1 Introduction

The square unit and its combination mechanism have many advantages, such as reusability, easy to assemble, small size after folding, and covering a large area after being unfolded. Therefore, since the 1960s, it has been widely used in aerospace^[1-3], architecture^[4-5], biology^[6], and other fields, specifically in spatial stretching arms, deploying antenna reflectors, deploying domes, contracting stomachs treatment instrument, and so on.

The movement process of the square unit and its combined mechanism includes three phases: the start-up phase, the unfolding moving phase, the unfolding finish and locking phase. In the second phase, since the deployable mechanism is composed of a large number of intermediate motion subunits, thus motion pair clearance, created by frictional wear and manufacturing errors, will cause contact collisions between the motion auxiliary elements. Such resulting contact collision force affects movement synchronization and deployment accuracy between different components, and eventually reduces the available life of mechanism. Therefore, in order to ensure the smooth movement and reliable loading of the deployable mechanism, it is necessary to explore its kinematics precision and dynamic characteristics. The Gonthier contact force model and the modified Coulomb friction model were utilized by Li^[7] to analyze the contact collision problem of a scissor-type linear array deployable structure. Li ^[8] considered the contact collision and friction problems of the deployable mechanism during the movement deploying process, and comprehensively analyzed the influence of the position of the clearance joint, the clearance value, and the number of clearance on the dynamic characteristics of the scissor-developing mechanism. In addition, based on linear elastic analysis and substructure methods, Li^[9] studied the effect of unit number, deployment angle, and rod length on structural instability after completion of deployment, and the analysis results are conducive to reducing budgets and improving computational efficiency. Hu^[10] studied the influence of the joint clearance and the alternating hot load on the dynamic characteristics of large scale space-based space deployable structures during the unfolding movement.

In this paper, the elastic deformation of the square expandable mechanism member was not taken into account. The clearance was introduced into the movement pair of the mechanism. The normal contact force between pin and the pin hole of the joint was established by using the Flours contact force model^[11-13], and the tangential contact force between them was obtained from the LuGre friction model^[14-15]. Finally, by using the dynamic equations of the square combination mechanism, a dynamic model of the square combined mechanism containing the joint clearance was established. The Baumgarte stability constraint method was used to avoid the problem of constraint violation in the integration process^[16-17].

2 Dynamic Equations of Square Combination Mechanism 2.1 Square Combination Mechanism

The square unit and its combination mechanism are composed of four scissor units, which are hinged by a movement joint. The Cartesian coordinate system of the square unit combinational mechanism is established, as shown in Fig 1.

The bars numbers are identified by 0 (frame), 1 (slide), 2, 3, 4, 5, 6, 7, 8, and 9.

Motion pair numbers are identified by o, a, b, c, d, e, f, g, h, k, m, n, p, q. Among these pairs, only pair a is a moving pair^[18-19], and the other pairs are rotating pairs.

Without affecting the mathematical description of the square combination mechanisms' model, in order to facilitate the investigation, the motion pair g is locally enlarged, rod 6 and rod 7 are relatively rotated to each other around pin h. Rod 6 and rod 7 are equally divided by o₁(o₂), and the lateral horizontal load of point q is F=100 N.

The length of all poles of the square combination mechanism is 350 mm. Assume the square combination mechanism shown in Fig. 1 is a rigid body, and rods are axis of the connected local coordinate system x_io_iy_i (i=1, 2, 3, 4). Define xoy as a global coordinate system, the center of mass of the rod and the slider o₀, o₁, o₂, o₃, o₄ as the origin of the local coordinate system, and φ_i as the positive angle between the vector x on the global coordinate system and the vector x_i on the local coordinate system. Then the position of component can be expressed as

$ {\mathit{\boldsymbol{q}}_i} = \left[ {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}},\phi } \right]_i^{\rm{T}} $

(1)

The complete general coordinates of all members of the square assembly are given as

$ \mathit{\boldsymbol{q}} = {\left[ {\left[ {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}},\phi } \right]_1^{\rm{T}},\left[ {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}},\phi } \right]_2^{\rm{T}}, \cdots ,\left[ {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}},\phi } \right]_n^{\rm{T}}} \right]^{\rm{T}}} $

