Analysis and modeling for the dynamics of the nipper mechanism considering jaw’s impacts

Abstract

The comber is very important in the spinning for high quality yarns, in which the nipper mechanism (NM) determines the quality of the yarn it combs. This paper is to connect topology and multi-body dynamics to reveal the principle of impact motion of the nipper mechanism. Firstly, the working process of the NM is analyzed and corresponding kinematic models of work sub-phases are derived. Subsequently, the transition from the work phase of the jaw closed to the opened is studied. The research results show that during the transformation process, the mechanism presented another work phase with multiple impacts, which is the main reason why the NM could not stably clamp the cotton clump and thus affect the quality of the yarn combed by the comber. According to the relative coordinate method, the dynamic model of the NM is set up. Combining the classical collision theory and the restitution coefficient equation, the acquisition of the impact impulse generated at the jaw and the subsequent motion of the mechanism are decided. Finally, the NM in the E62 comber is taken as an example to verify the correctness of the established dynamic model, and the influences of different input speeds, restitution coefficients, and stiffness coefficients of spring on the jaw’s impact are studied. This has certain theoretical value for improving the speed and efficiency of the comber.

Keywords

Comber nipper mechanism Dynamic Modeling impact dynamics jaw’s impact Restitution coefficient

Introduction

With the rapid development of human society and the progress of science and technology, people’s living standards continue to improve. The demand for high-grade textiles and special textiles is increasingly large. Without the combing process, it is difficult to meet the manufacturing requirements of yarn below 9.7tex which is used in the production of high-grade fabric and special fabric, such as medical suture thread, tire cord thread, high-speed sewing thread, and embroidery thread.¹ Therefore, the comber is the key equipment to produce yarn with high quality, in which the short fiber, neps, and impurities could be removed and cotton fibers can be straightened.²

Comber is mainly composed of five parts the nipper system, separating roller mechanism, feeding mechanism, top combing mechanism, and cylinder mechanism, among which the nipper system plays the most important role in the combing process with achieving complex motion including the jaw’s opening-closing motion and the nippers’ reciprocating swing motion. And meanwhile, the nipper mechanism (NM) in the system is the main part to produce noise and vibration.³ Therefore, the research on it has been attracting great attention. Zhao et al.^4,5 studied the low order resonance phenomenon of NMs based on the kineto-elasto-dynamics (KED) and established a nonlinear dynamic model suitable for the mechanism. Li et al.⁶ analyzed the carding force, separate drafting force, inertia force of the NM, and inertia force of the nipper pendulum shaft and investigated their relationships with the driving torque of the nipper pendulum shaft. Su et al.⁷ discussed the research on optimization of the nipper mechanism on CM500 series Combers and addressed the corresponding optimal crank radius. Liang et al.⁸ studied the stress and deformation on the working state of the bottom nipper in the comber and analyzed the dynamics of the nipper clamping process. But up till now, the dynamic model of NM considering the jaw’s impact and collision has not been established yet because the effective number of links and joints changed a few times during the working process.

As a high-speed mechanism, the quality of the yarn is greatly affected by the impact generated by NM during operation. Therefore, it is necessary to study the principles of impact and strategies for reducing impact. Li et al.⁹ proposed a deployable truss with multiple configurations and gave the optimal deploying-sequence scheme of the mechanism to minimize the deployment impact. Yang et al.¹⁰ proposed a reliability optimization design method for the stability of configuration transformation of constrained metamorphic mechanisms and optimized the tolerances of random variables of structural parameters. Song et al.¹¹ proposed two performance indices to quantitatively study how dynamic parameters such as mass, inertia, and stiffness affect the impact force reduction capability and maximum safe speed. Konno et al.¹² adopted sequential quadratic programming to solve a non-linear programming problem in the impact motion generated in humanoid robots. For multi-body systems considering impact, another urgent issue that needs to be addressed is the analysis of the motion reliability. For this, Wu et al.¹³ proposed a method to evaluate the mechanism reliability of the bistable compliant mechanism considering degradation and uncertainties of the parameters. Wang et al.¹⁴ proposed a reliability analysis of dynamics for a controllable metamorphic palletizing robot considering the multiple failure modes and the random and interval variables. To address the low computational efficiency and accuracy for reliability optimization of flexible mechanisms, Bai et al.¹⁵ proposed the mean-probability decomposition-coordination-based extreme support vector machine regression method based on the dynamics and uncertainty.

The impact problem caused in the NM belongs to the category of internal collision of multi-body systems, and a large number of novel and useful research results now have been achieved in this field. For the case of rigid impact, Hurmuzlu et al.¹⁶ developed two solution procedures that cast the impact equations in differential and algebraic forms are developed to solve the rigid body collisions of planar kinematic chains with multiple contact points, and elaborated on the rebounds at the noncolliding ends. Zahedi and Mata et al.^17,18 presented a method for solving the complete dynamic problem of multi-closed-loop mechanisms based on the Gibbs-Appell (G-A) and the Newton’s impact formulas. For the contact-impact problems of flexible multi-body systems, Shafei et al.¹⁹ investigated the effects of the mode shapes on the temporal response of these types of mechanical systems according to the Timoshenko beam theory and the assumed modes method. Chen et al.²⁰ presented a formulation based on the combination of the component mode synthesis method and the Lagrangian method. Dong et al.²¹ proposed a calculation method for collision dynamic model of flexible multi-body system based on variable topology theory by combining the floating coordinate method with the finite element method. Korayem et al.²² investigated the dynamic equations of an N-flexible link manipulator with revolute–prismatic joints via the Gibbs–Appell equations. For collision problems with friction considered, Natsiavas et al.²³ proposed a numerical integration method for determining the dynamics of a class of multibody systems. Ahmadizadeh et al.²⁴ developed a special computational algorithm to solve the dynamic model in which the contact force model is employed for modeling the impact-contact phenomenon. Shafei and Ahmadizadeh et al.^25,26 studied the multiple impacts of robotic systems, and derived the dynamic equations of motion for such systems by the Gibbs-Appell recursive algorithm and regularized method. Stronge²⁷ employed two impact parameters to treat terminal velocities and energy dissipation as functions of the normalized angle of incidence in the case of oblique planar impact.

