Cantilever Beam Design for Projectile Internal 

Moving Mass Systems 

by Jonathan Rogers and Mark Costello 

ARL-CR-658 September 2010 

prepared by 

Georgia Institute of Technology 

School of Aerospace Engineering 

Atlanta, GA 30332 

under contract 

W911QX-06-C-0013 

Approved for public release; distribution is unlimited. 
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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-CR-658 September 2010 

Cantilever Beam Design for Projectile Internal 

Moving Mass Systems 

Jonathan Rogers and Mark Costello 

Georgia Institute of Technology 

prepared by 

Georgia Institute of Technology 

School of Aerospace Engineering 

Atlanta, GA 30332 

under contract 

W911QX-06-C-0013 

Approved for public release; distribution is unlimited. 
ii

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4. TITLE AND SUBTITLE 

Cantilever Beam Design for Projectile Internal Moving Mass Systems 

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6. AUTHOR(S) 

Jonathan Rogers* and Mark Costello*

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AH80 

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School of Aerospace Engineering 

Georgia Institute of Technology 

Atlanta, GA 30332 

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*

 School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332 

14. ABSTRACT 

Internal masses that undergo controlled translation within a projectile have been shown to be effective control mechanisms for 

smart weapons. However, internal mass oscillation must occur at the projectile roll frequency to generate sufficient control 

force. This can lead to high power requirements and place a heavy burden on designers attempting to allocate volume within 

the projectile for internal mass actuators and power supplies. The work reported here outlines a conceptual design for an 

internal translating mass system using a cantilever beam and electromagnetic actuators. The cantilever beam acts as the moving

mass, vibrating at the projectile roll frequency to generate control force. First, a dynamic model is developed to describe the 

system. Then, the natural frequency, damping ratio, and length of the beam are varied to study their effects on force required

and total battery size. Trade studies also examine the effect on force required and total battery size of a roll-rate feedback

system that actively changes beam elastic properties. Results show that with proper sizing and specifications, the cantilever 

beam control mechanism requires relatively small batteries and low actuator control forces, with minimum actuator complexity 

and space requirements. 

15. SUBJECT TERMS

internal translating mass, ITM, smart weapons, cantilever beam 

16. SECURITY CLASSIFICATION OF: 

17. LIMITATION 

OF ABSTRACT 

UU 

18. NUMBER 

OF PAGES 

48 

19a. NAME OF RESPONSIBLE PERSON

Paul Weinacht 

a. REPORT 

Unclassified 

b. ABSTRACT 

Unclassified 

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Unclassified 

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Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18 
iii

Table of Contents 

List of Figures iv

List of Tables v

1. Introduction 1

2. Cantilever Beam Projectile Dynamic Model 2

2.1 Kinematics .......................................................................................................................4

2.2 Dynamics .........................................................................................................................5

2.3 Description of Controller.................................................................................................7

2.4 Description of Electromagnetic Actuator Control System ..............................................8

2.5 Description of Example Projectile ..................................................................................9

3. Results 10

4. Conclusion 23

5. References 25

Appendix A. A and B Matrix Terms 27

Appendix B. Translating Mass Projectile Schematic 31

List of Symbols, Abbreviations, and Acronyms 33

Distribution List 36 
iv

List of Figures 

Figure 1. The ITM-Beam projectile. ...............................................................................................2

Figure 2. Zoom view of the ITM-Beam system. ............................................................................8

Figure 3. Altitude vs. range for example trajectory. .....................................................................10

Figure 4. Cross range vs. range for example trajectory. ...............................................................11

Figure 5. Velocity (u) vs. time for example trajectory. ................................................................11

Figure 6. Roll rate vs. time for example trajectory. The thick lines represent high-frequency 

oscillations, which occur only for the ITM-Beam and translating mass case. ........................12

Figure 7. Selected time history of ITM displacement from projectile centerline. ........................13

Figure 8. Segment of current vs. time for ITM-Beam actuators. ..................................................14

Figure 9. Maximum angular displacement vs. beam length. ........................................................15

Figure 10. Average force required vs. beam length. .....................................................................15

Figure 11. Roll rate vs. time for example simulation. ..................................................................16

Figure 12. Average force required vs. torsional spring constant, partial flight profile. ................17

Figure 13. Average power required vs. torsional spring constant, partial flight profile. ..............18

Figure 14. Total charge required vs. torsional spring constant, partial flight profile. ..................18

Figure 15. Average force required vs. torsional spring constant, full flight profile. ....................19

Figure 16. Average power required vs. torsional spring constant, full flight profile. ..................19

Figure 17. Total charge required vs. torsional spring constant, full flight profile. .......................20

Figure 18. Current through actuators vs. time for example full flight trajectory. .........................21

Figure 19. Torsional spring constant vs. time for roll-rate feedback system, ζ = 0.05. ................22

Figure 20. Total charge required vs. damping ratio for constant and variable kT cases. ..............23 

Figure B-1. Translating mass projectile schematic. ......................................................................32
v

List of Tables 

Table 1. Relevant example projectile and ITM-Beam properties. ..................................................9

Table 2. Performance evaluation of roll-rate feedback system. ....................................................23
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

The past several decades have seen growing interest in developing of smart munitions resulting 

from attempts to increase accuracy and decrease collateral damage. Smart projectiles differ from 

other guided weapons, such as guided missiles, because their electronics and control mechanisms 

must be able to withstand extreme acceleration loads associated with launch and high spin rates. 

Control components must be relatively inexpensive since projectiles are typically fired in large 

quantities. Several different types of control mechanisms have been designed to meet these 

requirements—aerodynamic, thrust, and inertial load mechanisms. Common examples of 

aerodynamic mechanisms are canards, gimbaled nose configurations, and deflection of ram air 

through side ports. Examples of thrust mechanisms include cold gas jets and explosive thrusters. 

Examples of inertial load mechanisms are rotation of an unbalanced internal part and, of specific 

interest here, movement of an internal translating mass (ITM). Oscillation of an internal mass at 

the projectile roll frequency has been shown to produce useful control authority. A design that 

minimizes moving parts and substantially reduces power required is critical to physically 

implement an ITM control mechanism on board a smart munition. 

