Dynamic modeling of two-dimensional-deployable linked panels in space

Abstract

The objective of the present study is to derive a dominant equation of our previously proposed two-dimensional-deployable linked panel structure in space, which expresses the relationship between the deployment motion and input moment by an actuator. This article presents a derivation sequence of the dominant equation by the general dynamic modeling of square panels and demonstrates its numerical calculation examples. This numerical experimentation may be useful for designing an actuator considering deployment time in space, especially for realization of a future Solar Power Satellite which consists of square solar panels.

Keywords

deployable structures link mechanism solar cell panels solar power satellite space structures structural dynamics

Introduction

This article presents a derivation sequence of a dominant equation of our previously proposed two-dimensional (2D)-deployable “linked panel unit” in space,^1,2 which can be folded and deployed as shown in Figure 1 and extended two-dimensionally as shown in Figure 2. The objective of the present study is to derive the dominant equation expressing the relationship between the deployment motion and input moment by an actuator, and to show its numerical calculation examples for study of deployable structures^3,4 and adaptive structures,^5–7 especially for realization of a future Solar Power Satellite (SPS) consisting of square solar panels. The SPS was proposed by Dr Peter Glaser⁸ in 1968 as a concept to construct huge solar cell panels in space and transmit electric energy to the earth as a solution to the energy problem. Around 1980, its feasibility was investigated in the United States and Japan^9,10 as an efficient all day-and-night power generator without influences of weather and climate. In the present study, the problem is how to estimate the input moment required for the deployment of the “linked panel unit” in space in consideration of the deployment time. To address this problem, the present study reveals a dynamic modeling method^11–14 of the square panels and demonstrates the numerical calculation of the dominant equation derived by the modeling. This numerical experimentation may be useful when we design an actuator in future. The derived dominant equation can also be used for studies of damping control and deceleration control of an actuator so as not to cause structural vibration on post-deployment.^15,16

Figure 1.

2D-deployable “linked panel unit,”¹ consisting of square panels: (a) folded, (b) transition, and (c) deployed.

Figure 2.

2D-extended “linked panel units”²: (a) folded. (b) transition, and (c) deployed.

This kind of investigation was originally started with a scissor structure consisting of one-dimensional (1D) bars.¹⁷ Therefore, the present dynamic modeling of the 2D panels of the “linked panel unit” shown in the next section may be seen as an expansion from the 1D bars of the scissors system. As a numerical calculation example of the dominant equation derived by the modeling, section “Numerical calculation examples” examines some cases of positions and magnitude of the input moments.

Derivation sequence of dominant equation of “linked panel unit” in space

Figure 3 illustrates the composition of the “linked panel unit” shown in Figure 1 with node designations and 2D array of the square panels in a global coordinate system. Panels (0, 0) and (1, 1) are designated as square ABCD, and panels (0, 1) and (1, 0) are designated as square DABC, which means that the square ABCD is rotated counterclockwise at 90°. In Figure 3, the nodes indicated by arrows, such as nodes A of panels (0, 1) and (1, 1), are linked by a rotational axis perpendicular to the surface of the panel.

Figure 3.

Composition of “linked panel unit” shown in Figure 1: node designations, 2D array of square panels, and global coordinate system. Linked nodes are indicated by arrows.

Figure 4 shows transformation processes of the “linked panel unit” shown in Figures 1 and 3 with self-balancing moments $\bar{M}$ generated by actuators for the deployment and fold, which are assumed to be acting on the linked nodes. The positions of the input moments $\bar{M}$ will be examined in the next section. The square panels are assumed to be rigid bodies composed of a surface, frame, and brace as shown in Figure 5, where the surface and line densities are represented as ρ_S (kg/m²), ρ_F (kg/m), and ρ_B (kg/m), respectively. In Figure 5, a half side-length of the square is represented as L (m), and angular acceleration around the center of panel’s inertia in a local coordinate system is expressed as ${\overset{\cdot\cdot}{θ}}_{(i, j)}$ for each panel (i, j) (i, j = 0, 1) shown in Figure 4.

Figure 4.

Transformation processes of the “linked panel unit” shown in Figures 1 and 3 in a global coordinate system, which is assumed to be rigid bodies and deployed by self-balancing moments $\bar{M}$ at any linked nodes: (a) folded, (b) transition, and (c) deployed.

Figure 5.

Parameters for each panel (i, j) (i, j = 0, 1) assumed to be composed of a surface, frame, and brace: ρ_S, surface density; ρ_F and ρ_B, line density; L, a half side-length of square; $\overset{\cdot\cdot}{θ}$ , angular acceleration in a local coordinate system.

Figure 6 shows a general dynamic modeling of each panel (i, j) (i, j = 0, 1) shown in Figure 4, where each rotational motion is described by the rotational variable $θ_{(i, j)}$ . In reference to Figure 4, panels (0, 1) and (1, 0) are rotated counterclockwise during the deployment as shown in Figure 6(a), and panels (0, 0) and (1, 1) are rotated clockwise during the deployment as shown in Figure 6(b). Figure 6 also shows input moments $\bar{M}$ , frictional moments M, and internal forces F_x and F_y, which are suffixed by the node designations and (i, j). Moment components of the internal forces F_x and F_y around the center of panel’s inertia is represented as F_r. These forces and moments are acting on the linked nodes, and those on unconnected nodes will be assigned zero.

Figure 6.

General dynamic modeling of each panel (i, j) (i, j = 0, 1) shown in Figure 4: Rotational motion of (a) panel (0, 1) and (1, 0) and (b) panel (0, 0) and (1, 1). θ, rotational variable in the local coordinate system; $\bar{M}$ , input moments; M, frictional moments; F, internal forces, which are suffixed by the node designations and (i, j).

Based on this dynamic modeling, the dominant equation of the “linked panel unit” in space is derived by the following four steps.

