# Dynamics Analysis of the Rigid-Flexible Coupling Lifting Comprehensive Mechanism for a Rotary Dobby

Authors: Xiaoyong Wei; Guoguang Jin; Zhan Wei; Boyan Chang; Yanyan Song
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper addresses the problem of dynamics analysis of the rigid-flexible coupling lifting comprehensive mechanism for a rotary dobby, which is the important part of the loom. To provide a physical model basis for a precise dynamics model, the finite element method was used to discretize the bending arm of the rotary dobby effectively. Combining with the modal synthesis techniques, the dynamic model of the bending arm was established by using Kane’s formulation, and it laid a foundation for analyzing the dynamic performance of the heald frame. By comparing virtual prototype simulation results with the numerical calculation results of the bending arm, the correctness of this model was verified. Based on the established dynamic model, the modal truncation method is used to simplify the dynamic model; in addition, the influence of parameters such as the speed of the dobby, the warp tension, the movement distance of the heald frame, and the thickness of the bending arm on the dynamic characteristics of the heald frame was analyzed. Last, the sensitivity analysis (SA) method is used to analyze the effects of each parameter. The results show that it is appropriate to select the first four modes to calculate, and increasing the speed greatly or increasing the warp tension, the shedding performance is obviously worse, while the shedding performance of the loom can be optimized by reducing the shedding range or increasing the thickness of the bending arm.


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## Body

## 1. Introduction

The rotary dobby is the most advanced high-speed shedding device at present [1–3]. Its function is to form a passage where the weft can get through by dividing the warp yarns into two layers according to certain rules. The rotary dobby is made up of the lifting comprehensive mechanism, the heald selecting mechanism, the electronic control equipment, and so on, while the lifting comprehensive mechanism is the core part of the rotary dobby. It converts the uniform circular motion of the motor input into the linear motion of the heald frame in the vertical direction to ensure the normal operation of the loom. Consequently, studying the lifting comprehensive mechanism is of great significance to promote the overall performance of the rotary dobby.At present, the research on the rotary dobby is rare, and it is mainly focused on two aspects. One is the kinematics analysis of the mechanism, and the other is to do dynamic analysis of the mechanism by software so that its dynamic performance indicators can be obtained. Shen et al. [4] analyzed the motion characteristics of the rotary dobby and conducted the three-dimensional model of the main transmission mechanism. They used the vector triangle method to solve the kinematics of the main drive mechanism; moreover, the dynamic analysis of the mechanism was carried out by using the equivalent simplified model and the test system of the dobby is designed. Eren et al. [5] analyzed the kinematics of the rotary dobby; in addition, the influence of mechanism parameters on the motion of the heald frame was mainly discussed.The aforementioned research has laid the foundation for the analysis of motion characteristics and dynamics modeling for the rotary dobby [6, 7]. However, the speed of the modern loom is getting higher and higher with the desperate pursuit of productivity. Under this circumstance, problems such as the vibration of the machine, the reduction of stability, and the shortening of life span demand a prompt solution. In the case of high-speed motion, the deformations of links readily occur owing to the excitation forces arising from inertial and actuation forces. Studying the influence of the flexible deformation for the component to the shedding performance is of great significance to solve the above problems [8, 9]. The process of flexible analysis of the whole system is cumbersome. Moreover, it wastes the computing resources because of the complex structure of the lifting comprehensive mechanism. Consequently, building a universal rigid-flexible coupling dynamics model for the rotary dobby is of great significance to investigate the interaction effect between the rigid and elastic motions. The main objective of this paper is to address this critical issue.Since the mechanism is a rigid-flexible coupling dynamic system, it is much more difficult to establish the dynamic model for the flexible dynamic system compared with the rigid dynamic system, and much more factors need to be considered in the modeling process. With respect to the dynamic modeling of the flexible mechanism and rigid-flexible coupling mechanism, academia has made great contributions. Considering the flexible links of the planar parallel mechanism, Chen et al. [10] presented an improved curvature-based finite element method (ICFE) for discretization of the flexible links. On this basis, Kane’s formulation was integrated to formulate the dynamic model of a 3RRR planar parallel manipulator, and the correctness of the dynamic model was verified by comparing with the Abaqus model. Based on the theory of rigid-flexible coupled multibody dynamics, Liu et al. [11] studied the influence of driving motor speed, braking time, and rotational damping coefficient on the law of torsional vibration characteristics during braking by using the finite element method and multibody dynamics simulation software RecurDyn. Zheng et al. [12] described the deformation of the aeroengine blades by the assumed modal method considering the deformation coupling term, and the Lagrange dynamic equation was employed to derive the first-order approximation coupling model for the flexible blade with three-dimensional large overall motion. Based on this model, the blade’s vibration frequency, frequency veering, and mode shifting were also predicted, and the feasibility of the theory and method proposed was shown. On the basis of modal modes of support plates obtained in ANSYS, Cihan and Yunus [13] used MATLAB to obtain the mathematical model of state space in the main coordinate system, and they also did frequency response analysis of the system.Referring to the aforementioned review, to develop a complete dynamic model of a flexible mechanism and rigid-flexible coupling mechanism, many modeling approaches have been presented. Amongst them, some basic formulations, for instance, the Newton-Euler formulation, the Lagrangian formulation, and Kane’s formulation, were employed [14, 15]. To facilitate computation, some strategies are commonly utilized to discretize the flexible system from an infinite degrees-of-freedom system to a finite degrees-of-freedom system. The common discretization strategies mainly include the assumed mode method and finite element method.The motion of the bending arm for the rotary dobby is passed by the links to the heald frame. Its dynamic performance affects the shedding performance of the loom directly. The bending arm can be equivalent to a flexible curved beam model, and it can be discretized by using the finite element method considering its curve characteristics, which provides a physical model for dynamics modeling. It should be noted that, compared with other methods, Kane’s formulation possesses a brief deduction, and the final system equations for this system are very concise and have high calculation efficiency.In this paper, the bending arm of the rotary dobby is discretized into beam elements. Its precise dynamic model is established by using Kane’s formulation. The influence to the shedding performance of the dynamic performance of the bending arm under different working conditions and parameters is analyzed, which provides the premise for improving the dynamic performance of rotary dobby and provides a research foundation for improving the overall performance of the loom.