Previous investigation of projectiles equipped with loose or moving internal parts has revealed 

that these configurations can result in flight instabilities. Soper (1) considered the stability of a 

projectile with a cylindrical mass fitted loosely within a cavity. Using a similar configuration, 

Murphy (2) derived a quasi-linear solution for the motion of a projectile equipped with a moving 

internal part. A detailed set of experiments was later conducted by D’Amico (3) to model the 

motion of internal masses within spinning projectiles using a freely gimbaled gyroscope. 

Hodapp (4) further considered the effect of a small offset between the projectile body mass and 

ITM mass center. Hodapp’s analysis of the dynamic equations for this system showed that for 

small mass center offsets, slight movement could actually reduce the instability caused by the 

loose internal part. Using controlled movement of an ITM as a maneuver control mechanism 

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) examined ITM 

control for use on endo- and exoatmospheric interceptors using three orthogonal ITMs. Frost 

and Costello (8, 9) have studied the ability of an internal rotating mass 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 mass. It 

was determined that significant control authority could be created by oscillating the ITM at the 

projectile roll frequency. 
2

This report outlines a notional design of an ITM actuator that generates sufficient control 

authority using relatively low power. Control moment is generated due to an axial drag offset 

from the system mass center caused by the lateral motion of the ITM. The report begins with a 

description of the cantilever beam system that serves as the translating mass and derives the 

7-degrees-of-freedom (DOF) flight dynamic model used for trajectory predictions. A description 

of the control law and ITM electromagnetic actuators is also provided. The dynamic model is 

subsequently employed to demonstrate that the cantilever beam configuration provides sufficient 

control authority at reasonable power levels. Trade studies examine the effect of cantilever beam 

length on force required and battery size. The optimum natural frequency and damping ratio of 

the beam are determined to minimize actuator control effort. Finally, roll-rate feedback control 

is implemented to actively alter beam characteristics as the projectile roll rate changes during 

flight, further decreasing control effort required. A final example case demonstrates that 

sufficient control authority can be generated with relatively small battery sizes using the 

optimum spring and damper coefficients and the roll-rate feedback system. 

2. Cantilever Beam Projectile Dynamic Model 

The cantilever beam is a fixed-free elastic beam, with one end attached to the projectile at point 

L and the free end floating within the cavity, constrained to vibrate in the S S I J    plane. The 

beam’s first vibrational mode is the only mode considered significant to dynamic interaction 

with the projectile. For this reason, the cantilever beam can be accurately modeled as a rigid, 

massless beam, with a spherical mass (considered to be a permanent magnet) attached to the end. 

A torsional spring and damper are attached to the hinge point to simulate the elastic properties of 

the cantilever beam. This dynamically equivalent system is referred to as the ITM-Beam. A 

sketch of this configuration is shown in figure 1. Note that is defined as the angle between the 

ITM-Beam and centerline of the projectile. A permanent magnet is attached to the end of the 

beam and can swing freely about the hinge. Force is exerted on the magnet by electromagnets on 

both sides of the cavity to move the beam to a desired angle. 

Figure 1. The ITM-Beam projectile.
3

Five reference frames are used in the development of the equations of motion for this system, 

namely the inertial, projectile, translating mass, nonrolling, and projectile-fixed S reference 

frames. The projectile frame is obtained using the standard aerospace Euler angle sequence of 

rotations and is related to the inertial frame by 

B I

B I

B I

I cc cs s I

J ssc cs sss cc sc J

K K csc ss css sc cc

  

    

    

            

       

 

 

  . (1) 

The N frame is the standard nonrolling reference frame often used in projectile flight dynamics 

and is defined by a rotation of –φ along the B I

 axis. The S frame is also fixed to the projectile, 

with its origin at the hinge point. It is defined so that the ITM-Beam oscillates about the KS

 axis 

and S J

 points to the rear of the cavity exactly equidistant from both electromagnetic actuators. 

Therefore, the S frame can be related to the B frame by two constant Euler angles,  S and S  , so 

that 

0

SS SS S

S S

SS SS S

S B

S B

S B

I cc cs s I

J sc J

K K sc ss c

  

 

  

            

 

   

 

 

  . (2) 

Throughout the rest of this report, fixed angles of   90 S and 0 S   are used; thus, the S frame 

can be obtained by a single 90° rotation about the KB

 axis, resulting in the orientation shown in 

figure 1. The T frame is fixed to the ITM-Beam and is related to the S frame by the relationship 

0

0

0 01

T S

T S

T S

I c s I

J sc J

K K

 

 

                    

 

 

  . (3) 

Note that the T frame is aligned with the S frame when γ = 0 . All equations in this report use the 

following shorthand notation for trigonometric sine, cosine, and tangent functions: s sin    ,

cα = cos α , and tα = tan α . 

Throughout the development of the equations of motion, two operators will be used:  to 

denote components of a vector in a specific frame and  to denote the skew symmetric cross 

product operator. The vector component operator outputs a column vector comprised of the 

components of an input vector in a given frame. For example, if the position vector from α to β

is expressed in reference frame A as A A A α β αβ αβ αβ r = xI+yJ+zK       , then the vector 

component operator acting on this vector yields 
4

( ) A

x

r y

z



  





          

  . (4) 

Notice that the reference frame is denoted by the subscript on the operator. 

The cross-product operator outputs a skew symmetric matrix using the components of an input 

vector in the reference frame denoted in the subscript. For example, if the position vector from α

to β is expressed in reference frame A as A A A α β αβ αβ αβ r = xI+yJ+zK       , then the cross-product 

operator acting on α β r 

 expressed in reference frame A is 

0

() 0

0

A

z y

rz x

y x

 

   

 



               

  . (5) 

2.1 Kinematics 

The velocity of the composite body mass center can be described in the inertial frame or the 

projectile reference frame. 

v xI yJ zK uI vJ wK CI I B /     I IB B

        . (6) 

The translational kinematic differential equations relate these two representations of the mass 

center velocity components. 

x cc ssc cs csc ss u

y cs sss cc css sc v

z s sc cc w

    

    

  

             

     







 . (7) 

The angular velocity of the projectile with respect to the inertial reference frame can be written 

in terms of appropriate Euler angle time derivatives or in terms of projectile frame angular 

velocity components. 

