Nonlinear dynamic characteristics and control of planar linear array deployable structures consisting of scissor-like elements with revolute clearance joint

Abstract

This paper comprehensively deals with the parametric effects of the joint clearance and friction coefficient on the dynamics of planar deployable structures consisting of scissor-like elements (SLEs). The dynamic model for scissor deployable structure is based on a comprehensive consideration of the symmetry and array characteristics of this mechanism and on a Lagrange method, which represents the motion equations. A modified nonlinear contact-force model is employed to evaluate the intrajoint contact force, and the incorporation of the friction effect between the inter-connecting bodies is included in this study. The total impact forces produced in the real mechanical joint are embedded into the dynamics and the differential equations of motion are solved numerically based on a set of initial conditions. The clearance size, angle velocity, and friction coefficient are analyzed and discussed separately. Using Poincaré map, the regular and irregular responses of the deployable mulitibody systems are observed. Next, a control scheme is evaluated to maintain a more stable behavior and continuous contact between the clearance joints. The controlled results are compared with those without control, concluding that some undesired effects caused by the clearance joints can be prevented or reduced, resulting in continuous contact at the clearance joint.

Keywords

control deployable structure dynamic characteristics Poincaré map revolute clearance joint

Introduction

Deployable structures are a special kind of unfolding/folding structures, and they can generally be distinguished into three stages: folded state, unfolding and folding motion state, and configuration maintenance state. In the folded state, the structure is small in size and easy to store and transport; in the state of deployment, the relative motion of each element under the driving force exhibits the motion characteristics of a mechanism; In the configuration maintenance state, the number of degrees of freedom is zero, and the deployable structure can be used as a bearing structure to support external loads, exhibiting the structural characteristics. In order to address such diverse states, the research on deployable structures involves many disciplines and fields, including mechanism theory, structural analysis, materials, control, manufacturing and others, comprising cross- and multi-disciplinary, integrated system engineering.

After the 1970s, in order to continuously enhance the basic capabilities of the aerospace industry as well as to achieve in-depth space science research and promote the comprehensive development of space technology and applications, many scholars and research institutions have conducted the research on deployable structures (Chen, 2003; Chen et al., 2015; You and Pellegrino, 1997). Space stations need large solar arrays to provide operational energy; deep space detectors need ultralight thin film solar sails to provide power; navigation satellites and other high-precision earth observation systems need large satellite antennas to improve resolution. The implementation of the above-mentioned space science and technology projects depends significantly on the support of large scale deployable structures. Therefore, deployable structures have evolved into a research hot spot in the field of aerospace engineering. Moreover, they have received great attention from other scientific fields, such as biological organisms (tiger beetle wing, webbed duck feet (Hachem et al., 2004, 2005), architecture (foldable bridge (Lederman et al., 2014), retractable domes (Gantes et al., 1997)), aviation (foldable drones (Floreano and Wood, 2015), morphing wings (Di Luca et al., 2017)), and other areas (Felton et al., 2014; Peng et al., 2019).

In terms of basic configuration, different types of basic units, such as scissor-like element (SLE) (Li et al., 2016; Puig et al., 2010; Sun et al., 2013; Tibert, 2002), parallelogram element (Li et al., 2011; Wang et al., 2015), Bricard (Chen et al., 2005), and Bennet (Song et al., 2013) linkage mechanisms can form deployable structures with different configurations through different array combinations, such as satellite array antenna, solar panels or radar antenna array (Zhang et al., 2016; Zhao et al., 2019b). Among them, the SLE unit, which is any “X-shaped” unit that allows only the two bars to rotate freely around the connection point, has high research value because of its large shrinkage ratio, reliable unfolding action, high accuracy, and good stiffness. Also, the endpoints of this unit can be hinged to the endpoints of other units to form a more complex linear array expandable structure.

Due to the wide application of deployable structures in space, many theoretical problems in mechanism theory and structural behavior have been exposed, among which clearance dynamics and control are the two most important ones. In a real mechanism, the clearance between the bars of the SLE is inevitable due to the friction and wear, bar deformation and manufacturing error, which will aggravate the contact and collision phenomenon at the connections, produce obvious contact force, and increase the reaction force of the kinematic pair by more than 10 times compared to that without clearance (Li et al., 2011). In a deployable structure, clearance at the joints is the source of contact and collision, resulting in the relative motion of the joints, and consequently the asynchrony and failure of unfolding motion. Especially in the field of precision machinery, such as high-speed machinery and intelligent robots, the influence of clearance on mechanism motion has become one of the important bottlenecks restricting the development of the industry. In the aerospace field, the Hubble Space Telescope experienced periodic slip-stick phenomena because the effects of joint clearance at the connection of solar panels and the temperature difference during transitions in and out of Earth’s shadow (Nurre et al., 1971). Moreover, in the aviation field, the new morphing wing aircraft shows some faults during deployment, such as poor synchronization and inability to deploy to a certain position (Sofla et al., 2010). On the other hand, the joint clearance in the robotics has been observed to decrease the positioning accuracy, cause impact between joint elements, and produce much noise. This has led Bu et al. (2012) to propose an improved criterion based on trajectory planning to avoid the detachment between joint elements with clearance.