Although impact problems can be divided into various situations, as mentioned above such as considering rigidity, flexibility, friction, and studying the stability and reliability of mechanism operating, the means to analyze these situations all involve the establishment of dynamic models. At present, the commonly used methods for analyzing the dynamics of impact systems can be classified as: Kane’s method,^28,29 Newton-Euler method,^30-32 D’Alembert’s principle,³³ Lagrange method,^34-37 Gibbs-Appell (G-A) methodology,^38-41 and analytical algorithm.⁴² These methods have different modeling processes for motion equations, but the results obtained are similar. However, the difference from other studied objects is that the NM in the comber has two working phases, and the topological structure of the mechanism is inconsistent in both phases. The change in topology is caused by both force and position constraints, while the impact is generated during this changing process. Accordingly, the best strategy for dynamic analysis of the mechanism is to introduce a mathematic model that can reflect this topological change to disclose such phenomena, thereby minimizing the detrimental effects resulting from the impact. In this study, the working principle of the NM is analyzed based on the topological principle, and corresponding kinematic models are set up. According to the investigation of work sub-phases for the NM, the movement status of the NM during the process from jaw opening to stable closing is identified, and the third working phase that occurred in this process is defined. The research on the impact motion of this mechanism is based on the relative coordinate method, which uses graph theory to describe the structural characteristics and connection relationships between various bodies, reflecting effectively the changes in topological structure when the working sub-phases transform. Lastly, taking the NM in E62 comber as an example, influences of input speeds, restitution coefficients, and stiffness coefficients of spring on the jaw’s impact are researched, and some conclusions that contribute greatly to mechanical design are obtained.

Working principle of the nipper system

As shown in Figure 1, the nipper system is composed of the Whitworth mechanism, gear mechanism, and nipper mechanism. The system adopts a single driving scheme to satisfy 2 degrees of freedom (DOFs). Specifically, the cylinder shaft is driven by an electromotor and its rotation can be transmitted to the nipper pendulum shaft and tension shaft by the Whitworth mechanism and gear mechanism. When the cylinder shaft rotates one circle, the nipper pendulum shaft swings back and forth once and the jaw is opened and closed once, which is a complete working cycle of the NM. There are therefore two work phases in one cycle

Phase 1: pre-clamping state. Link 5 and link 6 in the steeve are stationary relatively, the length of the steeve remains unchanged and the jaw is opened. The mechanism is a 2-DOF planar seven-bar linkage with pure revolute joints, as shown in Figure 2.

Phase 2: clamping state. The jaw is closed stably and the link 2 and link 4 are stationary relatively, while the length of the steeve is variable because link 5 moves relative to link 6. The mechanism is a 2-DOF planar seven-bar linkage with revolute and prismatic joints, as shown in Figure 3.

Figure 1.

Mechanical model of the nipper system.

Figure 2.

Work phase 1 of the NM: (a) three-dimensional graph and (b) structural diagram.

Figure 3.

Work phase 2 of the NM: (a) three-dimensional graph and (b) structural diagram.

The achievement of transformation from phase 1 to phase 2 relies on the combination of the force and position constraints as shown in Figure 4, the compressed spring is installed between links 5 and 6. Where F is the internal force generated by the spring, and it is large enough to keep link 5 staying still relative to link 6 when the jaw is opened. When the top nipper contacts with the bottom nipper, the revolute joint between links 2 and 4 is defunct, and the prismatic joint between links 5 and 6 is active, the spring is further compressed and the jaw is closed steadily to realize the transformation.

Figure 4.

The structure of steeve in NM.

Kinematic analysis of the nipper system

Whitworth mechanism

The structural diagram of the Whitworth mechanisms is shown in Figure 5. Let x and y be the basis vectors in the absolute coordinate system attached on the cylinder shaft along the horizontal and vertical directions respectively. Link A is fixed on the cylinder shaft $O_{a}$ , and the motor drives the cylinder shaft to rotate it.

Figure 5.

Structural diagram of the Whitworth mechanism.

For this mechanism, the closed-loop vector equation is established in the form

l_{A} + l_{C} = l_{D}

(1)

Where $l_{γ} (γ = A, C, D)$ is the vector expression for link γ. According to equation (1), it can be deduced that the angle $θ_{c}$ between link C and the horizontal direction is expressed by

θ_{c} = \arctan (\frac{l_{D} \sin θ_{d} + l_{A} \sin θ_{a}}{l_{D} \cos θ_{d} + l_{A} \cos θ_{a}}) + π

(2)

Where $l_{γ} (γ = A, C, D)$ donates the length of link γ, $θ_{d}$ is the angle between link D and the horizontal direction which is a constant. $θ_{a}$ is the angle between the link A and the horizontal direction.

Nipper mechanism

The stable operation process of the NM consists of two phases in one work cycle. In phase 1 as shown in Figure 6, links 5 and 6 are treated as one link (5, 6). Let $θ_{i} (i = 1, \dots, 5, 7)$ be the angle between link i and the horizontal direction, $O_{i}$ be the joint between link i and its inboard link, $M_{1}$ and $M_{2}$ be the driving moments which come from the nipper pendulum shaft and tension shaft.

Figure 6.

Structural diagram of the NM in phase 1.

Since links C and 1 are fixed on the nipper pendulum shaft, the angle $θ_{1}$ can be founded in the form

θ_{1} = θ_{c} + θ_{e}

(3)

Where $θ_{e}$ is the angle between links C and 1 which is a constant.