/ B IB B N B B I        I J K pI qJ rK           . (8) 

The kinematic relationship between time derivatives of the Euler angles and projectile reference 

frame angular velocity components represents the rotational kinematic differential equations. 

1

0

0/ /

st ct p

c sq

sc cc r

 

 

 







               

       







 . (9) 
5

The final kinematic differential equation is the trivial relationship 

Beam   . (10) 

2.2 Dynamics 

The translational dynamic equations for the ITM-Beam projectile are derived through force 

balancing. Force-balance equations for the projectile and ITM-Beam are given as 

/      ma W F F F P PI P P C I (11) 

and 

/        ma W F F T XI T C I ,

 (12) 

respectively, where mP and mT are the masses of the projectile and ITM-Beam, P I/ a

 and X / I a are 

the accelerations of points P and X with respect to the inertial frame, and , ,,   

WW FF P T PI and FC



are the weights of the projectile and ITM-Beam, total aerodynamic force exerted on the 

projectile, input force exerted on the ITM-Beam by the actuators, and hinge constraint force, 

respectively. The definition of the system center of mass leads to 

///      ma m a m a CI P PI T XI . (13) 

Therefore, adding equations 11 and 12 and noting the relationship in equation 13, the 

translational dynamic equation for the system is formed. 

/        ma W W F CI P T P . (14) 

The aerodynamic forces given by FP

 in equation 14 are obtained using the standard aerodynamic 

expansion employed for projectile flight dynamic simulation. Both steady aerodynamic forces 

and Magnus forces are included as well as steady and unsteady aerodynamic moments. The 

aerodynamic coefficients and aerodynamic center distances used to generate these forces and 

moments are all a function of the local Mach number at the center of mass of the projectile. 

Computationally, these Mach number-dependent parameters are obtained by a table look-up 

scheme using linear interpolation. A full description of the weight force and body aerodynamic 

forces and moments is provided in reference 10. Writing equation 14 in the projectile reference 

frame yields 

0

0

0

B

B

B

X

m u rqu Y

v r pv

m

w qp w Z

m

 

                                     

   







 . (15) 
6

Note that , XB B Y , and B Z are projectile reference frame components of the sum of the three 

forces given in equation 14.

The rotational dynamic equations are obtained by first equating the I frame time rate of change 

of the system angular momentum about the system mass center to the total applied external 

moments on the system about the system mass center in the S I

 and S J directions, given by 

equations 16 and 17. Then, the same moment equation is used for each body separately, this 

time written in the KS

 direction. These four equations are given by 

/ /

/ /

IP IX

BI TI L

S L P P P I L X T X I S System

dH dH I r ma r ma I M

dt dt  

       

  

         , (16) 

/ /

/ /

IP IX

BI TI L

S L P P P I L X T X I S System

dH dH J r ma r ma J M

dt dt  

       

  

         , (17) 

    /

/

I X

T I

S L X T X I Input S L X S T T D

dH K r m a f BLc K r W k k dt     

         

 

         , (18) 

and 

   

 

/

/

,

I P

B I

S L P P P I Input S L P S P T D S

L

S P

dH K r m a f BLc K r W k k K dt

M

    

        

 



        





 (19) 

where, as before, = cos ( ) γ c γ . Note that equation 18 equates the time rate of change of the 

angular momentum of the ITM-Beam to the total moment applied to the ITM-Beam. Equation 

19 equates the time rate of change of the angular momentum of the projectile to the total moment 

applied to the projectile. 

Several intermediate expressions will be useful in deriving the rotational dynamic equations in 

the body-fixed S frame. First, note that the well-known two points fixed on a rigid body formula 

yields the relationship 

                  // / / / / / /  

        P

XI CI BI P L TI L X BI BI P L TI TI L X

m

aa r r r r

m

. (20) 

Equation 20 is used to expand 

αX/I in terms of known quantities and state derivatives. Also, 

using the definition of the system center of mass, it can be shown through algebraic manipulation 

that 

       / // /

     

L P P PI L X T XI L P CI P X T XI r m a r m a r ma r m a . (21) 
7

Equation 21 is also used to expand the cross-product terms on the left-hand side of equations 16 

and 17 in terms of known quantities and state derivatives. 

Equations 16–19 can be expanded using the expressions in equations 20 and 21 and rearranged 

to form a 4 × 4 system of equations given by 

11 12 13 14 1

21 22 23 24 2

31 32 33 34 3

41 42 43 44 4

AAAA p B

AAAA q B

AAAA r B

AAAA  B

                             









 . (22) 

In equation 22, rows 1–4 correspond to equations 18, 19, 17, and 16, respectively. The full 

expressions for the values of the A matrix and B vector are lengthy and provided in appendix A. 

The set of equations given by 7, 9, 10, 15, and 22 constitutes the equations of motion for the ITMBeam projectile. Given a known set of initial conditions, these 14 scalar equations are 

numerically integrated forward in time using a fourth-order Runge-Kutta algorithm to obtain a 

single trajectory. 

2.3 Description of Controller 

To create trajectory alterations, the ITM-Beam must be moved in a prescribed manner. The 

control law is formulated based on a feedback linearization technique (11), which assumes fullstate feedback. To compute the control force, input f , required to deflect the ITM-Beam to the 

desired angle, use the following: 

  

11 12 13 14 0

1 2

1

.



 

  

        

  

      

 

input FLC command

command command

f B Ap Aq Ar A K BLc

K K (23)

Note that BFLC is defined in appendix A and derived from equation 18. Likewise, 

11 12 13 14 AAA A , , , and are from equation 18 and provided in appendix A. Note that equation 18 is 

used to compute the feedback linearization control rather than equation 19 since equation 19 

would require feedback of aerodynamic loads. This is a complicated task and can be avoided by 

using equation 18 instead. 