Position vectors

In reference to Figures 3 –6, position vectors of all nodes A to D of each panel (i, j) in the global coordinate system are expressed as equations (1)–(4), using the rotational variable $θ_{(i, j)}$ and a half side-length L of the square panels

A_{(i, j)} = (\begin{matrix} X_{(i, j)} + \sqrt{2} L \sin θ_{(i, j)} \\ Y_{(i, j)} + \sqrt{2} L \cos θ_{(i, j)} \\ Z_{(i, j)} \end{matrix})

(1)

B_{(i, j)} = (\begin{matrix} X_{(i, j)} + \sqrt{2} L \cos θ_{(i, j)} \\ Y_{(i, j)} + \sqrt{2} L \sin θ_{(i, j)} \\ Z_{(i, j)} \end{matrix})

(2)

C_{(i, j)} = (\begin{matrix} X_{(i, j)} - \sqrt{2} L \sin θ_{(i, j)} \\ Y_{(i, j)} + \sqrt{2} L \cos θ_{(i, j)} \\ Z_{(i, j)} \end{matrix})

(3)

D_{(i, j)} = (\begin{matrix} X_{(i, j)} - \sqrt{2} L \cos θ_{(i, j)} \\ Y_{(i, j)} - \sqrt{2} L \sin θ_{(i, j)} \\ Z_{(i, j)} \end{matrix})

(4)

where $X_{(i, j)}$ , $Y_{(i, j)}$ , and $Z_{(i, j)}$ represent components of a position vector of the center of panel (i, j) in the global coordinate system.

Equations of motions

In reference to Figures 4 –6, equations of motion of panel (i, j) in X- and Y-direction in the global coordinate system are expressed as equations (5) and (6), and equation of its rotational motion in the local coordinate system is expressed as equation (7), respectively

(E q_{X (i, j)})

\begin{array}{l} {}^{A}{F_{x (i, j)}} + {}^{B}{F_{x (i, j)}} + {}^{C}{F_{x (i, j)}} + {}^{D}{F_{x (i, j)}} \\ + {({}^{C}{\bar{F}}_{S (i, j)} + {}^{C}{F_{S (i, j)}}) - ({}^{A}{\bar{F}}_{S (i, j)} + {}^{A}{F_{S (i, j)}})} \cos θ_{(i, j)} \\ + {({}^{B}{\bar{F}}_{S (i, j)} + {}^{B}{F_{S (i, j)}}) - ({}^{D}{\bar{F}}_{S (i, j)} + {}^{D}{F_{S (i, j)}})} \sin θ_{(i, j)} \\ = m \frac{d^{2} X_{(i, j)}}{d t^{2}} \end{array}

(5)

(E q_{Y (i, j)})

\begin{array}{l} {}^{A}{F_{y (i, j)}} + {}^{B}{F_{y (i, j)}} + {}^{C}{F_{y (i, j)}} + {}^{D}{F_{y (i, j)}} \\ + {({}^{D}{\bar{F}}_{S (i, j)} + {}^{D}{F_{S (i, j)}}) - ({}^{B}{\bar{F}}_{S (i, j)} + {}^{B}{F_{S (i, j)}})} \cos θ_{(i, j)} \\ + {({}^{C}{\bar{F}}_{S (i, j)} + {}^{C}{F_{S (i, j)}}) - ({}^{A}{\bar{F}}_{S (i, j)} + {}^{A}{F_{S (i, j)}})} \sin θ_{(i, j)} \\ = m \frac{d^{2} Y_{(i, j)}}{d t^{2}} \end{array}

(6)

(E q_{M (i, j)})

\begin{array}{l} \sqrt{2} L ({}^{A}{F_{r (i, j)}} + {}^{B}{F_{r (i, j)}} + {}^{C}{F_{r (i, j)}} + {}^{D}{F_{r (i, j)}}) \\ + ({}^{A}{\bar{M}}_{S (i, j)} + {}^{A}{M_{S (i, j)}}) + ({}^{B}{\bar{M}}_{S (i, j)} + {}^{B}{M_{S (i, j)}}) \\ + ({}^{C}{\bar{M}}_{S (i, j)} + {}^{C}{M_{S (i, j)}}) + ({}^{D}{\bar{M}}_{S (i, j)} + {}^{D}{M_{S (i, j)}}) \\ = J \frac{d^{2} θ_{(i, j)}}{d t^{2}} \end{array}

(7)

where

m = 4 L^{2} ρ_{S} + 8 L ρ_{F} + 4 \sqrt{2} L ρ_{B}

(8)

J = \frac{8}{3} L^{3} (L ρ_{S} + 4 ρ_{F} + \sqrt{2} ρ_{B})

(9)

{}^{A}{F_{r (i, j)}} = {}^{A}{F_{x (i, j)}} \cos θ_{(i, j)} + {}^{A}{F_{y (i, j)}} \sin θ_{(i, j)}

(10)

{}^{B}{F_{r (i, j)}} = - {}^{B}{F_{x (i, j)}} \sin θ_{(i, j)} + {}^{B}{F_{y (i, j)}} \cos θ_{(i, j)}

(11)

{}^{C}{F_{r (i, j)}} = - {}^{C}{F_{x (i, j)}} \cos θ_{(i, j)} - {}^{C}{F_{y (i, j)}} \sin θ_{(i, j)}

(12)

{}^{D}{F_{r (i, j)}} = {}^{D}{F_{x (i, j)}} \sin θ_{(i, j)} - {}^{D}{F_{y (i, j)}} \cos θ_{(i, j)}

(13)

\begin{array}{l} {}^{A}{\bar{F}}_{S (i, j)} = S_{F} \cdot {}^{A}{\bar{M}}_{(i, j)}, {}^{A}{F_{S (i, j)}} = S_{F} \cdot {}^{A}{M_{(i, j)}} \\ {}^{B}{\bar{F}}_{S (i, j)} = S_{F} \cdot {}^{B}{\bar{M}}_{(i, j)}, {}^{B}{F_{S (i, j)}} = S_{F} \cdot {}^{B}{M_{(i, j)}} \\ {}^{C}{\bar{F}}_{S (i, j)} = S_{F} \cdot {}^{C}{\bar{M}}_{(i, j)}, {}^{C}{F_{S (i, j)}} = S_{F} \cdot {}^{C}{M_{(i, j)}} \\ {}^{D}{\bar{F}}_{S (i, j)} = S_{F} \cdot {}^{D}{\bar{M}}_{(i, j)}, {}^{D}{F_{S (i, j)}} = S_{F} \cdot {}^{D}{M_{(i, j)}} \end{array}

