DYNAMICS OF A SPINNING MEMBRANE

Preprint AAS 1-601 DYNAMICS OF A SPINNING MEMBRANE Jer-Nan Juang Chung-Han Hung and William K. Wilkie INTRODUCTION A novel approach is introduced to conduct dynamic analysis and system identification of a spinning membrane. In this formulation inextensible membranes are modeled using a discrete set of lumped masses. Lagranges equations are used to derive the highly coupled ordinary differential equations for in-plane, out-plane, and twisting motions for the spinning membrane. The generalized and uncoupled linear equations for small motion are used to compute the vibration mode frequencies which are compared to results from an uncoupled analysis of blade motion using rotor dynamics. Numerical simulations along with 3-D animations are used to study the linear and nonlinear behavior of the spinning membrane. Linear and nonlinear system identification techniques will then be proposed for model verification and controller designs. The TA-0 In-Space Propulsion Technology Roadmap identifies solar sail systems as a missionenabling technology for continuous thrust applications and high delta-v destinations otherwise unreachable with chemical or solar electric propulsion SEP. The heliogyro is a high-performance, spinning solar sail architecture that uses long order of kilometers reflective membrane strips to produce thrust from solar radiation pressure. The membrane blades spin about a central hub and are stiffened by centrifugal forces only, making the design exceedingly light weight. The structural dynamics of the heliogyro was studied in 1970s, 1 using the techniques of helicopter rotor dynamics, to answer questions regarding concept feasibility that are raised by dynamic considerations. In spite of favorable performance estimates, the solar sail was not initially given serious consideration due to the absence of credible solutions for deploying, rigidizing, and orienting the extremely large surface areas. Instead of considering the heliogyro as a helicopter operating in a space environment, a simple approach is introduced to conduct dynamic analysis for a spinning membrane that is assumed inextensible. In this approach the membrane model is constructed as a discrete set of lumped masses. Lagranges equations 3 are applied to derive the highly coupled ordinary differential equations for in-plane, out-plane, and twisting motions for the spinning membrane. Furthermore, the nonlinear equations are linearized for small motion to yield a set of linear second-order matrix differential equation of motion. This second order spring-mass model provides the basis for quantitative and qualitative analyses for engineering development. With this model, the vibration mode frequencies are easily computed and were found to agree with the results from an uncoupled analysis of blade motion using rotor dynamics. Numerical simulations along with 3-D animations are Professor, Department of Engineering Science, National Cheng Kung University, Tainan City, Taiwan; Adjunct Professor, Aerospace Engineering Department, Texas A & M University, College Station, USA; Visiting Professor, National Institute of Aerospace, Hampton, Virginia, USA; Fellow AAS, AIAA, ASME. Graduate Student, Department of Engineering Science, National Cheng Kung University, Tainan City, Taiwan; Exchange Student, National Institute of Aerospace, Hampton, Virginia, USA Head, Structural Dynamics Branch, NASA Langley Research Center, Hampton, Virginia, USA 1

" r 1 m m 3 m 1 r!!3!1 r r i 3! m i r n! e z e y "! e x!i!n m n Figure 1. Discrete inextensible string model moving on the spinning plane provided to study the linear and nonlinear behavior of the spinning membrane. Linear and nonlinear system identification techniques will be proposed for model verification and controller designs. BASIC FORMULATION This section briefly describes an approach to derive dynamic equations of motion for a spinning membrane. Lagrange s equations are applied to formulate ordinary differential equations for inplane, out-plane, and twisting motions with the assumption that spinning membrane is inextensible. Pure in-plane motion Figure 1 shows a discrete inextensible string spinning on a plane. It is used to represent the side view of a spinning membrane model which is constructed as a discrete set of lumped masses m 1, m,..., m n experiencing in-plane motion. The quantities r i and θ i are the length of the ith mass m i and its angle relative to the horizontal axis e x of the body frame described by e x, e y, and e z. The spinning rate ω along the axis e z is assumed constant. The displacement vector R k of mass m k from the origin can be expressed by R k = r i cos θ i e x + r i sin θ i e y ; k = 1,,..., n 1 Taking derivative of the displacement vector yields the following velocity vector R k = k r i sin θ i θi e x + r i cos θ i ex + r i cos θ i θi e y + r i sin θ i ey = k r i θi sin θ i e x + cos θ i e y + r i cos θ i ω e z e x + r i sin θ i ω e z e y = k r i θi sin θ i e x + cos θ i e y + r i cos θ i ω e y r i sin θ i ω e x = k r i θi + ω sin θ i e x + r i θi + ω cos θ i e y where θ i means the angular velocity of the mass m i. Total kinetic energy of the string for all masses is T = 1 n m k Rk Rk 3 k=1