## 2. System Description

Figure1 shows an illustration of a rotary dobby. It can drive several healds to move together. Because of the consistency of the connection method for the heald frames, a set of heald frames is taken as an example to reduce the complexity of the study. The three-dimensional model of the lifting comprehensive mechanism is shown in Figure 2, which is the core mechanism of the rotary dobby. Figure 3 shows a working principle diagram of the lifting comprehensive mechanism. For the balance of the mechanism and to reduce the impact, two cam swing arms are installed on the disk symmetrically for 180°. For clarity, in Figure 3, one cam swing arm, the matching cam rollers, sliding block, and some other components are omitted.Figure 1
A rotary dobby legend.Figure 2
Three-dimension model of lifting comprehensive mechanism for a rotary dobby.Figure 3
Working principle diagram of lifting comprehensive mechanism for a rotary dobby. 0: dobby box body. 1: conjugate cam. 2: the big disc. 3: cam swing arm. 4: cam roller. 5: the slider. 6: the slider frame. 7: the axis of rotation. 8: eccentric disc. 9: ring connecting link. 10: the bending arm. 11: lifting comprehensive link. 12: the big blade. 13: link of tumbler. 14: link of heald frame. 15: the small blade. 16: the heald frame.The lifting comprehensive mechanism of the rotary dobby is composed of a rotating transmission mechanism for the conjugate cam and the planar linkage mechanism. The planar linkage mechanism consists of two planar four-bar mechanisms in series and the linkage mechanism of the heald frame, as shown in Figure3.The rotating transmission mechanism for the conjugate cam is composed of the conjugate cam (1), the big disc (2), the cam swing arm (3), the cam roller (4), the slider (5), the slider frame (6), and the axis of rotation (7). It is the most essential feature to convert the uniform circular motion of the large disk (2) to the nonuniform circular motion of the axis of rotation (7). The conjugate cam (1) is fixed on the dobby box body (0). An external input motor drives the large disk (2) to do uniform circular motion with the transmission ratio of 2:1. The cam swing arm (3) on the big disc (2) rotates with the large disk while it swings along the cam. The slider (5) is driven by the cam swing arm (3) to rotate around the center of the large disk; at the same time, it slides in the chute of the swing arm (3). The slider (5) is hinged with the slide frame (6). The slider frame (6) is driven by the slider (5) to do nonuniform circular motion around the axis of the big disc. The axis of rotation (7) is fixed with the slider frame (6), and it is connected with the eccentric disc (8) through the spline. The eccentric disc (8) rotates around the axis of the large disc.The crank and rocker mechanism is constituted with the eccentric disc (8), the ring connecting link (9), and the bending arm (10). The circular motion of the eccentric disc (8) drives the bending arm (10) to swing back and forth. The double-rocker mechanism is composed of the bending arm (10), the lifting comprehensive link (11), and the big blade (12). The bending arm (10) drives the big blade (12) to rotate around its rotation axis in a certain angle. Because the width of the heald frame is too large, in order to increase the system stiffness, two groups of the heald frame link (14) and small blade (15) are used to transfer the power. Large blades and small blades drive the links of the tumbler (13) to move in the plane. The heald frame (16) is driven by three parallel links of the tumbler to move up and down, so that the shedding movement of the loom can be finished.By changing the connection position of the lifting comprehensive link (11) and the bending arm (10), the shedding distance can be changed. The bending arm model is shown in Figure 4.Figure 4
Model of bending arm.The bending arm of the rotary dobby drives the heald frame through the links, so its motion performance affects the dynamic performance of the heald frame directly. In this paper, the influence of flexible deformation for the bending arm on the shedding performance is studied. When the dynamic model is established, it is assumed that the components except for the bending arm are rigid, and the stiffness and the friction resistance have no influence on the dynamic response of the system.