To address these issues, many theoretical and experimental works related to joint clearance have been investigated. Up to now, the influence of the clearance properties, such as clearance size, number of clearance joints, clearance location, different computation parameters and constitutive law models on the dynamic response of multi-body mechanical systems have been quantified by various references (Chen et al., 2018; Miao et al., 2019; Tian et al., 2018). Miao et al. (2019) established the mathematical model of a two-degrees-of freedom space arm system with clearance joints to study its dynamics and motion stability during the unfolding and locking phases. Tian et al. (2018) comprehensively investigated the modeling, numerical solutions, and experimental methods in mechanical systems with clearances, and discussed some common parameters that affect the dynamic behavior of mechanisms with clearances, such as contact force model, lubrication, wear, and control. Based on Lagrangian formulation and the finite element method, Chen et al. (2018) developed a rigid-flexible coupled dynamic model associated with the 2-DOF nine-bar mechanism including the clearance and flexible link. The obtained results show that flexible bars can partly reduce the collision between the contacting bodies compared to the rigid bars. In addition, the contact laws suitable for different contact conditions were developed to describe the impact events (Ambrósio and Pombo, 2018; Dubowsky and Freudenstein, 1971; Erkaya and Uzmay, 2010; Flores et al., 2011; Machado et al., 2012; Yaqubi et al., 2016; Zhao et al., 2019a), and the effects of link flexibility (Li et al., 2013; Sun et al., 2015), joint lubrication (Erkaya, 2018; Tian et al., 2013; Zheng et al., 2016), and joint wear (Xiang et al., 2015) on the mechanical system including revolute clearance joint have also been considered. Sun et al. (2015) used the absolute nodal coordinate formulation (ANCF) to model the flexible multibody system and applied the component mode synthesis to reduce the size of the matrices. Tian et al. (2013) proposed a new methodology for the dynamic analysis of rigid-flexible multibody systems with ElastoHydroDynamic (EHD) lubricated cylindrical joints using the Natural Coordinate Formulation (NCF) and the 20-node hexahedral elements of ANCF. The main outcomes, validated by the commercial software ADINA, show that the bearing flexibility extends the lubricant distribution space and reduces the lubricant pressure. Xiang et al. (2015) employed a modified contact force model to evaluate the joint reaction force where the nonlinear stiffness coefficient varies with the deformation. Furthermore, the amount of wear in the joint was quantified using Archard’s wear model and a parametric study associated with the slider-crank was performed.

On the other hand, the existence of revolute joint clearance will lead to a high degree of non-linearity and complexity of deployable structure’s motion. Slightest changes in values of parameters, such as clearance size or initial angle, may result in irregular behavior of the mechanical system. Also, clearance will produce a large contact impact force at the joint, which will cause the movement of the unit mechanism to be out of step, and lead to large bending and torsional deformation of the bar. As a result, the structural unfolding process will fail. Muvengei et al. (2012a) evaluated the effects of joint clearance at differently location on the dynamics of a slider-crank mechanism without considering friction. Li et al. (2018) extended these insights to deployable structures considering clearance and friction. Flores et al. (2007) analyzed the dynamic behavior of a slide-crank mechanism and a four-bar mechanism with different clearance size using the Poincaré map, concluding that when the points on the Poincaré map are scattered, the motion between the contacting bodies is irregular, and transition between continuous contact, free flight mode and collision mode. However, when the bearing and journal experienced periodic motion, they maintained continuous contact motion. In view of this, some undesirable results of mechanical systems with clearance, such as local deformation, wear and tear, and decrease of the positional accuracy should be reduced using appropriate design, which can prevent contact collision at the joint for maintaining continuous contact of the connecting components. Recently, based on Pyragas method, Olyaei and Ghazavi (2012) introduced an extended delayed feedback control (EDFC) unit into a slide-crank mechanism for stabilizing an unstable periodic orbit, which has been proven to be effective for different deterministic clearance sizes. Afterwards, Wang et al. (2016) used a modified EDFC to stabilize the chaotic motion of flexible multibody systems with uncertain joint clearance.

The main emphasis of the present work is on the nonlinear dynamic characteristics and control of planar scissor deployable structure including revolute joint with clearance and friction. In a real joint, when the journal and bearing come into contact the local deformation takes place, resulting in the intrajoint impact forces which are normal to the collision plane at the contact zone. In addition, the impact force is proportional to the relative penetration velocity, and inversely proportional to the initial impact velocity, which is described using a modified nonlinear contact force model. The friction effects, based on the LuGre friction model, are also modeled. Afterwards, the normal and tangential forces are embedded into the dynamics of the deployable systems modeled by Lagrange equations, and the effects of the clearance size and friction coefficient on the dynamic response of deployable structures with clearance are analyzed and discussed using Poincaré maps. Subsequently, to avoid discontinuous joint forces, a control scheme for maintaining continuous contact at a revolute clearance joint is presented, resulting in more stable dynamic behavior and higher positional accuracy. In short, the dynamic behavior of deployable structures with clearance is very sensitive to the joint conditions, even slight change to the clearance size or friction coefficient will lead to a transition of the mechanism’s response from regular to irregular or vice versa. The necessity to control the dynamic response of a mechanism with clearance depends on the specific design and requirements.

Modeling of joint clearance and contact force in planar deployable structures

Kinematic description of scissor deployable structures with joint clearance

The linear array deployable structure consists of many substructures with the same geometrical properties, and each sub structure consists of a pair of bars which are connected by a pivot at the midpoint. Figure 1 shows the deployable structure composed of two scissor units, where joint A is considered as the imperfect joint and $M xy$ is the global coordinate system. Bar 1 and bar 2 can rotate around the pin B, and similarly, bars 3 and 4 rotate with each other at the pin E. Bar 1 is hinged at the fixed point M and $φ_{i}$ (i = 1, 2, 3, 4) represents the angle between bar i and axis x in the global coordinate system. Hinges $o_{1}$ and $o_{2}$ (similarly, $o_{3}$ and $o_{4}$ ) are the midpoint of bars 1 and 2, respectively.

Figure 1.

Deployable structure consists of two SLEs with clearance in joint A.

$R_{B}$ and $R_{J}$ are used to indicate the radius of the bearing and journal, respectively, and $c = R_{B} - R_{J}$ is the clearance value, as shown in the upper left of Figure 1. The journal and bearing can move relative to each other unconstrained due to the revolute clearance, which leads to randomness of the system. According to the size of the clearance circle, the motion of the shaft in the bearing can be divided into three different modes: the free flight, the impact, and the contact mode, as shown in the lower left of Figure 1.