(14)

\begin{array}{l} {}^{A}{\bar{M}}_{S (i, j)} = S_{M} \cdot {}^{A}{\bar{M}}_{(i, j)}, {}^{A}{M_{S (i, j)}} = S_{M} \cdot {}^{A}{M_{(i, j)}} \\ {}^{B}{\bar{M}}_{S (i, j)} = S_{M} \cdot {}^{B}{\bar{M}}_{(i, j)}, {}^{B}{M_{S (i, j)}} = S_{M} \cdot {}^{B}{M_{(i, j)}} \\ {}^{C}{\bar{M}}_{S (i, j)} = S_{M} \cdot {}^{C}{\bar{M}}_{(i, j)}, {}^{C}{M_{S (i, j)}} = S_{M} \cdot {}^{C}{M_{(i, j)}} \\ {}^{D}{\bar{M}}_{S (i, j)} = S_{M} \cdot {}^{D}{\bar{M}}_{(i, j)}, {}^{D}{M_{S (i, j)}} = S_{M} \cdot {}^{D}{M_{(i, j)}} \end{array}

(15)

where

S_{F} = \frac{3 \sqrt{2} (L ρ_{S} + 2 ρ_{F} + \sqrt{2} ρ_{B})}{4 L (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B})}

(16)

S_{M} = \frac{L ρ_{S} + 4 ρ_{F} + \sqrt{2} ρ_{B}}{2 (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B})}

(17)

In this study, the frictional moment is expressed as equation (18) using a coefficient of rotational friction μ (N m s/rad) multiplied by relative angular velocity between the linked panels

{}^{A}{M_{(i, j)}} = {}^{B}{M_{(i, j)}} = {}^{C}{M_{(i, j)}} = {}^{D}{M_{(i, j)}} = - 2 μ \frac{d θ_{(i, j)}}{d t}

(18)

On the right-hand sides of equations (5) and (6), m is mass of the panel shown in Figure 5 as expressed in equation (8), which is multiplied by the acceleration of motion of the center of panel inertia, $X_{(i, j)}$ and $Y_{(i, j)}$ . On the right-hand sides of equation (7), J represents moment of inertia of the panel as shown in equation (9). The right-hand side of equation (7) is derived by adding the following right-hand sides of equations of rotational motion for each element (surface, frame, and brace), in reference to Figure 5

(Surface)

\begin{array}{l} 4 \int_{0}^{L} \int_{0}^{L} (x^{2} + y^{2}) \frac{d^{2} θ_{(i, j)}}{d t^{2}} ρ_{S} d x d y \\ = \frac{8}{3} L^{4} ρ_{S} \frac{d^{2} θ_{(i, j)}}{d t^{2}} \end{array}

(19)

(Frame)

\begin{array}{l} 4 \int_{0}^{L} (L^{2} + x^{2}) \frac{d^{2} θ_{(i, j)}}{d t^{2}} ρ_{F} d x \\ + 4 \int_{0}^{L} (L^{2} + y^{2}) \frac{d^{2} θ_{(i, j)}}{d t^{2}} ρ_{F} d y \\ = \frac{32}{3} L^{3} ρ_{F} \frac{d^{2} θ_{(i, j)}}{d t^{2}} \end{array}

(20)

(Brace)

4 \int_{0}^{\sqrt{2} L} w^{2} \frac{d^{2} θ_{(i, j)}}{d t^{2}} ρ_{B} d w = \frac{8 \sqrt{2}}{3} L^{3} ρ_{B} \frac{d^{2} θ_{(i, j)}}{d t^{2}}

(21)

On the left-hand sides of equations (5) and (6), the first four terms are the internal forces F acting on nodes A to D of panel (i, j) in X- and Y-direction as shown in Figure 6. The first four terms F_r on the left-hand side of equation (7) are the moment components of the internal forces F around the center of panel’s inertia, as expressed in equations (10)–(13). The other terms on the left-hand sides of equations (5)–(7) are the effects of the input moments $\bar{M}$ and frictional moments M at the corners of the panels, which are converted to substitute forces and moments acting on the panel’s inertia as shown in Figure 7 with equations (14)–(17). The following describes this substitute method using Figure 8.

Figure 7.

Conversion of input moments and frictional moments acting on the corners of the panels: (1) substitute forces and (2) substitute moments acting on the center of panel inertia. (a) panel (0, 1) and (1, 0) and (b) panel (0, 0) and (1, 1).

Figure 8.

(a) = (b) + (c): Acceleration of motion by (a) moment at a corner is divided into that by (b) force in a diagonal direction, and (c) moment around the center of panel inertia.

In the present method, acceleration of motion by a moment at a corner of the panel is divided into acceleration by a force in a diagonal direction and that by a moment around the center of the panel’s inertia, as shown in Figure 8. In reference to Figures 5 and 8(a), the equation of rotational motion by a moment ${\bar{M}}_{(i, j)}$ at a corner is expressed as equation (22), by adding equations (23)–(25) for each element

{\bar{M}}_{(i, j)} = \frac{16}{3} L^{3} (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B}) \frac{d^{2} φ}{d t^{2}}

(22)

where

(Surface)

\begin{array}{l} \int_{0}^{2 L} \int_{0}^{2 L} ({x^{'}}^{2} + {y^{'}}^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{S} d x^{'} d y^{'} \\ = \frac{32}{3} L^{4} ρ_{S} \frac{d^{2} φ}{d t^{2}} \end{array}

(23)

(Frame)

\begin{array}{l} \int_{0}^{2 L} (4 L^{2} + {x^{'}}^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{F} d x^{'} + \int_{0}^{2 L} {x^{'}}^{2} \frac{d^{2} φ}{d t^{2}} ρ_{F} d x^{'} \\ + \int_{0}^{2 L} (4 L^{2} + {y^{'}}^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{F} d y^{'} + \int_{0}^{2 L} {y^{'}}^{2} \frac{d^{2} φ}{d t^{2}} ρ_{F} d y^{'} \\ = \frac{80}{3} L^{3} ρ_{F} \frac{d^{2} φ}{d t^{2}} \end{array}