### 2.1. Kinematics Analysis

The bending arm can be assumed to be a curve bending beam for a wide range of motion, and the center line of the curved part is a circular arc curve. Kane’s formulation can be used to establish its dynamic model [16–19], and the finite element method is an effective way that can be used to discretize the physical model of curve bending beam components.As shown in Figure5, the bending arm is divided into beam elements. There are 11 nodes and 10 units. b1 and b2 are the unit vectors of the two coordinate axes along the body coordinate system. r1 and r2 are the unit vectors along the two coordinate axes of the inertial coordinate system.Figure 5
Diagram of the mesh for the bending arm.As shown in Figure6, the unit is analyzed. Each unit has two nodes and each node has three degrees of freedom, which are the movement along the two axes of the body coordinate system and the rotation of nodes.Figure 6
Unit model.There are many degrees of freedom when the bending arm is discretized by the finite element method; as a consequence, it is of great difficulty to do dynamical numerical integration, so the dynamic equation cannot be solved. To reduce the degrees of freedom for the system greatly, the physical coordinates of the system are transformed into modal coordinates by using modal synthesis technology, which is expressed as(1)δ=Φq,where q=[q1,q2,…,qN]T represents the modal coordinate array (N is the modal order number); Φ refers to modal shape matrix for the bending arm; δ refers to the displacement column vector of all nodes in the body coordinate system.Figure7 shows the motion diagram of the bending arm in the inertial coordinate system. Starting from the initial position, in the process of movement, the displacement of any point Pk on the bending arm in the inertial coordinate can be expressed as (2)UPk=r+u-r0+u0,where r and r0 are the position vector of the point Pk relative to the body coordinate in the motion process and initial time; u and u0 are the displacement vector because of the deformation for the bending arm.Figure 7
Displacement description of a bending arm in the inertial coordinate system.The components ofu in the two directions of the body coordinate system can be written as(3)uk1=N~1kquk2=N~2kq,where(4)N~1k=N1kBEkΦN~2k=N2kBEkΦ,where N1k and N2k are the unit shape function matrixes; BEk is the element position matrix.The velocity ofPk can be expressed as(5)VPkR=-Ωx2i+w21Tx+N~2kq+N~1kq˙b1+Ωx1i+w11Tx+N~1kq+N~2kq˙b2,where Ω represents the angular velocity of the body coordinate system B in the inertial coordinate system R; (x1i,x2i) is the coordinate of the unit node in the body coordinate system; wp1T is the element of row p, column 1 for the transformation matrix which transforms the element coordinate system to the body coordinate system.q ˙ i   ( i = 1,2 , ⋯ N ) is the derivative of the modal coordinate for the system, which is considered as a generalized velocity. The partial velocity vector VRiPk of Pk can be obtained by doing partial derivative of VRPk with respect to the generalized rate(6)VRiPk=∂VPkR∂q˙i=N~1ikb1+N~2ikb2,where N~1ik and N~2ik are the ith elements of N~1k and N~2k, respectively.The acceleration ofPk in the inertial coordinate system can be given by(7)aRPk=N~1kq¨-2ΩN~2kq˙+-Ω2N~1k-Ω˙N~2kq+-Ω2x1i+w11Tx-Ω˙x2i+w21Tx·b1+N~2kq¨+2ΩN~1kq˙+-Ω2N~2k+Ω˙N~1kq+-Ω2x2i+w21Tx+Ω˙x1i+w11Tx·b2.


### 2.2. Generalized Inertial Force and Generalized Active Force of the System

Based upon the aforementioned kinematic analysis, the inertia force of the unitEk relative to the generalized rate q˙i(i=1,2,⋯N) can be formulated as (8)Fi∗Ek=-∫0LkVRiPk·aRPkρAdx,where ρ is the density of the bending arm; A is the cross-sectional area; Lk is the length of the unit Ek.In conclusion, the generalized inertia force of the bending arm can be obtained:(9)F∗Γ=∑EkF∗Ek.The generalized active force corresponding to unit gravity can be described as(10)EkFi=∫0LkVRiPk·gρAdx.Therefore, the generalized active force corresponding to the overall gravity of the structure can be calculated as(11)FΓ=∑EkFEk.The generalized active force corresponding to the structural elasticity can be written as(12)Fk=-ΦTKSTRΦq=-K~STRq,where KSTR is the stiffness matrix of the finite element for the bending arm. K~STR is the structural modal stiffness matrix.The generalized active force corresponding to the external force is calculated. Node 10 and node 4 are loaded by two external loadsFN1 and FN2, respectively. The corresponding generalized active force can be expressed as (13)FN1=ViPkR·FN1(14)FN2=ViPkR·FN2,where the external force FN1=F1r1+F2r2, FN2=F3r1+F4r2, and F1 and F2 are the component forces of FN1 in the directions of r1 and r2, and F3 and F4 are the component forces of FN2 in the directions of r1 and r2.


### 2.3. Dynamical Equations of the System

The generalized inertial force of the system can be written as(15)F∗=F∗Γ.Likewise, the generalized active force of the system can be described as(16)F=FΓ+Fk+FN1+FN2.By Kane’s formulation(17)F∗+F=0.The dynamic equation described by the modal coordinates can be written in the compact form as(18)Mq¨+Cq˙+Kq=Q,where the parameters can be obtained from(19)M=∑Ek∫0LkN~1kTN~1k+N~2kTN~2kρAdx(20)C=∑Ek∫0Lk2ΩN~2kTN~1k-N~1kTN~2kρAdx(21)K=∑Ek-Ω2N~1kTN~1k-Ω˙N~1kTN~2k+Ω˙N~2kTN~1k-Ω2N~2kTN~2k+K~STR(22)Q=∑Ek-∫0Lk-Ω2x1i+w11TxN~1kT-Ω˙x2i+w21TxN~1kT-Ω2x2i+w21TxN~2kT+Ω˙x1i+w11TxN~2kTρAdx+∫0LkN~1kTg1+N~2kTg2ρAdx+cos⁡αBN~1aT-sin⁡αBN~2aTF1+sin⁡αBN~1aT+cos⁡αBN~2aTF2+cos⁡αBN~1bT-sin⁡αBN~2bTF3+sin⁡αBN~1bT+cos⁡αBN~2bTF4,whereg1 and g2 are the component products for acceleration of gravity g in the directions of b1 and b2. N~1a and N~2a denote the shape function matrix of node 10 in two directions of the body coordinate system, while N~1b and N~2b denote the shape function matrixes of node 4 in two directions of the body coordinate system. αB is the angle between b1 of the body coordinate system and r1 of the inertial coordinate system.So far, the dynamic model of the rigid-flexible coupling lifting comprehensive mechanism for the rotary dobby has been established.


