Cantilever Beam Design for Projectile Internal Moving Mass Systems

Transcription

1 Jonathan Rogers Graduate Research Assistant Mark Costello Sikorsky Associate Professor Me. ASME School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA Cantilever Bea Design for Projectile Internal Moving Mass Systes Internal asses that undergo controlled translation within a projectile have been shown to be effective control echaniss for sart weapons. However, internal ass oscillation ust occur at the projectile roll frequency to generate sufficient control force. This can lead to high power requireents and place a heavy burden on designers attepting to allocate volue within the projectile for internal ass actuators and power supplies. The work reported here outlines a conceptual design for an internal translating ass syste using a cantilever bea and electroagnetic actuators. The cantilever bea acts as the oving ass, vibrating at the projectile roll frequency to generate control force. First, a dynaic odel is developed to describe the syste. Then the natural frequency, daping ratio, and length of the bea are varied to study their affects on force required and total battery size. Trade studies also exaine the effect on force required and total battery size of a roll-rate feedback syste that actively changes bea elastic properties. Results show that, with proper sizing and specifications, the cantilever bea control echanis requires relatively sall batteries and low actuator control forces with iniu actuator coplexity and space requireents. DOI: / Introduction Contributed by the Dynaic Systes, Measureent, and Control Division of ASME for publication in the JOURNA OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON- TRO. Manuscript received October 29, 2008; final anuscript received May 4, 2009; published online August 18, Assoc. Editor: Ahet S. Yigit. The past several decades have seen growing interest in the developent of sart unitions resulting fro attepts to increase accuracy and to decrease collateral daage. Sart projectiles differ fro other guided weapons, such as guided issiles, in that their electronics and control echaniss ust be able to withstand extree acceleration loads associated with launch and high spin rates. Control coponents ust be relatively inexpensive, since projectiles are typically fired in large quantities. Several different types of control echaniss have been designed to eet these requireents, naely, aerodynaic echaniss, thrust echaniss, and inertial load echaniss. Coon exaples of aerodynaic echaniss are canards, gibaled nose configurations, and deflection of ra air through side ports. Exaples of thrust echaniss include cold gas jets and explosive thrusters. Exaples of inertial load echaniss are rotation of an unbalanced internal part and, of specific interest here, oveent of an internal translating ass ITM. Oscillation of an internal ass at the projectile roll frequency has been shown to produce useful control authority. A design that iniizes oving parts and substantially reduces power required is critical in order to physically ipleent an ITM control echanis on board a sart unition. Previous investigation of projectiles equipped with loose or oving internal parts has revealed that these configurations can result in flight instabilities. Soper 1 considered the stability of a projectile with a cylindrical ass fitted loosely within a cavity. Murphy 2, using a siilar configuration, derived a quasilinear solution for the otion of a projectile equipped with a oving internal part. A detailed set of experients was later conducted by D Aico 3 to odel the otion of internal asses within spinning projectiles using a freely-gibaled gyroscope. Hodapp 4 further considered the affect of a sall offset between the projectile body ass and the ITM ass center. Hodapp s analysis of the dynaic equations for this syste showed that, for sall ass center offsets, slight oveent could actually reduce the instability caused by the loose internal part. The use of controlled oveent of an ITM as a aneuver control echanis was considered by Petsopoulous et al. 5 for use on re-entry vehicles, while Robinett et al. 6 considered ITM control for ballistic rockets. More recently, Menon et al. 7 exained ITM control for use on endoand exoatospheric interceptors using three orthogonal ITMs. Frost and Costello 8,9 studied the ability of an internal rotating ass unbalance to actively control both fin and spin-stabilized projectiles. Most recently, Rogers and Costello 10 investigated control authority of a projectile equipped with a single internal translating ass. It was deterined that significant control authority could be created by oscillating the ITM at the projectile roll frequency. This paper outlines a notional design of an ITM actuator that generates sufficient control authority using relatively low power. Control oent is generated due to an axial drag offset fro the syste ass center caused by the lateral otion of the ITM. This paper begins with a description of the cantilever bea syste that serves as the translating