(24)

(Brace)

\begin{array}{l} 2 \int_{0}^{\sqrt{2} L} (2 L^{2} + {x^{″}}^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{B} d x^{″} \\ + \int_{0}^{2 \sqrt{2} L} {y^{″}}^{2} \frac{d^{2} φ}{d t^{2}} ρ_{B} d y^{″} \\ = \frac{32 \sqrt{2}}{3} L^{3} ρ_{B} \frac{d^{2} φ}{d t^{2}} \end{array}

(25)

From equation (22), angular acceleration of rotational angle φ by the moment ${\bar{M}}_{(i, j)}$ is expressed as follows

\frac{d^{2} φ}{d t^{2}} = \frac{3 {\bar{M}}_{(i, j)}}{16 L^{3} (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B})}

(26)

In reference to Figures 5 and 8(b), equation of motion by the substitute force ${\bar{F}}_{S (i, j)}$ in the diagonal direction is expressed as equation (27) using equation (26), by adding equations (28)–(30) which represent mass multiplied by the acceleration for each element

\begin{matrix} {\bar{F}}_{S (i, j)} = 4 \sqrt{2} L^{2} (L ρ_{S} + 2 ρ_{F} + \sqrt{2} ρ_{B}) \frac{d^{2} φ}{d t^{2}} \\ = \frac{3 \sqrt{2} (L ρ_{S} + 2 ρ_{F} + \sqrt{2} ρ_{B})}{4 L (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B})} {\bar{M}}_{(i, j)} \end{matrix}

(27)

where

(Surface)

4 L^{2} ρ_{S} \cdot \sqrt{2} L \frac{d^{2} φ}{d t^{2}}

(28)

(Frame)

8 L ρ_{F} \cdot \sqrt{2} L \frac{d^{2} φ}{d t^{2}}

(29)

(Brace)

4 \sqrt{2} L ρ_{B} \cdot \sqrt{2} L \frac{d^{2} φ}{d t^{2}}

(30)

On the other hand, in reference to Figures 5 and 8(c), equation of rotational motion by the substitute moment ${\bar{M}}_{S (i, j)}$ around the center of panel’s inertia is expressed as equation (31) using equation (26), by adding the right-hand sides of equations (32)–(34) which represent moment of inertia multiplied by the angular acceleration for each element

\begin{matrix} {\bar{M}}_{S (i, j)} = \frac{8}{3} L^{3} (L ρ_{S} + 4 ρ_{F} + \sqrt{2} ρ_{B}) \frac{d^{2} φ}{d t^{2}} \\ = \frac{L ρ_{S} + 4 ρ_{F} + \sqrt{2} ρ_{B}}{2 (2 L ρ_{S} + 5 ρ_{F} + 2 \sqrt{2} ρ_{B})} {\bar{M}}_{(i, j)} \end{matrix}

(31)

where

(Surface)

\begin{array}{l} 4 \int_{0}^{L} \int_{0}^{L} (x^{2} + y^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{S} d x d y \\ = \frac{8}{3} L^{4} ρ_{S} \frac{d^{2} φ}{d t^{2}} \end{array}

(32)

(Frame)

\begin{array}{l} 4 \int_{0}^{L} (L^{2} + x^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{F} d x \\ + 4 \int_{0}^{L} (L^{2} + y^{2}) \frac{d^{2} φ}{d t^{2}} ρ_{F} d y \\ = \frac{32}{3} L^{3} ρ_{F} \frac{d^{2} φ}{d t^{2}} \end{array}

(33)

(Brace)

4 \int_{0}^{\sqrt{2} L} w^{2} \frac{d^{2} φ}{d t^{2}} ρ_{B} d w = \frac{8 \sqrt{2}}{3} L^{3} ρ_{B} \frac{d^{2} φ}{d t^{2}}

(34)

In the same way as equations (27) and (31), equations (14)–(17) with Figure 7 are obtained as the conversion of all input moments and frictional moments acting on nodes A to D of any panel (i, j).

Relations between variables

In order to reduce many variables in equations (5)–(18) and to derive a single dominant equation, this step links the variables, based on the principle of action reaction and symmetricity of the analysis model.

Beforehand, we assign zero to the forces and moments on unconnected nodes as follows, in reference to Figures 3, 4, and 6

\begin{array}{l} {}^{A}{F_{x (0, 0)}} = {}^{A}{F_{y (0, 0)}} = 0, {}^{A}{\bar{M}}_{(0, 0)} = {}^{A}{M_{(0, 0)}} = 0 \\ {}^{A}{F_{x (1, 0)}} = {}^{A}{F_{y (1, 0)}} = 0, {}^{A}{\bar{M}}_{(1, 0)} = {}^{A}{M_{(1, 0)}} = 0 \end{array}

(35)

\begin{array}{l} {}^{B}{F_{x (0, 0)}} = {}^{B}{F_{y (0, 0)}} = 0, {}^{B}{\bar{M}}_{(0, 0)} = {}^{B}{M_{(0, 0)}} = 0 \\ {}^{B}{F_{x (0, 1)}} = {}^{B}{F_{y (0, 1)}} = 0, {}^{B}{\bar{M}}_{(0, 1)} = {}^{B}{M_{(0, 1)}} = 0 \end{array}

(36)

\begin{array}{l} {}^{C}{F_{x (0, 1)}} = {}^{C}{F_{y (0, 1)}} = 0, {}^{C}{\bar{M}}_{(0, 1)} = {}^{C}{M_{(0, 1)}} = 0 \\ {}^{C}{F_{x (1, 1)}} = {}^{C}{F_{y (1, 1)}} = 0, {}^{C}{\bar{M}}_{(1, 1)} = {}^{C}{M_{(1, 1)}} = 0 \end{array}