As for links 1, 2, 3, and frame, the corresponding closed-loop vector equation is established as

l_{D} + l_{1} = l_{3} + l_{2}

(4)

Where symbols represent the vector expressions for corresponding links. According to equation (4), the angle $θ_{3}$ between link 3 and the horizontal direction, and the angle $θ_{2}$ between link 2 and the horizontal direction can be obtained in forms

θ_{2} = π - 2 \arctan (\frac{K_{1} - 2 C_{2} l_{2}}{C_{1}^{2} + 2 C_{1} l_{2} + C_{2}^{2} + l_{2}^{2} - l_{3}^{2}})

(5)

θ_{3} = 2 \arctan (\frac{K_{2} + 2 C_{2} l_{3}}{C_{1}^{2} + 2 C_{1} l_{3} + C_{2}^{2} + l_{3}^{2} - l_{2}^{2}})

(6)

Where

K_{1} = \sqrt{4 l_{2}^{2} (C_{1}^{2} + C_{2}^{2}) - {(C_{1}^{2} + C_{2}^{2} + l_{2}^{2} - l_{3}^{2})}^{2}}

K_{2} = \sqrt{4 l_{3}^{2} (C_{1}^{2} + C_{2}^{2}) - {(C_{1}^{2} + C_{2}^{2} + l_{3}^{2} - l_{2}^{2})}^{2}}

C_{1} = l_{D} \cos (π - θ_{d}) + l_{1} \cos θ_{1}

C_{2} = l_{1} \sin θ_{1} - l_{D} \sin (π - θ_{d})

Similarly, the closed-loop vector equation of another loop in the mechanism is

l_{3} + l_{O_{3} O_{4}} + l_{O_{4} O_{5}} + l_{(5, 6)} = l_{O_{a} O_{c}} + l_{7}

(7)

Where $l_{O_{3} O_{4}}$ and $l_{O_{4} O_{5}}$ are vectors oriented from $O_{3}$ to $O_{4}$ and $O_{4}$ to $O_{5}$ respectively, $l_{O_{a} O_{c}}$ is the vector oriented from shafts $O_{a}$ to $O_{c}$ . Thus, the angle $θ_{4}$ between link 4 and the horizontal direction as well as the angle $θ_{5}$ between link (5, 6) and the horizontal direction can be derived as

θ_{4} = 2 \arctan (\frac{K_{3} + 2 C_{3} l_{4}}{C_{3}^{2} + 2 C_{3} l_{4} + C_{4}^{2} + l_{4}^{2} - l_{(5, 6)}^{2}})

(8)

θ_{5} = - 2 \arctan (\frac{K_{4} - 2 C_{3} l_{(5, 6)}}{C_{3}^{2} + 2 C_{3} l_{(5, 6)} + C_{4}^{2} + l_{(5, 6)}^{2} - l_{4}^{2}})

(9)

K_{3} = \sqrt{4 l_{4}^{2} (C_{3}^{2} + C_{4}^{2}) - {(C_{3}^{2} + C_{4}^{2} + l_{4}^{2} - l_{(5, 6)}^{2})}^{2}}

K_{4} = \sqrt{4 l_{(5, 6)}^{2} (C_{3}^{2} + C_{4}^{2}) - {(C_{3}^{2} + C_{4}^{2} + l_{(5, 6)}^{2} - l_{4}^{2})}^{2}}

C_{3} = l_{7} \cos θ_{7} - l_{3} \cos θ_{3} + l_{O_{a} O_{c}} \cos θ_{ac} - l_{O_{3} O_{4}} \cos θ_{2}

C_{4} = l_{7} \sin θ_{7} - l_{3} \sin θ_{3} + l_{O_{a} O_{c}} \sin θ_{ac} - l_{O_{3} O_{4}} \sin θ_{2}

Where $l_{O_{a} O_{c}}$ is the distance from $O_{a}$ to $O_{c}$ , $θ_{ac}$ is the angle between $l_{O_{a} O_{c}}$ and the horizontal direction, $l_{O_{3} O_{4}}$ is the distance from joints $O_{3}$ to $O_{4}$ .

Because the tension shaft and nipper pendulum shaft are connected by a gear train, the angle $θ_{7}$ that the eccentric wheel rotates is represented as

θ_{7} = θ_{c} / i_{g} - (θ_{ac} - \frac{π}{2}) / i_{g}

(10)

Where i_g is the transmission ratio of the gear mechanism.

Figure 7 shows the structural diagram of the NM in phase 2. In this state, links 2 and 4 are considered as one link (2, 4), link 6 moves relative to link 5 and the slid distance is s, $O_{6}$ is the prismatic joint between links 5 and 6.

Figure 7.

Structural diagram of the NM in phase 2.

Other angles are similar to that in phase 1, while angles $θ_{4}$ and $θ_{5}$ get new forms according to equation (7). These are

θ_{4} = θ_{2} - π - θ_{f}

(11)

θ_{5} = \arctan (\frac{K_{5}}{K_{6}})

(12)

K_{5} = l_{3} \sin θ_{3} - y_{O_{c}} - l_{7} \sin θ_{7} + l_{4} \sin θ_{4} + l_{O_{3} O_{4}} \sin θ_{2}

K_{6} = l_{3} \cos θ_{3} - x_{O_{c}} - l_{7} \cos θ_{7} + l_{4} \cos θ_{4} + l_{O_{3} O_{4}} \cos θ_{2}

Where $x_{O_{c}}$ and $y_{O_{c}}$ represent the projections on the x - and y -axes of the vector $l_{O_{a} O_{c}}$ , $θ_{f}$ is the angle between $l_{O_{4} O_{5}}$ and $l_{O_{4} O_{3}}$ which is a constant.

Dynamic analysis of the Nm considering jaw’s impact

System description

Due to the collision and impact generated during work phase transforming, the transformation process will take some time rather than complete immediately. This study proposes a dynamic modeling method to describe and analyze the corresponding complex motion that existed in the process. When the top nipper contacts with the bottom nipper, they will move in the opposite direction because of impact. Meanwhile, the spring will be compressed further and then recover gradually, which causes a variation in the length of the steeve. At this point, the mechanism presents a different structural topology compared with phases 1 and 2, and is defined as phase 3. The mechanism in this phase is a 3-DOF eight-bar planar linkage with revolute and prismatic joints. Its additional DOF is driven by the spring, as shown in Figure 8.