The commanded deflection angle, command  , is generated by synchronizing ITM-Beam movement 

with the projectile roll angle. This is done by setting 

  1            sin cos command T

A

BL , (24) 

where A is the magnitude of oscillation of point X from the cavity center and T  is a trim angle 

used to define the plane of control. Derivatives of equation 24 are computed analytically and 

used in equation 23. Note that the feedback linearization controller is developed and used 
8

within the simulation solely to create the prescribed motion of the ITM-Beam for control 

authority analysis and determination of power requirements. In this case, its purpose is to match 

the ITM-Beam oscillation frequency to the projectile roll rate within the simulation. 

2.4 Description of Electromagnetic Actuator Control System 

A zoom view of the ITM-Beam mechanism is shown in figure 2. Two electromagnets, each at 

opposite ends of the cavity, exert force on the fixed magnet at the end of the ITM-Beam. 

Figure 2. Zoom view of the ITM-Beam system. 

The force exerted on a fixed magnetic dipole DM is given by Purcell (12) as 

  

z

input

M

B

f DM

z

 . (25) 

The magnetic dipole moment per unit mass is a unique property of a material, with units 

joules/tesla/slug. For example, assuming the ITM is made of magnetized iron and using an ITMBeam mass of mt = 0.05 slugs, the dipole moment is found to be DM = 171.5 J/T. The quantity 

z

M

B

z



 can be found by first recognizing that the magnetic field, Bz , of an iron core solenoid is 

given by the expression 

  22 2 2 2



 

     

    

E M MA

z

M M A

I kn z z L B

z b zL b

, (26) 
9

where E I is the current through the electromagnet,  is the magnetic constant ( 7

4 10   N/A2

), Mz

is the distance from the endpoint of the ITM beam to the nearest electromagnet actuator, k is the 

dimensionless relative permeability of iron (200 at a magnetic flux density of 0.002 W/m2

), n is 

the number of coils per meter, b is the radius of the solenoid, and LA is the length of the solenoid. 

For all the following cases, values of 3, 2, and 10000 cm were used for b, L, and n, respectively. 

Taking the derivative of equation 26 with respect to Mz , 

   

 

   

2 2

32 32 22 2 2 2 2 2 2

1 1

2

 



 

   

  

       

/ /

z

M

E M M A

M M M A M A

B

z

I kn z z L

z b z b zL b zL b

 . (27) 

At each timestep, control force is computed using the feedback linearization. Knowing the 

magnetic dipole moment and required control force, the quantity z

M

B

z



 is computed at each 

timestep using equation 25. Then, knowing the position of the ITM-Beam with respect to the 

actuators, the current required can be computed at each timestep by rearranging equation 27 so 

that 

   

 

   

1

2 2

32 32 22 2 2 2 2 2 2

21 1









 

   

  

       

/ /

z

M

M M A

E

M M M A M A

B

z

z z L I

k n z b z b zL b zL b

 . (28) 

Note that the electromagnet dimensions used in equation 28 are comparable to dimensions for 

commercially available iron-core electromagnets. 

2.5 Description of Example Projectile 

The example projectile used in the following simulations is a representative fin-stabilized 

projectile. Relevant example projectile and ITM-Beam dimensional and mass properties are 

outlined in table 1. The hinge point is 0.9 ft behind the projectile mass center (P), and the ITMBeam oscillation amplitude is given by max BLsin( ) γ = 0.157ft, unless otherwise specified. In 

the following cases, the projectile is traveling through a standard atmosphere, with no 

atmospheric wind. 

Table 1. Relevant example projectile and ITM-Beam properties. 

Projectile Mass (slugs) 1.206 

ITM-Beam Mass (slugs) 0.05 

Projectile Reference Diameter (ft) 0.344 

Projectile Mass Center Position 

Measured Along Stationline (ft) 1.18 

Projectile Roll Inertia (slugs-ft2) 0.0278 

Projectile Pitch Inertia (slugs-ft2) 0.6291 

ITM-Beam Length (ft) (BL) 0.3 
10

3. Results 

An example trajectory of the ITM-Beam projectile is compared to an example trajectory of a 

projectile equipped with a strictly translating internal mass for model validation purposes. The 

translating mass projectile’s dynamic equations are given in reference 10, and a previously 

validated model of this system was used for trajectory predictions. A diagram of the translating 

mass projectile is provided in appendix B. Initial conditions used for the example trajectory 

were x = 0.0 ft, y = 0.0 ft, z = 0.0 ft, u = 2821.0 ft/s, v = 0.0 ft/s, w = 0.0 ft/s,  = 1.5707 rad, θ

= 0.05 rad, ψ = 0.0 rad, p = 5.0 rad/s, q = 0.0 rad/s, and r = 0.0 rad/s. The feedback linearization 

gains were 5 KK K 01 2 = 0.001, = 1000.0, and = 5 10 .  In the following cases, the ITM-Beam 

oscillation frequency is locked to the projectile roll rate. Figures 3–6 show the trajectories for 

the ITM-Beam projectile, translating mass projectile, and the rigid projectile with no internal 

moving mass (denoted “Rigid 6DOF”). The two translating mass trajectories are generated 

solely to demonstrate control authority and validate the ITM-Beam simulation; thus, both 

controlled rounds are commanded to maximum possible deflection. Notice that the trajectories 

of the internal mass projectiles match nearly identically, even though the dynamic equations for 

the two systems are significantly different. The correlation between these two models serves as 

validation of the ITM-Beam simulation. 

Figure 3. Altitude vs. range for example trajectory.

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

0

50

100

150

200

250

300

Range (ft)

Altitude (ft)

ITM-Beam

Translating Mass

Rigid 6DOF

ITM-Beam,

Translating Mass,

Rigid 6DOF
11

Figure 4. Cross range vs. range for example trajectory. 

Figure 5. Velocity (u) vs. time for example trajectory.

