Dynamic Analysis of Deployable Strut-Mechanism with Clearance Root-Joints for Spacial Flexible Solar Array

Abstract

With Kane dynamics theory, the multi-body dynamic models of deployable struts are established with considering the clearance root-joint and the deployable style and deployable process of the struts are studied. The dynamic performance of struts’ deployment is analyzed, such as geometric position, velocity, and acceleration, seeking the inherent regularity of each joint’s motion characteristics. The results show that root-joint’s clearance has almost no influence on the rotation angle and angular velocity of base strut, but to some extent effect on the angular acceleration of base strut. Amplitude of angular acceleration appears significant fluctuations in 0–2 s and stabilizes gradually, indicating the pin in the bush is in the state of constant contact and separation. The dynamic model of deployable struts considering root-joint’s clearance represents more realistically the dynamic characteristics of this system. The different hinges suffer different contact forces, and its impact how effects on the mechanism movement is different. Therefore, Analysis of the dynamic performance and design of deployable struts need to consider the impact of hinge clearance, which gain the motion characteristics of important parts during deploying process, providing guidance for future design of mechanisms and driving forces.

1 Introduction

Deployment mechanisms achieving folded or unfolded movement such as coilable mast, articulated mast, and pipe mast have been generally used in spacial flexible solar array. The truss which coilable mast becomes after deployment has obvious characteristics of nonlinearity; Deployment mechanism of articulated mast contains a lot of hinges, parts and more complicated folded mechanisms; and pipe mast has the relatively low stiffness and strength [1]. Multi-module deployable strut-mechanism of flexible solar array has been seen as a new type of deployable mechanism [2], having the advantages of low weight, high stiffness, low stowed volume, etc., which can be satisfied the requirements of accommodating other components such as flexible blankets and thin solar cells when stowed and providing the support and stiffness when deployed. In addition, the number of frame-modules can be flexibly adjusted according to the needs of different power.

The clearance joints of multi-module strut-mechanism have significant impact on constraints, number of degrees of freedom and topological structure of deployable system, which lead to the deviation between the actual movement and the ideal movement, increase dynamic stress of components, cause vibration and shock of assemblies, accelerate the abrasion of mechanism, and result in reduced accuracy of movement [3,4,5]. Therefore, carrying out dynamic analysis of deployable strut-mechanism with clearance joints is very important. Dynamic analysis of deployment and the influence law of joint clearance are the keys to master the movement characteristics of every joint and important components for multi-module deployable strut-mechanism, and the basis for optimization and the detailed design of deployment mechanism.

A.L. Adler from U.S. Air Force Research Laboratory conducted a preliminary dynamic analysis of the multi-module strut-mechanism, and analyzed the natural frequency and mode shapes under the action of gravity and different boundary conditions. In this paper, the result that the hinge clearance has a certain influence on the strut deployment process has been pointed out, but impact analysis has not been performed [6]. M. Eskenazi from American ABLE Engineering Research Institute used finite element method to analyze the natural frequency and deformation of the multi-module strut-mechanism with different number of modules and different deployment configurations [7]. However, the references [6, 7] has not been carried out the dynamic analysis of the deployable process of the strut-mechanism, and has not been considered the influence of hinge clearance on the dynamic characteristics for strut-mechanism with multiple degrees of freedom.

In this paper, the multi-module dynamic models of deployable struts are established with considering the clearance root-joint by Kane dynamics theory. The deployable mode and process of strut-mechanism are researched. The dynamic characteristics of every joint and important components for multi-module deployable strut-mechanism and the influence law of joint clearance are analyzed and simulated, which provide the basis for the design of deployment mechanism.

2 The Components of Multi-module Deployable Strut-Mechanism

Multi-module deployable strut-mechanism after unfolded becomes a large-area and modular configuration composed of several struts, shown in Fig. 1; then the flexible blanket unfolds (shown in Fig. 2) within each rectangular bay, which form a large-area, flexible solar array generating high power. The deployment mode is the same within each standard module for multi-module deployable strut-mechanism. Multi-module deployable strut-mechanism has the advantages of low weight, high stiffness, low stowed volume, etc. The standard number of modules can be increased or decreased according to the needs, which can be satisfied the power requirements of different spacecraft missions [5].

