 28th aerospace sciences meeting;1990 January 8–11;Reno,USA.Reston(VA):AIAA;1990.p.1–12.

26.Misterek D,Murdock J,Koshigoe S.Gas dynamic flow in a spinning coning solid rocket motor.AIAA 22nd fluid dynamicsplasma dynamicsamp;lasers conference;1991 June 24–26;Honolulu,USA.Reston(VA):AIAA;1991.p.1–9.

27.Cochran JE,Kang JY.Nonlinear stability analysis of the attitude motion of a spin stabilized upper stage.Proceedings of the 1st AAS/AIAA annual spaceflight mechanics meeting;1991 Feb 11–13;Houston,USA.Reston(VA):AIAA;1991.p.345–64.

28.Or AC.Rotor-pendulum model for the perigee assist module nutation anomaly.J Guid Control Dynam1992;15(3):297–303.

29.Or AC,Challonner DA.Stability of spinning spacecraft containing shallow pool of liquid under thrust.J Guid Control Dynam1994;17(5):1019–27.

30.Yam Y,Mingori DL,Halsmer DM.Stability of a spinning axisymmetric rocket with dissipative internal mass motion.J Guid Control Dynam1997;20(2):306–12.

31.Kang JY,Cochran JE.Resonant motion of a spin-stabilized thrusting spacecraft.J Guid Control Dynam2004;27(3):356–64.

32.Cooper GR,Costello M.Trajectory prediction of spin-stabilized projectiles with a liquid payload.J Spacecr Rockets2011;48(4):664–70.

33.Zhou W,Yang SX,Dong JL.Coning motion instability of spinning missiles induced by hinge moment.Aerosp Sci Technol2013;30(1):239–45.

34.Ayoubi MA,Goodarzi AF,Banergee A.Attitude motion of a spinning spacecraft with fuel sloshing and nutation damping.J Astronaut Sci2011;58(4):551–68.

26 January 2016;revised 19 October 2016;accepted 16 March 2017

Available online 21 June 2017

*Corresponding author at:Department of Mechanics,Huazhong University of Science and Technology,Wuhan 430074,China.

E-mail address:xylcfd@hust.edu.cn(Y.XIONG).

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Attitude control;

Nutation dampers;

Payloads;

Solid rocket booster;

Spin stabilization of spacecraft