where R k Rk = k = k j=1 j=1 r i r j θi + ω θj + ω sin θ i sin θ j + cos θ i cos θ j r i r j θi + ω θj + ω cos θ i θ j 4 Lagrangian with zero potential energy V becomes L = T V = 1 n m k k=1 j=1 r i r j θi + ω θj + ω cos θ i θ j 5 Substitution of the Lagrangian into Lagrange s equation produces equations of motion for the spinning string. For simplicity without loosing generality, let us consider the case where the string is discretized equally, i.e., m 1 = m = = m n = m and r 1 = r = = r n = r 6 We begin with a two-degree-of-freedom dof system. The Lagrangian shown in Eq. 4 for the two-mass system is L = 1 mr k=1 j=1 θi + ω θj + ω cos θ i θ j = 1 mr θ1 + ω + θ + ω + θ1 + ω θ + ω 7 cos θ 1 θ Application of Lagrange s equations of motion yields and d L dt θ L 1 θ 1 = τ θ1 mr θ 1 + θ cos θ 1 θ + d L dt θ L θ = τ θ mr θ + θ 1 cos θ 1 θ ω + θ + ω θ 8 sin θ 1 θ = τ θ1 ω + θ 1 + ω θ 9 1 sin θ 1 θ = τ θ where τ θ1 and τ θ are the toques applied to the first and second masses, respectively. Equations 8 and 9 may be written in the following matrix form as mr cos θ 1 θ θ1 cos θ 1 θ 1 θ ω + +mr θ + ω θ sin θ 1 θ ω + θ 1 + ω θ = 1 sin θ θ 1 τθ1 τ θ 10 3

Note that the nonlinear terms associated with ωθ 1 and ωθ will produce Coriolis effect which is a deflection of moving objects when they are viewed in a rotating reference frame. The Coriolis acceleration is perpendicular to both the the direction of the velocity of the moving mass and to the frame s rotation axis. Equation 10 can be linearized for small angles θ 1 0 and θ 0 such that cos θ 1 θ 1; sin θ 1 θ θ 1 θ to yield mr 1 1 1 θ1 θ + mr ω 1 1 1 1 θ1 θ = τθ1 τ θ 11 Solving eigenvalue problem of Eq. 11 produces two natural frequencies, 0 and 5ω. The zero frequency implies that there exists a rigid-body mode in the in-plane motion. Application of the Lagrange s derivation process to a three-degree-of-freedom 3dof system yields the equations of motion mr 3 cos θ 1 θ cos θ 1 θ 3 cos θ 1 θ cos θ θ 3 cos θ 1 θ 3 cos θ θ 3 1 ω + θ + ω θ sin θ 1 θ + ω + θ 3 + ω θ 3 sin θ 1 θ 3 +mr ω + θ 1 + ω θ 1 sin θ θ 1 + ω + θ 3 + ω θ 3 sin θ 3 θ ω + θ 1 + ω θ 1 sin θ 3 θ 1 + ω + θ + ω θ sin θ 3 θ θ 1 θ θ 3 = τ θ1 τ θ τ θ3 1 Its linearized version is mr 3 1 1 1 1 1 θ 1 θ θ 3 + mr ω 3 1 3 1 1 1 θ 1 θ θ 3 = τ θ1 τ θ τ θ3 13 Equation 13 contains three natural frequencies, 0, 5ω, and 14ω By induction, one would be able to derive equations of motion for any arbitrary degree-offreedom system by inserting the Lagrangian shown in Eq. 5 to Lagrange s equations d L dt θ L = τ θp 14 p θ p to yield mr +mr n n θ j cos θ p θ j mr n θp + ω θj + ω sin θ p θ j = τ θp ; θj + ω θp θ j sin θ p θ j p = 1,,..., n 15 4