Fig. 1

Multi-module deployment of deployable strut-mechanism for solar array

Fig. 2

Module of deployable strut-mechanism for solar array

Figure 2 shows the state of the fully deployment for a single-module deployable strut-mechanism. Base-strut, end strut and side strut are the main bearing units of solar array, generally made of carbon fiber–epoxy composites. Side struts are composed of two struts. Four side struts, one base-strut, one end strut constitute one regular bay, which is one of the modules of multi-module deployable strut-mechanism. The connection joints of base-strut, end strut and side strut are called driven units or slave driven units. Driven units provide power by a motor. Slave driven units don’t have a motor, but contain the lock-buffer mechanism. The connection joints of two side struts are called intermediate hinge units.

3 The Dynamic Modeling of Deployable Strut-Mechanism Considering Clearance Root-Joint

• A. The model of clearance root-joint

The dynamic models of deployable mechanism considering clearance root-joint are established based on Hertz contact theory [8] and Kane method. The important step in building joint-clearance mechanical models is setting up the relationship between the contact force and deformation. The contact configuration between the hinge pin and the bearing bush for root-joint is shown in Fig. 3. Given that the elastic modulus of the hinge pin is $E_{1}$, Poisson's ratio is $\nu_{1}$, the radius is $R_{1}$; and the elastic modulus of the bearing bush is $E_{2}$, Poisson's ratio is $\nu_{2}$, the radius is $R_{2}$. Dubowsky improved the Hertz contact law and proposed the nonlinear relationship between normal contact force $F_{N}$ and deformation δ:

$$ \delta = \frac{{\left( {k_{1} + k_{2} } \right)}}{2a}\left[ {\ln \left( {\frac{{8a^{3} }}{{F_{N} R\left( {k_{1} + k_{2} } \right)}}} \right) + 1} \right]F_{N} . $$

(1)

Fig. 3

Area-contact model between two cylinders

Above formula R, k₁ and k₂ are separately:

$$ R = \frac{{R_{1} R_{2} }}{{R_{1} + R_{2} }},\;k_{1} = \frac{{1 - \nu_{1}^{2} }}{{\pi E_{1} }},\;k_{2} = \frac{{1 - \nu_{2}^{2} }}{{\pi E_{2} }}. $$

(2)

The formula (1) is dealt with nonlinearly and considers the damping effect. The expression of the contact force [9] can be written as

$$ F_{N} = K\delta^{n} + D\dot{\delta } $$

(3)

The above formula $K$ $k$ is the linear contact stiffness coefficient; D is the damping coefficient; $\dot{\delta }$ is the contact velocity. The exponent n is generally 1.5 for the metal contact.

According to Coulomb friction law, the friction of the hinge pin at the contact points $F_{{\text{T}}}$ is

$$ F_{{\text{T}}} = - \frac{{v_{{\text{T}}} }}{{\left| {v_{{\text{T}}} } \right|}}\mu F_{{\text{N}}} , $$

(4)

where $v_{T}$ is the tangential velocity of the hinge pin relative to the bearing bush at contact points; $\mu$ is the Coulomb friction coefficient.

The joint-clearance mechanical model by Hertz contact law and Coulomb friction law is shown in Fig. 4. O-xy coordinate system is established in the bearing bush. Given that the center coordinate of the hinge pin is ($x_{1} ,y_{1}$); The clearance between hinge pin and bearing bush is $c = R_{2} - R_{1}$; The contact deformation is $\delta = \sqrt {x_{1}^{2} + y_{1}^{2} } - c$. Therefore, the contact force and friction can be expressed as follows:

$$ \begin{array}{*{20}c} \begin{gathered} F_{{\text{N}}} = \left( {K\delta^{n} + D\dot{\delta }} \right)S(\delta ) \hfill \\ F_{{\text{T}}} = - \frac{{v_{{\text{T}}} }}{{\left| {v_{{\text{T}}} } \right|}}\mu F_{{\text{N}}} S(\delta ) \hfill \\ \end{gathered} & {S(\delta ) = \left\{ {\begin{array}{*{20}c} 1 & {\delta > 0} \\ 0 & {\delta \le 0} \\ \end{array} } \right.} \\ \end{array} . $$

(5)

Fig. 4

Mechanical model of revolute joint with clearance

• B. The dynamic models of deployable strut-mechanism based on Kane method

(1) The definition of the system of generalized coordinates

Unit vectors of three directions in Cartesian coordinates are indicated separately with $e_{1}^{(0)}$, $e_{2}^{(0)}$, $e_{3}^{(0)}$, as shown in Fig. 5. The degrees of freedom for the system are 6, which $x_{01} ,y_{01} ,y_{02} ,z_{02} ,\theta_{1} ,\theta_{2}$ are chosen as generalized coordinates.