ass, and derives the seven-degree of freedo flight dynaic odel used for trajectory predictions. A description of the control law and ITM electroagnetic actuators are also provided. The dynaic odel is subsequently eployed to deonstrate that the cantilever bea configuration provides sufficient control authority at reasonable power levels. Trade studies exaine the effect of cantilever bea length on force required and battery size. The optiu natural frequency and daping ratio of the bea is deterined in order to iniize actuator control effort. Finally, roll-rate feedback control is ipleented to actively alter bea characteristics as the projectile roll rate changes during flight, further decreasing the control effort required. A final exaple case deonstrates that sufficient control authority can be generated with relatively sall battery sizes using the optiu spring and daper coefficients and the roll-rate feedback syste. 2 Cantilever Bea Projectile Dynaic Model The cantilever bea is a fixed-free elastic bea, with one end attached to the projectile at point and the free end floating within the cavity, constrained to vibrate in the I S J S plane. The bea s first vibrational ode is the only ode considered signifi- Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Copyright 2009 by ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

2 Fig. 1 The ITM-bea projectile cant to dynaic interaction with the projectile. For this reason, the cantilever bea can be accurately odeled as a rigid assless bea with a spherical ass considered to be a peranent agnet attached to the end. A torsional spring and daper are attached to the hinge point to siulate the elastic properties of the cantilever bea. This dynaically equivalent syste is referred to as the ITM-bea. A sketch of this configuration is shown in Fig. 1. Note that is defined as the angle between the ITM-bea and the centerline of the projectile. A peranent agnet is attached to the end of the bea and can swing freely about the hinge. Force is exerted on the agnet by electroagnets on both sides of the cavity to ove the bea to a desired angle. Five reference fraes are used in the developent of the equations of otion for this syste, naely, the inertial, projectile, translating ass, nonrolling, and projectile-fixed S reference fraes. The projectile frae is obtained using the standard aerospace Euler angle sequence of rotations, and is related to the inertial frae by I B c c c s s I J B s s c c s s s s + c c s c J I c s c + s s c s s s c c c I I 1 K The N frae is the standard nonrolling reference frae often used in projectile flight dynaics, and is defined by a rotation of along the I B axis. The S frae is also fixed to the projectile, with its origin at the hinge point. It is defined such that the ITM-bea oscillates about the K S axis, and J S points to the rear of the cavity exactly equidistant fro both electroagnetic actuators. Therefore, the S frae can be related to the B frae by two constant Euler angles S and S such that K B = I S J S K S = c S c S c S s S s S B s S c S 0 J I B B s S c S s S s S c S K Throughout the rest of this article, fixed angles of S =90 deg and S =0 are used, and thus the S frae can be obtained by a single 90 deg rotation about the K B axis, resulting in the orientation shown in Fig. 1. The T frae is fixed to the ITM-bea and is related to the S frae by the relationship I T J T K T = c s 0 S s c 0 J S I S K Note that the T frae is aligned with the S frae when =0. All equations in this paper use the following shorthand notation for trigonoetric sine, cosine, and tangent functions: s =sin, c =cos, and t =tan. Throughout the developent of the equations of otion, two operators will be used to denote coponents of a vector in a 2 3 specific frae and the skew syetric cross-product operator, respectively. The vector coponent operator outputs a colun vector coprised of the coponents of an input vector in a given frae. For exaple, if the position vector fro to is expressed in reference frae A as r = x I A + y J A + z K A then the vector coponent operator acting on this vector yields C A r = x y z 4 Notice that the reference frae is denoted by the subscript on the operator. The cross-product operator outputs a skew syetric atrix using the coponents of an input vector in the reference frae denoted in the subscript. For exaple, if the position vector fro to is expressed in reference frae A as r = x I A + y J A + z K A then the cross-product operator acting on r expressed in reference frae A is S A r = 0 z y z 0 x 5 y x Kineatics. The velocity of the coposite body ass center can be described in the inertial frae or the projectile reference frae v C/I = ẋi I + ẏj I + żk I = ui B + vj B + wk B 6 The translational kineatic differential equations relate these two representations of the ass center velocity coponents ẋ s s c c s c s c + s s u ẏ c s s s s + c c c s s s c v 7 ż = c c s s c c c w The angular velocity of the projectile with respect to the inertial reference frae can be written in ters of appropriate Euler angle tie derivatives or in ters of projectile frae angular velocity coponents B/I = I B + J N + K I = pi B + qj B + rk B 8 The kineatic relationship between tie derivatives of the Euler angles and projectile reference frae angular velocity coponents represents the rotational kineatic differential equations = 1 s t c t 0 c s 0 s /c c /c p q r The final kineatic differential equation is the trivial relationship = bea / Vol. 131, SEPTEMBER 2009 Transactions of the ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