Figure 8.

The topological structure of the phase 3: (a) three-dimensional graph and (b) structural diagram

The phase 3 is a closed-loop system with two loops. To facilitate the establishment of its dynamic model, two joints need to be cut off to reduce it to an open-loop system according to the reducing method,⁴³ as shown in Figure 9, in which link 0 is the frame.

Figure 9.

The phase 3 mechanism and its reduced system: (a) the reduced system and (b) the kinematic diagram of the phase 3.

Suppose the relative movements between adjacent links, shown in Figure 9(b), are selected as the generalized coordinates of the system, so the generalized coordinate vector is $q = {[q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}, q_{7}]}^{T}$ . Obviously, such coordinates can be expressed by $θ_{i}$ and s. Appear as

q = [\begin{array}{l} q_{1} \\ q_{2} \\ q_{3} \\ q_{4} \\ q_{5} \\ q_{6} \\ q_{7} \end{array}] = [\begin{matrix} - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & - 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 1 \end{matrix}] [\begin{matrix} θ_{1} \\ θ_{2} \\ θ_{3} \\ θ_{4} \\ θ_{5} \\ s \\ θ_{7} \end{matrix}] + [\begin{matrix} π \\ π \\ 0 \\ 0 \\ π \\ 0 \\ 2 π \end{matrix}]

(13)

Dynamics analysis

In order to produce an equivalence in dynamics after cutting two joints compared to before that, four constraint equations should be introduced. Let them be written in a matrix form

Φ \dot{q} = η

(14)

Φ = [\begin{array}{l} \frac{\partial f_{1}}{\partial q_{1}} & \frac{\partial f_{1}}{\partial q_{2}} & \frac{\partial f_{1}}{\partial q_{3}} & \frac{\partial f_{1}}{\partial q_{4}} & \frac{\partial f_{1}}{\partial q_{5}} & \frac{\partial f_{1}}{\partial q_{6}} & \frac{\partial f_{1}}{\partial q_{7}} \\ \frac{\partial f_{2}}{\partial q_{1}} & \frac{\partial f_{2}}{\partial q_{2}} & \frac{\partial f_{2}}{\partial q_{3}} & \frac{\partial f_{2}}{\partial q_{4}} & \frac{\partial f_{2}}{\partial q_{5}} & \frac{\partial f_{2}}{\partial q_{6}} & \frac{\partial f_{2}}{\partial q_{7}} \\ \frac{\partial f_{3}}{\partial q_{1}} & \frac{\partial f_{3}}{\partial q_{2}} & \frac{\partial f_{3}}{\partial q_{3}} & \frac{\partial f_{3}}{\partial q_{4}} & \frac{\partial f_{3}}{\partial q_{5}} & \frac{\partial f_{3}}{\partial q_{6}} & \frac{\partial f_{3}}{\partial q_{7}} \\ \frac{\partial f_{4}}{\partial q_{1}} & \frac{\partial f_{4}}{\partial q_{2}} & \frac{\partial f_{4}}{\partial q_{3}} & \frac{\partial f_{4}}{\partial q_{4}} & \frac{\partial f_{4}}{\partial q_{5}} & \frac{\partial f_{4}}{\partial q_{6}} & \frac{\partial f_{4}}{\partial q_{7}} \end{array}]

(15)

Where Φ $\in R^{4 \times 7}$ is the constrained matrix, $f_{j} (q_{1}, q_{2}, q_{3}, q_{4}, q_{5}, q_{6}, q_{7}) = 0 (j = 1, 2, 3, 4)$ is the constraint equation introduced, $\dot{q} \in R^{7 \times 1}$ is the generalized speed vector which is the first-order derivative of the matrix q with respect to time, $η$ is the 4-order column matrix.

Ignoring the influence of external disturbance such as friction, the dynamic equation of the phase 3 can be written by the relative coordinate method as a matrix form⁴⁴

A \overset{\cdot\cdot}{q} = B + Φ^{T} λ

(16)

Where $A \in R^{7 \times 7}$ is the inertia matrix which is symmetric positive definite, $\overset{\cdot\cdot}{q} \in R^{7 \times 1}$ is the generalized acceleration vector which is the second-order derivative of the matrix q with respect to time, $B \in R^{7 \times 1}$ is a column matrix relating to the gravity, the Coriolis, the centripetal force and the collision force, $λ \in R^{4 \times 1}$ is the matrix composed of Lagrangian multipliers, $λ = {[λ_{1}, λ_{2}, λ_{3}, λ_{4}]}^{T}$ . All unknown variables can be obtained by solving equations (14) and (16) simultaneously.

Let the collision force generated between top and bottom nippers at the contact point be $\tilde{F}$ . Collecting the terms related to the collision force, equation (16) can be written as the form of generalized acceleration with respect to $\tilde{F}$ . That is

\overset{\cdot\cdot}{q} = C \tilde{F} + D

(17)

Where $C \in R^{7 \times 1}$ is the coefficient matrix of $\tilde{F}$ , $D \in R^{7 \times 1}$ is the matrix independent of $\tilde{F}$ .

Assuming the shape, position, and orientation of every link and the external forces applied remain unchanged during colliding, because $C$ and $D$ are the functions of the generalized coordinate q, generalized speed $\dot{q}$ , and time t, which proves that their integrals over time are constants. When equation (17) is integrated within the collision time $t$ , the integral term containing $D$ is zero due to the short integration interval. It implies that the increment of generalized speed after colliding can be derived

Δ \dot{q} = \lim_{Δ t \to 0} \int_{t}^{t + Δ t} \overset{\cdot\cdot}{q} d t = \lim_{Δ t \to 0} \int_{t}^{t + Δ t} (C \tilde{F} + D) d t = \lim_{Δ t \to 0} (\int_{t}^{t + Δ t} C \tilde{F} d t + \int_{t}^{t + Δ t} D d t) = C F_{I}

(18)

Where $F_{I}$ is the impact impulse generated when top nipper collides with bottom nipper.