Fig. 5

Sketch of strut-mechanism’s forces and moments

(2) The kinematic analysis of deployable strut-mechanism

In this paper, the clearance influence of the rotating hinge including AB strut rotating around A point and AF strut rotating around A point are separately considered, as shown in Fig. 6.

Fig. 6

AB strut and AF strut with clearance joint

The displacement, velocity and acceleration of each strut are expressed as functions of the generalized coordinates. In Fig. 6, the external forces contain inertia forces and moments of inertia. To obtain inertia force and moment of inertia, the relationships among the displacement, velocity, acceleration of each strut’s centroid and generalized coordinates should be determined first. Given that the displacement and angular displacement vectors of each strut’s centroid are separately expressed as ${\varvec{L}}_{i} ,{\varvec{\theta}}_{i} ,(i = 1,2,3,4,5,6)$; the velocity and angular velocity vectors of each strut’s centroid are separately expressed as ${\varvec{V}}_{ci} ,{\varvec{\omega}}_{i} ,(i = 1,2,3,4,5,6)$; the acceleration and angular acceleration vectors of each strut’s centroid are separately expressed as ${\varvec{a}}_{ci} ,{\varvec{\varepsilon}}_{i} ,(i = 1,2,3,4,5,6)$; and the moments of inertia of each strut around centroid are separately expressed as $J_{i} ,(i = 1,2,3,4,5,6)$.

• (a) The Calculation of the Strut’s Displacement ${\varvec{L}}_{i}$

  $$ \left\{ \begin{gathered} {\varvec{L}}_{1} = \left( {x_{01} + l_{1} /2\sin \theta_{1} } \right){\varvec{e}}_{1}^{(0)} + \left( {y_{01} + l_{1} /2\cos \theta_{1} } \right){\varvec{e}}_{2}^{(0)} \\ {\varvec{\theta}}_{1} = - \theta_{1} {\varvec{e}}_{3}^{(0)} \\ {\varvec{L}}_{2} = \left( {x_{01} + (l_{1} + l_{2} /2)\sin \theta_{1} } \right){\varvec{e}}_{1}^{(0)} + \left( {y_{01} + l_{2} /2\cos \theta_{1} } \right){\varvec{e}}_{2}^{(0)} \\ {\varvec{\theta}}_{2} = \theta_{1} {\varvec{e}}_{3}^{(0)} \\ {\varvec{L}}_{3} = \left( {x_{01} + (l_{1} + l_{2} )\sin \theta_{1} } \right){\varvec{e}}_{1}^{(0)} + \left( {y_{02} + \frac{{l_{3} }}{2\cos \theta }_{2} } \right){\varvec{e}}_{2}^{(0)} + \left( {z_{02} + l_{3} /2\sin \theta_{2} } \right){\varvec{e}}_{3}^{(0)} \\ {\varvec{\theta}}_{3} = \theta_{2} {\varvec{e}}_{1}^{(0)} \\ {\varvec{L}}_{4} = \left( {y_{02} + l_{4} /2\cos \theta_{2} } \right){\varvec{e}}_{2}^{(0)} + \left( {z_{02} + l_{4} /2\sin \theta_{2} } \right){\varvec{e}}_{3}^{(0)} \\ {\varvec{\theta}}_{4} = \theta_{2} {\varvec{e}}_{1}^{(0)} \\ {\varvec{L}}_{5} = \left( {x_{01} + l_{5} /2\sin \theta_{1} } \right){\varvec{e}}_{1}^{(0)} + \left( {y_{01} + l_{5} /2\cos \theta_{1} } \right){\varvec{e}}_{2}^{(0)} + \left( {z_{02} + l_{4} \sin \theta_{2} } \right){\varvec{e}}_{3}^{(0)} \\ {\varvec{\theta}}_{5} = \theta_{1} {\varvec{e}}_{3}^{(0)} \\ {\varvec{L}}_{6} = \left( {x_{01} + (l_{5} + l_{6} /2)\sin \theta_{1} } \right){\varvec{e}}_{1}^{(0)} + \left( {y_{01} + \frac{{l_{6} }}{2\cos \theta }_{1} } \right){\varvec{e}}_{2}^{(0)} + \left( {z_{02} + l_{4} \sin \theta_{2} } \right){\varvec{e}}_{3}^{(0)} \\ {\varvec{\theta}}_{6} = - \theta_{1} {\varvec{e}}_{3}^{(0)} \\ \end{gathered} \right. $$