## 2.1. Kinematics Analysis

The bending arm can be assumed to be a curve bending beam for a wide range of motion, and the center line of the curved part is a circular arc curve. Kane’s formulation can be used to establish its dynamic model [16–19], and the finite element method is an effective way that can be used to discretize the physical model of curve bending beam components.As shown in Figure5, the bending arm is divided into beam elements. There are 11 nodes and 10 units. b1 and b2 are the unit vectors of the two coordinate axes along the body coordinate system. r1 and r2 are the unit vectors along the two coordinate axes of the inertial coordinate system.Figure 5
Diagram of the mesh for the bending arm.As shown in Figure6, the unit is analyzed. Each unit has two nodes and each node has three degrees of freedom, which are the movement along the two axes of the body coordinate system and the rotation of nodes.Figure 6
Unit model.There are many degrees of freedom when the bending arm is discretized by the finite element method; as a consequence, it is of great difficulty to do dynamical numerical integration, so the dynamic equation cannot be solved. To reduce the degrees of freedom for the system greatly, the physical coordinates of the system are transformed into modal coordinates by using modal synthesis technology, which is expressed as(1)δ=Φq,where q=[q1,q2,…,qN]T represents the modal coordinate array (N is the modal order number); Φ refers to modal shape matrix for the bending arm; δ refers to the displacement column vector of all nodes in the body coordinate system.Figure7 shows the motion diagram of the bending arm in the inertial coordinate system. Starting from the initial position, in the process of movement, the displacement of any point Pk on the bending arm in the inertial coordinate can be expressed as (2)UPk=r+u-r0+u0,where r and r0 are the position vector of the point Pk relative to the body coordinate in the motion process and initial time; u and u0 are the displacement vector because of the deformation for the bending arm.Figure 7
Displacement description of a bending arm in the inertial coordinate system.The components ofu in the two directions of the body coordinate system can be written as(3)uk1=N~1kquk2=N~2kq,where(4)N~1k=N1kBEkΦN~2k=N2kBEkΦ,where N1k and N2k are the unit shape function matrixes; BEk is the element position matrix.The velocity ofPk can be expressed as(5)VPkR=-Ωx2i+w21Tx+N~2kq+N~1kq˙b1+Ωx1i+w11Tx+N~1kq+N~2kq˙b2,where Ω represents the angular velocity of the body coordinate system B in the inertial coordinate system R; (x1i,x2i) is the coordinate of the unit node in the body coordinate system; wp1T is the element of row p, column 1 for the transformation matrix which transforms the element coordinate system to the body coordinate system.q ˙ i   ( i = 1,2 , ⋯ N ) is the derivative of the modal coordinate for the system, which is considered as a generalized velocity. The partial velocity vector VRiPk of Pk can be obtained by doing partial derivative of VRPk with respect to the generalized rate(6)VRiPk=∂VPkR∂q˙i=N~1ikb1+N~2ikb2,where N~1ik and N~2ik are the ith elements of N~1k and N~2k, respectively.The acceleration ofPk in the inertial coordinate system can be given by(7)aRPk=N~1kq¨-2ΩN~2kq˙+-Ω2N~1k-Ω˙N~2kq+-Ω2x1i+w11Tx-Ω˙x2i+w21Tx·b1+N~2kq¨+2ΩN~1kq˙+-Ω2N~2k+Ω˙N~1kq+-Ω2x2i+w21Tx+Ω˙x1i+w11Tx·b2.


## 2.2. Generalized Inertial Force and Generalized Active Force of the System

Based upon the aforementioned kinematic analysis, the inertia force of the unitEk relative to the generalized rate q˙i(i=1,2,⋯N) can be formulated as (8)Fi∗Ek=-∫0LkVRiPk·aRPkρAdx,where ρ is the density of the bending arm; A is the cross-sectional area; Lk is the length of the unit Ek.In conclusion, the generalized inertia force of the bending arm can be obtained:(9)F∗Γ=∑EkF∗Ek.The generalized active force corresponding to unit gravity can be described as(10)EkFi=∫0LkVRiPk·gρAdx.Therefore, the generalized active force corresponding to the overall gravity of the structure can be calculated as(11)FΓ=∑EkFEk.The generalized active force corresponding to the structural elasticity can be written as(12)Fk=-ΦTKSTRΦq=-K~STRq,where KSTR is the stiffness matrix of the finite element for the bending arm. K~STR is the structural modal stiffness matrix.The generalized active force corresponding to the external force is calculated. Node 10 and node 4 are loaded by two external loadsFN1 and FN2, respectively. The corresponding generalized active force can be expressed as (13)FN1=ViPkR·FN1(14)FN2=ViPkR·FN2,where the external force FN1=F1r1+F2r2, FN2=F3r1+F4r2, and F1 and F2 are the component forces of FN1 in the directions of r1 and r2, and F3 and F4 are the component forces of FN2 in the directions of r1 and r2.


## 2.3. Dynamical Equations of the System

The generalized inertial force of the system can be written as(15)F∗=F∗Γ.Likewise, the generalized active force of the system can be described as(16)F=FΓ+Fk+FN1+FN2.By Kane’s formulation(17)F∗+F=0.The dynamic equation described by the modal coordinates can be written in the compact form as(18)Mq¨+Cq˙+Kq=Q,where the parameters can be obtained from(19)M=∑Ek∫0LkN~1kTN~1k+N~2kTN~2kρAdx(20)C=∑Ek∫0Lk2ΩN~2kTN~1k-N~1kTN~2kρAdx(21)K=∑Ek-Ω2N~1kTN~1k-Ω˙N~1kTN~2k+Ω˙N~2kTN~1k-Ω2N~2kTN~2k+K~STR(22)Q=∑Ek-∫0Lk-Ω2x1i+w11TxN~1kT-Ω˙x2i+w21TxN~1kT-Ω2x2i+w21TxN~2kT+Ω˙x1i+w11TxN~2kTρAdx+∫0LkN~1kTg1+N~2kTg2ρAdx+cos⁡αBN~1aT-sin⁡αBN~2aTF1+sin⁡αBN~1aT+cos⁡αBN~2aTF2+cos⁡αBN~1bT-sin⁡αBN~2bTF3+sin⁡αBN~1bT+cos⁡αBN~2bTF4,whereg1 and g2 are the component products for acceleration of gravity g in the directions of b1 and b2. N~1a and N~2a denote the shape function matrix of node 10 in two directions of the body coordinate system, while N~1b and N~2b denote the shape function matrixes of node 4 in two directions of the body coordinate system. αB is the angle between b1 of the body coordinate system and r1 of the inertial coordinate system.So far, the dynamic model of the rigid-flexible coupling lifting comprehensive mechanism for the rotary dobby has been established.