F_{I} = \lim_{Δ t \to 0} \int_{t}^{t + Δ t} \tilde{F} d t

(19)

On the other hand, increments in the angular velocity and the centroid velocity as well as the increments in velocities of top and bottom nippers at the impact point after colliding can be obtained

△ ω = G △ \dot{q}

(20)

△ \dot{r} = H △ \dot{q}

(21)

△ V = △ v_{2} - △ v_{4} = S^{T} (△ \dot{r} - ρ \times △ ω)

(22)

Where $△ \dot{r} \in R^{7 \times 1}$ and $△ ω \in R^{7 \times 1}$ are matrices of increments in the centroid velocity and the angular velocity, $G \in R^{7 \times 7}$ and $H \in R^{7 \times 7}$ are the coefficient matrices of $△ \dot{r}$ and $△ ω$ with respect to the generalized speed increment, $△ v_{2}$ and $△ v_{4}$ are velocity increments of top and bottom nippers at the collision point, $S = {[0, 1, 0, - 1, 0, 0, 0]}^{T}$ , $ρ = diag (0, ρ_{2}, 0, ρ_{4}, 0, 0, 0)$ , $ρ_{2}$ and $ρ_{4}$ are the position vector oriented from centroids of top nipper and bottom nipper to the contact point.

Substituting equation (18), (20), and (21) into equation (22), the relationship between the difference in velocity increments as shown in equation (22) and the impact impulse can be established

△ V = Q F_{I}

(23)

Where

Q = S^{T} (H - ρ \times G) C

According to the relevant theory of linear algebra, it can be proved that the matrix $Q$ is invertible. Most generally, the impact is partially elastic, and the restitution coefficient e is between 0 and 1. When link 2 collides with link 4, the restitution coefficient equation should be satisfied. That is

(△ v_{2} - △ v_{4}) n = - (1 + e) (v_{2}^{+} - v_{4}^{+}) n

(24)

Where $n$ is the normal vector at the collision point, $v_{2}^{+}$ and $v_{4}^{+}$ are the instantaneous velocities at the collision point of top and bottom nippers when the collision occurs. Let such velocities be called velocities before colliding, because the vector $n$ is in the same direction as the impact impulse $F_{I}$ , the modulus of the impact impulse can be obtained by substituting equation (23) into equation (24)

F_{I} = - Q^{- 1} (1 + e) (v_{2}^{+} - v_{4}^{+}) n

(25)

Therefore, the centroid velocity and angular velocity of each link at the moment of the collision ending, or calling them velocities after colliding, can be deduced by substituting equations (25) and (18) into equations (20) and (21)

{\begin{matrix} v^{-} = v^{+} + △ v \\ ω^{-} = ω^{+} + △ ω \end{matrix}

(26)

Where $v^{+}$ and $ω^{+}$ are the centroid velocity and angular velocity of every link before colliding, $v^{-}$ and $ω^{-}$ are the centroid velocity and angular velocity of every link after colliding correspondingly.

Running of the Nm during Phase 3

Let the contact between top nipper and bottom nipper at the beginning of the process form phase 1 transforming to phase 2 be recorded as the first collision. After that, the NM transforms from phase 1 into phase3. Subsequently, it operates continuously to the second collision under the effects of spring and driving forces. Let the operation process be named the first operation of the phase 3, in which the motion state after the first collision is taken as the initial motion state of phase 3, and the final motion of its first operation is treated as the initial motion of the second collision. Consequently, during the transformation from phase 1 to phase 2, the motion of NM can be divided into two running stages.

Internal collision stage: Collision occurred between top and bottom nippers of the phase 3. The motion after colliding can be obtained based on equation (26).

Continuous operation stage: The mechanism in phase 3 is driven by $M_{1}$ , $M_{2}$ and spring force. The motion of the mechanism can be obtained by dynamic modeling.

Assuming the transformation of the work phase occurs at time $t_{1}$ , the relationships between the motion state values of during such two stages are set up by

{\begin{array}{l} {{}^{p}v}_{s}^{+} = {{}^{p}v}^{-} \\ {{}^{p}ω}_{s}^{+} = {{}^{p}ω}^{-} \end{array}

(27)

{\begin{matrix} {{}^{p + 1}v}^{+} = {{}^{p}v}_{s}^{-} \\ {{}^{p + 1}ω}^{+} = {{}^{p}ω}_{s}^{-} \end{matrix}

(28)

Where collision sequence p is a positive integer starting from 1, ${{}^{p}v}^{-}$ and ${{}^{p}ω}^{-}$ are the centroid velocity matrix and angular velocity matrix of links after the p-th collision in which both ${{}^{1}v}^{-}$ and ${{}^{1}ω}^{-}$ can be obtained by equation (26), ${{}^{p}v}_{s}^{+}$ and ${{}^{p}ω}_{s}^{+}$ are the initial centroid velocity matrix and angular velocity matrix of the p-th operation of the phase 3 respectively, ${{}^{p}v}_{s}^{-}$ and ${{}^{p}ω}_{s}^{-}$ are respectively the final centroid velocity matrix and final angular velocity matrix of the p-th operation of the phase 3, ${{}^{p + 1}v}^{+}$ and ${{}^{p + 1}ω}^{+}$ are the centroid velocity matrix and angular velocity matrix of links before the p+1-th collision.

During the transformation from phase 1 to phase 2, the motion state values after the p-th collision are regarded as the initial state values of the p-th operation of the phase 3, and the final motion state values of its p-th operation are used as the initial motion state values of the next collision and so on until the relative velocity between top and bottom nippers reaches the preset threshold. At this time, it is believed that the NM has completed the transformation and entered the phase 2. This process can be analyzed through numerical simulation, as shown in Figure 10.

Figure 10.

The flow chart of numerical simulation.

