Advertisement

A Multi-Blade Model for Heliogyro Solar Sail Structural Dynamic Analysis

  • Jer-Nan JuangEmail author
  • Jerry E. Warren
  • Lucas G. Horta
  • William K. Wilkie
Article
  • 7 Downloads

Abstract

An analytical model is derived to study the structural dynamic stability behavior for the free-flying heliogyro solar sail consisting of a finite set of symmetric or evenly distributed thin blades on a flat surface. Key properties of the flutter instability are described in terms of the relationship among the eigenvalues and eigenvectors associated with a flutter instability frequency. Various simulation cases for an idealized single blade, fixed rotational speed heliogyro model, and a generalized multi-bladed, freely spinning heliogyro model are presented to show how the flutter instability frequencies change as a function of the solar radiation pressure and the size of the dynamic model. Convergence of the flutter instability frequency versus the system order is demonstrated.

Keywords

Structural dynamics Solar sail Flutter instability Heliogyro solarelasticity 

Nomenclature

c

Blade width

E

Youngs modulus

ekx,ekη,ekζ

Orthogonal unit vectors of the k th blade frame

eIx,eIy,eIz

Orthogonal unit vectors of the inertia frame

ehx,ehy,ehz

Orthogonal unit vectors of the hub frame

G

Torsional rigidity; GE/[2(1+)ν]

Ixx,Iyy,Izz

Hub moment of inertia

\(\bar {I}_{xx},\bar {I}_{yy},\bar {I}_{zz}\)

\(\bar {I}_{xx}=I_{xx}/m_{b}\ell ^{2},\bar {I}_{yy}=I_{yy}/m_{b}\ell ^{2},\bar {I}_{zz}=I_{zz}/m_{b}\ell ^{2}\)

i,j,k

Integers for indexing mode shapes

Blade length

m,mb

Blade density per unit length (kg/m), total mass of a blade (mb = m for a constant m)

\(m_{h},\bar {m}_{h}\)

Hub mass (kg), \(\bar {m}_{h}=m_{h}/m_{b}\)

p0

Solar radiation pressure; N/m2

\(\overline {p}\)

Non-dimensional p0; \(\overline {p}=(p_{0} c)/(m_{b} {\Omega }_{0}^{2})\)

Rx,Ry,Rz

Translational displacements of the hub frame

\(\overline {R}_{x}, \overline {R}_{y}, \overline {R}_{z}\)

\(\overline {R}_{x}=R_{x}/\ell , \overline {R}_{y}=R_{y}/\ell , \overline {R}_{z}=R_{z}/\ell \)

vk,wk,uk,ϕk

In-plane, out-of-plane, elongation elastic displacements, and twist angle of the k th blade

\(\overline {u}_{k}, \overline {v}_{k}, \overline {w}_{k}\)

\(\overline {u}_{k}=u_{k}/\ell , \overline {v}_{k}=v_{k}/\ell , \overline {w}_{k}=w_{k}/\ell \)

\({u}_{k}^{\prime }, {v}_{k}^{\prime }, {w}_{k}^{\prime }\)

\({u}_{k}^{\prime } = \partial u_{k}/\partial x, {v}_{k}^{\prime } = \partial v_{k}/\partial x, {w}_{k}^{\prime } = \partial w_{k}/ \partial x\)

x,y,z

Hub coordinates along the principal axes

xk,ξk,ηk

Blade coordinates of the k th blade

𝜗x,𝜗y,𝜗z

Rotational angles of the hub frame

Ω0

Nominal spin rate of heliogyro, rad/sec

\(\varphi _{u_{k}i}, \varphi _{v_{k}i}, \varphi _{w_{K}i},\varphi _{\phi _{k} i}\)

i th mode shape for elastic displacements of the k th blade

ν

Poison ratio

ρ

Density per unit volume for blades; kg/m3

Notes

Acknowledgments

This work was sponsored in part under contract (Sub-award Number C17-2B53-JJ) from the National Aeronautics and Space Administration.