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

Range (ft)

Cross Range (ft)

ITM-Beam

Translating Mass

Rigid 6DOF

ITM-Beam,

Translating Mass

0 1 2 3 4 5 6 7 8

1500

2000

2500

3000

Time (s)

u (ft/s)

ITM-Beam

Translating Mass

Rigid 6DOF

ITM-Beam,

Translating Mass,

Rigid 6DOF
12

Figure 6. Roll rate vs. time for example trajectory. The thick lines represent highfrequency oscillations, which occur only for the ITM-Beam and translating 

mass case. 

Figure 7 shows a selected time history of ITM displacement from the projectile centerline. For 

the ITM-Beam projectile, this displacement, x s , is given by 

s BL x  sin  . (29) 

Notice that the two time histories for ITM displacement shown in figure 7 are nearly identical. A 

control-force time history for an example ITM-Beam projectile simulation can be used together 

with an ITM displacement time history to generate a time history of current required for each 

electromagnetic actuator. This is accomplished using the procedure outlined in equations 25–28. 

Furthermore, this current time history can be integrated to produce the total charge required for a 

given example flight in A/s. This value for total charge can be used to size the battery for the 

ITM-Beam control system. The total charge required for the example flight shown previously 

was 13.5 A/s. Note that this number is relatively large since the spring and damper coefficient 

values have not been optimized for this preliminary example case. 

0 1 2 3 4 5 6 7 8

10

20

30

40

50

60

70

80

90

100

110

120

Time (s)

Roll Rate (rad/s)

ITM-Beam

Translating Mass

Rigid 6DOF

ITM-Beam,

Translating Mass,

Rigid 6DOF
13

Figure 7. Selected time history of ITM displacement from projectile centerline. 

Figure 8 shows a segment of the current time history for the example ITM-Beam simulation used 

previously. Notice that input f (labeled “Force” in the plot) and the ITM displacement, x s , are 

shown on the same plot as the current time history to demonstrate the phase relationships 

between current, input force, and ITM displacement. When the ITM-Beam is displaced in the 

positive S

I

 direction ( x s positive), electromagnet EM 1 shuts off, and EM 2 is responsible for 

control. Likewise, when the ITM-Beam is displaced in the negative s



I direction ( x s negative), 

electromagnet EM 2 shuts off, and EM 1 is responsible for control. This scheme takes advantage 

of the fact that the electromagnets are more effective when the ITM is at close range. Since 

current required is a nonlinear function of distance to the ITM and control force required, the 

current time history is not sinusoidal like the ITM displacement and control-force time histories. 

The length of the ITM-Beam has a significant impact on the force required to move the beam in 

a prescribed fashion. From equation 18, the external moment exerted on the ITM-Beam by the 

actuators in the KS

 direction about point L is given by 

L K M f BLc S Beam Input   

 

 . (30) 

2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Time (s)

ITM Displacement (ft)

ITM-Beam

Translating Mass
14

Figure 8. Segment of current vs. time for ITM-Beam actuators. 

Therefore, the input force required to exert a given control moment on the ITM-Beam varies 

inversely with beam length. In addition, for a given maximum ITM displacement, x s , the 

maximum angular displacement, max γ , varies inversely with beam length. Trade studies verified 

these results using example simulations with various beam lengths. The maximum ITM 

displacement from the projectile centerline was = 0.157 x s ft for all cases. Figures 9 and 10 

show maximum angular displacement and average force required, respectively, as a function of 

beam length. Notice that as beam length increases, maximum angular displacement and average 

force required decrease. The average control-force reduction with increased beam length occurs 

due to the increased “efficiency” of the ITM-Beam actuators (i.e., the same control moment 

requires less control force for a larger beam length). 

4.25 4.26 4.27 4.28 4.29 4.3 4.31 4.32 4.33 4.34 4.35

-1

-0.5

0

0.5

1

Time (s)

Current (Amps)

Current EM 1

Current EM 2

ITM Displacement

Force

-0.4

-0.32

-0.24

-0.16

-0.08

0.0

0.08

0.16

0.24

0.32

0.4

ITM Displacement (ft)

-300

-200

-100

0

100

200

300Force (lbs)
15

Figure 9. Maximum angular displacement vs. beam length. 

Figure 10. Average force required vs. beam length.

2 3 4 5 6 7 8 9 10

10

15

20

25

30

35

40

45

50

55

60

Beam Length (in)

Maximum Angular Displacement (deg)

2 3 4 5 6 7 8 9 10

130

140

150

160

170

180

190

200

Beam Length (in)

Average Force Required (lbs)
16

As outlined previously, the ITM-Beam system is a dynamic model used to represent a fixed-free 

elastic beam. The rigid ITM-Beam is attached to the projectile at point L with a torsional spring 

and torsional damper to model the elastic beam’s vibrational properties. A trade study examines 

how force and power requirements vary with different spring and damping coefficients. Once an 

optimum spring and damper coefficient are determined, they can be used to identify the proper 

elastic properties of the fixed-free beam for a prototype system. 

The performance of the system is examined for a range of torsional spring constants and 

damping ratios for the example projectile rolling at a steady-state rate of ~128 rad/s. The 

projectile trajectory is simulated for a 2-s flight with no gravity. This simplified flight profile is 

used solely to establish the correlation between spring and damper parameters and average force, 

average power, and total battery charge required. Figure 11 shows the projectile roll-rate time 

history for this flight profile. The high-frequency oscillation of the roll rate occurs at the mass 

oscillation frequency. This is due to the continually changing axial moment of inertia of the 

projectile as the mass translates. When the translating mass travels farther from the centerline, 

axial inertia grows and roll rate decreases due to conservation of angular momentum. Likewise, 

when the translating mass returns toward the centerline, axial inertia decreases and the roll rate 

increases. 

Figure 11. Roll rate vs. time for example simulation.

