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A Study on Dynamic Errors of High Speed Press Mechanism with Clearance

Jia Fang, Wang Lei

Abstract — The present paper focuses upon the dynamic errors of high speed press mechanism with clearance. Based on a continuous contact model, the clearance at joint is treated as a rigid linkage, whose length equals the radial clearance and has no mass. The dynamic equation for high speed press is developed via the kinetics universal equation. By comparison of the results between the computer simulation and experimentation computer simulation, clearance and its size as well as the spinning speed of crank are further analyzed concerning the impacts on both the output error and the kinetic responses of high speed press. The investigation is of great significance to the accuracy design of high speed press. Keywords—clearance; characteristics high speed press; dynamic

of high speed press with clearance. Based on the continuous contact model, the clearance at joint is treated as a rigid linkage, whose length equals the radial clearance and has no mass. The dynamic equation for high speed press derives from the kinetics universal equation. Clearance and its size as well as the spinning speed of crank are further analyzed through simulation focusing upon the impacts on the output errors and kinetic responses of high speed press, which contributes significantly to the accuracy design of high speed press. II. MODELING OF MECHANISM-WITH-CLEARANCE

MOVEMENT KINETICS

I. . INTRODUCTION Joint clearances exist inevitably in mechanical systems due to manufacturing error and rigging error. Initially, a clearance size is little and its effect can be neglected. However, when the machine runs, the clearance size will increase, and the collision and vibration in the joint occurs, which leads to the wear and tear of the joint, eventually resulting in the deterioration of the precision of the machine. Over the last years, investigations have been made to estimate the effects on mechanical errors from clearance. Dhande & Chakraborty[1] proposed a stochastic model for mechanical error analysis in several four-bar function generators. This model is known as the equivalent linkage model and takes the impacts of tolerance and clearances into consideration as the net impacts on equivalent link length. Choi et al[2] proposed a clearance vector model and performed the mechanical error analysis of a four-bar path generator with lubricated joints based on the clearance vector model and the sensitivity analysis of eccentricities obtained by the finite difference method. Meng Xianju, Zhang Ce, etc. [3] presented a probabilistic model of motion accuracy for linkages with clearances and performed the analysis and synthesis of reliability of mechanisms with clearance. Song Li & Yang Jian [4] used the kinematics analysis and considered the impacts of error as a random process, and emulated the output motion of planar linkages by Monte-Carlo method. Dong Xia & Wang Kedian [5][6], when considering revolute clearances and linkage length error, applied the concept of probability and statistics theory to the analysis of the movement precision of the complex linkage system of the vacuum circuit breaker. This paper deals with the dynamic mechanical accuracy

A. Modelling of mechanism with clearance at joints In the light of the fact that the time of separation and collision is short, the mechanism is described as a continuous contact model. Clearance at joint is treated as a rigid linkage, whose length equals to the radial clearance and has no mass. The clearance vector model is shown in Fig. 1. In the model, the clearance vector rij represents the relative position between two adjacent links in a joint, and it can be presented as

Fig.1 Clearance vector model

Fig2. Press mechanism with a revolute

(1) where rij=Rj - Ri, Ri and Rj are the radius of the axis shaft and axis sleeve respectively. α is the internal angle between Line OiOj and Axis x.

rij = rij e

jα

1-4244-1358-3/07/$25.00 2007 IEEE

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Based on the clearance vector model described above, the mechanism with joint clearances can be modelled to the multi-bar mechanism without joint clearance showed in Fig2. In the model, the state change of the contacting between the journal and the sleeve is neglected when the equals are derived and numerical integration is performed, then the calculation is simplified. B. Mechanical Kinetics Analysis As shown in Fig.2, there is a high speed press with clearance in the joint of Linkage 2 and Slider 3. l1 and l3 are the length of the crank, linkage and displacement of the slider respectively. ls1, ls2 are the distance from the crank and linkage top to the axes. θ1, θ2 respectively represent the angle of between crank and x axis and the angle between positive direction of the length vector linkage and x axis. ω is the crank angle speed. The seal vector equation and the displacement output expression go as follow:

F si = mi r si M si = J iθi

(i = 1 ~ 3) (i = 1 ~ 2)

(9)

Force of action exists in the high speed press:

(10) Therefore, five forces are in the high speed press: active moment, dissipative moments, gravitational force, inertia forces (moments), and working force. D. The establishment and solution of mechanism movement differential equation Through the kinematical and kinetic analysis on the mechanism, with kinetics universal function, dynamic difference equation function for high speed press is thus developed:

0, 11π θ1 , 2π 6 F (θ1 ) = Fmax , θ ∈ 11π , 2π 1 6

l3 = ∑ li + r

i =1 2

2

(2)