References

  1. 1.
    MacNeal, R.H.: The heliogyro: an interplanetary flying machine, NASA contractor report CR 84460 (1967)Google Scholar
  2. 2.
    MacNeal, R.H., Hedgepeth, J.: Helicopters for interplanetary space flight. In: 34th National Forum of the American Helicopter Society, Washington, D.C. (1978)Google Scholar
  3. 3.
    Hodges, D.H., Dowell, E.H.: Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA TN D-7818 (1974)Google Scholar
  4. 4.
    Juang, J.-N., Warren, J.E., Horta, L.G., Wilkie, W.K.: Progress in NASA heliogyro solar sail structural dynamics and solarelastic stability research. In: The 4th International Symposium on Solar Sailing, 17Th-20Th January, Kyoto Research Park, Kyoto, Japan (2017)Google Scholar
  5. 5.
    Wilkie, W.K., Warren, J.E., Horta, L.G., Lyle, K.H., Juang, J.-N., Gibbs, S.C., Dowell, E.H., Guerrant, D.V., Lawrence, D.A.: Recent Advances in heliogyro solar sail structural dynamics, stability, and control research, AIAA SciTech 2015, Kissimee, Florida, January, 2015 (2015)Google Scholar
  6. 6.
    Wilkie, W.K., Warren, J.E., Horta, L.G., Juang, J.-N., Gibbs, S.C., Dowell, E.H., Guerrant, D.V., Lawrence, D.A.: Recent progress in heliogyro solar sail structural dynamics. Presented at the European Conference on Spacecraft Structures, Materials, and Environmental Testing, Braunschweig, Germany, April 1–4, 2014 (2014)Google Scholar
  7. 7.
    Wilkie, W.K., Warren, J.E., Juang, J.-N., Horta, L.G., Lyle, K.H., Littell, J.D., Bryant, R.G., Thomson, M. W., Walkemeyer, P. E., Guerrant, D. V., Lawrence, D. A., Gibbs, S. C., Dowell, E. H., Heaton, A. F.: Heliogyro Solar Sail Research at NASA. In: Macdonald, M (ed.) Advances in Solar Sailing, pp 631–650. Springer-Praxis, Berlin (2014)Google Scholar
  8. 8.
    Juang, J.-N., Hung, C.-H., Wilkie, W. K.: Dynamics of a spinning membrane, The Journal of the Astronautical Sciences, published online: 30 October 2015, vol. 60, No. 3-4, pp. 494–516; Special Issue: the Jer-Nan Juang Astrodynamics Symposium (2013).  https://doi.org/10.1007/s40295-015-0062-0
  9. 9.
    Gibbs, S.C., Dowell, E.H.: Solarelastic stability of the heliogyro. In: Macdonald, M (ed.) Advances in Solar Sailing, pp 661–665. Springer-Praxis, Berlin (2014)Google Scholar
  10. 10.
    Dowell, E.H.: Can solar sails flutter. AIAA J. 49, 1305–1307 (2011)CrossRefGoogle Scholar
  11. 11.
    Meirovitch, L., Juang, J.-N.: Dynamics of a gravity-gradient stabilized flexible spacecraft, NASA contrator report for research grant NGR 47-004-098, NASA CR-2456 (1974)Google Scholar
  12. 12.
    Meirovitch, L., Juang, J.: On the natural modes of oscillation of rotating flexible structure about nontrivial equilibrium. J. Spacecr. Rocket. 13, 33–44 (1976)CrossRefGoogle Scholar
  13. 13.
    Meirovitch, L., Juang, J.-N.: Effect of the mass center shift for force-free flexible spacecraft. AIAA J. 13, 1535–1536 (1975)CrossRefGoogle Scholar
  14. 14.
    Meirovitch, L.: Hybrid state equations of motion for flexible bodies in terms of quasi-coordinates. J. Guid. Dyn. Control 14, 1008–1013 (1991)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, S., Junkins, J.L.: Explicit generalization of lagrange’s equations for hybrid coordinate dynamical systems. J. Guid. Dyn. Control 15(6), 1443–1452 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Junkins, J.L., Kim, Y.: Introduction to dynamics and control of flexible structures, American institute of aeronautics and astronautics, Washington, DC, pp. 452 (1993)Google Scholar
  17. 17.
    Natori, M., Nemat-Nasser, S., Mitsugi, J.: Instability of a rotating blade subjected to solar radiation pressure. In: AIAA 30th Structures, Structural Dynamics and Materials Conference, p 1989 (1989)Google Scholar
  18. 18.
    Natori, M., Nenat-Nasser, S.: Application of a mixed variational approach to aeroelastic stability analysis of a nonuniform blade. J. Struct. Mech. l4, 5–31 (1986)CrossRefGoogle Scholar
  19. 19.
    Gibbs, S.C., Dowell, E.H.: Membrane paradox for solar sails. AIAA J. 52, 2904–2906 (2014)CrossRefGoogle Scholar
  20. 20.
    Dowell, E.: On asymptotic approximations to beam mode shapes. J. Appl. Mech. 5(12), 439 (1984)CrossRefGoogle Scholar

Copyright information

© This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection 2019

Authors and Affiliations

  • Jer-Nan Juang
    • 1
    • 2
    Email author
  • Jerry E. Warren
    • 3
  • Lucas G. Horta
    • 3
  • William K. Wilkie
    • 3
  1. 1.National Institute of AerospaceHamptonUSA
  2. 2.Department of Engineering Science, National Cheng Kung UniversityTainanTaiwan
  3. 3.Structural Dynamics BranchNASA Langley Research CenterHamptonUSA

Personalised recommendations