(37)

\begin{array}{l} {}^{D}{F_{x (1, 0)}} = {}^{D}{F_{y (1, 0)}} = 0, {}^{D}{\bar{M}}_{(1, 0)} = {}^{D}{M_{(1, 0)}} = 0 \\ {}^{D}{F_{x (1, 1)}} = {}^{D}{F_{y (1, 1)}} = 0, {}^{D}{\bar{M}}_{(1, 1)} = {}^{D}{M_{(1, 1)}} = 0 \end{array}

(38)

On the other hand, the forces and moments acting on the linked nodes are expressed as follows, based on the principle of action and reaction

\begin{array}{l} {}^{B}{F_{x (1, 1)}} = - {}^{B}{F_{x (1, 0)}}, {}^{B}{F_{y (1, 1)}} = - {}^{B}{F_{y (1, 0)}} \\ {}^{B}{\bar{M}}_{(1, 1)} = - {}^{B}{\bar{M}}_{(1, 0)}, {}^{B}{M_{(1, 1)}} = - {}^{B}{M_{(1, 0)}} \end{array}

(39)

\begin{array}{l} {}^{C}{F_{x (0, 0)}} = - {}^{C}{F_{x (1, 0)}}, {}^{C}{F_{y (0, 0)}} = - {}^{C}{F_{y (1, 0)}} \\ {}^{C}{\bar{M}}_{(0, 0)} = - {}^{C}{\bar{M}}_{(1, 0)}, {}^{C}{M_{(0, 0)}} = - {}^{C}{M_{(1, 0)}} \end{array}

(40)

\begin{array}{l} {}^{D}{F_{x (0, 1)}} = - {}^{D}{F_{x (0, 0)}}, {}^{D}{F_{y (0, 1)}} = - {}^{D}{F_{y (0, 0)}} \\ {}^{D}{\bar{M}}_{(0, 1)} = - {}^{D}{\bar{M}}_{(0, 0)}, {}^{D}{M_{(0, 1)}} = - {}^{D}{M_{(0, 0)}} \end{array}

(41)

\begin{array}{l} {}^{A}{F_{x (0, 1)}} = - {}^{A}{F_{x (1, 1)}}, {}^{A}{F_{y (0, 1)}} = - {}^{A}{F_{y (1, 1)}} \\ {}^{A}{\bar{M}}_{(0, 1)} = - {}^{A}{\bar{M}}_{(1, 1)}, {}^{A}{M_{(0, 1)}} = - {}^{A}{M_{(1, 1)}} \end{array}

(42)

Here, we take into account the symmetricity of the forces and moments with respect to X-axis as follows, in reference to Figures 3, 4, and 6

\begin{array}{l} {}^{A}{F_{x (1, 1)}} = {}^{C}{F_{x (1, 0)}}, {}^{A}{F_{y (1, 1)}} = - {}^{C}{F_{y (1, 0)}} \\ {}^{A}{\bar{M}}_{(1, 1)} = - {}^{C}{\bar{M}}_{(1, 0)}, {}^{A}{M_{(1, 1)}} = - {}^{C}{M_{(1, 0)}} \end{array}

(43)

We also take into account the symmetricity of the geometry of the analysis model with respect to X- and Y-axis as follows, using the X- and Y-components of position vectors of the center of panel (1, 0), $X_{(1, 0)}$ and $Y_{(1, 0)}$ , and its rotational variable $θ_{(1, 0)}$

\begin{array}{l} X_{(1, 1)} = X_{(1, 0)} \\ X_{(0, 1)} = X_{(0, 0)} = - X_{(1, 0)} \end{array}

(44)

\begin{array}{l} Y_{(0, 0)} = Y_{(1, 0)} \\ Y_{(0, 1)} = Y_{(1, 1)} = - Y_{(1, 0)} \end{array}

(45)

\begin{array}{l} θ_{(0, 1)} = θ_{(1, 0)} \\ θ_{(1, 1)} = θ_{(0, 0)} = - θ_{(1, 0)} \end{array}

(46)

Under the X- and Y-axis symmetricity, node C_(1,0) moves on the Y-axis in the global coordinate system as shown in Figure 4. Therefore, we can express $X_{(1, 0)}$ as follows using $θ_{(1, 0)}$ , from the X-component of equation (3) for node C_(1,0)

\begin{array}{l} C_{X (1, 0)} = X_{(1, 0)} - \sqrt{2} L \sin θ_{(1, 0)} = 0 \\ X_{(1, 0)} = \sqrt{2} L \sin θ_{(1, 0)} \end{array}

(47)

In the same way, node B_(1,0) moves on the X-axis under the X- and Y-axis symmetricity. Therefore, we can express $Y_{(1, 0)}$ as follows using $θ_{(1, 0)}$ , from the Y-component of equation (2) for node B_(1,0)

\begin{array}{l} B_{Y (1, 0)} = Y_{(1, 0)} + \sqrt{2} L \sin θ_{(1, 0)} = 0 \\ Y_{(1, 0)} = - \sqrt{2} L \sin θ_{(1, 0)} \end{array}

(48)

From equations (47) and (48), the second derivatives of $X_{(1, 0)}$ and $Y_{(1, 0)}$ are expressed as follows

\begin{array}{l} \frac{d^{2} X_{(1, 0)}}{d t^{2}} = - \frac{d^{2} Y_{(1, 0)}}{d t^{2}} = \sqrt{2} L {- \sin θ_{(1, 0)} {(\frac{d θ_{(1, 0)}}{d t})}^{2} \\ + \cos θ_{(1, 0)} \frac{d^{2} θ_{(1, 0)}}{d t^{2}}} \end{array}

(49)

Relational equation between input moment and rotational variable

Using the above equations (35)–(46), equations (5)–(7) with equations (8)–(18) are expressed as follows, using a formula manipulation system such as the Mathematica¹⁸ of Maple¹⁹

(E q_{X (0, 0)})

\begin{array}{l} - {}^{C}{F_{x (1, 0)}} + {}^{D}{F_{x (0, 0)}} - S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{D}{\bar{M}}_{(0, 0)} \sin θ_{(1, 0)} = - m \frac{d^{2} X_{(1, 0)}}{d t^{2}} \end{array}