The clearance exists in the real joint to ensure the relative movement between the connecting components, which is also the source of the impact forces resulting the friction and wear, uncertainty in motion position, and degradation of the system performance. A fundamental requirement of dynamic analysis of a deployable structure with clearance is to accurately determine whether the bearing boundaries and journal are in contact with each other. In Figure 2, points $P_{b}$ and $P_{j}$ represent the centers of bearing and journal, respectively, and $e_{bj}$ denotes the eccentricity vector connecting the centers of bearing and journal.

Figure 2.

Determination of contact state of deployable structure with clearance.

In Figure 2, $r_{j}^{P}$ and $r_{b}^{P}$ respectively represent the position vectors of the geometric center of journal and bearing in the global coordinate system, which can be defined in equation (1):

e_{bj} = r_{j}^{P} - r_{b}^{P}

(1)

The unit normal and tangential vectors at the point of collision is defined by

{\begin{matrix} n = \cos α i + \sin α j \\ t = - \sin α i + \cos α j \end{matrix}

(2)

where $α$ is the angle between $e_{bj}$ and the x-axis. The relative penetration between two contact bodies can be expressed as:

δ = e_{bj} - c,

(3)

where $e_{bj}$ is the modulus of vector $e_{bj}$ .

The relative indentation can be used to determine whether there is contact between the bearing and journal (Flores and Ambrósio, 2010), and the change between different contact states can be calculated by equation (4).

{\begin{matrix} δ (q_{t}) δ (q_{t - 1}) > 0, δ (q_{t}) < 0 free flight \\ δ (q_{t}) δ (q_{t - 1}) > 0, δ (q_{t}) > 0 continuous contact \\ δ (q_{t}) δ (q_{t - 1}) \leq 0, δ (q_{t}) \leq 0 from contact to seperation \\ δ (q_{t}) δ (q_{t - 1}) \leq 0, δ (q_{t}) \geq 0 from seperation to contact \end{matrix}

(4)

Figure 3 shows the position relationship when the journal and bearing in the clearance joint are about to contact.

Figure 3.

Contact points detection in the general clearance joint.

$Q_{b}$ and $Q_{j}$ are the candidate contact points when the contact bodies are about to impact. In the dynamic analysis of clearance mechanism, the potential contact points detection is the basis for calculating contact deformation and relative contact velocity, which determines the accuracy and precision of the calculation. For the general clearance joint, the positional relationship between potential contact points should meet the following geometric constraint conditions:

δ = r_{b}^{Q} - r_{j}^{Q},

(5)

{\begin{matrix} n_{j} \times n_{b} = 0 \\ δ \times n_{b} = 0 \end{matrix},

(6)

where $r_{b}^{Q}$ and $r_{j}^{Q}$ are the position vectors of potential contact points in the global coordinate systems, $n_{j}$ and $n_{b}$ are the normal vectors of the possible contact points (Li et al., 2018).

When the potential contact points are determined, then the contact penetration and relative contact velocity can be obtained. Figure 4 shows the position relationship between the bearing and the shaft during impact. For convenience of display, the second unit is not drawn, where the contact region is indicated in blue.

Figure 4.

Local deformation of deployable structure during impact.

The impact points are denoted as $Q_{b}$ and $Q_{j}$ , their position vectors in the global coordinates are represented as $r_{b}^{Q}$ and $r_{j}^{Q}$ . ${\overset{\cdot}{r}}_{b}^{Q}$ and ${\overset{\cdot}{r}}_{j}^{Q}$ are the velocity of the contact points, which can be obtained by differentiating the position vectors with respect to time:

{\begin{matrix} r_{b}^{Q} = r_{b} + R_{B} n \\ r_{j}^{Q} = r_{j} + A_{j} s_{j}^{P} + R_{J} n \end{matrix}

(7)

{\begin{matrix} {\overset{\cdot}{r}}_{b}^{Q} = {\overset{\cdot}{r}}_{b} + R_{B} \overset{\cdot}{n} \\ {\overset{\cdot}{r}}_{j}^{Q} = {\overset{\cdot}{r}}_{j} + {\overset{\cdot}{A}}_{j} s_{j}^{P} + R_{j} \overset{\cdot}{n} \\ \overset{\cdot}{n} = \overset{\cdot}{α} t \end{matrix}

(8)

where $r_{b}$ and $r_{j}$ are the geometric centers of joint A and bar 2, $s_{j}^{P}$ is the position vector of the center of the journal in the body-fixed coordinate $o_{1} x_{1} y_{1}$ . $A_{j}$ is the transformation matrix of coordinates $o_{1} x_{1} y_{1}$ and $M xy$ , and $\overset{\cdot}{n}$ is the derivatives of the vector $n$ with respect to time.

The relative velocity between the contact bodies is projected onto the collision plane yielding the relative normal velocity $v_{N}$ and relative tangential velocity $v_{t}$ .

{\begin{matrix} v_{N} = [{({\overset{\cdot}{r}}_{j}^{Q} - {\overset{\cdot}{r}}_{b}^{Q})}^{T} n] n \\ v_{t} = {({\overset{\cdot}{r}}_{j}^{Q} - {\overset{\cdot}{r}}_{b}^{Q})}^{T} - v_{N} \end{matrix},

(9)

Contact force model in revolute clearance joint

When the bearing and journal are in contact and experiencing relative motion, a contact force and a friction force develop which are applied to the contact plane as force constraints. It is one of the important tasks of dynamic analysis of a mechanical system with clearance to establish a suitable contact force model for describing the contact-impact phenomenon.

