High Speed of Tufting Machine Based on Transfer Matrix Method

DING Caihong()， TONG Yun( )

Colleage of Mechanical Engineering， Donghua University， Shanghai 201620， China

Abstract: In order to achieve high speed of tufting machine， the oval gear is used as the transmission mechanism between the main shaft and the slave shaft. And then，mathematical model of tufting machine spindle system is established by transfer matrix method.Finally， the dynamic reaction force of the connection and the unbalanced response of the joint between the general tufting machine and the improved spindle system are studied by using Matlab numerical simulation.The analysis results show that when the spindle speed reaches 1 000 r/min， the dynamic reaction force of the improved spindle system at the joint is far less than that of the general tufting machine， and the unbalanced response is reduced from 0.22 mm to 0.10 mm.

Key words: tufting machine；high speed； oval gear；dynamic reaction force； unbalanced response；vibration

Introduction

With the development of social economy， the demand for carpets is increasing. The production of carpets depends on the carpet tufting machine， and the most critical part of the speed increase is the spindle system[1].

Jiao and Ren[2]analyzed the influencing factors affecting the fan rotor， and adjust the influencing factors， namely the weight，the position and numerical value of the damper， to reduce the vibration of the fan rotor. Binetal.[3]put forward the whole machine balance method for improving the speed of turbomachinery shafting. This method is based on the multi-plane influence coefficient balance method. It does not need the test mass， only needs the initial response of synchronous vibration， and calculates the multi-plane influence coefficient at different speeds according to the steady-state response prediction of finite element method.Huang[4]used the oval gear instead of the linkage mechanism in the improved tobacco packaging machine， and used the stability of the oval gear to reduce the vibration of the system.Tang and Lu[5]used a balance shaft when balancing the engine crank linkage. Since the crank-link mechanism itself has an imbalance， a corresponding vibration force is generated during operation， and the balance shaft is eccentrically designed to achieve an unbalanced inertial force balance with respect to the link mechanism. Zhang and Cai[6]used the online dynamic balancing technique to increase the rotor speed. Multiple measuring surfaces and alignment faces are selected for multiple operation and correction on the rotor.The unit correction amount on the calibration surface causes the vibration of a certain measurement surface to be an influence coefficient at a certain rotation speed.Thus，the magnitude and position of the weights added to the correction surfaces can be determined according to the vibration caused by the imbalance amount， thereby limiting the vibration value of the measurement surface to a certain value. In summary， Huangetal.[4-5]used the improved structural form approach to achieve good dynamic balance. Jiaoetal.[2-3]achieved dynamic balance by adding weights or reducing mass on the shaft.

This paper analyzes the reasons that affect the high speed of general tufting machine. By establishing the model， the relationship between the magnitude of the reaction force in the tufting machine spindle system and the working speed，and the relationship between the vibration displacement and the working speed are obtained. And also we propose the use of over gear as transmission parts to improve the structure of the spindle system.

1 Structure of General Tufting Machine Spindle System

The spindle system of the carpet tufting machine is the core component of the carpet tufting machine[7]. General tufting machine means the spindle system only includes a needle linkage mechanism， a hook linkage mechanism， a shaft support， a rocker slider mechanism，a main shaft， a needle shaft， and a hook shaft. As shown in Fig.1， one end of the needle link mechanism is fixed to the main shaft， and the other end is fixed to the needle shaft. The main shaft transmits power to the rocking shaft by rotation， and a rocker slider mechanism is fixedly mounted on the needle shaft. The tufting needle at the end of the rocker slider mechanism is driven up and down by the rotation of the needle shaft. Similarly， the hook linkage mechanism is also attached to the main shaft and the hook shaft. Therefore， the rotation of the main shaft also causes the hook shaft to rotate[8].

Fig.1 Schematic diagram of the spindle system

2 Dynamic Response Analysis of New Scheme and Old Scheme

2.1 Analysis of dynamic reaction force of crank connecting rod mechanism

Since the crank mechanism generates a varying inertial force during the movement of the crank-link mechanism， inertial force magnitude and direction are constantly changing，acting on the main shaft， resulting in changes in the dynamic reaction force at the junction between the crank linkage and the main shaft.

The parameters of the existing crank mechanism of the 2-meter machine are shown in Table 1.

Table1 Crank linkage mechanism parameter

The Assur Groups Method[9-10]is used to calculate the dynamic reaction force of the spindle rotation speed of 500 r/min and 1 000 r/min respectively.Therefore， from formula (1)， the reaction force at the joint can be obtained.

CFR=DF，

(1)

whereFandFRare known forces and dynamic reaction forces，DandCare the coefficient matrix of known force and dynamic reaction force.

Figure 2 shows the dynamic reaction force of the linkage between the link mechanism and the spindle at 500 r/min and1 000 r/min.