0 0.5 1 1.5 2

114

116

118

120

122

124

126

128

130

132

Time (s)

Roll Rate (rad/s)
17

Figures 12–14 show the effect of spring and damper coefficients on average force, average 

power, and total battery charge required. In figure 12, it can be clearly seen that an optimum 

torsional spring constant exists in which the spring-mass-damper system of the ITM-Beam 

operates near resonance with the projectile roll rate. These peaks are not as sharp as typical 

spring-mass-damper resonant peaks due to the fact that the projectile roll rate varies over time. 

However, note that significant reductions in force are achieved if the spring constant is placed 

near its optimum value and damping is lowered as much as possible. These reductions in force 

are mirrored by reductions in average power and total charge required, resulting in significantly 

smaller battery sizes. 

A similar study examines the same spring-mass-damper parameters for a full flight profile of the 

example projectile using the same initial conditions as those used in the first example study. 

Figures 15–17 show that as in the partial flight profile case, optimal spring coefficients can be 

found. 

Figure 12. Average force required vs. torsional spring constant, partial flight profile.

0 20 40 60 80 100 120 140 160 180 200

0

50

100

150

Torsional Spring Constant (ft-lbs/rad)

Average Force Required (lbs)

 = 0.02

 = 0.05

 = 0.1

 = 0.2
18

Figure 13. Average power required vs. torsional spring constant, partial flight profile. 

Figure 14. Total charge required vs. torsional spring constant, partial flight 

profile.

0 20 40 60 80 100 120 140 160 180 200

0

0.5

1

1.5

2

2.5

3

Torsional Spring Constant (ft-lbs/rad)

Average Power Required (hp)

 = 0.02

 = 0.05

 = 0.1

 = 0.2
19

Figure 15. Average force required vs. torsional spring constant, full flight 

profile. 

Figure 16. Average power required vs. torsional spring constant, full flight 

profile.

30 40 50 60 70 80 90 100

20

25

30

35

40

45

50

55

60

Torsional Spring Constant (ft-lbs/rad)

Average Force Required (lbs)

 = 0.02

 = 0.05

 = 0.1

 = 0.2

30 40 50 60 70 80 90 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Torsional Spring Constant (ft-lbs/rad)

Average Power Required (hp)

 = 0.02

 = 0.05

 = 0.1

 = 0.2
20

Figure 17. Total charge required vs. torsional spring constant, full flight profile. 

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 figure 6, roll rate of the projectile varies 

between 5 rad/s initially and a final value of ~80 rad/s. This large variation in roll rate means 

that the spring coefficient is only optimized for a very short period of the overall flight, and the 

broad peaks shown in figures 15–17 result. To demonstrate this, figure 18 shows a current time 

history for an example full flight trajectory using = 70.0 T k lb/rad and = 0.05 D k lb/rad/s. Note 

that the spring constant is optimized for the projectile roll rate ~1 s into flight and once again 

after spin decay occurs ~7 s into the flight. 

Despite the broad nature of the peaks shown in figures 15–17, significant size and weight savings 

can be achieved using the proper spring constants in the form of smaller batteries. As shown in 

figure 17, batteries with a total charge of less than 5 A/s may be used for systems with optimal 

spring coefficients and low damping ratios. Furthermore, figure 18 shows that reasonable 

maximum current levels, on the order of 150 mA, can be expected with an optimized system. 
21

Figure 18. Current through actuators vs. time for example full flight 

trajectory. 

Average force levels and, therefore, total charge required can be decreased even further by 

actively changing the elastic properties of the beam during flight. A fixed-free cantilever beam 

like that used in this system has a first vibrational mode shape of 

2

        sin x

Bx

BL

s , (31) 

where B x is the distance along the beam. The natural frequency of the first vibrational mode is 

2





  n

E

BL

, (32) 

where E is the modulus of elasticity of the material and  is the density of the beam (13). By 

changing the modulus of elasticity, it is therefore possible to tune the natural frequency of the 

cantilever beam to a desired value. Recent investigations into smart materials (14–17), 

specifically, materials used in tunable vibration absorbers, have shown that various methods can 

be used to actively alter a material’s modulus of elasticity, allowing the beam’s torsional spring 

constant to be actively optimized during flight as the projectile roll rate changes. This would 

allow the ITM-Beam system to operate with the lowest possible power through the entire flight, 

yielding further reductions in battery size. 

1 2 3 4 5 6 7 8

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Current (Amps)

Time (s)
22

To investigate this, several example simulations were run. The first set simulated the projectile 

for the full flight using the optimum spring constants obtained from figure 17. This produced the 

least possible battery charge required for the ITM-Beam system, with a fixed spring constant for 

each damping ratio considered. The second set of simulations included a roll-rate feedback 

mechanism. In these cases, at specific points throughout the flight, the torsional spring and 

damping coefficients of the ITM-Beam were adjusted to match the roll rate. Figure 19 shows 

how the torsional spring constant was adjusted for an example flight with ζ = 0.05. Note that the 

curve in figure 19 has the same qualitative shape as the roll rate time history shown in figure 6. 

Table 2 summarizes the results of the two sets of simulations with the fixed torsional spring and 

variable torsional spring constants. Note that in the variable torsional spring cases, the torsional 

damping coefficient D k was adjusted slightly to keep the damping ratio ζ constant. Figure 20 

demonstrates that implementing the roll-rate feedback system saves ~1 A/s of charge for all 

damping ratios considered (34.4% decrease in battery size for ζ = 0.02, 28.1% decrease for ζ

= 0.05, 20.4% for ζ = 0.1, and 11.2% for ζ = 0.2). 

Figure 19. Torsional spring constant vs. time for roll-rate feedback system, ζ = 0.05.

0 1 2 3 4 5 6 7 8

0

10

20

30

40

50

60

70

80

90

Time (s)

Torsional Spring Constant (lbs/rad)
23

Table 2. Performance evaluation of roll-rate feedback system. 

Damping Ratio ζ

Optimum kT for No 

Feedback Case

Charge Required (A/s) Percentage Decrease 

in Charge Required 

With Feedback 

No Feeback 

(Constant kT) 

Feedback 

(Variable kT) 

0.02 71 3.924 2.576 34.4% 

0.05 69 4.576 3.287 28.1% 

0.1 67 5.895 4.692 20.4% 

0.2 58 8.786 7.798 11.2% 

Figure 20. Total charge required vs. damping ratio for constant and variable kT cases. 