3 2.2 Dynaics. The translational dynaic equations for the ITM-bea projectile are derived through force balancing. Force balance equations for the projectile and ITM-bea are given, respectively, as P a P/I = W P + F P F C F I 11 T a X/I = W T + F C + F I 12 where P and T are the asses of the projectile and ITM-bea, respectively; a P/I and a X/I are, respectively, the accelerations of points P and X with respect to the inertial frae; and W P, W T, F P, F I, and F C are the weights of the projectile and ITM-bea, the total aerodynaic force exerted on the projectile, the input force exerted on the ITM-bea by the actuators, and the hinge constraint force, respectively. The definition of the syste center of ass leads to a C/I = P a P/I + T a X/I 13 Therefore, adding Eqs. 11 and 12 and noting the relationship in Eq. 13, the translational dynaic equation for the syste is fored a C/I = W P + W T + F P 14 The aerodynaic forces given by F P in Eq. 14 are obtained using the standard aerodynaic expansion eployed for projectile flight dynaic siulation. Both steady aerodynaic forces and Magnus forces are included, as well as steady and unsteady aerodynaic oents. The aerodynaic coefficients and aerodynaic center distances used to generate these forces and oents are all a function of the local Mach nuber at the center of ass of the projectile. Coputationally, these Mach nuber dependent paraeters are obtained by a table look-up schee using linear interpolation. A full description of the weight force and body aerodynaic forces and oents are provided in Ref. 10. Writing Eq. 14 in the projectile reference frae yields ẇ = XB u v Y B Z 0 r q B w r 0 p q p 0 u v 15 Note that X B, Y B, and Z B are projectile reference frae coponents of the su of the three forces given in Eq. 14. The rotational dynaic equations are obtained by first equating the I frae tie rate of change of the syste angular oentu about the syste ass center to the total applied external oents on the syste about the syste ass center in the I S and J S directions, given by Eqs. 16 and 17. Then, the sae oent equation is used for each body separately, this tie written in the K S direction. These four equations are given by I S P P dh I X B/I dh T/I + + r P P a P/I + r X T a dt dt X/I = I S M syste J S I P dh B/I + dt I X dh T/I dt = J S M syste + r P P a P/I + r X T a X/I K S I X dh T/I + r X T a dt X/I = f input Bc + K S S r X C S W T + k T + k D 18 K S I P dh B/I + r P P a dt P/I = f input Bc + K S S r P C S W P k T k D + K S C S M P 19 whereas before c =cos. Note that Eq. 18 equates the tie rate of change of the angular oentu of the ITM-bea to the total oent applied to the ITM-bea. Equation 19 equates the tie rate of change of the angular oentu of the projectile to the total oent applied to the projectile. Several interediate expressions will be useful in deriving the rotational dynaic equations in the body-fixed S frae. First, note that the well-known two points fixed on a rigid body forula yields the relationship a X/I = a C/I + P B/I r P + T/I r X + B/I B/I r P + T/I T/I r X 20 Equation 20 is used to expand a X/I in ters of known quantities and state derivatives. Also, using the definition of the syste center of ass, it can be shown through algebraic anipulation that r P P a P/I + r X T a X/I = r P a C/I + r P X T a X/I 21 Equation 21 is also used to expand the cross-product ters on the left-hand side of Eqs. 16 and 17 in ters of known quantities and state derivatives. Equations can be expanded using the expressions in Eqs. 20 and 21 and rearranged to for a 4 4 syste of equations given by A11 A12 A13 A14 A 21 A 22 A 23 A 24 q B 2 22 A 31 A 32 A 33 A 34 r A 41 A 42 A 43 A 44 p B 3 B 4 = B1 In Eq. 22, rows 1 4 correspond to Eqs , respectively. The full expressions for the values of the A atrix and the B vector are lengthy and are provided in Appendix A. The set of equations given by Eqs. 7, 9, 10, 15, and 22 constitute the equations of otion for the ITM-bea projectile. Given a known set of initial conditions, these 14 scalar equations are nuerically integrated forward in tie using a fourth-order Runge Kutta algorith to obtain a single trajectory. 2.3 Description of Controller. In order to create trajectory alterations, the ITM-bea ust be oved in a prescribed anner. The control law is forulated based on a feedback linearization technique 11, which assues full state feedback. Equation 23 is used to copute the control force, f input, required to deflect the ITM-bea to the desired angle. f input = 1 B FC A 11 p A 12 q A 13 r A 14 K 0 coand Bc K 1 coand K 2 coand 23 Note that B FC is defined in Appendix A and is derived fro Eq. 18. ikewise, A 11, A 12, A 13, and A 14 are fro Eq. 18 and provided in Appendix A. Note that Eq. 18 is used to copute the feedback linearization control, rather than Eq. 19, since Eq. 19 would require feedback of aerodynaic loads. This is a copli- Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