As shown in Fig.2，the dynamic reaction force when the main shaft rotates at 1 000 r/min is much greater than the dynamic reaction force at 500 r/min.Therefore， when the speed of the tufting machine main shaft is increased， the frequency of the dynamic reaction force is also increased， which will increasethe vibration of the spindle system.In the conventional tufting machine， since the crank-link mechanism used as the transmission member， the vibration of the spindle system is inevitably increased due to the increase in the main shaft rotation speed.

Fig.2 Dynamic reaction of 500 r/min and 1 000 r/min

As the rotational speed increases， the imbalance force becomes larger and larger， and the imbalance of the link mechanism always exists[11].And with the increase of the speed of the tufting machine， the unbalanced excitation of the spindle system will increase.Therefore， long term operation will cause spindle deformation， wear and fatigue damage which may limits the speed increase. Therefore， in order to increase the speed of the tufting machine， a new transmission member can be designed instead of the link mechanism as a transmission member between the main shaft and the slave shaft.

2.2 Analysis of dynamic reaction force of oval gear transmission scheme

In the general tufting machine spindle system， since the crank and rod mechanism is used as the transmission member between the main shaft and the driven shaft， the inertial force of the crank-link mechanism itself causes the dynamic reaction force to increase with the spindle speed. The dynamic reaction force becomes larger and larger， which makes the vibration and noise of the tufting machine larger and larger. From Ref.[12]， it can be seen that the working speed of the ordinary tufting machine， that is， the tufting machine with crank and connecting rod mechanism as transmission parts， is about 500-600 r/min[12].

Therefore， in order to improve the working speed of the tufting machine， an oval gear scheme is adopted instead of the link mechanism as the transmission between the main shaft and the slave shaft. The elliptical gear transmission is stable in transmission at high speed due to the smooth transmission， the position of the meshing point， and the compact structure[13-14].

As shown in Fig.3， it is a schematic diagram of a new type of improved spindle system.In the oval gear transmission scheme， since the oval gear transmission is employed， the rotational speeds of the main shaft and the slave shaft are changed. At the same time， the needle is changed from the original elongated shaft of the shaft to a number of short slave shafts. The oval gear 3 is mounted on the intermediate shaft， and the hook linkage mechanism is mounted on the intermediate shaft and the hook shaft. The hook is connected to the hook shaft， so that the hook reciprocates back and forth. Hence， the power source of the tufting machine is to transmit power to the main shaft through a belt.

For the oval gear transmission scheme， the parameters of the oval gear mounted on the main shaft are shown in Table 2.

Table 2 Oval gear scheme equivalent parameters

Therefore， the dynamic sequence solving[15]is used to analyze the dynamics of the oval gear. Hence， the dynamic reaction curve of the joint at the working speed of 1 000 r/min is obtained， compared with the dynamic reaction force at the connection of the general tufting machine.

Therefore， from formula (2)， the dynamic reaction force at the joint can be obtained.

(2)

By using MATLAB simulation， the dynamic reacting force at the hinge isobtained.As shown in Fig.4， when the working speed of the tufting machine is 1 000 r/min， the dynamic reaction force of the oval gear mechanism is much lower than that of the crank and rod mechanism. The maximum dynamic reaction force is reduced from the original 3 226 N to 1 750 N. This means that the dynamic reaction force between the spindle and the elliptical gear becomes smaller and the acting force on the spindle at the hinge becomes smaller. In conlude， compared with the spindle system of the general tufting machine， the alternating load of the improved spindle system is obviously reduced at high speed.

Fig.4 Dynamic reaction force at the joint

3 Unbalanced Response Analysis

In section 2.1， the dynamics analysis of the linkage mechanism is carried out by using the assur group method， and the relationship between the working speed of the tufting machine and the dynamic reaction force is obtained.In order to more directly express the vibration of the spindle system， the transfer matrix method[16-17]is used to establish a mathematical model for the spindle system which is shown in Fig.1.

The transfer matrix method is to simplify the shafting into lumped parameter model. It is necessary to discretize the continuously distributed elastic system intoNmass plates and massless shaft segments，multi degree of freedom system composed ofLelastic supports.That is to say， the quality of the shafting is concentrated on the nodes to be divided.Division of spindle by disk and lumped mass， from left to right is 1，2，…N， and there are altogetherN+1 sections.The lumped mass unit or disk and the mass-free axis segment are composed of a unit，and the relationship between the left state vector and the right state vector is established，then the transfer matrix is extracted.Finally， the relationship between the state variables of any section and the initial section of the rotor is established by using the continuous condition.The method of establishing the model is called transfer matrix method.