Research on impact characteristics

Taking the NM in E62 Comber as an example,¹ its virtual model is shown in Figure 1. The geometrical parameters and motion parameters are shown in Tables 1 and 2. The initial angular $θ_{a}$ of the crank is 1.391 rad. The stiffness coefficient of the pressurized spring is 7 N/mm, the spring is compressed 7.9 mm before working, the gear transmission ratio is 1, and the collision restitution coefficient e between the top and bottom nippers is 0.2. When the generalized coordinate $q_{4}$ is 1.021 rad, the first contact between the bottom and top nippers occurs. The transformation of work phase is completed when the normal velocity difference between the collision points is less than or equal to 0.01 m/s. This study is conducted on the jaw’s impact characteristics of the NM for three different input speeds of 450, 550, and 650 r/min.

Table 1.

Physical parameters of the NM.

Parameter	Value	Parameter	Value
m₁/kg	2.837	l₅/m	0.03
m₂/kg	4.058	l₆/m	0.096
m₃/kg	0.647	L₇/m	0.009
m₄/kg	0.719	$l_{O_{3} O_{4}}$ /m	0.075
m₅/kg	0.066	$x_{O_{c}}$	0.088
m₆/kg	0.154	$y_{O_{c}}$	0.201
m₇/kg	0. 16	J₁/(kg·m²)	3.989 × 10⁻³
l_a/m	0.065	J₂/(kg·m²)	7.06 × 10⁻³
l_d/m	0.207	J₃/(kg·m²)	0.901 × 10⁻³
l₁/m	0.082	J₄/(kg·m²)	0.658 × 10⁻³
l₂/m	0.187	J₅/(kg·m²)	0.013 × 10⁻³
l₃/m	0.074	J₆/(kg·m²)	0.125 × 10⁻³
l₄/m	0.072	J₇/(kg·m²)	0.022 × 10⁻³

Table 2.

Motion parameters of the NM for simulating.

Parameter	Value	Parameter	Value
$q_{1}^{(0)}$ /(rad)	0.326	$q_{6}^{(0)}$ /(m)	0
$q_{2}^{(0)}$ /(rad)	2.931	$q_{7}^{(0)}$ /(rad)	0.934
$q_{3}^{(0)}$ /(rad)	0.588	$θ_{d}$ /(rad)	3.002
$q_{4}^{(0)}$ /(rad)	1.381	$θ_{e}$ /(rad)	0.646
$q_{5}^{(0)}$ /(rad)	3.714	$θ_{ac}$ /(rad)	1.158

The simulation time step is set to 0.01µ $s$ . Figure 11 is the curve of angular displacement $q_{4}$ under three speeds. The simulation results show that the jaw rebounds several times rather than closing instantaneously during work-phase transformation, and the rebound decreases gradually to 0 with the increase in rebound times. In addition, the faster speed increases the rebound peak of the jaw and prolongs the duration of transformation process as shown in the partially enlarged image P, which will lead to the jaw clamping the cotton clump unsteadily and affecting the combing quality. It is also the key problem that must be solved in the development of higher-speed combers. On the other hand, the curve of the virtual simulation coincides with the curve of the numerical computation, verifying the correctness of the impact dynamic model established for the NM in Section 4.

Figure 11.

Angular displacement of the jaw under different input speed: (a) 450 r/min, (b) 550 r/min, and (c) 650 r/min.

Figure 12 shows the impact impulse generated by each collision of the jaw at different speeds. It can be seen from the diagram that the impact impulse gradually decreases to 0 as the collision times increase, which is consistent with the trend of the jaw’s rebound and manifests that the rebound of the jaw is caused by impact. The higher the speed, the larger the impulse generated by per collision, resulting in more rebounds and collisions of the jaw, thereby extending the duration of work-phase transformation. When the impulse is 0, there is no force driving the jaw to rebound and the top nipper remains stationary relative to the bottom nipper and the transformation is finished. Therefore, to make the jaw clamp the cotton clump quickly and steadily, it is necessary to reduce the impact impulse generated by each collision.

Figure 12.

Impulse peaks of the jaw under different input speeds.

Another interesting phenomenon shown in Figures 11 and 12 is that the jaw rebounds significantly only in the first two times. The amplitude and time consumed for the first and second rebounds are shown in Table 3. Under three speeds, the times consumed by the first two rebounds account for 96.47%, 96.73%, and 96.75% of the times consumed for the process of work-phase transforming respectively. When the speed increases by 44.4%, the first rebound amplitude of the jaw increases by 118.4%, and the second increases by 113.6%, which indicated that the speed has an obvious influence on the impact motion of the jaw. Since the first two rebounds almost reflect all the motion characteristics of the jaw during transformation, we focus on these two rebounds in the following analysis.

Table 3.

Effect of different speeds on jaw’s impact motion.

Rotating speed (r/min)	1st rebound amplitude × 10⁻³ (rad)	2nd rebound amplitude × 10⁻⁴ (rad)	1st rebound time (ms)	2nd rebound time (ms)
450	2.91	1.11	2.55	0.46
550	4.97	1.77	3.27	0.57
650	6.36	2.35	3.56	0.61

Taking e = 0.05, 0.1, 0.15, 0.2, and 0.25 respectively to study further the influence of the restitution coefficient on jaw’s impact when input speed is 650 r/min and other conditions remain unchanged. The research results are shown in Table 4 and Figure 13, obviously, both convey an important conclusion that the restitution coefficient has a great influence on the impact motion of the jaw because the 4-fold increase in restitution coefficient results in a 19.8-fold increase in the amplitude of the first jaw’s rebound and a 551-fold increase in the second. In fact, when the variety of cotton clumps is changed, the flexibility is also different, resulting in a slight change in the restitution coefficient for the colliding between the top and bottom nippers. Although this change is minimal, it can also affect the jaw’s impact, causing unexpected vibrations. Besides, Figure 13 shows that the 4-fold increase in restitution coefficient leads to a 3.4-fold increase in the time consumed for work-phase transforming and a 3-fold increase in the number of collisions, which illustrates that the NM made from larger restitution coefficient materials takes more time to change its work phase and is more unstable during the period.