## 3. Dynamic Response Calculation and Simulation

It should be mentioned that the law of motion for the heald frame is affected by the elastic force of the warp yarns during its movement, so the force size depends on the number of warp yarns, the shedding distance, and some other factors. The calculation process for the force is complex. According to the motion law of the heald frame, it is assumed that the size of the force loaded on the heald frame varies with time, as shown in Figure8. The elastic force loaded on the heald frame reaches the maximum at the extreme position, whose peak value is 690 N.Figure 8
Diagram of the force on the heald frame.The numerical simulation experiment is implemented in the environment of the commercial software MATLAB [20, 21]. Meanwhile, under the same conditions, the commercial software ADAMS is used to do virtual prototype model flexible simulation in terms of the mechanism [22]. They are performed on a computer with an Intel (R) Core (TM) i5-3210M processor (2.50 GHz), 8.00 G system memory, and Windows 10 operating system.Herein, the speed of the loom is 800 r/min, which is the input speed of the dobby at the same time. The thickness of the bending arm for a certain type of rotary dobby is 6 mm. The material is steel, the density is 7800 kg/m3, the modulus of elasticity is 2.07×1011 N/m2, and Poisson's ratio is 0.3. Based on (18), in the numerical calculation, it is assumed that the first four modes can meet the requirements. A flexible bending arm model is established in commercial software ANSYS, and then it is imported into ADAMS to get the dynamic response [23, 24].Node 10 is in the position where the bending arm is connected with the follow-up links; therefore, the motion law of the follow-up links can be obtained by studying the motion law of node 10. The deviation of numerical calculation results and ADAMS simulation results of node 10 in the horizontal and vertical directions can be obtained, which are shown in Figure9. In two cases, the speed changes of node 10 in the horizontal and vertical directions are shown in Figure 10.Figure 9
Displacement deviation of node 10.
(a)
Horizontal displacement deviation (b)
Vertical displacement deviationFigure 10
Velocity contrast of node 10 for two approaches.
(a)
Horizontal velocity contrast of node 10 (b)
Vertical velocity contrast of node 10As we can see from Figures9 and 10, comparisons manifest that the results calculated from the two approaches are close. It can test and verify the correctness of the proposed dynamic model and the reliability of the calculation results is illustrated. The curve of dynamic response calculated by the constructed dynamic model can reflect the vibration nature of the bending arm better.The motion of the bending arm is passed by the links to the heald frame. Node 10 is located at the position of the connection point for the bending arm and the link. Thus, the motion rule of the heald frame can be obtained from the curve of the motion rule for node 10. By numerical calculation, the moving distance of the heald frame is about 140 mm. Figure11 shows the displacement deviation of the heald frame for the rigid bending arm and considering the flexible deformation of the bending arm. Figures 12 and 13 show the speed and acceleration of the heald frame, respectively.Figure 11
Displacement deviation of the heald frame.Figure 12
Change of speed for the heald frame.Figure 13
Change of acceleration for the heald frame.The amplitude of displacement deviation for node 10 in the rigid condition and considering its flexible deformation is 0.33 mm and 0.48 mm in the horizontal and vertical orientation, while the amplitude of displacement deviation for the heald frame is 1.89 mm. It can be seen that when the motion of the bending arm is passed through the links to the heald frame, its motion deviation has great influence on the shedding precision of the mechanism. Therefore, it is of great significance to study the flexible deformation of the bending arm.Due to the inertial force and flexible deformation of the mechanism at high speed, the heald frame will vibrate when it moves. In the initial stage of shedding motion, the displacement deviation is negative, which shows that the shedding motion of the heald frame is delayed compared with the theoretical time due to the flexible nature of the mechanism. Under the influence of inertia force and the warp elastic force on the heald frame, there are positive and negative deviation for the displacement with time going by, and there is great fluctuation for the acceleration all the time.


## 4. Effect of Modal Truncation Order on Dynamic Performance

According to the established dynamic equations, to facilitate computation, different modal orders are intercepted, and the influence of different modal truncation numbers on the motion law of the heald frame is discussed. The speed of the loom is set to be 800 r/min.Compared with the results when the bending arm is rigid, the maximum displacement deviation for the heald frame from the first-order mode to the first-eight-order mode is calculated, respectively.As shown in Figure14, the displacement deviation of the heald frame is generally small. When the modal truncation order number is less than 4, the maximum deviation of the displacement for the heald frame varies greatly. On the contrary, when the modal truncation order number increases from 4 to 8, the maximum displacement deviation is stable.Figure 14
The maximum displacement deviation of different modal truncation order.Intercepting different modal orders, we can get the velocity deviation, the amplitude of velocity deviation, acceleration deviation, and the amplitude of acceleration deviation for the heald frame.As can be seen from Figures15–18, the flexible deformation of the bending arm has a significant effect on the velocity and acceleration of the heald frame. When the mode interception order number increases from 1 to 8, the trend of speed deviation for the heald frame is basically unchanged. After the mode interception order number reaches 4, the velocity deviation amplitude of the heald frame does not change significantly. After the mode interception order number reaches 3, the trend of acceleration deviation for the heald frame is basically unchanged. The amplitude for acceleration deviation is increased by 1.06 times when the mode interception order number increases from 3 to 4. When the mode interception order is greater than 4, the modal truncation order number has little effect on the amplitude for acceleration deviation.Figure 15
Velocity deviation of different modal truncation order.Figure 16
Amplitude of velocity deviation for different modal truncation order.Figure 17
Acceleration deviation of different modal truncation order.Figure 18
Amplitude of acceleration deviation for different modal truncation order.On the basis of the above statements, it is concluded that, in the study of the motion law of the heald frame, the higher the truncation order number is, the more accurate the result is. And in order to save computing resources, the first four modes should be intercepted for the dynamic solution. Moreover, this verifies the correctness of the calculation results for part 3 in this paper to select the first four modes.