4 Table 1 Relevant exaple projectile and ITM-bea properties Projectile ass slugs ITM-bea ass slugs 0.05 Projectile reference diaeter ft Projectile ass center position easured along the stationline ft 1.18 Projectile roll inertia slugs ft Projectile pitch inertia slugs ft ITM-bea length B ft 0.3 Fig. 2 Zoo view of the ITM-bea syste cated task and can be avoided through the use of Eq. 18 instead. The coanded deflection angle coand is generated by synchronizing ITM-bea oveent with the projectile roll angle. This is done by setting coand = sin T 1 A B cos + 24 where A is the agnitude of oscillation of point X fro the cavity center, and T is a tri angle used to define the plane of control. Derivatives of Eq. 24 are coputed analytically and used in Eq. 23. Note that the feedback linearization controller is developed and used within the siulation solely to create the prescribed otion of the ITM-bea for control authority analysis and the deterination of power requireents. Its purpose in this case is to atch the ITM-bea oscillation frequency to the projectile roll rate within the siulation. 2.4 Description of Electroagnetic Actuator Control Syste. A zoo view of the ITM-bea echanis is shown in Fig. 2. Two electroagnets, each at opposite ends of the cavity, exert force on the fixed agnet at the end of the ITM-bea. The force exerted on a fixed agnetic dipole DM is given by 12,13 f input = DM B z z M 25 The agnetic dipole oent per unit ass is a unique property of a aterial, with units joules/tesla/slug. For exaple, assuing the ITM is ade of agnetized iron and using an ITM-bea ass of T =0.05 slugs, the dipole oent is found to be DM =171.5 J/T. The quantity B z / z M can be found by first recognizing that the agnetic field B z of an iron-core solenoid is given by the expression B z = I E kn 2 z M zm 2 + b z M + A zm + A 2 + b 26 where I E is the current through the electroagnet, is the agnetic constant N/A 2, z M is the distance fro the end point of the ITM-bea to the nearest electroagnet actuator, k is the diensionless relative pereability of iron 200 at a agnetic flux density of W/ 2, n is the nuber of coils per eter, b is the radius of the solenoid, and A is the length of the solenoid. For all cases used below, values of 3 c, 2 c, and 10,000 were used respectively for b,, and n. Taking the derivative of Eq. 26 with respect to z M B z z M = I E kn 2 1 zm 2 + b 2 + 3/2 z M + A 2 z M + A 2 + b 2 2 z M z 2 M + b 2 3/2 + 1 zm + A 2 + b 2 27 At each tiestep, the control force is coputed using the feedback linearization. Knowing the agnetic dipole oent and the required control force, the quantity B z / z M is coputed at each tiestep using Eq. 25. Then, knowing the position of the ITMbea with respect to the actuators, the current required can be coputed at each tiestep by rearranging Eq. 27 such that I E = 2 k n 1 zm 2 + b z M z 2 M + b 2 3/2 + 1 zm + A 2 + b 2 z M + A 2 3/2 1 B z z M + A 2 + b 2 28 z M Note that the electroagnet diensions used above are coparable to diensions for coercially-available iron-core electroagnets. 2.5 Description of Exaple Projectile. The exaple projectile used in all exaple siulations below is a representative finstabilized projectile. Relevant exaple projectile and ITM-bea diensional and ass properties are outlined in Table 1. The hinge point is 0.9 ft behind the projectile ass center P, and the ITM-bea oscillation aplitude is given by B sin ax =0.157 ft unless otherwise specified. In all the following cases the projectile is traveling through a standard atosphere with no atospheric wind. 3 Results An exaple trajectory of the ITM-bea projectile is copared with an exaple trajectory of a projectile equipped with a strictly translating internal ass for odel validation purposes. The translating ass projectile s dynaic equations are given in Ref. 10, and a previously validated odel of this syste was used for trajectory predictions. A diagra of the translating ass projectile is provided in Appendix B. Initial conditions used for the exaple trajectory were x=0.0 ft, y =0.0 ft, z=0.0 ft, u= ft/ s, v =0.0 ft/s, w=0.0 ft/s, = rad, =0.05 rad, =0.0 rad, p=5.0 rad/ s, q=0.0 rad/ s, and r=0.0 rad/ s. The feedback linearization gains were K 0 =0.001, K 1 =1000.0, and K 2 = In all cases shown below, the ITM-bea oscillation frequency is locked to the projectile roll rate. Figures 3 6 show the trajectories for the ITM-bea projectile, the translating ass projectile, and the rigid projectile with no internal oving ass denoted Rigid 6DOF. The two translating ass trajectories are generated solely to deonstrate control authority and to validate the ITMbea siulation, and thus both controlled rounds are coanded to axiu possible deflection. Notice that the trajectories of the internal ass projectiles atch nearly identically, even though the dynaic equations for the two systes are significantly different / Vol. 131, SEPTEMBER 2009 Transactions of the ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