Performing linearization of Eq. 15 for small angles yields n n mr θ j + mr ω θ p θ j = τ θp ; p = 1,,..., n 16 Equation 16 can be written in a conventional matrix form M θ θ + Kθ θ = τ θ 17 where the mass matrix M θ is M θ = mr n n 1 n 1 n 1 n 1 n 1 n n n 1....... 1 1 1 1 the stiffness matrix K θ is n 1 i n 1 n 1 n 1 n 1 i n 1 K θ = mr ω n 1 n n i 1 i=....... n 1 1 1 1 i and the torque vector τ θ is τ θ = τ θ1 τ θ τ θ3 Equation 17 produces a set of n natural frequencies. τ θn i=n 1 18 19 0 Pure out-plane motion ii + 1 ω 1, i = 1, 3, 5,..., n 1 1 For simplicity, assume that the model of an inextensible membrane is constructed as a discrete set of lumped masses. A four-mass example model is shown in Fig.. The displacement vectors R k1 of mass m k1 and R k of mass m k from the origin can be expressed by R k1 = k r k cos φ k e x + s e y + k r k sin φ k e z R k = k r k cos φ k e x s e y + k r k sin φ k e z 5

! e y! e z r 1 "!1 m11 r s m 1! m 1 s m! e x Figure. Discrete inextensible membrane model moving on the plane orthogonal to the spinning plane where s is half of the length between the masses m k1 and m k for k = 1,,..., n. The velocities of the displacement vectors shown in Eq. are R k1 = ωs e x + k r k φk sin φ k e x + k ωr k cos φ k e y + k r k φk cos φ k e z R k = ωs e x + k r k φk sin φ k e x + k ωr k cos φ k e y + k 3 r k φk cos φ k e z To gain insight, let us begin with the two-mass model. Kinetic energy for the two Masses is T = 1 m 11 R11 R11 + m 1 R1 R1 Assume that m 11 = m 1 = m 1. Lagrangian with zero potential energy V = 0 becomes L = T V = m 1 r1 φ 1 + ω cos φ 1 + m 1 s ω 5 Application of Lagrange s equation to produce equation of motion d L dt φ L = τ φ1 m 1 r1 1 φ φ 1 + ω sin φ 1 = τ φ1 6 1 where τ φ1 is the external torque for the angle φ 1. The two-mass out-plane motion acts like a pendulum model where the gravity is replaced by lω. Its linearized equation is m 1 r1 φ1 + ω φ 1 = τ φ1 7 The natural frequency for the two-mass membrane model described by Eq. 7 is ω. Now, consider the four-mass model having two degrees of freedom dof. Kinetic energy for the four-mass membrane model is T = 1 m 11 R11 R11 + m 1 R1 R1 + m 1 R1 R1 + m R R 8 Assume that 4 m 11 = m 1 = m 1 = m = m and r 1 = r = r; 9 6