The spindle system is simplified into a lumped parameter model. In the simplification of the spindle system， an equivalent model is required for each module of the tufting machine spindle system and then spliced together to form a simplified overall model.According to the principle of kinetic energy equality and center of mass theorem，the crank and rod mechanism is simplified into two equivalent discs and elastic fixing.They are respectively assigned to the main shaft and the slave shaft.The hook linkage mechanism simplifies it into a single degree of freedom equivalent wheel， and the spindle and slave shaft are simplified into a number of concentrated mass points and massless shaft segments.The rocker slider mechanism is a single degree of freedom equivalent disc according to the principle of equal kinetic energy and the principle of center of mass theorem.In addition，the pulley equates it into an equivalent disc，which is simplified by the bearing into an elastic support.Therefore，the lumped parameter model of the spindle system of tufting machine is shown in Fig.5，and the simplified parameters are shown in Table 3.

Table 3 Spindle system equivalent parameters

In the spindle system of the oval gear transmission， as shown in Fig.6， the simplified method is the same as that of the general tuftingmachine.In the spindle system of an oval gear， the needle shaft is a series of short shaft， and its deformation is extremely small. Therefore， it is only necessary to establish a mathematical model for the spindle and components connected to the spindle when building the mathematical model. Therefore， in the spindle system modeling process， the processing of the spindle is the same as the previous one， and the lumped parameter model is also established for the spindle system.The main shaft is simplified to a number of concentrated mass points and a massless shaft section. The oval gear is simplified to an equivalent wheel and the bearing simplifies an elastic support.The vibration simulation analysis of the mathematical model of the oval gear scheme is carried out by using Matlab.

Fig.5 Lumped parameter model of universal tufting machine spindle system

Fig.6 Lumped parameter model for spindle system of oval gear mechanism

For the spindle system，it is simplified to a total of 23 nodes.A basic element consists of a massless shaft segment and a disk or a massless shaft segment and a lumped mass point.Therefore， total element number is 23.Since the vibration displacement in theYdirection is to be studied， for the general tufting machine， the unbalance forceFiin theYdirection is added at the node 5 and node 19.For an improved spindle system，the unbalanced forceFipositions are node 7，node 10，node 13，node 16，and node 19.The magnitude of the imbalance force has been found in section 2.1.Therefore， according to the transfer matrix method， the transfer relation between the two sides of the disk state vector is[18]

(3)

In this case， the state vector contains vertical displacementy， cross section rotationθ， cross section bending momentMand shear forceQ. Furthermore， the field transfer relationship is

(4)

where，Bireprents field transfer.

The element transfer matrix consisting of formulas (3) and (4) is

(5)

whereTireprents element matrix;Hireprents unbalanced force matrix.

Therefore， the transmission relationship of the whole spindle system is

(6)

whereNreprents total element number;ANreprents total transfer matrix.

Considering the terminal boundary conditions， the terminal boundary state parametersM1andQ1are 0， Then there is an equation the displacementy1of the initial surface 1 can be obtained from Eq. (8)， and returning the result to Eq. (7)， the unbalanced response of each node can be solved.

(7)

At the same time， becauseMN+1=0，QN+1=0.

(8)

According to the mathematical model of the simplified spindle system， the radial vibration displacement of the node under unbalanced excitation is obtained by Matlab simulation.Therefore， node 3 can be taken as the research object， where node 3 is the connection point between the main shaft and the hook shaft， and the corresponding radial displacement of node 3 at 1 000 r/min is extracted.

As shown in Fig.7，it is the vibration displacement of the general tufting machine spindle system and the oval gear spindle system at the lower joint at 1 000 r/min.It can be seen from Fig.7 that when the spindle system of a general tufting machine is used， the maximum deformation displacement of the spindle is about 0.22 mm， which far exceeds the maximum deformation displacement that the spindle can withstand.When using an oval gear spindle system， the maximum deformation displacement of the spindle is approximate 0.1 mm.Therefore， when the working speed of the tufting machine is 1 000 r/min， the oval gear spindle system is superior to the general tufting machine spindle system. Therefore，the oval gear spindle system is a new type of spindle system capable of achieving high speed.

Fig.7 Vibration displacement in Y direction at 1 000 r/min

4 Conclusions

In this paper， the vibration analysis of the transmission mechanism in the general tufting machine is carried out. The dynamic reaction force at the speed of the tufting machine spindle of 500 r/min and 1 000 r/min is obtained， which indicates that when the spindle rotation speed is increased to 1 000 r/min， the reaction force will increase by more than 2 times.Because the imbalance of the linkage mechanism cannot be completely balanced， the oval gear transmission is used instead of the linkage mechanism.It is verified by Matlab that the dynamic reaction force is much smaller than the dynamic reaction force of the link mechanism at 1 000 r/min， and the radial vibration displacement of the main shaft is also smaller than that of the general tufting machine. Therefore， oval gear transmission instead of linkage transmission is a feasible method.