Given that the generalized speed is derivative of generalized coordinate, so there are $u_{x01} = \dot{x}_{01}$, $u_{y01} = \dot{y}_{01}$, $u_{y02} = \dot{y}_{02}$, $u_{z02} = \dot{z}_{02}$, $u_{1} = \dot{\theta }_{1}$, $u_{2} = \dot{\theta }_{2}$.

By taking the derivative of ${\varvec{L}}_{i}$ with respect to time, the centroid velocity and the angular velocity of each strut are obtained as follows:

$$ \left\{ \begin{gathered} {\varvec{V}}_{ci} = \frac{{{\text{d}}{\varvec{L}}_{i} }}{{{\text{d}}t}} \hfill \\ {\varvec{\omega}}_{i} = \frac{{{\text{d}}{\varvec{\theta}}_{i} }}{{{\text{d}}t}} \hfill \\ \end{gathered} \right.\;\;(i = 1,2,3,4,5,6). $$

(6)

By taking the derivative of velocity and angular velocity with respect to time, the centroid acceleration and angular acceleration of each strut are obtained as follows:

$$ \left\{ \begin{gathered} {\varvec{a}}_{ci} = \frac{{{\text{d}}^{2} {\varvec{L}}_{i} }}{{{\text{d}}t^{2} }} = \frac{{{\text{d}}{\varvec{V}}_{ci} }}{{dt^{2} }} \hfill \\ {\varvec{\varepsilon}}_{ci} = \frac{{{\text{d}}^{2} {\varvec{\theta}}_{i} }}{{{\text{d}}t^{2} }} = \frac{{{\text{d}}{\varvec{\omega}}_{i} }}{{{\text{d}}t^{2} }} \hfill \\ \end{gathered} \right.\;\;(i = 1,2,3,4,5,6). $$

(7)

• (b) Active forces and inertia forces of struts

Due to the root-joint clearance, the hinge pin of AB and AF strut generates active forces from the bearing bush, which contain friction and contact forces. Therefore, the active forces effecting on AB strut include driven torque ${\varvec{M}}_{1}$, friction ${\varvec{F}}_{N1}$ and contact forces ${\varvec{F}}_{T1}$ of hinge pin; and the active forces effecting on AF strut include driven torque ${\varvec{M}}_{4}$, friction ${\varvec{F}}_{N2}$ and contact forces ${\varvec{F}}_{T2}$ of hinge pin. The radius of the bearing bush and hinge pin are separately defined as $r_{1} ,r_{2}$ for A rotating point of AB strut; and the radius of the bearing bush and hinge pin are separately defined as $r_{3} ,r_{4}$ for A rotating point of AF strut. Given that contact stiffness, stiffness and friction coefficient of the rotating hinge for AB strut are separately defined as $K_{1} ,D_{1} ,\mu_{1}$; and contact stiffness, stiffness and friction coefficient of the rotating hinge for AF strut are separately defined as $K_{2} ,D_{2} ,\mu_{2}$:

$$ \begin{aligned} \delta_{1} & = \sqrt {x_{01}^{2} + y_{01}^{2} } - (r_{1} - r_{2} ) \\ F_{N1} & = \left( {K_{1} \delta_{1} + D_{1} \dot{\delta }_{1} } \right)S(\delta_{1} )\\ & = \left\{ {K_{1} \left( {\sqrt {x_{01}^{2} + y_{01}^{2} } - (r_{1} - r_{2} )} \right) + D_{1} \frac{{x_{01} \dot{x}_{01} + y_{01} \dot{y}_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }}} \right\}S(\delta_{1} ) \\ F_{T1} & = - \frac{{v_{T1} }}{{\left| {v_{T1} } \right|}}\mu_{1} F_{N1} S(\delta_{1} ) \\ \delta_{2} & = \sqrt {y_{02}^{2} + z_{02}^{2} } - (r_{3} - r_{4} ) \\ F_{N2} & = \left( {K_{2} \delta_{2} + D_{2} \dot{\delta }_{2} } \right)S(\delta_{2} )\\ & = \left\{ {K_{2} \left( {\sqrt {y_{02}^{2} + z_{02}^{2} } - (r_{3} - r_{4} )} \right) + D_{2} \frac{{y_{02} \dot{y}_{02} + z_{02} \dot{z}_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }}} \right\}S(\delta_{2} ) \\ F_{T2} & = - \frac{{v_{T2} }}{{\left| {v_{T2} } \right|}}\mu_{2} F_{N2} S(\delta_{2} ). \\ \end{aligned} $$