Inserting the velocity vectors shown in 3 into Eq. 8 yields the Lagrangian for the four-mass spinning model L = T V = mr φ 1 + φ + φ 1 φ cosφ 1 φ + ω cos φ 1 + ω cos φ 1 + cos φ + ms ω Applying the Lagrange s equation d L dt φ L 1 φ 1 d L dt φ L = τ φ1 30 φ = τ φ 31 produces the nonlinear equations of motion mr cosφ 1 φ cosφ 1 φ 1 φ1 φ + mr φ sinφ 1 φ + ω sin φ 1 cos φ 1 + cos φ φ 1 sinφ φ 1 + ω sin φ cos φ 1 + cos φ = τφ1 τ φ 3 where τ φ1 and τ φ are the external toques for the angles φ 1 and φ, respectively. Note that the out-plane motion does not have nonlinear terms associated with ωφ 1 and/or ωφ as opposed to the case for the in-plane motion see Eq. 10. Thus no Coriolis effect exists in the out-plane motion. For small angles φ 1 0 and φ 0, these nonlinear equations reduce to the linear matrix equation mr 1 φ1 + mω 1 1 φ r 3 0 φ1 τφ1 = 33 0 φ τ φ which possesses two natural frequencies, i.e., ω and 6ω. The same derivation process is applied for the six-mass out-plane model to yield the nonlinear equations of motion mr + mr = 3 cosφ 1 φ cosφ 1 φ 3 cosφ 1 φ cosφ φ 3 cosφ 1 φ 3 cosφ φ 3 1 τ φ1 τ φ τ φ3 ω 3 cos φ 1 + cos φ + cos φ 3 sin φ 1 + φ sin φ 1 φ + φ 3 sin φ 1 φ 3 ω cos φ 1 + cos φ + cos φ 3 sin φ + φ 1 sin φ φ 1 + φ 3 sin φ φ 3 ω cos φ 1 + cos φ + cos φ 3 sin φ 3 + φ 1 sin φ 3 φ 1 + φ sin φ 3 φ For small angles φ 1 0, φ 0, and φ 3 0, Eq. 34 reduces to 3 1 φ 1 6 0 0 mr 1 φ + mr ω 0 5 0 1 1 1 φ 3 0 0 3 φ 1 φ φ 3 φ 1 φ φ 3 = τ φ1 τ φ τ φ3 34 35 7

! e y! e z " m r!1 r m! r! e x!3!i m r m r!n m Figure 3. Discrete inextensible string model moving on the plane orthogonal to the spinning plane which has three natural frequencies, ω, 6ω, and 15ω. By induction, the generalized nonlinear equations of motion are mr n n +mr ω φ j cos φ p φ j mr cos φ j sin φ p = τ φp ; n φ j sin φ p φ j p = 1,,..., n 36 where τ φp is the external torque for the angle φ p, and m = m. Note that Eq. 36 may also be derived by considering the membrane as a string shown in Fig. 3 with m = m, because the outplane motion takes place only on the x z plane. It is equivalent to view the membrane from the direction of y axis. Note that the inertia terms associated with the angular acceleration φ j shown in Eq. 36 for the out-plane motion are identical to those associated with θ j shown in Eq. 15 for the in-plane motion. However, Coriolis force does not appear in the out-plane motion, i.e., no coupling term ωφ j appears in Eq. 36. Centrifugal forces given by rω and r φ j provide the stiffness for the out-plane motion. Linearizing Eq. 36 for small angles are mr n φ j + mr ω n which may be written in second-order matrix form φ p = τ φp ; p = 1,,..., n 37 where the mass matrix M φ is M φ φ + Kφ φ = τ φ 38 M φ = mr n n 1 n 1 n 1 n 1 n 1 n n n 1....... 1 1 1 1 39 8

e x ey ω r 1 m 11 ψ 1 ϕ 1 s e z m 1 Figure 4. Twisting motion for a 1-dof discrete inextensible membrane the stiffness matrix K φ is K φ = mr ω n k 0 0 0 k=1 0 n k 0 0 k= 0 0 n k 0 k=3....... 0 0 0 n 40 and the torque vector τ φ is τ φ = where n is the degrees of freedom for the system, i.e., half of the number of masses. The matrix M φ shown in Eq. 39 for the out-plane motion is identical to M θ shown in Eq. 18 for the in-plane motion. However, the equality does not apply to the stiffness matrices K φ shown in Eq. 40 and K θ shown in Eq. 19. Equation 39 produces a set of n natural frequencies i i + 1 ω, i = 1, 3, 5,..., n 1 4 Pure twisting motion The one-degree-of-freedom twisting motion is sketched in Fig. 4 where ϕ 1 is the twisting angle of the x y plane and ψ 1 is the angle of the length r 1 away from its original position. The position vectors of the masses m 11 and m 1 are τ φ1 τ φ τ φ3. τ φn 41 R 11 = r 1 cos ψ 1 e x + s cos ϕ 1 e y + s sin ϕ 1 e z R 1 = r 1 cos ψ 1 e x s cos ϕ 1 e y s sin ϕ 1 e z 43 9