4. Conclusion 

A conceptual design for a projectile equipped with a controllable internal mass is proposed using 

a cantilever beam configuration. The system is designed to minimize moving parts and exploit 

the beam’s elastic properties, thereby creating a robust mechanism that minimized power 

requirements. A dynamic model of the configuration is developed, as is a model of the 

electromagnetic actuator system to study current requirements and battery size. Example 

simulations show that useful control authority is generated with this projectile control 

mechanism, matching the results from earlier studies of projectiles with internal translating mass 
24

control. Furthermore, by tuning the natural frequency of the vibrating beam to the projectile roll 

rate, significant reductions in power requirements are achieved on the order of 60%. Power 

reductions can be further realized with active tuning of the beam’s elastic properties during 

flight. Battery size requirements to power the actuator are modest, with several commercial-offthe-shelf options available. The internal oscillating beam configuration shows promise as a 

viable, cost-effective, reliable projectile control mechanism.
25

5. References 

1. Soper, W. Projectile Instability Produced by Internal Frictio. AIAA Journal 1978, 16 (1), 

8–11. 

2. Murphy, C. Influence of Moving Internal Parts on Angular Motion of Spinning Projectiles. 

J. Guid. Control 1978, 1 (2), 117–122. 

3. D’Amico, W. Comparison of Theory and Experiment for Moments Induced by Loose 

Internal Parts. J. Guid. Control 1987, 10 (1), 14–19. 

4. Hodapp, A. Passive Means for Stabilizing Projectiles with Partially Restrained Internal 

Members. J. Guid. Control 1989, 12 (2), 135–139. 

5. Petsopoulos, T.; Regan, F.; Barlow, J. Moving Mass Roll Control System for Fixed-Trim 

Re-Entry Vehicle. J. Spacecraft Rockets 1996 33 (1), 54–61. 

6. Robinett, R.; Sturgis, B.; Kerr, S. Moving Mass Trim Control for Aerospace Vehicles. J. 

Guid. Control Dynam. 1996 19 (5), 1064–1071. 

7. Menon, P.; Sweriduk, G.; Ohlmeyer, E.; Malyevac, D. Integrated Guidance and Control of 

Moving Mass Actuated Kinetic Warheads. J. Guid. Control Dynam. 2004, 27 (1), 118–127. 

8. Frost, G.; Costello, M. Linear Theory of a Projectile with a Rotating Internal Part in 

Atmospheric Flight. J. Guid. Control Dynam. 2004, 27 (5), 898–906. 

9. Frost, G.; Costello, M. Control Authority of a Projectile Equipped With an Internal 

Unbalanced Part. J. Dy. Syst. T ASME 2006, 128 (4), 1005–1012. 

10. Rogers, J.; Costello, M. Control Authority of a Projectile Equipped with an Internal 

Translating Mass. J. Guid. Control Dynam. 2008, 31, (5), 1323–1333. 

11. Slotine, J.; Li, W. ISBN 0-13-040890-5. Applied Nonlinear Control; Prentice Hall: New 

York, 1991, 207–236. 

12. Purcell, E. ISBN 0-07-004908-4. Electricity and Magnetism; McGraw-Hill: New York, 

1985, 227, 412-138. 

13. Inman, D. ISBN 0-13-726-142-X. Engineering Vibration; Prentice Hall: New York, 2001, 

451.
26

14. Flatau, A.; Dapino, M.; Calkins, F. High Bandwidth Tunability in a Smart Vibration 

Absorber. J. Intell. Mat. Syst. Str. 2000, 11, 923–929. 

15. Varga, Z.; Filipcsei, G.; Zrinyi, M. Smart Composites with Controlled Anisotropy. Polymer

2005, 46, 7779–7787. 

16. Varga, Z.; Filipcsei, G.; Zrinyi, M. Magnetic Field Sensitive Functional Elastomers With 

Tuneable Elastic Modulus. Polymer 2006, 47, 227–233. 

17. Davis, C.; Lesieutre, G. An Actively Tuned Solid-State Vibration Absorber Using 

Capacitive Shunting of Piezoelectric Stiffness. J. Sound Vib. 2000, 232 (3), 601–617.
27

Appendix A. A and B Matrix Terms
28

The terms of equation 22 are given as follows: 

 

11 31 3

12 32 3

2

13 33 2 1

2

14 33





 

 

 

  

 

P T

T

P T

T

PT PT

T

P T

T

m m A I BL s r

m

m m A I BL c r

m

mm mm A I BL BL c r s r

m m

m m A I BL

m

   

 

21 31 1 3

22 32 2 3

2 2

23 33 2 1 1 2

24 2 1

P T

P

P T

P

PT PT

P

P T

m m A I rr

m

m m A I rr

m

mm mm A I BL r c r s r r

m m

m m A BL r c r s

m

 

 

 

 

   

 

 

31 21 21 4 2 4

32 22 22 4 1 4 3 6

33 23 23 6 2 6

34 23 6

PT PT

P T

PT PT

P T

PT PT

P T

P T

T

mm mm A I I BLr c r r

m m

mm mm A I I BLr s r r r r

m m

mm mm A I I BLr c r r

m m

m m A I BLr c

m









 

  

 

 

41 11 11 5 2 5 3 6  

42 12 12 5 1 5

43 13 13 6 1 6

44 13 6

PT PT

P T

PT PT

P T

PT PT

P T

P T

T

mm mm A I I BLr c r r r r

m m

mm mm A I I BLr s r r

m m

mm mm A I I BLr s r r

m m

m m A I BLr s

m









  

 

 

 
29

     

         

     

      

1 / //

/ /

/ /

/ /

S L X T S C I Input T D S B I T S T I

P T

S L X S T S L X TI TI T L X

P T

S L X BI BI S P L

P T

S L X BI S TI S L X

B K r m a f BLc k k K I

m m Kr W Kr r

m

m m Kr r

m

m m Kr r

m

   