## 5. Influence of Different Parameters on the Dynamic Performance

The established rigid-flexible coupling lifting comprehensive mechanism for the rotary dobby can be used to research the dynamic performance. Changing the speed of the dobby, the warp tension, the movement distance of the heald frame, and the thickness of the bending arm, selecting the first four modes to calculate, the influence of the working parameters and structural parameters on the dynamic characteristics of the heald frame is analyzed because of the flexible deformation of the bending arm.


### 5.1. Influence of the Speed for the System on the Dynamic Performance

In order to make comparative analysis about the dynamic performance of the system under different speeds, taking 100 r/min as the distance, the dynamic response of the heald frame is simulated when the input speed is between 400 r/min and 1200 r/min. It is manifested in Figures19 and 20; when the speed is 1200 r/min, the amplitude of displacement deviation of the heald frame is 4.3 mm, which is about 2.3 times as big as the results of the speed of 800 r/min. The change amplitude of acceleration is 19945 m/s2, which is about 1.6 times as big as the results of the speed of 800 r/min.Figure 19
Displacement deviation of the heald frame for 1200 r/min.Figure 20
Acceleration of the heald frame for 1200 r/min.As can be seen from Figures21 and 22, the displacement deviation amplitude of the heald frame increases 0.5 mm approximately when the input speed increases 100 r/min. However, the amplitude of the acceleration for the heald frame is small when the rotational speed is between 400 r/min and 1100 r/min. Even the amplitude of the acceleration is reducing when the speed is about 900 r/min. This is because the acceleration fluctuations also have a complex relationship with the natural frequency of the system. So, on the premise of ensuring the accuracy of the shedding motion of the loom, to avoid the damage to the mechanism because of large acceleration, the loom speed of the rotary dobby machine should not exceed 1100 r/min.Figure 21
Displacement deviation amplitude of the heald frame for different speed.Figure 22
Acceleration amplitude of the heald frame for different speed.


### 5.2. Influence of the Elastic Force for the Warp Yarns on the Dynamic Performance

Under the actual working conditions, the maximum value of the stress on the heald frame increases with the increasing of the number of warp yarns. To study the applicability of the rotary dobby for different types of fabrics, considering Figure8, it is assumed that the force increases to 2 times and 4 times as big as before, respectively. When the input speed of the dobby machine is 800 r/min, the dynamic response of the heald frame is simulated when the peak value for the force from the warp yarns are 1380 N and 2760 N, respectively.From Figure23, and taking Figure 11 into consideration, the conclusion can be obtained: when the peak warp force is 690 N, 1380 N, and 2760 N, the displacement deviation fluctuation range of the heald frame is –0.66 mm~1.23 mm, –1.21 mm~0.92 mm, and –2.19 mm~0.35 mm. As the warp density increases, the fluctuation amplitude of the heald frame displacement deviation does not change much, but the positive deviation gets smaller, and the negative gets bigger. This is because when the warp force gets larger, the obstruction to the heald frame is more obvious.Figure 23
Displacement deviation of the heald frame for different warp force.
(a)
When the peak of force is 1380 N (b)
When the peak of force is 2760 NFrom Figure24, and taking Figure 13 into consideration, we can further discover that when the warp force is doubled, the acceleration amplitude of the frame is 2.1 times as big as before. When the warp force is four times as much, the acceleration amplitude of the frame is 6.7 times as big as before.Figure 24
Acceleration of the heald frame for different warp force.
(a)
When the peak of force is 1380 N (b)
When the peak of force is 2760 NBased on the above analysis, in order to prevent the impact damage to the mechanism or the broken warp yarns because of the excessive acceleration, the warp yarns should not be too dense.


### 5.3. Influence of the Shedding Distance on the Dynamic Performance

The shedding distance of the heald frame can be changed by changing the connection position of the bending arm and the links. Different shedding distance has some influence on the motion law of the heald frame.The dynamic response of the heald frame is calculated, respectively, when the shedding distance of the heald frame is 115 mm and 80 mm, the input speed of the rotary dobby is 800 r/min, and the peak value of the force is 690 N.From the results shown in Figures25 and 26, meanwhile, taking Figures 11 and 13 into consideration, it is observed that when the shedding distance is reduced from 140 mm to 115 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.36 times as against before, while the acceleration amplitude has reduced about 0.26 times. When the shedding distance is decreased from 140 mm to 80 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.86 times as against before; at the same time, the acceleration amplitude has reduced about 0.63 times. Therefore, based on the above analysis, it makes remarkable effects by reducing the moving distance of the heald frame to optimize the shedding performance of the rotary dobby mechanism.Figure 25
Displacement deviation of the heald frame for different shedding distance.
(a)
When the shedding distance is 115 mm (b)
When the shedding distance is 80 mmFigure 26
Acceleration of the heald frame for different shedding distance.
(a)
When the shedding distance is 115 mm (b)
When the shedding distance is 80 mm