The Hertz theory is the basic equation for studying fatigue, friction, and the interaction between any contacting bodies (Machado et al., 2012). Based on this theory, Hertz (Gourgiotis et al., 2019; Hertz, 1881), Brändlein (Brändlein et al., 1998), and Goldsmith (Goldsmith, 1960) evaluated the relationship between the material properties and contact stress. However, they did not consider the relationship between the kinematic energy of the contacting bodies and the elastic potential energy and the dissipated energy during the contact event, resulting in a low calculation accuracy and a limited application range. Therefore, this issue has led some scholars to extend the dissipated energy into the Hertz model to make this model closer to the nature of the impact, thus developing some more accurate models, such as Hunt-Crossley model (Hunt and Crossley, 1975), Lee-Wang model (Lee and Wang, 1983), Lankarani-Nikravesh model (Lankarani and Nikravesh, 1990), Zhiying-Qishao model (Qing and Lu, 2006), Gonthier model (Gonthier et al., 2004), and Flores contact model (Flores et al., 2006; Flores and Ambrósio, 2004). Among them, the Hunt-Crossley, L-N, and Zhiying-Qishao models are limited by the recovery coefficient; the Lee-Wang and the Gonthier models are limited by the contact form; the Flores model is suitable for general mechanical collisions without obvious restriction, and performs well in solving stability. However, in the above contact laws the contact stiffness is calculated simply according to the shape and material properties of the contact bodies and taken as a constant. It is not consistent with the actual situation because the stiffness coefficient is related to the physical and geometrical properties of contact bodies and varies with the deformation (Li et al., 2016). Thus, in this paper, a modified contact force model, based on the improved elastic foundation model (Liu et al., 2007) and Flores model (Flores and Ambrósio, 2004), is employed to calculate the contact force in a revolute clearance joint, which extends the scope of Flores model and is applicable to the complicated contact conditions of the problem at hand.

The modified contact force model can be expressed as:

F_{N} = K_{g} δ^{n} [1 + \frac{8 (1 - c_{r})}{5 c_{r}} \frac{\overset{\cdot}{δ}}{{\overset{\cdot}{δ}}^{(-)}}]

(10)

K_{g} = \frac{1}{8} π E^{*} \sqrt{\frac{2 δ {(3 (R_{B} - R_{J}) + 2 δ)}^{2}}{{(R_{B} - R_{J} + δ)}^{3}}}

(11)

where $c_{r}$ is the coefficient of restitution, n is the nonlinear power exponent typically taken as 1.5. $\overset{\cdot}{δ}$ and ${\overset{\cdot}{δ}}^{(-)}$ are separately the magnitude of the relative penetration velocity and initial impact velocity, the latter is obtained at the end of the free flight mode and remains intact during contact. If $δ < 0, (at t_{i - 1})$ before impact and $δ \geq 0, (at t_{i})$ after the transition, the continuous contact mode has occurred within this time period, and the initial impact velocity ${\overset{\cdot}{δ}}^{(-)}$ should be updated. In addition, $K_{g}$ is the nonlinear stiffness coefficient related to the material property, clearance size and indentation of contact bodies, in which $E^{*}$ is the compound elastic modulus represented as:

\frac{1}{E^{*}} = \frac{1 - ν_{i}^{2}}{E_{i}} + \frac{1 - ν_{j}^{2}}{E_{j}},

(12)

where $ν$ and E are Poisson ration and Young modulus, respectively.

LuGre friction model

Friction in joints is inevitable in mechanical dynamics, which affects the efficiency and motion accuracy of the mechanical system (Hao et al., 2019). The LuGre friction model (Marques et al., 2019; Muvengei et al., 2012b; Tan et al., 2020; Xiang et al., 2019), based on the Dahl model, can capture the most of characteristics of friction in a test, such as viscous and slip phenomenon, stribeck effect, stiction and detached force (Chen et al., 2016; Pennestrì et al., 2016). This dynamic friction model can be described as

{\begin{matrix} F_{t} = μ F_{N} \\ μ = σ_{0} z + σ_{1} z^{.} + σ_{2} v_{t} \end{matrix},

(13)

z^{.} = \frac{dz}{dt} = v_{t} - \frac{σ_{0} | v_{t} |}{g (v_{t})} z,

(14)

g (v_{t}) = μ_{k} + (μ_{s} - μ_{k}) e^{- {| \frac{v_{t}}{v_{s}} |}^{2}},

(15)

where $F_{t}$ is friction force, and $F_{N}$ is the normal contact force obtained by the equation (10), $μ$ is the instantaneous coefficient of friction. z represents the microscopic average bristle deflection, $σ_{0}$ , $σ_{1}$ , and $σ_{2}$ are separately the bristle stiffness, the microscopic damping and the viscous friction coefficient. $v_{s}$ and $v_{t}$ are the Stribeck velocity and the tangential velocity between the connecting components. Additionally, $μ_{k}$ and $μ_{s}$ are the kinetic friction coefficient and static friction coefficient, respectively. The specific values of the above parameters can refer to Table 1.

Table 1.

LuGre friction model parameters identification.

Designation	Value
Bristle stiffness coefficient, $σ_{0}$	1.0 × 10⁵ N/m
Microscopic damping coefficient, $σ_{1}$	400 N s/m
Viscous friction coefficient, $σ_{2}$	0
Kinetic friction coefficient, $μ_{k}$	0.1
Static friction coefficient, $μ_{s}$	0.2
Stribeck velocity, $v_{s}$	0.001 m/s

After some further mathematical calculations, the magnitude of contact force and its orientation are obtained as:

{\begin{matrix} Q_{C} = \bar{K} δ^{n} [1 + \frac{8 (1 - c_{r})}{5 c_{r}} \frac{\overset{\cdot}{δ}}{{\overset{\cdot}{δ}}^{(-)}}]; γ = α + ε \\ ε = \tan^{- 1} (μ sign (v_{t})) \end{matrix},

(16)

where $ε$ is the angle between the contact force vector $Q_{C}$ and the normal direction $n$ , $α$ is the angle between the clearance vector and the axis $M x$ , while the relevant forces, directions and angles are shown in Figure 5. $\bar{K}$ is calculated by adding the resulting tangential element from equation (13) to the normal component of the contact force from equation (10), which can be expressed as:

\bar{K} = \sqrt{1 + μ^{2}} K_{g}

(17)

Figure 5.