(50)

(E q_{X (0, 1)})

\begin{array}{l} - {}^{C}{F_{x (1, 0)}} - {}^{D}{F_{x (0, 0)}} - S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{D}{\bar{M}}_{(0, 0)} \sin θ_{(1, 0)} = - m \frac{d^{2} X_{(1, 0)}}{d t^{2}} \end{array}

(51)

(E q_{X (1, 0)})

\begin{array}{l} {}^{B}{F_{x (1, 0)}} + {}^{C}{F_{x (1, 0)}} + S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{B}{\bar{M}}_{(1, 0)} \sin θ_{(1, 0)} = m \frac{d^{2} X_{(1, 0)}}{d t^{2}} \end{array}

(52)

(E q_{X (1, 1)})

\begin{array}{l} - {}^{B}{F_{x (1, 0)}} + {}^{C}{F_{x (1, 0)}} + S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{B}{\bar{M}}_{(1, 0)} \sin θ_{(1, 0)} = m \frac{d^{2} X_{(1, 0)}}{d t^{2}} \end{array}

(53)

(E q_{Y (0, 0)})

\begin{array}{l} - {}^{C}{F_{y (1, 0)}} + {}^{D}{F_{y (0, 0)}} + S_{F} \cdot {}^{D}{\bar{M}}_{(0, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \sin θ_{(1, 0)} = m \frac{d^{2} Y_{(1, 0)}}{d t^{2}} \end{array}

(54)

(E q_{Y (1, 0)})

\begin{array}{l} {}^{B}{F_{y (1, 0)}} + {}^{C}{F_{y (1, 0)}} - S_{F} \cdot {}^{B}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ + S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \sin θ_{(1, 0)} = m \frac{d^{2} Y_{(1, 0)}}{d t^{2}} \end{array}

(55)

(E q_{M (0, 0)})

\begin{array}{l} \sqrt{2} L {({}^{C}{F_{x (1, 0)}} - {}^{D}{F_{y (0, 0)}}) \cos θ_{(1, 0)} \\ - ({}^{D}{F_{x (0, 0)}} + {}^{C}{F_{y (1, 0)}}) \sin θ_{(1, 0)}} + 8 S_{M} \cdot μ \frac{d θ_{(1, 0)}}{d t} \\ - S_{M} \cdot {}^{C}{\bar{M}}_{(1, 0)} + S_{M} \cdot {}^{D}{\bar{M}}_{(0, 0)} = - J \frac{d^{2} θ_{(1, 0)}}{d t^{2}} \end{array}

(56)

(E q_{M (0, 1)})

\begin{array}{l} \sqrt{2} L {(- {}^{C}{F_{x (1, 0)}} + {}^{D}{F_{y (0, 0)}}) \cos θ_{(1, 0)} \\ - ({}^{D}{F_{x (0, 0)}} - {}^{C}{F_{y (1, 0)}}) \sin θ_{(1, 0)}} - 8 S_{M} \cdot μ \frac{d θ_{(1, 0)}}{d t} \\ + S_{M} \cdot {}^{C}{\bar{M}}_{(1, 0)} - S_{M} \cdot {}^{D}{\bar{M}}_{(0, 0)} = J \frac{d^{2} θ_{(1, 0)}}{d t^{2}} \end{array}

(57)

(E q_{M (1, 0)})

\begin{array}{l} \sqrt{2} L {(- {}^{C}{F_{x (1, 0)}} + {}^{B}{F_{y (1, 0)}}) \cos θ_{(1, 0)} \\ - ({}^{B}{F_{x (1, 0)}} + {}^{C}{F_{y (1, 0)}}) \sin θ_{(1, 0)}} - 8 S_{M} \cdot μ \frac{d θ_{(1, 0)}}{d t} \\ + S_{M} \cdot {}^{B}{\bar{M}}_{(1, 0)} + S_{M} \cdot {}^{C}{\bar{M}}_{(0, 0)} = J \frac{d^{2} θ_{(1, 0)}}{d t^{2}} \end{array}

(58)

(E q_{M (1, 1)})

\begin{array}{l} \sqrt{2} L {({}^{C}{F_{x (1, 0)}} - {}^{B}{F_{y (1, 0)}}) \cos θ_{(1, 0)} \\ - ({}^{B}{F_{x (1, 0)}} - {}^{C}{F_{y (1, 0)}}) \sin θ_{(1, 0)}} + 8 S_{M} \cdot μ \frac{d θ_{(1, 0)}}{d t} \\ - S_{M} \cdot {}^{B}{\bar{M}}_{(1, 0)} - S_{M} \cdot {}^{C}{\bar{M}}_{(1, 0)} = - J \frac{d^{2} θ_{(1, 0)}}{d t^{2}} \end{array}

(59)

From equations (50) and (51), we can derive the following equation

{}^{D}{F_{x (0, 0)}} = 0

(60)

In the same way, we can obtain the following equation from equations (52) and (53)

{}^{B}{F_{x (1, 0)}} = 0

(61)

Using equations (50)–(53), the following equation is derived

\begin{matrix} {}^{C}{F_{x (1, 0)}} = - \frac{S_{F}}{2} ({}^{B}{\bar{M}}_{(1, 0)} - {}^{D}{\bar{M}}_{(0, 0)}) \sin θ_{(1, 0)} \\ - S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} + m \frac{d^{2} X_{(1, 0)}}{d t^{2}} \end{matrix}

(62)

We can also derive the following equation from equations (54) and (55)

\begin{array}{l} {}^{D}{F_{y (0, 0)}} = - {}^{B}{F_{y (1, 0)}} + S_{F} \cdot {}^{B}{\bar{M}}_{(1, 0)} \cos θ_{(1, 0)} \\ - S_{F} \cdot {}^{C}{\bar{M}}_{(1, 0)} \sin θ_{(1, 0)} \\ - S_{F} \cdot {}^{D}{\bar{M}}_{(0, 0)} \cos θ_{(1, 0)} + 2 m \frac{d^{2} Y_{(1, 0)}}{d t^{2}} \end{array}