### 5.4. Influence of the Thickness for the Bending Arm on Dynamic Performance

The change of the structural parameters of the bending arm can affect its mass, stiffness, moment of inertia of an area, and so on; furthermore, the dynamic performance of the mechanism is changed. The influence of the thickness for the bending arm on the dynamic performance of the heald frame is simulated when the shedding distance of the frame is 140 mm, the input speed of the rotary dobby is 800 r/min, and the peak value of the force is 690 N.From the results shown in Figures27 and 28, taking Figures 11 and 13 into consideration, we can discover that when the thickness of the bending arm is increased from 6 mm to 7 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.14 times as against before, while the acceleration amplitude has reduced about 0.13 times. In addition, when the thickness of the bending arm is increased from 6 mm to 8 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.24 times as against before, while the acceleration amplitude has reduced about 0.24 times. Consequently, taking the above analysis into consideration, the shedding performance of the rotary dobby can be optimized by increasing the thickness of the bending arm under the premise of ensuring the reasonable distance between the adjacent bending arms of the rotary dobby.Figure 27
Displacement deviation of the heald frame for different thickness of the bending arm.
(a)
When the thickness of the bending arm is 7 mm (b)
When the thickness of the bending arm is 8 mmFigure 28
Acceleration of the heald frame for different thickness of the bending arm.
(a)
When the thickness of the bending arm is 7 mm (b)
When the thickness of the bending arm is 8 mm


## 5.1. Influence of the Speed for the System on the Dynamic Performance

In order to make comparative analysis about the dynamic performance of the system under different speeds, taking 100 r/min as the distance, the dynamic response of the heald frame is simulated when the input speed is between 400 r/min and 1200 r/min. It is manifested in Figures19 and 20; when the speed is 1200 r/min, the amplitude of displacement deviation of the heald frame is 4.3 mm, which is about 2.3 times as big as the results of the speed of 800 r/min. The change amplitude of acceleration is 19945 m/s2, which is about 1.6 times as big as the results of the speed of 800 r/min.Figure 19
Displacement deviation of the heald frame for 1200 r/min.Figure 20
Acceleration of the heald frame for 1200 r/min.As can be seen from Figures21 and 22, the displacement deviation amplitude of the heald frame increases 0.5 mm approximately when the input speed increases 100 r/min. However, the amplitude of the acceleration for the heald frame is small when the rotational speed is between 400 r/min and 1100 r/min. Even the amplitude of the acceleration is reducing when the speed is about 900 r/min. This is because the acceleration fluctuations also have a complex relationship with the natural frequency of the system. So, on the premise of ensuring the accuracy of the shedding motion of the loom, to avoid the damage to the mechanism because of large acceleration, the loom speed of the rotary dobby machine should not exceed 1100 r/min.Figure 21
Displacement deviation amplitude of the heald frame for different speed.Figure 22
Acceleration amplitude of the heald frame for different speed.


## 5.2. Influence of the Elastic Force for the Warp Yarns on the Dynamic Performance

Under the actual working conditions, the maximum value of the stress on the heald frame increases with the increasing of the number of warp yarns. To study the applicability of the rotary dobby for different types of fabrics, considering Figure8, it is assumed that the force increases to 2 times and 4 times as big as before, respectively. When the input speed of the dobby machine is 800 r/min, the dynamic response of the heald frame is simulated when the peak value for the force from the warp yarns are 1380 N and 2760 N, respectively.From Figure23, and taking Figure 11 into consideration, the conclusion can be obtained: when the peak warp force is 690 N, 1380 N, and 2760 N, the displacement deviation fluctuation range of the heald frame is –0.66 mm~1.23 mm, –1.21 mm~0.92 mm, and –2.19 mm~0.35 mm. As the warp density increases, the fluctuation amplitude of the heald frame displacement deviation does not change much, but the positive deviation gets smaller, and the negative gets bigger. This is because when the warp force gets larger, the obstruction to the heald frame is more obvious.Figure 23
Displacement deviation of the heald frame for different warp force.
(a)
When the peak of force is 1380 N (b)
When the peak of force is 2760 NFrom Figure24, and taking Figure 13 into consideration, we can further discover that when the warp force is doubled, the acceleration amplitude of the frame is 2.1 times as big as before. When the warp force is four times as much, the acceleration amplitude of the frame is 6.7 times as big as before.Figure 24
Acceleration of the heald frame for different warp force.
(a)
When the peak of force is 1380 N (b)
When the peak of force is 2760 NBased on the above analysis, in order to prevent the impact damage to the mechanism or the broken warp yarns because of the excessive acceleration, the warp yarns should not be too dense.


## 5.3. Influence of the Shedding Distance on the Dynamic Performance

The shedding distance of the heald frame can be changed by changing the connection position of the bending arm and the links. Different shedding distance has some influence on the motion law of the heald frame.The dynamic response of the heald frame is calculated, respectively, when the shedding distance of the heald frame is 115 mm and 80 mm, the input speed of the rotary dobby is 800 r/min, and the peak value of the force is 690 N.From the results shown in Figures25 and 26, meanwhile, taking Figures 11 and 13 into consideration, it is observed that when the shedding distance is reduced from 140 mm to 115 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.36 times as against before, while the acceleration amplitude has reduced about 0.26 times. When the shedding distance is decreased from 140 mm to 80 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.86 times as against before; at the same time, the acceleration amplitude has reduced about 0.63 times. Therefore, based on the above analysis, it makes remarkable effects by reducing the moving distance of the heald frame to optimize the shedding performance of the rotary dobby mechanism.Figure 25
Displacement deviation of the heald frame for different shedding distance.
(a)
When the shedding distance is 115 mm (b)
When the shedding distance is 80 mmFigure 26
Acceleration of the heald frame for different shedding distance.
(a)
When the shedding distance is 115 mm (b)
When the shedding distance is 80 mm