Normal and tangential contact force components.

Dynamic model and control in planar deployable structure

Dynamic model of planar scissor deployable structure including clearance

The examined scissor deployable structure consists of four bars and a slider, in which bar 1 is the active bar and a constant angle velocity is applied. It has the clearance in joint A connecting bar 1 to the slide. Obviously, the revolute clearance introduces two additional DOFs to the mechanical system, which consist of the horizontal and vertical displacements of the center of the journal with respect to the bearing center. Thus, the general coordinates of the deployable structure are selected as follows: $φ_{1}, φ_{2}, y$ .

According to the Lagrange equation, the dynamic model of the scissor deployable structure with revolute clearance joint is given by:

\frac{d}{dt} (\frac{\partial T}{\partial {\overset{\cdot}{q}}_{j}}) - \frac{\partial T}{\partial q_{j}} + \frac{\partial U}{\partial q_{j}} = Q_{j} (j = y, 1, 2),

(18)

where T, U, $Q_{j}$ represent the kinetic energy, potential energy and general force of the system, respectively, while $q_{j}$ is the generalized coordinate.

The kinetic energy of the deployable structure can be expressed as:

T = \frac{1}{2} \sum_{i = 1}^{5} m_{i} v_{si}^{2} + \frac{1}{2} \sum_{i = 1}^{4} J_{i} {\overset{\cdot}{φ}}_{i}^{2}

(19)

From equation (19), the kinetic energy can be further written as:

\begin{matrix} T = \frac{1}{2} m_{1} l^{2} {\overset{\cdot}{φ}}_{1}^{2} + \frac{1}{2} J_{1} {\overset{\cdot}{φ}}_{1}^{2} + \frac{1}{2} m_{2} l^{2} {\overset{\cdot}{φ}}_{1}^{2} + \frac{1}{2} J_{2} {\overset{\cdot}{φ}}_{2}^{2} \\ + \frac{1}{2} m_{3} (4 l^{2} {\overset{\cdot}{φ}}_{1}^{2} + l^{2} {\overset{\cdot}{φ}}_{2}^{2} + 4 l^{2} {\overset{\cdot}{φ}}_{1} {\overset{\cdot}{φ}}_{2} \cos (φ_{1} - φ_{2})) + \frac{1}{2} J_{3} {\overset{\cdot}{φ}}_{3}^{2} \\ + \frac{1}{2} m_{4} (4 l^{2} {\overset{\cdot}{φ}}_{1}^{2} + l^{2} {\overset{\cdot}{φ}}_{2}^{2} + 4 l^{2} {\overset{\cdot}{φ}}_{1} {\overset{\cdot}{φ}}_{2} \cos (φ_{1} + φ_{2})) + \frac{1}{2} J_{4} {\overset{\cdot}{φ}}_{4}^{2} \\ + \frac{1}{2} m_{5} {\overset{\cdot}{y}}^{2} \end{matrix}

(20)

where $m_{i} (i = 1, 2, 3, 4)$ is the mass of bar $i$ , while $m_{5}$ is the mass of the slide. $J_{i}$ is the moment of inertia of each bar. l represents the distance between the center of mass of link i and the joint which connects link i and link $i - 1$ . $φ_{i}$ and y separately represent the rotation angle of each bar and the displacement of the slide. ${\overset{\cdot}{φ}}_{i}$ and $\overset{\cdot}{y}$ denote the angular speed of each bar and the velocity of the slide.

The potential energy of the scissor deployable structure with clearance is denoted as:

\begin{matrix} U = m_{1} gl \sin φ_{1} + m_{2} gl \sin φ_{1} \\ + m_{3} gl (\sin φ_{3} + \sin φ_{1} + \sin φ_{2}) \\ + m_{4} gl (\sin φ_{3} + \sin φ_{1} + \sin φ_{2}) + m_{5} gy \end{matrix}

(21)

The generalized force corresponding to the generalized coordinate $q_{j}$ can be expressed as:

Q_{j} = \sum_{i = 1}^{5} (F_{i} \cdot \frac{\partial r_{i}}{\partial q_{j}} + M_{i} \frac{\partial φ_{i}}{\partial q_{j}})

(22)

In equation (22), $F_{i}$ and $M_{i}$ are the resultants of external forces and moments acting on body i, respectively, while $r_{i}$ is the position vector at the point of external force.

The general forces can be expressed as:

{\begin{matrix} Q_{y} = - Q_{C} \cdot \sin ε \\ Q_{φ_{1}} = - l \cdot Q_{C} \cdot \sin (φ_{1} - ε) + M_{1} \\ Q_{φ_{2}} = l \cdot Q_{C} \cdot \sin (φ_{2} - ε) \end{matrix},

(23)

where $Q_{C}$ and $ε$ can be obtained from equation (16), and $M_{1}$ is the external moment applied to ensure that the deployable structure moves at a constant angular velocity.

As the deployable structure moves with a constant angular velocity $ω$ , its equation of motion is: $φ_{1} = ω \cdot t + φ_{0}$ , where $φ_{0}$ represents the initial angular position of bar 1. Then, substituting equations (20), (21), and (23) into equation (18), the second-order nonlinear differential equations (equation (24)) with initial conditions and variable coefficients are derived. Through MATLAB programming, Runge-Kutta method is adopted to calculate the differential equation.

{\begin{matrix} {\overset{\cdot\cdot}{φ}}_{2} = f_{1} (t, φ_{2}, y, {\overset{\cdot}{φ}}_{2}, \overset{\cdot}{y}) \\ \overset{\cdot\cdot}{y} = f_{2} (t, φ_{2}, y, {\overset{\cdot}{φ}}_{2}, \overset{\cdot}{y}) \end{matrix}

(24)

Integrating equation (24), the position and velocity in the next step can be obtained, and then substituting the results into equation (18), $M_{1}$ in the next step can be calculated. It should be noted that $M_{1}$ in the first step can be obtained using the initial conditions.

