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

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


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.


Structural dynamics Solar sail Flutter instability Heliogyro solarelasticity 



Blade width


Youngs modulus


Orthogonal unit vectors of the k th blade frame


Orthogonal unit vectors of the inertia frame


Orthogonal unit vectors of the hub frame


Torsional rigidity; GE/[2(1+)ν]


Hub moment of inertia

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

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


Integers for indexing mode shapes

Blade length


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


Solar radiation pressure; N/m2

\(\overline {p}\)

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


Translational displacements of the hub frame

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

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


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

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

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

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

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


Hub coordinates along the principal axes


Blade coordinates of the k th blade


Rotational angles of the hub frame


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


Poison ratio


Density per unit volume for blades; kg/m3



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


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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

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