(2)

2.2 Establishment of Dynamic Equations

The complete constraint equation for all motion pairs of square combination can be expressed as

$ \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( {\mathit{\boldsymbol{q}},t} \right) = 0 $

(3)

The speed equation of the square combination mechanism can be obtained by the first derivative of the time t in Eq. (3), and the acceleration equation can be computed by the second order of the time t in Eq.(3). The augmentation method can be used to describe motion pair's counter-force constraints of square combination mechanism, and then determine its kinematic differential equation:

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{M\ddot q}} + \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_q^{\rm{T}}\mathit{\boldsymbol{\lambda }} = \mathit{\boldsymbol{Q}}\\ \mathit{\boldsymbol{ \boldsymbol{\varPhi} }}\left( {\mathit{\boldsymbol{q}},t} \right) = 0 \end{array} \right. $

(4)

where Φ_q is Jacobian matrix of the combined mechanism, M is the mass matrix of the square combined mechanism, ${\mathit{\boldsymbol{\ddot{q}}}}$ is the acceleration vector, λ is the Lagrange multiplier corresponding to the constrain counter-force of mechanism's motion pair, and Q is the generalized external force matrix received by the square combined mechanism. Such external forces include gravity, friction, externally applied loads, and internal impact forces, etc.

$ \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{M}}&{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_q^{\rm{T}}}\\ {{\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}_q}}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\ddot q}}}\\ \lambda \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \mathit{\boldsymbol{Q}}\\ {\gamma - 2\alpha \mathit{\boldsymbol{ \boldsymbol{\dot \varPhi} }} - {\beta ^2}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}} \end{array}} \right] $

(5)

In the Eq.(5), $\mathit{\boldsymbol{ \boldsymbol{\dot \varPhi} = }}\frac{{{\rm{d}}\mathit{\boldsymbol{ \boldsymbol{\varPhi} }}}}{{{\rm{d}}t}}$, α and β are greater than zero. If values fall in the range of 5 to 50, better modification effect can be expected.

3 Dynamic Model of Revolute Pair with Clearance 3.1 Kinematic Relation

The kinematics characteristics of the rotating pair with clearance considered is an important prerequisite to explore the impact of contact collision on the dynamic performance of the square combination mechanism.

If the clearance is taken into account in the joint g of square combination mechanism, when the contact between pin hole i and pin j is in contact, as shown in Fig. 2, the center of local coordinate system x_io_iy_i and x_jo_jy_j coincide with the center of pin hole i and the centroid of pin j respectively.

In Fig. 2, r_j^Q and r_i^Q are the position vectors of the center of pin hole o_i and the center of pin o_j in the global coordinate system, and e represents the eccentricity of the two center points.

$ \mathit{\boldsymbol{e}} = \mathit{\boldsymbol{r}}_j^Q - \mathit{\boldsymbol{r}}_i^Q $

(6)

When the pin hole contacts the pin, the normal vector of the contact surface between the pin hole and the pin is given by

$ \mathit{\boldsymbol{n}} = \frac{\mathit{\boldsymbol{e}}}{e} $

(7)

where $e = \sqrt {{\mathit{\boldsymbol{e}}^{\rm{T}}}\mathit{\boldsymbol{e}}} $.

Once c represents the clearance between the pin hole and the pin, then c=R_B－R_J. Here R_B and R_J are the radius of the pin hole and the pin respectively.

The contact deformation amount between the pin hole and the pin is

$ \delta = e - c $

(8)

In order to more clearly express the contact state between the pin hole and the pin, the total contact deformation δ is used to facilitate judgment. When δ≥0, the pin hole and the pin element are in separated state; when δ≥0, the pin hole and the pin are in contact. For the respective contact points Q_i and Q_j of pin hole and the pin in a joint with clearance, after contact, their position vector at the global coordinate system xoy can be described by the following formula:

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{r}}_i^Q = {\mathit{\boldsymbol{r}}_i} + {T_i}\mathit{\boldsymbol{s}}_i^Q + {R_i}\mathit{\boldsymbol{n}}\\ \mathit{\boldsymbol{r}}_j^Q = {\mathit{\boldsymbol{r}}_j} + {T_j}\mathit{\boldsymbol{s}}_j^Q + {R_j}\mathit{\boldsymbol{n}} \end{array} \right. $