Table 4.

Effect of different restitution coefficients on jaw’s impact motion.

Restitution coefficient	1st rebound amplitude × 10⁻³ (rad)	2nd rebound amplitude × 10⁻⁴ (rad)	1st rebound time (ms)	2nd rebound time (ms)
0.05	0.46	0.01	1.11	0.05
0.1	1.73	0.16	2.03	0.18
0.15	3.73	0.78	2.84	0.37
0.2	6.36	2.35	3.56	0.61
0.25	9.57	5.52	4.21	0.89

Figure 13.

Effect of different restitution coefficients on (a) time consumed, (b) collision frequency during work-phase transformation.

To study the influence of spring stiffness on the jaw’s impact, the stiffness coefficients of 7, 8, 9, 10, and 11 are taken respectively with input speed 650 r/min and other conditions unchanged. As shown in Table 5 and Figure 14, when spring stiffness increases by 57.1%, the amplitudes of the first and second jaw’s rebound decrease by 31% and 30.2% respectively, and the time consumed decreases by 28.4% and 26.2%. Furthermore, the increase in spring stiffness has no obvious influence on the number of collisions between the top and bottom nippers during the transformation process, which suggests that a reasonable increase in spring stiffness will restrain the impact effect of the jaw.

Table 5.

Effect of different spring stiffness on jaw’s impact motion.

Spring stiffness (N/mm)	1st rebound amplitude × 10⁻³ (rad)	2nd rebound amplitude × 10⁻⁴ (rad)	1st rebound time (ms)	2nd rebound time (ms)
7	6.36	2.35	3.56	0.61
8	5.72	2.13	3.24	0.56
9	5.19	1.93	2.97	0.52
10	4.76	1.79	2.75	0.48
11	4.39	1.64	2.55	0.45

Figure 14.

Effect of different stiffness coefficients of spring on (a) time consumed, (b) collision frequency during work-phase transformation.

From the aforementioned analytical results, there is a noteworthy point that the input speed and the restitution coefficient between the top and bottom nippers have greater effects on the jaw’s impact. In order to reduce the jaw’s impact, improve the stability and efficiency of the NM clamping the cotton clump and thus improve the combing quality of the comber, materials with large restitution coefficients should be avoided in the design stage to produce the top and bottom nippers, a lower speed input and larger stiffness coefficient of spring should be adopted within a reasonable range.

Conclusion

The impact generated by the NM during work-phase transformation has a great effect on the quality of the yarn combed by the comber. For that, this paper provides a method to analyze the effect of this impact on the stability and efficiency of NM. The traditional analysis method for the NM is to identify it as two working phases. However, this work considers the process of transforming from work phase 1 to phase 2, defining the third working phase of the NM, and analyzed the movement of the mechanism during this phase, pointing out that this movement process is essentially composed of two alternating movements. Based on the methods proposed in this paper, the NM in E62 comber is chosen as a specific example for analyzing its impact dynamic characteristics, and the effects of rotating speed, restitution coefficient, and spring stiffness on work-phase transformation are analyzed. The research results imply that a higher speed and larger restitution coefficient prolong the time consumed by transforming process and exacerbate instability in clamping cotton clumps, the increase of such two parameters will intensify the jaw’s impact effect, while an increase in spring stiffness within a reasonable range will dilute the impact. This work provides certain theoretical guidance for the optimization design and stability analysis of combers.

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: This study was supported by the National Natural Science Foundation of China (No. 52005368, 52175243).

ORCID iD

Yang Zhou

References

Ren

Jia

GX.

Key Technologies for efficient cotton spinning combing. Beijing: Textile & Apparel Press, 2017.

Liu

XJ.

Dynamic balance optimisation for the middle shaft system of a cotton comber based on ADAMS. Fibres & Textiles in Eastern Europe 2012; 20(3): 43-45.

Liu

Jia

Ren

JZ.

Simulation and vibration of combing nipper mechanism. Journal of Textile Research 2015; 36(9): 108-113.

Zhao

Tang

WC.

Research on lower order resonance phenomenon of nipper mechanism on comber. Engineering Mechanics 2012; 29(4): 251-256.

Zhao

Tang

Wang

Dynamics modeling of nipper mechanism on comber with non-linear finite element method. China Mechanical Engineering 2009; 20(6): 678-682.

Ren

Liang

, et al. Dynamic analyses on nipper pendulum shaft in cotton combers. Journal of Textile Research 2021; 42(6): 160-165.

Gao

Liu

, et al. Research on optimization of nipper mechanism on cm500 combers using adams. Tekstil Ve Konfeksiyon 2015; 25(2): 151-154.

Liang

Jia

Ren

, et al. Deformation and stress analysis on nippers of cotton combing machine. Journal of Textile Research 2021; 42(12): 145-150.

Huang

, et al. Design and analysis of a foldable metamorphic mechanism with flat panel. Journal of Northwestern Polytech University 2018; 36(S1): 33-40.

10.

Yang

Sun

, et al. Reliability optimization design method of constrained metamorphic mechanism oriented to the stability of configuration transformation. Journal of Mechanical Engineering 2023; 59(3): 22-37.

11.

Song

She

Wang

, et al. Toward Tradeoff Between Impact Force Reduction and Maximum Safe Speed: Dynamic Parameter Optimization of Variable Stiffness Robots. Journal of Mechanisms and Robotics-Transactions of the Asme 2020; 12(5): 054503.

12.

Konno

Myojin

Matsumoto

, et al. An impact dynamics model and sequential optimization to generate impact motions for a humanoid robot. International Journal of Robotics Research 2011; 30(13): 1596-1608.

13.

Yan

, et al. Mechanism reliability of bistable compliant mechanisms considering degradation and uncertainties: Modeling and evaluation method. Applied Mathematical Modelling 2016; 40(23-24): 10377-10388

14.