where s is half of the length between m 11 and m 1. The relationship between the angles ψ and ϕ is approximated by ψ 1 = sϕ 1 r 1 44 Accurate relation for large angles ψ 1 and ϕ will be provided later for the general case of twisting motion with n degrees of freedom. Substituting Eq. 44 into Eq. 43 and taking derivatives of the resulting equations yields the following velocity vectors R 11 = s ϕ 1 sin sϕ 1 /r 1 ωs cos ϕ 1 e x + s ϕ 1 sin ϕ 1 + ωr 1 cos sϕ 1 /r 1 e y + s ϕ 1 cos ϕ 1 e z R 1 = s ϕ 1 sin sϕ 1 /r 1 + ωs cos ϕ 1 e x + s ϕ 1 sin ϕ 1 + ωr 1 cos sϕ 1 /r 1 e y s ϕ 1 cos ϕ 1 e z 45 Lagrangian for the 1dof twisting motion with the assumption that m 11 = m 1 = m 1 becomes L = T V = m 1 R11 R11 + R1 R1 = m 1 r1 ω cos sϕ 1 /r 1 + m 1 s { ϕ 1 cos sϕ 1 /r 1 + ω cos } 46 ϕ 1 Application of Lagrange s equation of motion yields d L dt ϕ 1 L ϕ 1 = τ ϕ1 m 1 s ϕ 1 cos sϕ 1 /r 1 + m 1 s/r 1 s ϕ 1 + r 1 ω sin sϕ 1 /r 1 47 + m 1 s ω sin ϕ 1 = τ ϕ1 where τ ϕ1 is the external torque for the twisting angle ϕ 1. Equation 47 can be linearized for small angle ϕ 1 0 to become This linear equation has a natural frequency ω. m 1 s ϕ 1 + 4 m 1 s ω ϕ 1 = τ ϕ1 48 Generalization of the above 1dof to ndof begins with the definition of the position vectors at the kth degree of freedom R k1 = r cos ψ i e x + s cos ϕ n e y + s sin ϕ n e z 49 R k1 = r cos ψ i e x s cos ϕ n e y s sin ϕ n e z where the angles ψ k and ϕ k are related by r sin ψ 1 = s sin ϕ 1 ; r sin ψ ϕ ϕ 1 ϕk ϕ k 1 = s sin ; ; r sin ψ k = s sin 50 The above relation is exact without any approximation. Figure 5 shows the schematic diagram of the relationship between ψ 1 and ϕ 1. The set of points c, b, and d forms an isosceles triangle whereas the set of points b, d, and a forms a right triangle. From the triangle geometry, the length between the points b and d can be computed by s sinϕ 1 / or r sinψ 1 that yields Eq. 50. 10