(9)

where r_i and r_j are position vectors of centroids of pin holes i and pins j in the global coordinate system respectively, T_i and T_j are transfer matrix between local coordinates and global coordinate system, s_i^Q and s_j^Q are position vectors of contact points Q_i and Q_j for pin holes and pins at the local coordinate system x_io_iy_i and x_jo_jy_j respectively, and r_i^Q and r_j^Q represent position vectors of contact points Q_i and Q_j at the global coordinate system xoy respectively.

Calculating the first derivative of time t with respect to Eq.(9), the relative contact speed between pin hole and pin when considering clearance can be obtained. ν_N and ν_t are the normal and tangent vectors of relative contact velocity, respectively:

$ \left\{ \begin{array}{l} {v_N} = {\left( {\mathit{\boldsymbol{\dot r}}_j^Q - \mathit{\boldsymbol{\dot r}}_i^Q} \right)^{\rm{T}}}\mathit{\boldsymbol{n}}\\ {v_t} = {\left( {\mathit{\boldsymbol{\dot r}}_j^Q - \mathit{\boldsymbol{\dot r}}_i^Q} \right)^{\rm{T}}}\mathit{\boldsymbol{t}} \end{array} \right. $

(10)

where the tangent vector t of the contact surface is obtained from the normal vector n by counterclockwise rotating 90°, $\mathit{\boldsymbol{\dot r}}_i^Q$ and $\mathit{\boldsymbol{\dot r}}_j^Q$ are contact speed of the pin and the pin hole respectively.

3.2 Flores Contact Force Model

Hertz's contact theory has a disadvantage of failing to express the energy transfer during contact collision^[20]. Flores solved this problem and proposed a nonlinear contact-damping collision force model:

$ {F_N} = K{\delta ^n}\left[ {1 + \frac{{8\left( {1 - {c_r}} \right)}}{{5{c_r}}}\frac{{\dot \delta }}{{{{\dot \delta }_0}}}} \right] $

(11)

where δ represents contact deformation, ${\dot \delta }$ is contact deformation speed, ${\dot \delta_0 }$ means initial contact velocity, c_r is recovery factor, K is Hertz contact stiffness coefficient. K's expression is

$ K = \frac{4}{{3{\rm{ \mathit{ π} }}\left( {{\sigma _i} + {\sigma _j}} \right)}}{\left( {\frac{{{R_i}{R_j}}}{{{R_i} - {R_j}}}} \right)^{\frac{1}{2}}} $

(12)

where ${\sigma _{\left( {k = i, j} \right)}} = \frac{{1 - \nu _k^2}}{{{\rm{ \mathit{ π} }}{E_k}}}$, ν_i and ν_j are Poisson ratio of pin hole and pin respectively, E_i and E_j are elastic modulus of pin hole and pin respectively, R_i and R_j are contact radius of pin hole and the pin respectively.

3.3 Friction Force Model

The friction phenomenon is indispensable when analyzing the contact collision between relative moving bodies. In order to capture the slip phenomenon in the friction between the contact bodies, as well as to avoid the energy increase phenomenon during the contact process of the contact body, LuGre friction model was adopted. The contact force calculation formula for this model is

$ \begin{array}{l} {F_t} = \left\{ {{\sigma _0}z\left[ {1 - \frac{{{\sigma _1}\left| {{\nu _t}} \right|}}{{{\mu _k} + \left( {{\mu _s} - {\mu _k}} \right){e^{ - \left| {\frac{{{\nu _t}}}{{{\nu _s}}}} \right|\gamma }}}}} \right] + } \right.\\ \;\;\;\;\;\;\;\left. {\left( {{\sigma _1} + {\sigma _2}} \right){\nu _t}} \right\}{F_n} \end{array} $

(13)

In Eq.(13)^[21-22], F_n represents normal contact force, σ₀ is stiffness coefficient of the bristles, σ₁ denotes microscopic damping coefficient, σ₂ is viscous friction coefficient, z is average deformation of the bristles, μ_k is dynamic friction coefficient, μ_s is static friction coefficient, ν_s is Stribeck velocity, ν_t is tangent vectors of relative contact velocity, and γ denotes index parameters which is generally taken as 2.