(3) Using equation (2) and equation (3), calculate the time first-order and second-order derivatives. The speed and acceleration equations are

l3 = ∑ li cos θi + r cos α

i =1

∑ Cθ iθiδθi + Cααδα + ∑ (mi gj) δ rsi

i =1 i =1

2

2

l3 = ∑ li + r

i =1

2

(4) (5)

l3 = l1θ1 sin(θ1 θ 2 ) + rα sin(α θ 2 )

l3 = ∑ li + r

i =1

2

(11) where δθi angle of virtual displacement of component i δα angle of virtual displacement of clearance linkage δrsi virtual displacement of the centroids of component i δli virtual displacement of the slider Together with the above equations, equation with too many items can be expressed through combination and simplification as follows:

i =1 i =1

+ ∑ ( mi rsi ) δ rsi + ∑ J iθ iδθ i = T1δθ1 + F (θ1 )δ l3

2

2

(6)

1 l3 = l1θ12 cos θ12 cos θ 2

(

+l2θ 22 + r (α sin αα 2 + α 2 cos αα 2 )

)

(12) The equation is strong non-linear and can only be solved using numerical methods. In this paper, Runge2-kutta is used. III. DYNAMIC MODELING AND RESULTS ANALYSIS Suppose the crank moves with uniform speed and the relevant parameters are as shown in Table 1. The following analysis is on the impacts of the displacement, speed, acceleration of press sliders, when clearances (0.3mm, 0.5mm, 0.8mm) exist at the joint between linkage and slider, crank speed is set as ω =300,500 rpm, and the press is spinning wherein maxim force is achieved. The time steplength is t=0.000005s.

α = fi ( t , θ1 ,θ 2 , α ,θ1 , θ 2 , α )

(7)

C. Mechanical Kinetics Analysis The active moment T1 is applied to the high speed press, and the dissipative moments Tr1, Tr2, Tα are correspondingly applied to crank, linkage and clearance linkage expressed as following:

(8) where Cθi , Cα are corresponding dissipative coefficients. Meanwhile, the inertia forces (moments) are applied to the centroids of the crank, linkage and slide. In the centroids S, the inertia force and inertia moment are as follow:

Tri = Cθ iθ i Tα = Cα α

(i = 1 ~ 2)

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TABLE 1: TABLE I BASIC PARAMETERS OF THE HIGH SPEED PRESS MECHANISM Parameter m i(Kg) Jsi(Kgm2 lsi(mm) l1(mm) ) Crank 1 15 106 14.5 Linkage 2 350 266 14.0319 60 Slider 3 2200

working time of the high speed press, the error change is unstable, sometimes bigger than the other two, while sometimes smaller. That is to say, when the size of the clearance reaches a certain extent, the precision of the working position of the high speed press will become unstable.

A. Impacts of the size of clearance on mechanism error and dynamic mechanical properties Analysis is on impacts of the size of clearance on dynamic mechanical error, when the formula speed of the crank is set as ω=500 rpm, the press is in idle spinning and the maxim force vary, and the clearance r changes.

(a) idle spinning

(a) idle spinning

(b)maxim force Fig.4 Curve of slider speed error

(b)maxim force Fig.3 Curve of slider displacement error

As shown in Fig.3a, the change tendencies of slider displacement error caused by various clearances are basically the same. The absolute values vary, with greater clearance causing greater variation range of slider displacement error and various differences when θ is in different part. For example, around the angles of 70°,88°,276°and 300°, the three curves have greater distance while in other angles, they are closer. As shown in Fig.3b, when the clearance r is different, the change tendencies of the slider displacement error thus caused are basically the same. But under the condition r=0.3mm, in 300°~ 360°work period of the press, when r=0.8mm, the speed error change of the slider is quite stable, while in the other two, violent vibration occurs. When r=0.8mm, the changes of the slider displacement errors are not stable in various cycles, with more complicated unstable factors. Meanwhile, the slider displacement error has changes. Under general conditions, bigger clearance causes bigger slider displacement error. When r equals 0.8mm, during and a short period after the

As shown in Fig.4a, the change tendencies of slider speed error caused by various clearances are basically the same. The peak values around the angles of 70°, 88°, 276° and 300°, vary due to the difference of clearances. The slider speed error caused by greater clearance is thus greater. As shown in Fig.4b, when the clearances are different, the change tendencies of the caused slider speed errors are basically the same. In 300°~ 360° work period of the press, when r=0.3mm, the speed error change of the slider is quite stable, while in the other two, violent vibration occurs. In the range from 50° to 100°, the slider speed errors caused by various clearances have a sudden change in various positions. Attention should be given to the fact that when the high speed press starts and ends its work, the slider speed error caused by various clearances is: r=0.3mm, wherein the slider speed error is big. Under the conditions r=0.5mm and r=0.8mm, the slider speed errors are relatively smaller and they are close to each other.

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speed, the peak values of the corresponding near slider speed are contrary between positive and negative. And greater spinning speed brings about acute peak values.