e x ey ω a r ψ 1 b h d s c ϕ 1 e z ϕ 1 Figure 5. Triangle geometry for angles ψ 1 and ϕ 1 Differentiating the postion vectors, Eq. 49, yields the velocity vectors R k1 = R k = ωs cos ϕ k r k ωs cos ϕ k r k ψ i sin ψ i e x + s ϕ k sin ϕ k + rω k cos ψ i e y + s ϕ k cos ϕ k e z ψ i sin ψ i e x + s ϕ k sin ϕ k + rω k cos ψ i e y + s ϕ k cos ϕ k e z The Lagrangian for n masses is 51 L = T V n = m ω s cos ϕ k + s ϕ k + r k=1 j=1 ψ i ψ j sin ψ i sin ψ j + ω cos ψ i cos ψ j 5 Differentiating the Lagrangian with respect to ϕ p for p = 1,,... n yields L ϕ p = m n s ϕ k ϕ p + r k k=1 j=1 { = m n s ϕ k δ k p + rs k k=1 mrs n k=1 j=1 = ms ϕ p + mrs n mrs n k=p+1 j=1 ψ i ϕ p j=1 ψ j cos ϕi ϕ i 1 ψ j sin ψ i sin ψ j + ψ j ϕ p ψ i sin ψ i sin ψ j } ψ j cos ϕi ϕ i 1 δi p cos ψ i sin ψ i sin ψ j δi 1 p cos ψ i sin ψ i sin ψ j ψ j sin ψ j tan ψ p cos ϕp ϕ p 1 ψ j sin ψ j tan ψ p+1 cos ϕp+1 ϕ p 53 where the second equality is obtained by using the following formulation ψ i = s ϕ p r cos ϕi ϕ i 1 δ i p s cos ψ i r cos ϕi ϕ i 1 δ i 1 p 54 cos ψ i 11

The delta δ function has the property that δi p = 1 if i = p and δi p = 0 if i p, and δi 1 p = 1 if i = p + 1 and δi 1 p = 0 if i p + 1. Note that ψ n+1 = 0. Similarly, differentiating the Lagrangian with respect to ϕ p yields L ϕ p = m ϕ p n ω s cos ϕ k + s ϕ k + r k=1 = m n { ω s } δ k p sin ϕ k k=1 + mrs n mrs n k=1 j=1 k=1 j=1 cos ϕi ϕ i 1 cos ϕi ϕ i 1 = mω s sin ϕ p n rs mω tan ψ p cos ψ j cos n +rs m δi p cos ψ i δi p cos ψ i ψ j ψ p sin ψ j cos ϕp ϕ p 1 k j=1 cos cos ψ i ψ j sin ψ i sin ψ j + ω cos ψ i cos ψ j ϕi ϕ i 1 ϕp ϕp 1 n n δi 1 p cos ψ i ψ i ψ j cos ψ i sin ψ j ϕi ϕ i 1 δi 1 p cos ψ i ω sin ψ i cos ψ j k=p+1 j=1 k=p+1 j=1 tan ψ p+1 cos ψ j cos ψ j ψ p+1 sin ψ j cos ϕp+1 ϕ p ϕp+1 ϕ p 55 for p = 1,,..., n. The second equality is obtained by inserting the following equation. ψ i = s ϕ p r cos ϕi ϕ i 1 δ i p s cos ψ i r cos ϕi ϕ i 1 δ i 1 p 56 cos ψ i Note that the right-hand sides of Eqs. 54 and 56 are identical implying that ψ i ϕ p Lagrange s equations of motion = ψ i ϕ p. Applying d L L = τ ϕp 57 dt ϕ p ϕ p 1

yields the equations of twisting motion for ndof spinning membrane ms ϕ p + mω s sin ϕ p + rs m n rs m +rs m n rs m +rs m n rs m rs m n +rs m n k=p+1 j=1 n k=p+1 j=1 n ψ j sin ψ j tan ψ p+1 cos ϕp+1 ϕ p ψ j + ω cos ψ j tan ψ p cos k=p+1 j=1 n ψ j + ω cos ψ j tan ψ p+1 cos ϕp ϕ ψ j sin ψ j tan ψ p cos p 1 ϕp ϕp 1 ϕp+1 ϕ p ψ j ψ p sec ψ p 1 ϕp ϕ sin ψ j cos p 1 ψ j ψ p+1 sec ψ p 1 ϕp+1 ϕ sin ψ j cos p k=p+1 j=1 ψ j ϕp ϕ p 1 sin ψ j tan ψ p sin ϕp ϕp 1 ϕp+1 ϕ ψ p ϕp+1 ϕ j sin ψ j tan ψ p+1 sin p = τ ϕp for p = 1,,..., n. For small angles ϕ p 0, Eq. 58 reduces to 58 M ϕ ϕ + K ϕ ϕ = τ ϕ 59 where the mass matrix M ϕ is M ϕ = ms I n n 60 the quantity I n n is an identity matrix of order n, the stiffness matrix K ϕ is 1 + n k + n k n k 0 0 k=1 k= k= n k 1 + n k + n k n k 0 k= k= k=3 k=3 K ϕ = ms ω 0 n k 1 + n k + n k 0 k=3 k=3 k=4....... 0 0 0 1 + n k and the state vector ϕ and the torque vector τ ϕ are ϕ 1 ϕ ϕ = ϕ 3 ; τ ϕ =. ϕ n τ ϕ1 τ ϕ τ ϕ3. τ ϕn k=n 61 6 13