Control procedure of planar scissor deployable structure with clearance

In order to avoid discontinuous forces, reduce friction and have more stable dynamic behavior, the scissor deployable structure with clearance should maintain continuous contact in the joint. For this purpose, the continuous contact in the revolute clearance joint is considered as a system constraint and the necessary input torque for establishing this dynamical behavior is obtained. At the beginning, the bearing and journal are in contact, and under this constraint, they remain in contact throughout the motion. To this end, the following constraint is considered:

g = e^{2} - (R_{B} - R_{J})^{2} = 0,

(25)

where e is the distance between the center of the journal and the center of the bearing when they keep in contact. By expressing e using the generalized coordinates y, $φ_{1}$ and $φ_{2}$ , equation (25) becomes:

2 l^{2} + y^{2} + 2 yl (\sin φ_{2} - \sin φ_{1}) - 2 l^{2} \cos (φ_{1} - φ_{2}) - (R_{B} - R_{J})^{2} = 0

(26)

Since the journal and bearing remain in continuous contact, the intra-joint force acts as an internal force on the connecting bodies. Using the generalized coordinate y and angle $ε$ to represent the generalized force $Q_{C}$ , along with using the constraint equation (26), the governing equations for the scissor deployable structure with continuous contact can be derived:

- 4 m l^{2} {\overset{\cdot\cdot}{φ}}_{2} \cos (φ_{1} + φ_{2}) + 4 m l^{2} \sin (φ_{1} + φ_{2}) + 6 mgl \cos φ_{1} = l \cdot \frac{m_{5} g + m_{5} \overset{\cdot\cdot}{y}}{\sin ε} \cdot \sin (φ_{1} - ε) + M_{1}

(27)

(J_{2} + J_{4} + 2 m l^{2}) {\overset{\cdot\cdot}{φ}}_{2} + 4 m l^{2} {\overset{\cdot}{φ}}_{1}^{2} \sin (φ_{1} + φ_{2}) - 2 mgl \cos φ_{2} = - l \cdot \frac{m_{5} g + m_{5} \overset{\cdot\cdot}{y}}{\sin ε} \cdot \sin (φ_{2} + ε)

(28)

\begin{matrix} (2 y + 2 l (\sin φ_{2} - \sin φ_{1})) \overset{\cdot\cdot}{y} + 4 l \overset{\cdot}{y} (\cos φ_{2} \cdot {\overset{\cdot}{φ}}_{2} - \cos φ_{1} \cdot {\overset{\cdot}{φ}}_{1}) \\ + 2 {\overset{\cdot}{y}}^{2} + (2 ly \cos φ_{2} - 2 l^{2} \sin (φ_{1} - φ_{2})) \cdot {\overset{\cdot\cdot}{φ}}_{2} \\ + (- 2 ly \cos φ_{1} + 2 l^{2} \sin (φ_{1} - φ_{2})) \cdot {\overset{\cdot\cdot}{φ}}_{1} \\ + (2 ly \sin φ_{1} + 2 l^{2} \cos (φ_{1} - φ_{2})) \cdot {\overset{\cdot}{φ}}_{1}^{2} \\ + (- 2 ly \sin φ_{2} + 2 l^{2} \cos (φ_{1} - φ_{2})) \cdot {\overset{\cdot}{φ}}_{2}^{2} \\ - 4 l^{2} \cos (φ_{1} - φ_{2}) \cdot {\overset{\cdot}{φ}}_{1} {\overset{\cdot}{φ}}_{2} = 0 \end{matrix}

(29)

m is the mass of the bar, and $m = m_{1} = m_{2} = m_{3} = m_{4}$ . In the case that bar 1 in the deployable structure rotates with constant angular velocity, equations (28) and (29) are used to control the system for continuous contact, and equation (27) is used to obtain the required input torque for maintaining the desired motion.

Simulation and discussion

In this section, the deployable structure consisting of two SLEs is used as an illustrative example to demonstrate how a revolute clearance joint affect the dynamics of a mechanism and validate the control procedure presented in the preceding sections, where joint A is a real joint, as depicted in Figure 1.

In Figure 1, each bar has the same length and mass, 2 m and 5.68 kg, respectively, and the mass of the slide is 5 kg. The initial deployment angle $φ_{1}$ is set as 30°, and the bar 1 rotates at a constant angular speed of 150 rpm. The journal and bearing centers concentric at the beginning of the simulation, and the initial conditions of the simulation can be obtained from the movement of the ideal mechanism. Also, the numerical parameters used in dynamic solution are listed in Table 2.

Table 2.

Numerical parameters used in the dynamic simulation.

Bearing radius	10.0 mm	Poisson’s ratio	0.29
Restitution coefficient	0.7	Density	7.8 g/cm³
Young’s modulus	206.8 GPa	Nonlinear power exponent	3/2

The steps of the computational methodology for dynamic analysis in planar deployable structure with clearance and friction are shown as follows:

Define the initial configuration of the system and the initial conditions for the simulation.

Define the model parameters of gap joint and the structural parameters of deployable structure, such as the bearing and journal radii, the restitution coefficient and material parameters.

Establish the kinematic model of the motion pair with clearance. If the contact between the bearing and journal takes place, evaluate the normal and tangential force using equations (10) and (13) respectively. Otherwise, continue to the motion equations of the system given in equation (18).

Integrate the dynamic model of the deployable structure with joint clearance and output the new positions and velocities of the mechanism in the next step.

Update the time, restart the calculation from step (3) with new joint parameters until the simulation is complete.

For the control process, equations (27)–(29) are substituted into kinematic model of the gap joint (step (3)) to control the bearing and journal for remaining in contact all the time, other processes are the same as steps (1), (2), (4), and (5).