The LuGre model not only describes the slip phenomenon in friction, but also avoids the increase in energy of the contact body during the collision process, which can reflect the hinge gap friction phenomenon more truthfully.

4 Dynamic Analysis of Square Combination Mechanism

In order to verify the rationality of the dynamics model and the mechanical model of the square combined mechanism with clearance, a numerical example was demonstrated to verify it. The square combination mechanism consists of homogenous alloy rods and remarks the length of each rod by l=350 mm. The mass of each rod is 2.8 kg, and the mechanism applies a driving force at the hinge point q.

Radius of the pin hole and the pin is 3.75 mm, elastic modulus of the pin hole and the pin's material is E_i=E_j=206.8 GPa, Poisson's ratio is υ_i=υ_j=0.32, density is ρ=7.8 g/cm³, c_f=0.12.

In order to predict the dynamic characteristics of square combination mechanism with clearance, four physical quantities of square combination mechanism member can be used to determine the dynamic characteristic change, which are displacement, speed, acceleration, and counter-force of joint.

The dynamic analysis process of the square combination mechanism is shown in Fig. 3. Firstly, define initial conditions of movement and properties of the motion pair clearance on the basis of giving structural parameters of the square combination mechanism. Then, build construct kinematics model of the square combination mechanism that considers clearance. Finally, determine the contact state of pin hole and pin, and select corresponding model.

When the clearance values fall within the 0.25 mm-0.5 mm interval, the variation trends of dynamic characteristics of joint g are as shown in Figs. 4-9.

As shown in Fig. 4, for the displacement curve, the trajectories of ideal square deployable mechanism and clearance-containing square mechanism basically coincide. From Fig. 5, it can be seen that, if consider the clearance effect, the centroid velocity of bar 6 transiently fluctuates at the initial movement stage, but the overall trend of movement is stable, and the speed of the square combination member mechanism is less affected by the joint clearance. It can be seen from Fig. 6, with respect to the ideal mechanism, high-frequency oscillation occurs in the acceleration curve of mechanisms with clearances. This feature is correlated with mechanisms′ stability and movement accuracy. In Figs. 4-6, the external load is acting in x direction. Compared with x direction, the physical quantities in y direction, such as displacement, velocity, and acceleration of the centroid of connecting rod, are much more consistent with the ideal curves.

Differences can be identified from the comparison of the square mechanisms′ dynamic characteristics when the clearance value is 0.25 mm and 0.5 mm: as shown in Fig. 7 and Fig. 8, the influence of the clearance value on displacement and speed is not significant; in Fig. 9 and Fig. 10, as the clearance value increases, the acceleration amplitude and the contact force amplitude of the square mechanism gradually increase, notably, the value increases about 3 times in the initial stage. It can be concluded that, with continuing increase of clearance value, the dynamic performance of the square mechanism will subject to further deterioration until failure.

5 Conclusions

(1) Based on the contact-separation-collision process, the clearance effect of joint was successfully introduced into the dynamic model of square deployable mechanism. The contact collision force was calculated by using the Flores model and the LuGre friction model respectively. In addition, the Baumgarte stability constraint method was utilized to effectively avoid constraint violations produced in integration process.

(2) The investigation of the dynamic characteristics of square deployable mechanism with clearance shows that the effect pairs of joint clearance needs to be take into account in mechanism dynamics calculation. Fig. 6 displays that a high-frequency oscillation characteristic appears on the acceleration curve of a mechanism with clearance rather than an ideal mechanism, and then the oscillation process converts to a smooth movement between 0.34 s and 0.42 s. This phenomenon indicates that the element movement of revolute pair with clearance converts from continuous collision to separate contact in this period of time.

(3) By the analysis of joints′ clearance values at 0.25 mm and 0.5 mm respectively, it can be seen that, with the increase of the clearance value, the dynamic curve of square expandable mechanism varied in much wide range, and the fluctuation amplitude was much larger. Moreover, it can be predicted that with further increase of clearance value, the collision process will be more severe until structural fail.