The second-order matrix equation of motion, Eq. 59, possesses a set of n natural frequencies. i i + 1 ω + 1, i = 1, 3, 5,..., n 1 63 In summary, natural frequencies of linear models for a spinning membrane are shown in Table 1. The first in-plane mode is the rigid-body motion with zero frequency. Only in-plane motion will experience Coriolis effect. Table 1. Natural Frequencies of a Spinning Membrane Motion Mode Hz n Formula In-Plane Out-Plane Twisting 1st 0 1 nd 5ω 3 3rd 14ω 5 1st ω 1 nd 6ω 3 3rd 15ω 5 1st ω 1 nd 7ω 3 3rd 16ω 5 ω n n +1 1 ω n n +1 ω n n +1 +1 NUMERICAL SIMULATION This section demonstrates angular responses of in-plane, out-plane, and twisting motions. Numerical simulations along with 3D animations are provided to show the nonlinear and linear behavior of the spinning membrane. Baseline design parameters A six-mass discrete inextensible spinning membrane was examined. Figure 6 shows the membrane model used for the following simulations and animations. Every two discrete mass particles form a mass set locating at the edges of the membrane chord. The following model is a spinning membrane with three discrete mass sets. For simplicity, each mass set has an identical distance, i.e., r 1 = r = r 3 = r. The membrane has zero pitch at the root. It rotates about the z axis at the center of the root and perpendicular to the membrane. The rotating speed is assumed to be a constant ω. The baseline parameters of the spinning membrane are summarized in Table. We use wx- Maxima 11.04.0 to derive the equations of motion, which are second-order ordinary differential equations. Transfer them into Scilab 5.3 to do numerical simulations, and then make 3D animations to show three types of motions, in-plane, out-plane, and twisting motion. 14

Figure 6. Six-mass spinning membrane Table. Spinning membrane baseline parameters and data Feature Dimension Each mass Number of discrete mass, N 6 Membrane length, L 00 m r = 00/6 Membrane chord width, c 0 m s = 0/ Mass of each blade, M 1.05 kg m = 1.05/6 Revolution period, T 180 s ω = π/180 To see the behavior of each type of motion, the six-mass model starts with initial angles only, without initial angular velocity. We numerically simulate the model up to five of the revolution period which is nine hundred seconds. The angular responses are provided to see the behavior of the membrane. 3D animations were made to study the linear and nonlinear behavior of the membrane. Pure in-plane motion Pure in-plane motion is examined. The membrane model is given in-plane initial angles only, i.e., θ 1 = 5 o, θ = o, θ 3 = 4 o. Figure 7 shows the angular responses of the three mass sets. The abbreviation revs means revolutions with 180 seconds per revolution See Table. The blue solid curves represent the nonlinear responses, while the red dotted curves illustrate the linear angle responses. The nonlinear terms associated with the in-plane angles θ 1, θ, and θ 3 have Coriolis effects due to the reference coordinate fixed at the rotating frame. In Fig. 7, the mean value of the blue curve drifts downward because the spinning speed is positive. On the other hand, if the spinning speed is negative, the mean value of the nonlinear response will drift upward. For this reason, we observed the relative responses of the in-plane motion. Figure 8 shows the relative responses of the in-plane motion including the angular response of the mass set relative to the mass set 1 and the angle response of mass set 3 relative to mass set 1. In order to avoid the Coriolis effect for comparison 15