5 Fig. 3 Altitude versus range for exaple trajectory The correlation between these two odels serves as validation of the ITM-bea siulation. Figure 7 shows a selected tie history of ITM displaceent fro the projectile centerline. For the ITM-bea projectile, this displaceent, denoted s x, is given by Fig. 6 Roll rate versus tie for exaple trajectory. The thick lines represent high frequency oscillations, which occur only for the ITM-bea and translating ass case. s x = B sin 29 Notice that the two tie histories for ITM displaceent shown in Fig. 7 are nearly identical. A control force tie history for an exaple ITM-bea projectile siulation can be used, together with an ITM displaceent tie history, as shown in Fig. 7, to generate a tie history of current required for each electroagnetic actuator. This is accoplished using the procedure outlined in Eqs Furtherore, this current tie history can be integrated to produce the total charge required for a given exaple flight in A sec. This value for total charge can be used to size the battery for the ITMbea control syste. The total charge required for the exaple flight shown above was 13.5 A sec. Note that this nuber is relatively large since the spring and daper coefficient values have not been optiized for this preliinary exaple case. Figure 8 shows a segent of the current tie history for the exaple ITM-bea siulation used above. Notice that f input labeled Force in the plot and the ITM displaceent s x are shown on the sae plot as the current tie history to deonstrate the phase relationships between current, input force, and ITM displaceent. When the ITM-bea is displaced in the positive I S Fig. 4 Cross range versus range for exaple trajectory Fig. 5 u Velocity versus tie for exaple trajectory Fig. 7 Selected tie history of ITM displaceent fro projectile centerline Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

6 Fig. 8 Segent of current versus tie for ITM-bea actuators Fig. 10 Average force required versus bea length direction s x positive, electroagnet EM 1 shuts off and EM 2 is responsible for control. ikewise, when the ITM-bea is displaced in the negative I S direction s x negative, electroagnet EM 2 shuts off and EM 1 is responsible for control. This schee takes advantage of the fact that the electroagnets are uch ore effective when the ITM is at close range. Since the current required is a nonlinear function of both distance to the ITM and control force required, the current tie history is not sinusoidal like the ITM displaceent and the control force tie histories. The length of the ITM-bea has a significant ipact on the force required to ove the bea in a prescribed fashion. Fro Eq. 18, the external oent exerted on the ITM-bea by the actuators in the K S direction about point is given by K S M bea = f input Bc 30 Therefore, the input force required to exert a given control oent on the ITM-bea varies inversely with bea length. In addition, for a given axiu ITM displaceent s x, the axiu angular displaceent ax varies inversely with bea length. Trade studies verified these results using exaple siulations with various bea lengths. The axiu ITM displaceent fro the projectile centerline was s x =0.157 ft for all cases. Figures 9 and 10 show axiu angular displaceent and average force required as a function of bea length, respectively. Notice that as bea length increases, axiu angular displaceent and average force required both decrease. The average control force reduction with increased bea length occurs due to the increased efficiency of the ITM-bea actuators i.e., the sae control oent requires less control force for a larger bea length. As outlined above, the ITM-bea syste is a dynaic odel used to represent a fixed-free elastic bea. The rigid ITM-bea is attached to the projectile at point with a torsional spring and torsional daper to odel the elastic bea s vibrational properties. A trade study exaines how force and power requireents vary with different spring and daping coefficients. Once optiu spring and daper coefficient are deterined, they can be used to identify the proper elastic properties of the fixed-free bea for a prototype syste. The perforance of the syste is exained for a range of torsional spring constants and daping ratios for the exaple projectile rolling at a steady-state rate of approxiately 128 rad/s. The projectile trajectory is siulated for a 2 s flight with no gravity. This siplified flight profile is used solely to establish the correlation between spring and daper paraeters and average force, average power, and total battery charge required. Figure 11 shows the projectile roll-rate tie history for this flight profile. The high frequency oscillation of the roll rate occurs at the ass oscillation frequency. This is due to the continually-changing axial oent of inertia of the projectile as the ass translates. When the translating ass travels farther fro the centerline, the axial inertia grows and the roll rate decreases due to conservation of angular oentu. ikewise, when the translating ass returns toward the centerline, the axial inertia decreases, and the roll rate increases. Figures show the effect of spring and daper coefficients on average force, average power, and total battery charge Fig. 9 Maxiu displaceent versus bea length Fig. 11 Roll rate versus tie for exaple siulation / Vol. 131, SEPTEMBER 2009 Transactions of the ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