Prior to obtaining the results concerning to the clearance effect, the dynamic simulation was performed to study the clearance influence in the global dynamics of the scissor deployable structure. At this stage, two different results associated with this structure are presented and discussed here. Firstly, the deployable structure was modeled considering all joints as ideal, where the bearing and journal fit perfectly with each other. Secondly, the mechanism was simulated with a revolute clearance joint between the slide and bar 1. The dynamic response of the scissor deployable structure with clearance is represented in Figure 6 by the time plots of the displacement, angle, velocity, angle velocity, acceleration and angle acceleration, and the contact force developed by the impact components and the driving torques at different constant angle velocity are represented in Figure 7. In all, these cases the clearance size is equal to 0.5 mm.

Figure 6.

Dynamic response of the scissor deployable structure with clearance. Response of the slider: (a), (b) and (c); Response of the center of mass of connecting bar 2: (d), (e), and (f).

Figure 7.

Contact force in clearance joint and input torque provided on bar 1 at different angle velocity: (a) contact impact force and (b) driving torque providing constant angle velocity.

From Figure 6(a) and (d), it is observed that the displacement and angle curves of the centers of mass of slide and bar 2 are affected by the existence of the joint clearance, but the difference from the ideal values is not discernable. Indeed, there is no significant deviation in the displacement and angle curves on the long time variable when the deployable structure is simulated with both the ideal and the clearance joints. In contrast to the displacement curves, the velocity and acceleration curves in Figure 6(b) and (c) present differences between the dynamic response of the deployable structure modeled with and without clearance, which is observed from the fluctuation of the curves within 0.005 s. The fluctuation and high frequency oscillation prove that the dynamic behavior of this system tends to be non-linear and is related to the contact between the two components in the clearance joint. Also, it propagates through the whole mechanism, which will lead to the vibration of the system. The same trend can also be seen in the curves of angular velocity and angle acceleration shown in Figure 6(e) and (f). Furthermore, Figure 7(a) illustrates that the mechanism with clearance bears discontinuous contact force, reducing its efficiency and lifetime, and a larger motor is needed to provide appropriate input torque to maintain bar 1 to rotate with constant angular velocity, depicted in Figure 7(b). Also, it can be seen that the magnitude of contact force and driving moment varies greatly with the increase of the angle velocity. When it increases from 100 rpm to 250 rpm, the max contact force increases five times. The reason is that the angular velocity of bar 1 implies the input energy for the system, so the larger the angular velocity, the greater the contact force and input torque.

The dynamic characteristics of the mechanism system are very sensitive to the clearance size, especially for deployable structures which have array characteristics. Because each bar hinges with other bars at its midpoint and endpoint respectively, and each bar has the same length, the motion of each bar is constrained by other bars simultaneously. At the same time, the movement of any bar will not only affect the movement of the SLE including this bar, but also affect the movement of other SLEs articulated with the bar at its end points. In view of this the dynamics of deployable structure present some unique characteristics, which are different from the general linear mechanism, such as crank-slide mechanism. Thus, in order to understand the influence of the existing clearance on the dynamic behavior of deployable structures, the contact forces evaluated by the modified nonlinear model and input torques are illustrated in Figure 8, in which the clearance sizes are 0.6 mm, 0.4 mm, and 0.2 mm, respectively. Also, the trajectory of the journal inside the bearing boundaries, as well as the Poincaré maps, are plotted in Figures 9 and 10, in which the slider velocity and the slider acceleration are chosen to plot the Poincaré maps and the range of the clearance used in the present work is 0.7 mm to 0.2 mm spaced at 0.1 mm.

Figure 8.

Contact force and input torque at different clearance size in joint A: (a) contact impact force and (b) driving torque providing constant angle velocity.

Figure 9.

Journal center orbit for different clearance sizes: (a) c = 0.7 mm, (b) c = 0.6 mm, (c) c = 0.5 mm, (d) c = 0.4 mm, (e) c = 0.3 mm, and (f) c = 0.2 mm.

Figure 10.

Poincaré maps for different clearance sizes: (a) c = 0.7 mm, (b) c = 0.6 mm, (c) c = 0.5 mm, (d) c = 0.4 mm, (e) c = 0.3 mm, and (f) c = 0.2 mm.

Figure 8 shows the variation of contact force and input moment with clearance size where the constant angular velocity is 150 rpm. It can be seen that with the decrease of clearance size, the amplitude of contact force and driving moment in the clearance joint decreases gradually, but the number of contact collision and the impact frequency between the journal and the bearing increase gradually. It should be highlighted that when the working time exceeds 0.03s, the magnitude of contact force and impact frequency between the two components with 0.2 mm clearance value are larger than those with 0.6 mm and 0.4 mm clearance values, which can be confirmed by the journal trajectory illustrated in the following Figure 9. The results prove that the dynamic characteristics of the deployable structure with clearance are very complex and different from those of a simple mechanism. In practical work, it is necessary to investigate the change of deployable structure relative to clearance size in order to design more stable mechanisms.

Figure 9 shows a nonlinear motion in the clearance joint between bar 2 and the slider, since the motion is unpredictable. The contact is very discontinuous and the free flight mode can be observed frequently in the revolute clearance joint. More importantly, the motion of the deployable structure is very sensitive to the clearance size due to its array characteristics. When the clearance size decreases, the contact collision frequency between bearing and journal is higher and the transition between the free flight mode and impact mode is faster. However, the collision frequency of the deployable structure does not fully decrease with the decrease of the clearance size exactly as we expected. On the contrary, when the clearance sizes are 0.3 mm and 0.2 mm, the contact frequency of the mechanism are higher than that of 0.4 mm, and the change trend can also be noticed from the Poincaré maps illustrated in Figure 10. Moreover, the dynamic behavior of this system tends to be regular when the clearance size is 0.4 mm since the journal and bearing are in continuous contact, as shown in Figure 9(d) where the free flight and impact motion do not exist. This behavior is expected because the dynamic response of the system repeats itself from cycle to cycle in Figure 10(d).