Active torques are as follows:

$$ \begin{aligned} {\varvec{F}}_{N1} & = - F_{N1} \frac{{x_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }}{\varvec{e}}_{1}^{(0)} - F_{N1} \frac{{y_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }}{\varvec{e}}_{2}^{(0)} \\ {\varvec{M}}_{T1} & = - F_{T1} r_{2} {\varvec{e}}_{3}^{(0)} ,\;{\varvec{M}}_{1} = - M_{1} {\varvec{e}}_{3}^{(0)} \\ {\varvec{F}}_{N2} & = - F_{N2} \frac{{y_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }}{\varvec{e}}_{2}^{(0)} - F_{N2} \frac{{z_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }}{\varvec{e}}_{3}^{(0)} \\ {\varvec{M}}_{T2} & = F_{T2} r_{4} {\varvec{e}}_{1}^{(0)} ,\;{\varvec{M}}_{4} = M_{4} {\varvec{e}}_{1}^{(0)} . \\ \end{aligned} $$

Moments of inertia are as follows:

$$ {\varvec{F}}_{i}^{*} = - m_{i} {\varvec{a}}_{ci} ,{\varvec{M}}_{i}^{*} = - J_{i} {\varvec{\varepsilon}}_{i} \;(i = 1,2,3,4,5,6) $$

(8)

• (c) Generalized active forces and generalized inertial forces

The partial velocities constitute a transformation matrix, which are the partial differential of the centroid velocity and angular velocity with respect to generalized velocity. Then the matrix multiplying by force vector can obtain the generalized forces:

$$\begin{aligned} & {\varvec{U}}_{F} = \left( {\begin{array}{*{20}c} {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{1} }}} \\ {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{2} }}} \\ {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{x01} }}} \\ {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{y01} }}} \\ {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{y02} }}} \\ {\frac{{\partial {\varvec{V}}_{c1} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{V}}_{c2} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{V}}_{c3} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{V}}_{c4} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{V}}_{c5} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{V}}_{c6} }}{{\partial u_{z02} }}} \\ \end{array} } \right)\\ & {\varvec{U}}_{M} = \left( {\begin{array}{*{20}c} {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{1} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{1} }}} \\ {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{2} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{2} }}} \\ {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{x01} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{x01} }}} \\ {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{y01} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{y01} }}} \\ {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{y02} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{y02} }}} \\ {\frac{{\partial {\varvec{\omega}}_{1} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{\omega}}_{2} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{\omega}}_{3} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{\omega}}_{4} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{\omega}}_{5} }}{{\partial u_{z02} }}} & {\frac{{\partial {\varvec{\omega}}_{6} }}{{\partial u_{z02} }}} \\ \end{array} } \right)\end{aligned} $$

Generalized active forces are as follows:

$$\begin{aligned} {\varvec{F}} & = {\varvec{U}}_{F} [{\varvec{F}}_{N1} ,0,0,{\varvec{F}}_{N2} ,0,0]^{{\text{T}}}\\ &\quad + {\varvec{U}}_{M} [{\varvec{M}}_{1} + {\varvec{M}}_{T1} ,0,0,{\varvec{M}}_{4} + {\varvec{M}}_{T2} ,0,0]^{{\text{T}}}.\end{aligned} $$

(9)

Generalized inertial forces are as follows:

$$\begin{aligned} {\varvec{F}}^{*} & = {\varvec{U}}_{F} [{\varvec{F}}_{1}^{*} ,{\varvec{F}}_{2}^{*} ,{\varvec{F}}_{3}^{*} ,{\varvec{F}}_{4}^{*} ,{\varvec{F}}_{5}^{*} ,{\varvec{F}}_{6}^{*} ]^{{\text{T}}}\\ &\quad + {\varvec{U}}_{M} [{\varvec{M}}_{1}^{*} ,{\varvec{M}}_{2}^{*} ,{\varvec{M}}_{3}^{*} ,{\varvec{M}}_{4}^{*} ,{\varvec{M}}_{5}^{*} ,{\varvec{M}}_{6}^{*} ]^{{\text{T}}}.\end{aligned} $$