(63)

From equations (57) and (58) with equations (49) and (60)–(63), we finally obtain the single dominant equation as follows, which expresses the relationship between the input moments $\bar{M}$ and the rotational variable $θ_{(1, 0)}$

α_{(θ)} \frac{d^{2} θ_{(1, 0)}}{d t^{2}} + β_{(θ)} {(\frac{d θ_{(1, 0)}}{d t})}^{2} + γ \frac{d θ_{(1, 0)}}{d t} = δ_{(θ)}

(64)

where

α_{(θ)} = 8 m L^{2} \cos^{2} θ_{(1, 0)} + 2 J

(65)

β_{(θ)} = - 8 m L^{2} \cos θ_{(1, 0)} \sin θ_{(1, 0)}

(66)

γ = 16 S_{M} \cdot μ

(67)

\begin{array}{l} δ_{(θ)} = ({}^{B}{\bar{M}}_{(1, 0)} - {}^{D}{\bar{M}}_{(0, 0)}) \cdot (\sqrt{2} L S_{F} \cos^{2} θ_{(1, 0)} \\ + \sqrt{2} L S_{F} \cos θ_{(1, 0)} \sin θ_{(1, 0)} + S_{M}) \\ + 2 {}^{C}{\bar{M}}_{(1, 0)} (\sqrt{2} L S_{F} \cos^{2} θ_{(1, 0)} \\ - \sqrt{2} L S_{F} \cos θ_{(1, 0)} \sin θ_{(1, 0)} + S_{M}) \end{array}

(68)

Numerical calculation examples

This section presents numerical calculation examples of the dominant equation derived as in equation (64), using three cases of positions of the input moments as shown in Figure 9. In order to calculate the dominant equation numerically, we arrange the following equations for iterative calculation during the deployment and fold

\overset{\cdot\cdot}{θ} (t_{n}) = {δ_{(θ)} - β_{(θ)} \cdot \overset{\cdot}{θ} {(t_{n})}^{2} - γ \cdot \overset{\cdot}{θ} (t_{n})} \cdot α_{(θ)}^{- 1}

(69)

\overset{\cdot}{θ} (t_{n + 1}) = \overset{\cdot\cdot}{θ} (t_{n}) \cdot Δ t + \overset{\cdot}{θ} (t_{n})

(70)

θ (t_{n + 1}) = \overset{\cdot}{θ} (t_{n}) \cdot Δ t + θ (t_{n})

(71)

where

n = 0, 1, 2, \dots, Δ t = t_{n + 1} - t_{n}, t_{0} = 0

(72)

\overset{\cdot}{θ} (t_{0}) = 0, θ (t_{0}) = R

(73)

where $\overset{\cdot}{θ}$ and $\overset{\cdot\cdot}{θ}$ represent the first and second derivative of $θ$ ( $= θ_{(1, 0)}$ ) with respect to time t. The parameters in equations (64)–(73) with equations (8), (9), (16), and (17) are assumed to be as follows as an example: a half-length of the side of square L = 0.5 m, surface density of the panel ρ_S = 1 kg/m², line density of the frame and brace ρ_F = ρ_B = 1 kg/m, coefficient of rotational friction μ = 0, 0.1, and 0.2 N m s/rad, and time increment Δt = 0.01 s. In equation (73), parameter R represents an initial angle of θ, which is assumed to be as follows for deployment and fold: (deploy) R = 0° = 0 rad and (fold) R = 45° = π/4 ≈ 0.785 rad.

Figure 9.

Three cases of positions of input moments for deployment and fold.

In order to compare the three cases shown in Figure 9 under the same benchmark, the present study assumes the total amount of input moments ${\bar{M}}_{a l l}$ for deployment and fold as follows: (deploy) ${\bar{M}}_{a l l} = 1, \dots, 4$ (N m = kg m²/s²) and (fold) ${\bar{M}}_{a l l} = - 1, \dots, - 4$ (N m). Then, a percentage of ${\bar{M}}_{a l l}$ or zero is assigned to each input moment shown in equation (68) as follows for each case

Case 1

{}^{B}{\bar{M}}_{(1, 0)} = - 0.5 {\bar{M}}_{a l l}

(74)

{}^{D}{\bar{M}}_{(0, 0)} = - 0.5 {\bar{M}}_{a l l}

(75)

{}^{C}{\bar{M}}_{(1, 0)} = 0

(76)

Case 2

{}^{B}{\bar{M}}_{(1, 0)} = 0

(77)

{}^{D}{\bar{M}}_{(0, 0)} = 0

(78)

{}^{C}{\bar{M}}_{(1, 0)} = 0.5 {\bar{M}}_{a l l}

(79)

Case 3

{}^{B}{\bar{M}}_{(1, 0)} = 0.25 {\bar{M}}_{a l l}

(80)

{}^{D}{\bar{M}}_{(0, 0)} = - 0.25 {\bar{M}}_{a l l}

(81)

{}^{C}{\bar{M}}_{(1, 0)} = 0.25 {\bar{M}}_{a l l}

(82)

Figures 10 –12 show the numerical calculation result of time change of the acceleration, velocity, and angle of $θ$ ( $= θ_{(1, 0)}$ ) for each case during the deployment, and Figures 13 –15 show those of the fold. The vertical axes in Figures 10 –15(a)–(c) represent angular acceleration $\overset{\cdot\cdot}{θ}$ (rad/s²), angular velocity $\overset{\cdot}{θ}$ (rad/s), and angle $θ$ (rad), respectively, and the horizontal axes represent time t (s). In Figures 10 –12, the total input moment ${\bar{M}}_{a l l}$ for the deployment is changed from 1 to 4, and ${\bar{M}}_{a l l}$ for the fold is changed from −1 to −4 as shown in Figures 13 –15. In Figures 10 –15(a)–(c), the coefficient of rotational friction μ is changed from 0 to 0.2, where the tendency of the variation in the graphs is the same as that of ${\bar{M}}_{a l l} = 1 or - 1$ in the graph (a).

Figure 10.