 

 

 



 

 

 

    

 



     

         

        

      

      

 

  

 

 

     

       

     

      

 

2 / //

/ /

/ /

/ /

S P L P S C I Input T D S B I P S B I

L P T

S S P S L P TI TI S L X

P T

S L P BI BI S P L

P T

S L P BI S TI S L X

S LP S

B K m r a f BLc k k K I

m m K M Kr r

m

m m Kr r

m

m m Kr r

m

K r

   

 

 

 



 

 

 



    

 



     





         

        

      

      

    

 





 

   WP

         

     

     

       

3 // //

/ /

/ /

/ /

S BI P S BI S TI T S TI S L X S T

S L P S CI S T P X S CI

P T

S P X TI TI S L X

P T L

S P X B I B I S P L S S system

P T

S P

B J I J I Jr W

Jm r a Jm r a

m m Jr r

m

m m Jr r J M

m

m m J r

m

 

 

 



 

 

 



  

 



 





          

       

      

        

  



 

 

 

       X BI S TI S L X / /        r 

     

         

     

     

       

4 // //

/ /

/ /

/ /

S BI P S BI S TI T S TI S L X S T

S L P S CI S T P X S CI

P T

S P X TI TI S L X

P T L

S P X B I B I S P L S S system

P T

S P

B I I I I Ir W

Im r a Im r a

m m Ir r

m

m m Ir r I M

m

m m I r

m

 

 

 



 

 

 



  

 



 





          

       

      

        

  



 

 

 

       X BI S TI S L X / /        r 

     
30

      

         

     

      

/ //

/ /

/ /

/ /

FLC S L X T S C I T D S B I T S T I

P T

S L X S T S L X TI TI T L X

P T

S L X BI BI S P L

P T

S L X BI S TI S L X

B Kr m a kkK I

m m Kr W Kr r

m

m m Kr r

m

m m Kr r

m

  

 

 

 



 

 

 

   

 



     

         

        

      

      

 

  

 

 
31

Appendix B. Translating Mass Projectile Schematic
32

Figure B-1. Translating mass projectile schematic. 
33

List of Symbols, Abbreviations, and Acronyms 

123 aa a , , Components in the S frame of the acceleration of the system mass center with 

 respect to the inertial frame 

P I/ a

 Acceleration of the projectile mass center with respect to the inertial frame 

X / I a

 Acceleration of the end of the ITM-Beam with respect to the inertial frame 

A Maximum magnitude of ITM-Beam displacement from center of cavity 

b Radius of the electromagnetic actuator 

BL Length of the ITM-Beam assembly 

Bz

 Magnetic field produced by the electromagnetic actuators in the S I direction 

C Composite body center of mass 

DM Magnitude of the magnetic dipole moment of the fixed magnet at the end of the 

 ITM-Beam 



FC Hinge constraint force on the internal translating mass 



FI Input force exerted by electromagnetic actuators on ITM-Beam 



FP Total aerodynamic force exerted on the projectile 

input f Scalar value of the total force applied to ITM-Beam by electromagnetic actuators 

/

P HB I

 Angular momentum of the projectile with respect to the inertial frame about the 

 projectile center of mass 

/

X HT I

 Angular momentum of the ITM-Beam with respect to the inertial frame about 

 point X

Txx I The xx component of the ITM-Beam moment of inertia matrix about point X 

Pxx I The xx component of the projectile moment of inertia matrix 

, ,

 

I I I I J K Inertial frame unit vectors 

, , 

BB B I J K Projectile reference frame unit vectors 

, ,  

NN N I J K Nonrolling reference frame unit vectors
34

, ,  

TT T I J K ITM-Beam fixed frame unit vectors 

, ,  

S S S I J K S frame unit vectors 

EI Current through the electromagnetic actuator 

D k Torsional damper coefficient 

T k Torsional spring coefficient 

L Junction of between the ITM-Beam and the projectile, referred to as the “hinge 

 point” 

LA Length of the electromagnetic actuator 

mP Mass of the projectile with the cavity 

mT Mass of the ITM-Beam 

m Total mass of the system 

L MP

 External moments applied to the projectile about the hinge point L

L M

system

 External moments applied to the projectile-ITM system about the hinge point L 

 L MBeam External moment exerted on the ITM-Beam by the actuators in the KS direction 

 about the hinge point L

P Projectile center of mass 

L X r 

 Distance vector from the hinge point L to point X at the end of the ITM-Beam 

L P r 

 Distance vector from the hinge point L to the projectile center of mass P

P X r 

 Distance vector from projectile center of mass P to point X at the end of the ITM- 

 Beam 

123 rrr , , Components in the S frame of P L r 



456 rrr , , Components in the S frame of P X r 



pqr   , , Components of B I/

 in the S frame 

pqr , , Components of B I/

 in the projectile reference frame
35

uvw , , Translational velocity components of the composite body center of mass resolved 

 in the projectile reference frame 



vC/I Velocity of the system mass center with respect to the inertial frame 

WP

 Weight of the projectile (without the ITM-Beam) 

WT

 Weight of the ITM-Beam 

X Point at the end of the ITM-Beam 

x, , y z Position vector components of the composite body center of mass expressed in 

 the inertial reference frame 

, , XBB B Y Z Total external force components on the projectile and ITM-Beam system 

 expressed in the projectile reference frame 

 B I/

 Angular acceleration of the projectile body with respect to the inertial frame 

T I/

 Angular acceleration of the ITM-Beam with respect to the inertial frame 

 Deflection angle of the ITM-Beam 

, ,   Euler roll, pitch, and yaw angles 

,  T T  Euler pitch and yaw angles for the orientation of the S frame with respect to the B 

 frame 

B I/

 Angular velocity of the projectile body with respect to the inertial frame 

T I/

 Angular velocity of the ITM-Beam with respect to the inertial frame 

Beam Magnitude of the angular velocity of the ITM-Beam with respect to the S frame 
NO. OF 

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