(10)

Dynamic equation of strut is represented as follows:

$$ {\varvec{F}} + {\varvec{F}}^{*} = 0 $$

(11)

Dynamic equations of struts are represented as follows:

$$ \begin{aligned} &\left( - m_{1} \frac{{l_{1} }}{2} - m_{2} \left( {l_{1} + \frac{{l_{2} }}{2}} \right) - m_{3} (l_{1} + l_{2} ) - m_{5} \frac{{l_{5} }}{2}\right.\\ &\quad\left.- m_{6} \left( {l_{5} + \frac{{l_{6} }}{2}} \right) \right)\cos \theta_{1} \dot{u}_{x01} + \left( {m_{1} \frac{{l_{1} }}{2} + m_{2} \frac{{l_{2} }}{2} + m_{5} \frac{{l_{5} }}{2} + m_{6} \frac{{l_{6} }}{2}} \right) \sin \theta_{1} \dot{u}_{y01} \hfill \\ &\quad- \left( {m_{1} \frac{{l_{1}^{2} }}{4} + m_{2} \frac{{l_{2}^{2} }}{4} + m_{5} \frac{{l_{5}^{2} }}{4} + m_{6} \frac{{l_{6}^{2} }}{4}} \right)\dot{u}_{1}\\ &\quad - \left( {m_{2} \left( {l_{1}^{2} + l_{1} l_{2} } \right) + m_{3} \left( {l_{1} + l_{2} } \right)^{2} + m_{6} \left( {l_{5}^{2} + l_{5} l_{6} } \right)} \right) \hfill \\ &\left( {\dot{u}_{1} \cos^{2} \theta_{1} - u_{1}^{2} \sin \theta_{1} \cos \theta_{1} } \right) - \left( {J_{1} + J_{2} + J_{5} + J_{6} } \right)\dot{u}_{1}\\ &\quad - \frac{{l_{1} }}{2}F_{N1} \left( {\cos \theta_{1} \frac{{x_{01} }} {{\sqrt {x_{01}^{2} + y_{01}^{2} } }} + \sin \theta_{1} \frac{{y_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }}} \right) + M_{1} + F_{T1} r_{2} = 0. \hfill \\ \end{aligned} $$

(12)

$$ - \left( {m_{3} + m_{4} } \right)\dot{u}_{y02} + \left( {m_{3} \frac{{l_{3} }}{2} + m_{4} \frac{{l_{4} }}{2}} \right)\left( {\dot{u}_{2} \sin \theta_{2} + u_{2}^{2} \cos \theta_{2} } \right) - F_{N2} \frac{{y_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }} = 0 $$

(13)

$$ \begin{gathered} \left( {m_{3} \frac{{l_{3} }}{2} + m_{4} \frac{{l_{4} }}{2}} \right)\sin \theta_{2} \dot{u}_{y02}\\ - \left( {m_{3} \frac{{l_{3} }}{2} + m_{4} \frac{{l_{4} }}{2} + m_{5} l_{4} + m_{6} l_{4} } \right)\cos \theta_{2} \dot{u}_{z02} \hfill \\ - \left( {m_{3} \frac{{l_{3}^{2} }}{4} + m_{4} \frac{{l_{4}^{2} }}{4}} \right)\dot{u}_{2}\\ - \left( {m_{5} + m_{6} } \right)l_{4}^{2} \left( {\dot{u}_{2} \cos^{2} \theta_{2} - u_{2}^{2} \sin \theta_{2} \cos \theta_{2} } \right)\\ - \left( {J_{3} + J_{4} } \right)\dot{u}_{2} \hfill \\ + \frac{{l_{4} }}{2}F_{N2} \left( {\sin \theta_{2} \frac{{y_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }} + \cos \theta_{2} \frac{{z_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }}} \right)\\ + M_{4} + F_{T2} r_{4} = 0. \hfill \\ \end{gathered} $$