On the other hand, when the friction effect is included in the dynamic model of the system and the clearance size in the revolute joint is determined as 0.5 mm, the evolution of the kinematic friction coefficient is investigated using the journal center trajectory and Poincaré maps, where the four friction coefficients used for comparison are 0.01, 0.03, 0.05, and 0.1, respectively. Figure 11 shows the journal center trajectory relative to bearing, which exhibits complex profiles. Also, it is obvious that the contact location is mainly concentrated in a quarter of the area between the bearing and the journal. Moreover, in Figure 11(a), the rebounds in the clearance joint often take place after impacts instead of continuous contact. In contrast, the rebounds gradually diminish and eventually become continuous contact, as illustrated in Figure 11(b) to (d). It can be reasonably inferred that the dynamic response of the mechanism system tends to be simply regular with the kinematic friction coefficient increases, which can also be confirmed by Figure 12. For smaller friction coefficient, the response of the system is irregular, and the transition between different motion modes is very frequent. However, for larger coefficient, the response of the system tends to be regular. When the friction coefficient is 0.1, the Poincaré map is regular and the perturbation is very small relative to other friction coefficients.

Figure 11.

Journal center orbit for different friction coefficients: (a) u = 0.01, (b) u = 0.03, (c) u = 0.05, and (d) u = 0.1.

Figure 12.

Poincaré maps for different friction coefficients: (a) 0.01, (b) 0.03, (c) 0.05, and (d) 0.1.

In short, it can be concluded that the dynamics of the revolute clearance joint is quite sensitive to the friction coefficient and its increase leads to a better response of the system, that is, the motion of the mechanism becomes more predictable and regular.

From the above dynamic analysis, it was observed that the SLE deployable structure exhibits complex behavior due to the existence of the clearance, which results in more notable discontinuities in contact force and deteriorate the mechanism. In order to prevent the contact loss in the clearance joint to improve the performance of the mechanism, a control scheme presented in equations (27) and (28) is applied for maintaining the components in continuous contact. In the initial motion state of the mechanism, the bearing and the journal are in contact, and the mechanism is simulated for several values of clearance size, 0.7 mm, 0.5 mm, 0.3 mm, and 0.1 mm, as shown in Figure 13.

Figure 13.

Journal center orbit for different clearance sizes: (a) 0.7 mm, (b) 0.5 mm, (c) 0.3 mm, and (d) 0.1 mm.

In Figure 13, it can be seen that the bearing and journal are always in contact, and the continuous contact mode has been established for different clearance sizes. Since the free flight mode has vanished and no impact takes place in the clearance joint, fatigue and the caused noise will be reduced. Furthermore, it is predictable that a higher amount of input torque is needed to maintain continuous contact in mechanism with clearance joint. This cannot be always easily provided, because a larger input torque means a larger engine, which in practical engineering will be limited by the cost and size of the motor. Therefore, in the actual mechanism with joint clearance, we need to compare comprehensively the two situations with or without control, and accordingly choose the most suitable scheme to implement.

Conclusion

The nonlinear dynamic characteristics and control method of deployable structures consisting of SLEs with revolute clearance joints was presented in this work, where the dynamic model of the mechanism system was established using Lagrange equation. By means of a continuous contact force model, joint clearance has been introduced in the multibody system. When the contact is detected, a modified contact impact model, based on the improved elastic foundation model and Flores model, together with the LuGre friction model are used to evaluate the intra-joint forces produced by the contact deformation between the connecting components, where the stiffness coefficient is nonlinear and related to the clearance size and material properties of the contact bodies. Afterwards, a control procedure is employed to maintain continuous contact between bearing and journal in the clearance joint.

From the numerical simulations it can be concluded that deployable structures with clearance exhibit complex dynamic behavior and the effect of clearance on the dynamic behavior of the system cannot be ignored. Main parameters, such as the driving angle velocities, clearance value and kinematic friction coefficient were investigated. Regarding the driving angle velocities, high values cause a more deteriorated response compared with low driving velocity. On the other hand, the trajectory of the journal inside the bearing was used to denote the influence of the clearance size on the dynamics of the mechanism system, which is demonstrated through Poincaré maps. It can be concluded that the dynamic response of deployable structures with array characteristics is very sensitive to the clearance size. When the clearance value is reduced, the dynamic response changes from irregular to periodic to irregular and finally to quasi-periodic. When the friction effect is included, the degree of non-linearity of the multibody system increases, and the dynamic response changes from irregular to regular behavior with the increase of the kinematic friction coefficient. These results indicate that the dynamic behavior of the deployable structure with clearance varies greatly, even if only a small change of the initial parameters is introduced, it will affect the motion of the mechanism from periodic to irregular, or vice-verse. In view of this a control method is employed to maintain the continuous contact in the clearance joint for improving the performance of the mechanism and ensuring more stable dynamic behavior. It should be noted that some penalties in the clearance joints are expected because larger input torque is required to limit the movement of the journal within the bearing which may exceed the saturation limit of the actuator.

Summarizing, joint clearance can not be ignored in the dynamic analysis of the mechanism. Whether it is necessary to control the dynamic behavior of the mechanism needs to be determined according to the situation at hand. However, when the clearance is unavoidable and the control cost is high, the appropriate clearance size can be chosen according to the analysis results to ensure the optimum stable motion of the deployable structure. Finally, it should be noted that the flexibility of the connecting bars and the lubrication effect in the joints are not included in this work, but these phenomena may reduce the energy consumption of the system. Therefore, it is necessary to consider these factors in future work in order to more realistically model the dynamics of the deployable structure.

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 author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 51175422) and Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2019JQ-753).

ORCID iD

Bo Li

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