7 Fig. 12 Average force required versus torsional spring constant partial flight profile Fig. 14 Total charge required versus torsional spring constant partial flight profile required. In Fig. 12, it can be clearly seen that an optiu torsional spring constant exists, in which the spring-ass-daper syste of the ITM-bea operates near resonance with the projectile roll rate. These peaks are not as sharp as typical spring-assdaper resonant peaks due to the fact that the projectile roll rate varies over tie. Note, however, that significant reductions in force are achieved if the spring constant is placed near its optiu value and daping is lowered as uch as possible. These reductions in force are irrored by reductions in average power and total charge required, resulting in significantly saller battery sizes. A siilar study exaines the sae spring-ass-daper paraeters for a full flight profile of the exaple projectile using the sae initial conditions as those used in the first exaple study above. Figures show that, as in the partial flight profile case, optial spring coefficients can be found. However, the results for the partial flight profile have a significantly sharper peak than the results for the full flight profile. This is because, as shown in Fig. 6, the roll rate of the projectile varies between 5 rad/s initially and a final value of approxiately 80 rad/s. This large variation in roll rate eans that the spring coefficient is only optiized for a very short period of the overall flight, and the broad peaks shown in Figs result. To deonstrate this, Fig. 18 shows a current tie history for an exaple full flight trajectory using k T =70.0 lb/rad and k D =0.05 lb/ rad/ s. Note that the spring constant is optiized for the projectile roll rate approxiately 1 s into flight, and once again after spin decay occurs approxiately 7 s into the flight. Despite the broad nature of the peaks shown in Figs , significant size and weight savings can be achieved using the proper spring constants in the for of saller batteries. As shown in Fig. 17, batteries with a total charge of less than 5 A sec ay be used for systes with optial spring coefficients and low daping ratios. Furtherore, Fig. 18 shows that reasonable axiu current levels, on the order of 150 A, can be expected with an optiized syste. Average force levels, and therefore total charge required, can be decreased even further by actively changing the elastic properties of the bea during flight. A fixed-free cantilever bea like that used in this syste has a first vibrational ode shape of s x = sin x B 31 2B where x B is the distance along the bea. The natural frequency of the first vibrational ode is Fig. 13 Average power required versus torsional spring constant partial flight profile Fig. 15 Average force required versus torsional spring constant full flight profile Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

8 Fig. 16 Average power required versus torsional spring constant full flight profile Fig. 18 Current through actuators versus tie for exaple full flight trajectory n = 2B E 32 where E is the odulus of elasticity of the aterial and is the density of the bea 14. By changing the odulus of elasticity, it is therefore possible to tune the natural frequency of the cantilever bea to a desired value. Recent investigations into sart aterials 15 18, specifically aterials used in tunable vibration absorbers, have shown that various ethods can be used to actively alter a aterial s odulus of elasticity, allowing the bea s torsional spring constant to be actively optiized during flight as the projectile roll rate changes. This would allow the ITM-bea syste to operate with the lowest possible power through the entire flight, yielding further reductions in battery size. To investigate this, several exaple siulations were run. The first set siulated the projectile for the full flight using the optiu spring constants obtained fro Fig. 17. This produced the least possible battery charge required for the ITM-bea syste with a fixed spring constant for each daping ratio considered. The second set of siulations included a roll-rate feedback echanis. In these cases, at specific points throughout the flight the torsional spring and daping coefficients of the ITM-bea were adjusted to atch the roll rate. Figure 19 shows how the torsional spring constant was adjusted for an exaple flight with =0.05. Note that the curve in Fig. 19 has the sae qualitative shape as the roll-rate tie history shown in Fig. 