(14)

$$ \begin{gathered} - \left( {m_{1} + m_{2} + m_{5} + m_{6} } \right)\dot{u}_{y01}\\ + \left( {m_{1} \frac{{l_{1} }}{2} + m_{2} \frac{{l_{2} }}{2} + m_{5} \frac{{l_{5} }}{2} + m_{6} \frac{{l_{6} }}{2}} \right)\left( {\dot{u}_{1} \sin \theta_{1} + u_{1}^{2} \cos \theta_{1} } \right) \hfill \\ - F_{N1} \frac{{y_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }} = 0. \hfill \\ \end{gathered} $$

(15)

$$ \begin{gathered} - \left( {m_{1} + m_{2} + m_{3} + m_{5} + m_{6} } \right)\dot{u}_{x01}\\ - \left( {m_{1} \frac{{l_{1} }}{2} + m_{2} \left( {l_{1} + \frac{{l_{2} }}{2}} \right) + m_{3} \left( {l_{1} + l_{2} } \right) + m_{5} \frac{{l_{5} }}{2} + m_{6} \left( {l_{5} + \frac{{l_{6} }}{2}} \right)} \right) \hfill \\ \left( {\dot{u}_{1} \cos \theta_{1} - u_{1}^{2} \sin \theta_{1} } \right) - F_{N1} \frac{{x_{01} }}{{\sqrt {x_{01}^{2} + y_{01}^{2} } }} = 0 \hfill \\ \end{gathered} $$

(16)

$$ - \left( {m_{3} + m_{4} + m_{5} + m_{6} } \right)\dot{u}_{z02} - \left( {m_{3} \frac{{l_{3} }}{2} + m_{4} \frac{{l_{4} }}{2} + m_{5} l_{4} + m_{6} l_{4} } \right)\left( {\dot{u}_{2} \cos \theta_{2} + u_{2}^{2} \sin \theta_{2} } \right) - F_{N2} \frac{{z_{02} }}{{\sqrt {y_{02}^{2} + z_{02}^{2} } }} = 0. $$

(17)

4 Results of Dynamic Analysis of Deployable Strut-Mechanism

According to the dynamic equations of deployable strut-mechanism, the numerical solutions of dynamic parameters are obtained by making use of the classic fourth-order Runge–Kutta method. In this case the input parameters used in calculations are shown in Table 1.

Table 1 Parameters of deployable strut-mechanism for dynamic analysis

• A. Dynamic analysis of deployable strut-mechanism without considering clearance root-joint

The dynamic responses of parameters of deployable strut-mechanism without considering clearance root-joint, such as the rotating angle of AB strut $\theta_{1}$, the rotating angle of AF strut $\theta_{2}$ are shown in Figs. 7 and 8.

• (a) Time-dependence of $\theta_{1} ,\dot{\theta }_{1} ,\ddot{\theta }_{1}$ for AB strut

Fig. 7

$\theta_{1} ,\dot{\theta }_{1} ,\ddot{\theta }_{1}$ of AB strut

Fig. 8

$\theta_{2} ,\dot{\theta }_{2} ,\ddot{\theta }_{2}$ of AF strut

• (b) Time-dependence of $\theta_{2} ,\dot{\theta }_{2} ,\ddot{\theta }_{2}$ for AF strut

Figures 7 and 8 show:

• (1) $\theta_{1} ,\dot{\theta }_{1} ,\ddot{\theta }_{1}$ and $\theta_{2} ,\dot{\theta }_{2} ,\ddot{\theta }_{2}$ increase with time. However, $\ddot{\theta }_{1}$ and $\ddot{\theta }_{2}$ increase slowly with time, the trend of which is not obvious;

• (2) When $M_{1} = 1.0,M_{4} = 0.3$, $\theta_{1}$ 和 $\theta_{2}$ can rotate 90° simultaneously after 13.5 s. And then, the angle of AB strut and AF strut is 90° in the xoz plane.

  • B. Dynamic analysis of deployable strut-mechanism with considering clearance root-joint

The dynamic responses of parameters of deployable strut-mechanism with considering clearance root-joint such as the rotating angle of AB strut $\theta_{1}$, the x direction of B point of AB strut, $x_{01}$, $y_{01}$, $F_{N1}$ and $F_{N2}$ are shown in Figures 9, 10, 11, 12, 13 and 14. The initial values of $x_{01}$, $y_{01}$ are separately 0–0.0001 m; and the initial values of $y_{02} ,z_{02}$ are separately − 0.0001 m, 0 m.