## 5.4. Influence of the Thickness for the Bending Arm on Dynamic Performance

The change of the structural parameters of the bending arm can affect its mass, stiffness, moment of inertia of an area, and so on; furthermore, the dynamic performance of the mechanism is changed. The influence of the thickness for the bending arm on the dynamic performance of the heald frame is simulated when the shedding distance of the frame is 140 mm, the input speed of the rotary dobby is 800 r/min, and the peak value of the force is 690 N.From the results shown in Figures27 and 28, taking Figures 11 and 13 into consideration, we can discover that when the thickness of the bending arm is increased from 6 mm to 7 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.14 times as against before, while the acceleration amplitude has reduced about 0.13 times. In addition, when the thickness of the bending arm is increased from 6 mm to 8 mm, the fluctuation amplitude of the displacement deviation has been decreased 0.24 times as against before, while the acceleration amplitude has reduced about 0.24 times. Consequently, taking the above analysis into consideration, the shedding performance of the rotary dobby can be optimized by increasing the thickness of the bending arm under the premise of ensuring the reasonable distance between the adjacent bending arms of the rotary dobby.Figure 27
Displacement deviation of the heald frame for different thickness of the bending arm.
(a)
When the thickness of the bending arm is 7 mm (b)
When the thickness of the bending arm is 8 mmFigure 28
Acceleration of the heald frame for different thickness of the bending arm.
(a)
When the thickness of the bending arm is 7 mm (b)
When the thickness of the bending arm is 8 mm


## 6. SA of the Displacement Deviation and the Acceleration for the Heald Frame

The SA method is used to improve the performance of the system and the results will be considered to optimize the dobby. Sobol’s sensitivity analysis is one of the well-known statistical methods which is used successfully in mathematical models. The introduction of this method can be found in [25], and it is not described in detail here because of the space limitation. It is reasonable to use this method to make the best decision to optimize and also to improve the performance by analyzing the behavior of the system.In this paper, the SA of the parameters in Section5 towards the displacement deviation and acceleration of the heald frame is conducted by using Sobol’s method, including the speed of the dobby, the warp tension, the movement distance of the heald frame, and the thickness of the bending arm.Table1 presents the intervals of each parameter. Sobol’s sampling method is applied to generate uniform random numbers on intervals presented in Table 1. Using the established dynamic model discussed in the previous section, the displacement deviation and acceleration of the heald frame are obtained with respect to each extracted random number. The results of the SA of the displacement deviation and acceleration are shown in Tables 2 and 3. Also, the pie chart diagrams of the total sensitivity for the displacement deviation and acceleration are illustrated in Figures 29 and 30, respectively.Table 1
Properties of the system.
Parameter Specifications Unit Speed [400,1200] r/min Warp tension [1,4] times Movement distance of the heald frame [80,140] mm Thickness of the bending arm [6,8] mmTable 2
The SA results of the displacement deviation.
Parameter First-order sensitivity Total sensitivity Speed 0.9631 0.34068 Warp tension -0.1897 0.26368 Movement distance of the heald frame 0.7052 0.2110 Thickness of the bending arm 0.0588 0.0015Table 3
The SA results of the acceleration.
Parameter First-order sensitivity Total sensitivity Speed 0.8282 0.2133 Warp tension 2.0116 1.3612 Movement distance of the heald frame 0.9317 0.1953 Thickness of the bending arm 0.1493 0.0064Figure 29
The results of the total sensitivity for the displacement deviation.Figure 30
The results of the total sensitivity for the acceleration.Due to Tables2 and 3, it is understood that the influence of different parameters on the dynamic performance is different. By studying the numerical results of the SA, the relative size of first-order sensitivity and total sensitivity for different parameters is basically the same. But due to Figure 29, we can see that the most effective parameter for the displacement deviation is the speed of the system, and then it is the warp tension and the movement distance of the heald frame. As shown in Figure 30, the most sensitive parameter for the acceleration corresponds to the warp tension, followed by the speed and the movement distance of the heald frame. To discover how each parameter influences the displacement deviation and acceleration for the heald frame, simulations are done and the results are presented in Section 5 of this paper. The results provide an important reference for the optimization of the system.


## 7. Conclusion

In this paper, the rigid-flexible coupling dynamic modeling and analysis of the rigid-flexible coupling lifting comprehensive mechanism for the rotary dobby have been investigated. Some conclusions are drawn as follows:(1)
It is an effective method to analyze the dynamic performance of the rigid-flexible coupling lifting comprehensive mechanism for the rotary dobby by using Kane’s formulation when the beam element is used to discretize the bending arm, and it is convenient to research the dynamic performance of the lifting comprehensive mechanism under different working conditions and parameters by using the constructed dynamical model. It also lays a theoretical foundation for optimizing the dynamic performance of the rotary dobby accurately(2)
According to the established dynamic model, the influence of different modal truncation order number on the dynamic response of the heald frame is studied. The results show that, to balance efficiency and precision, it is appropriate to select the first four modes to calculate(3)
By studying the influence of different parameters on the dynamic performance, it is concluded that the mechanism is suitable for the loom with the speed less than 1100 r/min. The dynamic performance for different number of warp yarns at constant speed shows that too many warp yarns are not conducive to the stability of the heald frame. Decreasing shedding distance can optimize the dynamic performance of the lifting comprehensive mechanism obviously on the premise of satisfying the technological requirements. Increasing the thickness of the bending arm can enhance the stability of the heald frame to some extent(4)
SA is done by using Sobol’s method. The effects of each parameter are studied. It is shown that the most sensitive parameter for the displacement deviation corresponds to the speed. Moreover, the most percentage of sensitivity for the acceleration among all the other sensitivity indices is corresponded to the warp tensionThe abovementioned numerical simulation results are of great significance to the dynamic optimization design of the rotary dobby, and it lays the foundation to improve the overall weaving performance of the loom. With respect to the dynamic modeling method, we are inspired to build the dynamic model of other curved beams.



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*Source: 4896162-2019-06-04.xml*