6. Table 2 suarizes the results of the two sets of siulations with the fixed torsional spring and variable torsional spring constants. Note that in the variable torsional spring cases, the torsional daping coefficient k D was adjusted slightly as well so as to keep the daping ratio constant. Figure 20 deonstrates that ipleentation of the roll-rate feedback syste saves approxiately 1 A sec of Fig. 19 Torsional spring constant versus tie for roll-rate feedback syste, =0.05 Table 2 Perforance evaluation of roll-rate feedback syste Daping ratio Optiu k T for no feedback case Charge required A h No feedback constant k T Feedback variable k T Percentage decrease in charge required with feedback % Fig. 17 Total charge required versus torsional spring constant full flight profile / Vol. 131, SEPTEMBER 2009 Transactions of the ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

9 Fig. 20 Total charge required versus daping ratio for constant and variable k T cases charge for all daping ratios considered 34.4% decrease in battery size for =0.02,28.1% decrease for =0.05,20.4% for =0.1, and 11.2% for = Conclusion A conceptual design for a projectile equipped with a controllable internal ass is proposed using a cantilever bea configuration. The syste is designed to iniize oving parts and to exploit the bea s elastic properties, thereby creating a robust echanis that iniizes power requireents. A dynaic odel of the configuration is developed, as is a odel of the electroagnetic actuator syste to study current requireents and battery size. Exaple siulations show that useful control authority is generated with this projectile control echanis, atching the results fro earlier studies of projectiles with internal translating ass control. Furtherore, by tuning the natural frequency of the vibrating bea to the projectile roll rate, significant reductions in power requireents are achieved on the order of 60%. Power reductions can be further realized with active tuning of the bea s elastic properties during flight. Battery size requireents to power the actuator are odest, with several COTS options available. The internal oscillating bea configuration shows proise as a viable, cost effective, reliable projectile control echanis. Noenclature a 1, a 2, and a 3 coponents in the S frae of the acceleration of the syste ass center with respect to the inertial frae a P/I acceleration of the projectile ass center with respect to the inertial frae a X/I acceleration of the end of the ITM-bea with respect to the inertial frae A axiu agnitude of ITM-bea displaceent fro center of cavity b radius of the electroagnetic actuator B length of the ITM-bea assebly B z agnetic field produced by the electroagnetic actuators in the I S direction C coposite body center of ass DM agnitude of the agnetic dipole oent of the fixed agnet at the end of the ITM-bea F C hinge constraint force on the internal translating ass F I input force exerted by electroagnetic actuators on ITM-bea F P total aerodynaic force exerted on the projectile f input scalar value of the total force applied to ITM-bea by electroagnetic actuators P H B/I X H T/I angular oentu of the projectile with respect to the inertial frae about the projectile center of ass angular oentu of the ITM-bea with respect to the inertial frae about point X I Txx the xx coponent of the ITM-bea oent of inertia atrix about point X I Pxx the xx coponent of the projectile oent of inertia atrix I I, J I, and K I inertial frae unit vectors I B, J B, and K B projectile reference frae unit vectors I N, J N, and K N nonrolling reference frae unit vectors I T, J T, and K T ITM-bea fixed frae unit vectors I S, J S, and K S S frae unit vectors I E current through the electroagnetic actuator k D torsional daper coefficient k T torsional spring coefficient junction of between the ITM-bea and the projectile, referred to as the hinge point A length of the electroagnetic actuator P ass of the projectile with the cavity T ass of the ITM-bea total ass of the syste M P external oents applied to the projectile about the hinge point M syste external oents applied to the projectile- ITM syste about the hinge point external oent exerted on the ITM-bea by the actuators in the K S direction about the hinge point P projectile center of ass r X distance vector fro the hinge point to point X at the end of the ITM-bea r P distance vector fro the hinge point to the projectile center of ass P r P X distance vector fro projectile center of ass P to point X at the end of the ITM-bea M bea r 1, r 2, and r 3 coponents in the S frae of r P r 4, r 5, and r 6 coponents in the S frae of r P X p, q, and r coponents of B/I in the S frae p, q, and r coponents of B/I in the projectile reference frae u, v, and w translational velocity coponents of the coposite body center of ass resolved in the projectile reference frae v C/I velocity of the syste ass center with respect to the inertial frae W P weight of the projectile without the ITM-bea W T weight of the ITM-bea X point at the end of the ITM-bea x, y, and z position vector coponents of the coposite body center of ass expressed in the inertial reference frae X B, Y B, and Z B total external force coponents on the projectile and ITM-bea syste expressed in the projectile reference frae Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