• (a) Time-dependence of $\theta_{1} ,\dot{\theta }_{1} ,\ddot{\theta }_{1}$

Fig. 9

$\theta_{1} ,\dot{\theta }_{1} ,\ddot{\theta }_{1}$ of AB strut

Fig. 10

$x_{01}$ of A point’s revolute joint with clearance on AB strut

Fig. 11

$y_{01}$ of A point’s revolute joint with clearance on AB strut

Fig. 12

$F_{N1}$ of A point’s revolute joint with clearance on AB strut

Fig. 13

$\theta_{2} ,\dot{\theta }_{2} ,\ddot{\theta }_{2}$ of AF strut

Fig. 14

T ${\mathrm{F}}_{\mathrm{N}2}$ $F_{N2}$ of A point’s revolute joint with clearance on AF strut

• (b) Time-dependence of $x_{01} ,y_{01}$

• (c) Time-dependence of $F_{N1}$

• (d) Time-dependence of $\theta_{2} ,\dot{\theta }_{2} ,\ddot{\theta }_{2}$

• (e) Time-dependence of $F_{N2}$

Figures 9, 10, 11, 12, 13 and 14 show:

• (1) Root-joint clearance has almost no influence on the time-dependence of ${\dot{\uptheta }}_{1}\mathrm{and}$ and $\theta_{2} ,\dot{\theta }_{2}$, but to some extent effect on the time-dependence of $\ddot{\theta }_{1}$ and $\ddot{\theta }_{2}$. Amplitude of angular acceleration appears significant fluctuations in 0–2 s and stabilizes gradually, indicating the hinge pin in the bearing bush is in the state of constant contact and separation;

• (2) In the interval of $\theta_{1}$ rotated 90°, $\left| {x_{01} } \right|$ appears the trend of increasing first and then decreasing. The maximum of $\left| {x_{01} } \right|$ is 0.00054 m. $\left| {y_{01} } \right|$ varies within 0.0001 m. When $\theta_{1} = 90^\circ$, $\left| {y_{01} } \right|$ appears the trend of slightly increasing;

• (3) $F_{N1}$ and $F_{N2}$ gradually stabilizes with time. The stabilized value of $F_{N1}$ is about 0.48 N, and the stabilized value of $F_{N2}$ is about 0.4 N;

• (4) The deployable time appears different for strut-mechanism when considering and without considering clearance root-joint. AF strut can rotate 90° after 9 s with considering clearance root-joint. In addition, AF strut can rotate 90° after 13.5 s without considering clearance root-joint. Thus, the existence of clearance root-joint will lead to the different rotating degrees of $\theta_{1}$ and $\theta_{2}$ at the same time.

5 Conclusions

With Kane dynamics theory, the multi-body dynamic models of deployable struts have been established with considering the clearance root-joints, and the deployable style and deployable process of the struts have been studied. The conclusions are as follows:

• (1) Root-joint clearance has almost no influence on the rotation angle and angular velocity of base-strut, but to some extent effect on the angular acceleration of base-strut. Amplitude of angular acceleration appears significant fluctuations in 0–2 s and stabilizes gradually, indicating the hinge pin in the bearing bush is in the state of constant contact and separation.

• (2) The absolute value of the displacement in the X direction for the hinge pin connected to AB side-strut first increases then decreases in the period of AB strut rotating 90 degrees, which maximum value is 0.0005 m; and the displacement in the Y direction appears substantially sinusoidal trend, which is in the range of [− 0.0001 m, 0.0001 m], illustrating that the hinge pin has certain impact on the bearing bush in the process of deployable strut-mechanism.

• (3) The value of the contact force $F_{N1}$ for AB side-strut is gradually stabilized in 0.45 N considering root-joint clearance; and the value of the contact force $F_{N2}$ for AF base-strut is gradually stabilized in 0.4 N. The reason why $F_{N1}$ is larger is that the AB side-strut needs to rotate BC side-strut, meaning the joint of AB strut generating greater impact load.

• (4) The dynamic models of deployable struts considering root-hinge clearance describe more actually the dynamic characteristics of system. The fluctuation of the angular acceleration and joint’s impact in the deployable process because of joint clearance are amplified along with long dimension of solar array, giving rise to the larger tip displacement of the solar array, which have adverse effect on the structure. To this end, the clearance should be controlled.

• (5) Based on the model of root-joint clearance, the driven moment should be analyzed to determine a reasonable ratio of ${\varvec{M}}_{1}$ and ${\varvec{M}}_{4}$.