Figure 7. Angular responses of the in-plane motion: nonlinear blue solid line, linear red dotted line; 5 revs 900 seconds. with linear-model response, we move the base frame to the first mass set to observe the rest of mass sets. The nonlinear responses without Coriolis effect are quite close to its linear response under small angle conditions. Pure out-plane motion The membrane model is given with out-plane initial angles only, i.e., φ 1 = 3 o, φ = 5 o, φ 3 = 4 o. Figure 9 shows the angular responses of the out-plane motion system, confirming that the linear model is a good approximation of the nonlinear model for small angles. There is no nonlinear coupling term associated with ω φ, thus no Coriolis effect exists. Pure twisting motion The membrane model is given with twisting initial angles only with ϕ 1 = 4 o, ϕ = 5 o, ϕ 3 = o. Figure 10 shows the angular responses of the twisting motion system, indicating that the nonlinear responses are almost identical to the linear responses for small-angle initial conditions. For larger-angle initial conditions such as ϕ 1 = 15 o, ϕ = 15 o, ϕ 3 = 1 o, the linear responses appear to apart in frequency from the nonlinear ones as shown in Fig. 11. However, both linear and nonlinear responses are quite similar in motion. Hence, one may consider to use the linear model to approximate its nonlinear version. Three-dimensional 3D animations are made to demonstrate the responses for certain initial conditions such as only angular displacements, both angular displacements and velocities, and/or input torques applied to the system. The purpose of these animations are to gain insights how the linear and nonlinear models behave. Frequencies of a nonlinear model are time varying in nature. It is observed that nonlinearities tend to lower time-varying frequencies and amplitudes. 16

Figure 8. Relative angular responses of in-plane motion: nonlinear blue solid line, linear red dotted line; 5 revs 900 seconds Figure 9. Angular responses of out-plane motion: nonlinear blue solid line, linear red dotted line; 5 revs 900 seconds 17

Figure 10. Angular responses of twisting motion: nonlinear blue solid line, linear red dotted line; 5 revs 900 seconds Figure 11. Angular responses of twisting motion: nonlinear blue solid line, linear red dotted line; 5 revs 900 seconds 18

CONCLUDING REMARKS A derivation method is developed to study structural dynamics of a spinning membrane which is proposed to be a conceptual design of a solar sail to travel in space for continuous thrust application. The first step is construct a membrane model as a discrete set of arbitrary number of lumped masses. The second step is to derive nonlinear ordinary differential equations for pure in-plane, out-plane, and twisting motions. Lagranges equations are applied to derive highly coupled ordinary differential equations for in-plane, out-plane, and twisting motions for the spinning membrane. The third step is to linearize these equations to yield second-order matrix equations of motion for small angular displacements. These linear models are used to solve for natural frequencies to gain physical insight of the dynamic behavior of the spinning membrane. They are consistent with the frequencies derived by considering the spinning membrane as a helicopter operating in a space environment, that leads to a set of partial differential equations. The proposed approach allows us to perform linear and nonlinear structural dynamic analysis for feasibility studies of a solar sail architecture that uses long spinning membrane to produce thrust from solar radiation pressure. Future studies include system identification and control analyses based on the discretized linear and nonlinear models. REFERENCES 1 MacNeal, R. H., Structural Dynamics of the Heliogyro, NASA Contrator Report, NASA CR-1745, May 1971. Botter, T. B., Coverstone, V. L., and Burton, R. L., Structural Dynamics of Spin-Stablilized Solar Sails with Applications to UltraSail, Journal of Guidance, Control, and Dynamics, Vol. 31, No., March-April 008. 3 Juang, J-N. and Phan, M. Q., Identification and Control of Mechanical Systems, Cambridge University Press, New York, NY 10011-411, 001, ISBN 0-51-78355-0. 4 Juang, J.-N., Applied System Identification, Prentice Hall, Inc., Englewood Cliffs, New Jersey 0763, 1994, ISBN 0-13-07911-X. 19