10 B/I angular acceleration of the projectile body with respect to the inertial frae T/I angular acceleration of the ITM-bea with respect to the inertial frae deflection angle of the ITM-bea,, and Euler roll, pitch, and yaw angles T and T Euler pitch and yaw angles for the orientation of the S frae with respect to the B frae B/I angular velocity of the projectile body with respect to the inertial frae T/I angular velocity of the ITM-bea with respect to the inertial frae bea agnitude of the angular velocity of the ITM-bea with respect to the S frae Appendix A The ters of Eq. 22 are given as follows: A 11 = I T31 + B P T s r 3 A 12 = I T32 B P T c r 3 A 13 = I T33 + B 2 P T A 23 = I P33 + P T + B P T c r 2 s r 1 A 14 = I T33 + B 2 P T A 21 = I P31 P T r 1r 3 A 22 = I P32 P T r 2r 3 B r 2c r 1 s + P T r r 2 2 A 24 = P T B r 2c r 1 s A 31 = I P21 + I T21 P T Br 4c P T r 2r 4 A 32 = I P22 + I T22 P T Br 4s + P T r 1r 4 + r 3 r 6 A 33 = I P23 + I T23 P T Br 6c P T r 2r 6 A 41 = I P11 + I T11 + P T A 34 = I T23 P T Br 6c Br 5c + P T r 2r 5 + r 3 r 6 A 42 = I P12 + I T12 + P T Br 5s P T r 1r 5 A 43 = I P13 + I T13 + P T Br 6s P T r 1r 6 A 44 = I T13 + P T Br 6s B 1 = K S S r X T C S a C/I f input Bc k T k D K S S B/I I T C S T/I + K S S r X C S W T P T K S S r X S T/I S T/I C T r X P T K S S r X S B/I S B/I C S r P P T K S S r X S B/I C S T/I C S r X B 2 = K S P S r P C S a C/I + f input Bc + k T + k D K S S B/I I P C S B/I + K S C S M P + P T K S S r P S T/I S T/I C S r X + P T K S S r P S B/I S B/I C S r P + P T K S S r P S B/I C S T/I C S r X + K S S r P C S W P B 3 = J S S B/I I P C S B/I J S C T/I I T C S T/I + J S S r X C S W T J S S r P C S a C/I J S T S r P X C S a C/I P T J S S r P X S T/I S T/I C S r X P T J S S r P X S B/I S B/I C S r P + J S C S M syste C S r X P T J S S r P X S B/I C S T/I B 4 = I S S B/I I P C S B/I I S S T/I I T C S T/I + I S S r X C S W T I S S r P C S a C/I I S T S r P X C S a C/I P T I S S r P X S T/I S T/I C S r X P T I S S r P X S B/I S B/I C S r P + I S C S M syste C S r X P T I S S r P X S B/I C S T/I B FC = K S S r X T C S a C/I k T k D K S S B/I I T C S T/I + K S S r X C S W T P T K S S r X S T/I S T/I C T r X / Vol. 131, SEPTEMBER 2009 Transactions of the ASME Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see

11 Fig. 21 Translating ass projectile scheatic Appendix B P T K S S r X S B/I S B/I C S r P P T K S S r X S B/I C S T/I C S r X Figure 21 is a scheatic of the translating ass projectile. References 1 Soper, W., 1978, Projectile Instability Produced by Internal Friction, AIAA J., 16 1, pp Murphy, C., 1978, Influence of Moving Internal Parts on Angular Motion of Spinning Projectiles, J. Guid. Control, 1 2, pp D Aico, W., 1987, Coparison of Theory and Experient for Moents Induced by oose Internal Parts, J. Guid. Control, 10 1, pp Hodapp, A., 1989, Passive Means for Stabilizing Projectiles With Partially Restrained Internal Mebers, J. Guid. Control, 12 2, pp Petsopoulos, T., Regan, F., and Barlow, J., 1996, Moving Mass Roll Control Syste for Fixed-Tri Re-Entry Vehicle, J. Spacecr. Rockets, 33 1, pp Robinett, R., Sturgis, B., and Kerr, S., 1996, Moving Mass Tri Control for Aerospace Vehicles, J. Guid. Control Dyn., 19 5, pp Menon, P., Sweriduk, G., Ohleyer, E., and Malyevac, D., 2004, Integrated Guidance and Control of Moving Mass Actuated Kinetic Warheads, J. Guid. Control Dyn., 27 1, pp Frost, G., and Costello, M., 2004, inear Theory of a Projectile With a Rotating Internal Part in Atospheric Flight, J. Guid. Control Dyn., 27 5, pp Frost, G., and Costello, M., 2006, Control Authority of a Projectile Equipped With an Internal Unbalanced Part, ASME J. Dyn. Syst., Meas., Control, 128 4, pp Rogers, J., and Costello, M., 2008, Control Authority of a Projectile Equipped With an Internal Translating Mass, J. Guid. Control Dyn., 31 5, pp Slotine, J., and i, W., 1991, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, pp Purcell, E., 1985, Electricity and Magnetis, McGraw-Hill, New York, pp Purcell, E., 1985, Electricity and Magnetis, McGraw-Hill, New York, pp Inan, D., 2001, Engineering Vibration, Prentice Hall, p Flatau, A., Dapino, M., and Calkins, F., 2000, High Bandwidth Tunability in a Sart Vibration Absorber, J. Intell. Mater. Syst. Struct., 11, pp Varga, Z., Filipcsei, G., and Zrinyi, M., 2005, Sart Coposites With Controlled Anisotropy, Polyer, 46, pp Varga, Z., Filipcsei, G., and Zrinyi, M., 2006, Magnetic Field Sensitive Functional Elastoers With Tuneable Elastic Modulus, Polyer, 47, pp Davis, C., and esieutre, G., 2000, An Actively Tuned Solid-State Vibration Absorber Using Capacitive Shunting of Piezoelectric Stiffness, J. Sound Vib., 232 3, pp Journal of Dynaic Systes, Measureent, and Control SEPTEMBER 2009, Vol. 131 / Downloaded 19 Aug 2009 to Redistribution subject to ASME license or copyright; see