Wang

Chen

Dong

, et al. Reliability analysis and optimization of dynamics of metamorphic mechanisms with multiple failure modes. Applied Mathematical Modelling 2023; 117: 431-450.

15.

Bai

Zhou

, et al. Reliability optimization of two-link flexible manipulator. Applied Mathematical Modelling 2021; 101: 76-95.

16.

Hurmuzlu

Marghitu

, et al. Rigid-body collisions of planar kinematic chains with multiple contact points. International Journal of Robotics Research 1994; 13(1): 82-91.

17.

Zahedi

Shafei

Shamsi

On the dynamics of multi-closed-chain robotic mechanisms. International Journal of Non-Linear Mechanics 2022; 147: 104241.

18.

Mata

Provenzano

Valero

, et al. Serial-robot dynamics algorithms for moderately large numbers of joints. Mechanism and Machine Theory 2002; 37(8): 739-755.

19.

Shafei

Riahi

MM.

The effects of mode shapes on the temporal response of flexible closed-loop linkages under the impulse excitation. Mechanical Systems and Signal Processing 2022; 178: 109256.

20.

Chen

Liu

Hong

JZ.

An efficient formulation based on the Lagrangian method for contact-impact analysis of flexible multi-body system. Acta Mechanica Sinica, 2016, 32(2): 326-334.

21.

Dong

Hong

Zhu

, et al. Numerical and experimental studies on impact dynamics of a planar flexible multibody system. Acta Mechanica Sinica, 2010, 26(04): 635-642.

22.

Korayem

Dehkordi

SF.

Derivation of motion equation for mobile manipulator with viscoelastic links and revolute-prismatic flexible joints via recursive Gibbs-Appell formulations. Robotics and Autonomous Systems 2018; 103: 175-198.

23.

Natsiavas

Passas

Paraskevopoulos

A time-stepping method for multibody systems with frictional impacts based on a return map and boundary layer theory. International Journal of Non-Linear Mechanics, 2021, 131: 103683.

24.

Ahmadizadeh

Shafei

Fooladi

Dynamic modeling of closed-chain robotic manipulators in the presence of frictional dynamic forces: A planar case. Mechanics Based Design of Structures and Machines 2023; 51(8): 4347-4367.

25.

Ahmadizadeh

Shafei

Jafari

Frictional Impact-Contacts in Multiple Flexible Links. International Journal of Structural Stability and Dynamics 2021; 21(6): 2150075.

26.

Ahmadizadeh

Shafei

Fooladi

Dynamic analysis of multiple inclined and frictional impact-contacts in multi-branch robotic systems. Applied Mathematical Modelling 2021; 91: 24-42.

27.

Stronge

WJ.

Smooth dynamics of oblique impact with friction. International Journal of Impact Engineering 2013; 51: 36-49.

28.

Chang

Peng

ST.

Impulsive motion of multibody systems. Multibody System Dynamics 2007; 17(1): 47-70.

29.

Glocker

Energetic consistency conditions for standard impacts Part I: Newton-type inequality impact laws and Kane’s example. Multibody System Dynamics 2013; 29(1): 77-117.

30.

Song

Chang

Jin

, et al. Analysis and simulation of impact problem of metamorphic mechanism considering configuration transformation. Journal of Mechanical Engineering 2020; 56(17): 48-58.

31.

Song

Chang

Jin

, et al. An approach for the impact dynamic modeling and simulation of planar constrained metamorphic mechanism. Shock and Vibration 2020; 2020: 8837838.

32.

Gan

Dai

Dias

, et al. Joint force decomposition and variation in unified inverse dynamics analysis of a metamorphic parallel mechanism. Meccanica 2016; 51(7): 1583-1593.

33.

Peng

Song

Kan

ZY.

A novel nonsmooth dynamics method for multibody systems with friction and impact based on the symplectic discrete format. International Journal for Numerical Methods in Engineering, 2019, 121(7): 1530-1557.

34.

Nurahmi

Gan

Setya

WTA

. Unified singularity crossing of a 3-(rr)ps metamorphic parallel mechanism through dynamic modeling. Machines 2023; 11(3): 361.

35.

Gao

Zhang

Kang

, et al. Dynamic analysis of a three-fingered deployable metamorphic robotic grasper. Mechanism and Machine Theory 2023; 180: 105140.

36.

Yang

Jin

Gong

, et al. Research on impact motion in metamorphic mechanism. China Mechanical Engineering 2009; 20(13): 1608-1612.

37.

Rong

Zhang

MK.

Unified inverse dynamics for a novel class of metamorphic parallel mechanisms. Applied Mathematical Modelling 2019; 74: 280-300.

38.

Shafei

Sadeghi

The kinematics and kinetics of multi-closed-chain mechanisms in the impact and non-impact stages. Meccanica 2022; 57(10): 2591-2608.

39.

Mata

Provenzano

Valero

, et al. Serial-robot dynamics algorithms for moderately large numbers of joints. Mechanism And Machine Theory 2002; 37(8): 739-755.

40.

Zahedi

Shafei

Shamsi

Kinetics of planar constrained robotic mechanisms with multiple closed loops: An experimental study. Mechanism And Machine Theory 2023; 183: 105250.

41.

Zahedi

Shafei

Shamsi

Application of hybrid robotic systems in crop harvesting: Kinematic and dynamic analysis. Computers and Electronics in Agriculture 2023; 209: 107724.

42.

Anderson

Critchley

JH.

Improved ‘Order-N’ performance algorithm for the simulation of constrained multi-rigid-body dynamic systems. Multibody System Dynamics 2003; 9(2): 185-212.

43.

Zhou

Chang

B Y

Jin

G G

, et al. A mathematic description method of variable topology configuration for dynamic modeling of metamorphic mechanism. Journal of Mechanical Engineering, 2022, 58(9): 49-61.

44.

Liu

Pan

XS.

Dynamics of multibody systems. Beijing: Higher Education Press, 2014.