(Cases 1, Deploy) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

Figure 11.

(Case 2, Deploy) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

Figure 12.

(Case 3, Deploy) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

Figure 13.

(Cases 1, Fold) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

Figure 14.

(Case 2, Fold) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

Figure 15.

(Case 3, Fold) Calculation result: (a) acceleration, (b) velocity, and (c) angle.

From these graphs, we can estimate the relationship between the magnitude of the input moments and the transformation time. Figures 10 –12 show that it takes about 5–6 s for the deployment by the input moment ${\bar{M}}_{a l l} = 1$ in all cases. On the other hand, fold time depends on the position of the input moments. The fold time shown in Figures 13 and 15 by the input moment ${\bar{M}}_{a l l} = - 1$ is roughly the same as the deployment time shown in Figures 10 and 12 by the input moment ${\bar{M}}_{a l l} = 1$ . However, Figures 14 shows that it takes about 8 s for the fold by the input moment ${\bar{M}}_{a l l} = - 1$ , which is obviously different from the deployment time shown in Figure 11 by the input moment ${\bar{M}}_{a l l} = 1$ . Therefore, we need to study the positions of the input moments. In any cases, the transformation time by ${\bar{M}}_{a l l} = 4$ or −4 becomes half that by ${\bar{M}}_{a l l} = 1$ or −1. This numerical experimentation may be useful for designing an actuator in consideration of the transformation time.

Figures 16 and 17 show the trajectory of motion of the “linked panel unit” during the deployment and fold, using the result of Case 3 and equations (1)–(4) with equations (44)–(48), which indicate that the transformation process is accelerated, respectively. These results may be useful when we design a lock mechanism to fix the structure in consideration of the impulse force generated by the acceleration and velocity at the end of the transformation shown in Figures 10 –15(a) and (b).

Figure 16.

(Case 3, Deploy) Trajectory of accelerated motion of “linked panel unit”: (a) each panel and (b) all panels.

Figure 17.

(Case 3, Fold) Trajectory of accelerated motion of “linked panel unit”: (a) each panel and (b) all panels.

Conclusion

This article presented a derivation sequence of the dominant equation of the “linked panel unit” in space by its general dynamic modeling and showed its numerical calculation examples. The derived dominant equation expresses the relationship between the deployment motion and the input moment by an actuator. In the present study, the problem is how to estimate the input moment required for the deployment of the “linked panel unit” in space in consideration of the deployment time. For this problem, this article presented a continuum modeling method of the square panels and examined some cases of the positions of the input moments, using the dominant equation derived by the modeling. This numerical experimentation may be useful for designing the actuator considering the deployment time, including a lock mechanism in consideration of the impulse force generated by the acceleration and velocity at the end of the deployment. The derived dominant equation can also be used for studies of damping control and deceleration control of an actuator so as to avoid structural vibration on post-deployment.^15,16 On the other hand, the validation of this theoretical method demands considerable physical experimentation, including final test in space. In our future research, we plan to produce a physical model with an extent of several meters so as to construct it in space eventually. Our future study also requires extension of the analysis model as shown in Figure 2 for larger space structures such as the SPS, which may be feasible by the general dynamic modeling shown in Figure 6.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the JSPS (Japan Society for the Promotion of Science) (grant-in-aid for JSPS research fellows: 26.5965).

References

Takatsuka

Deployable linked panel unit. JP Patent No. 4908660, Japan, 2012.

Takatsuka

2D-extension of deployable linked panel unit. Int J Space Struct 2014; 29(4): 171–179.

Pellegrino

Guest

(eds). IUTAM-IASS symposium on deployable structures: theory and applications. Dordrecht: Kluwer Academic Publishers, 2000.

Pellegrino

(ed.). Deployable structures. New York: Springer, 2001.

Miura

Furuya

Adaptive structure concept for future space applications. AIAA J 1988; 26(8): 995–1002.

Utku

Ramesh

Das

et al . Control of a slow-moving space crane as an adaptive structure. AIAA J 1991; 29(6): 961–967.

Akgün

Gantes

Sobek

et al . A novel adaptive spatial scissor-hinge structural mechanism for convertible roofs. Eng Struct 2011; 33(4): 1365–1376.

Glaser

PE.

Power from the sun: its future. Science 1968; 162(3856): 857–861.

DOE/NASA. Program assessment report statement of findings: satellite power systems concept development and evaluation program. Report no. DOE/ER-0085, November 1980. Washington, DC: US Department of Energy.

10.

ISAS. Report of institute of space and astronautical science, special issue no. 43. Report no. SPS 2000, 2001, http://www.isas.jaxa.jp/publications/hokokuSP/hokokuSP43/houkokuSP43.pdf#page=0

11.

Norton

RL.

Kinematics and dynamics of machinery (SI units). 1st ed. New York: McGraw-Hill Higher Education, 2008.

12.

Meriam

Kraige

LG.

Engineering mechanics: dynamics. 5th ed. Hoboken, NJ: John Wiley & Sons, 2003.

13.

Craig

JJ.

Introduction to robotics: mechanics and control. 2nd ed. Boston, MA: Addison-Wesley, 1989.

14.

Wie

Space vehicle dynamics and control. 2nd ed. Reston, VA: AIAA, 2008.

15.

Takatsuka

. Inverse dynamics for deceleration control of deployment of linked panel structure in space. In: Proceedings of the AIAA SciTech forum, San Diego, CA, 4 January 2016, pp. 1–12. Reston, VA: AIAA.

16.

Takatsuka

. Vibration theory for damping control of 2D-deployable linked panel structures in space. In: Proceedings of the AIAA SciTech forum, Grapevine, TX, 9 January 2017, pp. 1–14. Reston, VA: AIAA.

17.

Takatsuka

General dynamic modeling of a scissor structure for its deployment control in space. Int J Space Struct 2015; 30(3–4): 245–259.

18.

Wolfram

The Mathematica book. 5th ed. Champaign, IL: Wolfram Media, Inc., 2003.

19.

Fox

WP.

Mathematical modeling with Maple. 1st ed. Boston, MA: Cengage, 2011.