(a) idle spinning

(a) idle spinning

(b)maxim force Fig. 5 Curve of slider speed error

As shown in Fig. 5a, when the press is in idle spinning, the change tendencies of slider acceleration error caused by various sizes of clearance are basically the same, with a negative and a positive maximum at 88° and 276°. The larger the size of the clearance is, the bigger is the absolute value of the slider acceleration error. As shown in Fig. 5b, when the press is operating with the biggest force, the slider acceleration errors caused by various clearances engender impact maximum value in the angles of 330° and 360°. But when the clearance becomes bigger, the peak value of the slider acceleration error decreases. Under the condition r=0.3mm mm, the variation of clearance angle acceleration is relatively stable in the angles of 300°~ 360° when the press is operating. The other two angles cause violent changes of vibration. Relatively smaller error peak values appear in the angles of 60° and 150°. When the clearances are different, the corresponding error peak values and θ1 the relative angles of crank are also different. B. Impacts of crank speed on the errors of mechanism with clearance and dynamic errors The impacts of crank speed on mechanism dynamic errors is investigated when r equals 0.5mm, the press is in idle spinning and biggest force, and the crank angle speed is set as ω=300,500 rpm. Fig. 6 to Fig. 8 present the error curves of the slider displacement, speed and acceleration caused by the clearances, when the high speed press is under various conditions and the crank is spinning at various speeds. As shown in Fig. 8a, when the press is in idle spinning, ω=300 rpm, the peak values of the slider acceleration error appear in the angles of 120° and 245°. Under the condition ω=500, the peak values of the slider acceleration error appear in the angles of 88° and 276°. In different spinning

(b)maxim force Fig. 6 Curves of slider displacement error at different speeds

As shown in Fig. 6a, when the crank speed is different, the change tendencies of the slider displacement errors are basically the same, with only the difference between absolute values. The greater the crank speed is, the greater the value of slider displacement error and its range are. Meanwhile, when θ1 is in different part, the differences among them also vary. For instance, between the angles from 80° and 300°, the two curves are at a relatively greater distance from each other, while in other parts, they are closer. As shown in Fig. 6b, when the crank speed is different, the change tendencies of the slider displacement errors are basically the same. There is only difference in terms of the absolute values. The greater the crank speed is, the greater the change difference of the slider displacement error and its range is. Meanwhile, when θ1 is in different parts, their value differences vary. For instance, in the parts from 170° to 240°the two curves are closer, while from 110° to 180°, and from 310° to 330° when ω=500 rpm, the slider displacement error has little vibration change, when ω=300 rpm, there is no such change. As shown in Fig. 7a, when the crank speed is different, the change tendencies of the slider speed errors are basically the same. The peak value will change due to the different speed of the crank. Greater speed will cause correspondingly greater value of slider speed error and the range.

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(a) idle spinning

(a) idle spinning

(b)maxim force Fig.7 Curves of slider speed error at different speeds

As shown in Fig. 7a, when the crank speed is different, the change tendencies of the slider speed errors are basically the same. The peak value will change due to the different speed of the crank. Greater speed will cause correspondingly greater value of slider speed error and the range. As shown in Fig. 7b, when the crank speed is different, the change tendencies of the slider speed errors are basically the same. But in the parts of sudden change from 50° to 100° and from 250° to 300°, the greater the crank speed is, the more acute the change process is, with shorter period of time. In the working time of the press, the greater the crank speed is, the bigger the corresponding change difference of the slider speed error is. In the processes from 110°to 180°, and from 310° to 330°, when ω=500 rpm, the slider speed error has a small vibration change, while when ω=300 rpm, there is no such change. As shown in Fig. 8a, when the press is in idle spinning, ω=300 rpm, the peak values of the slider acceleration error appear in the angles of 120° and 245°. Under the condition ω=500 rpm, the peak values of the slider acceleration error appear in the angles of 88° and 276°. In different spinning speed, the peak values of the corresponding near slider speed are contrary between positive and negative. And greater spinning speed brings about acute peak values. As shown in Fig. 8b, when the press is operating with greatest working force, the greater the speed of the crank is, the greater the error difference of the slider acceleration displacement is. In the ranges from 110°to 180° and from 310° to 330°,ω=500 rpm, the slider displacement error has little change of vibration, while when ω=300 rpm,, there is no change.

(b)maxim force Fig.8 Curves of slider acceleration error at different speeds

C. Comparison of the results between the simulation and the experiment A measurement system is adapted to capture the acceleration sign of the slider. The acceleration sign is inverted to the charge sign by the acceleration transducer, then amplified to certain multiple by the electricity amplifier, and analyzed by CRAS signal analysis machine. A JF75G-125A press was measured in the experiment. When the crank speed was set as ω=300 rpm, the acceleration signs of the slider were colleted and analyzed during both idle spinning and pile driving.

(a) idle spinning

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[2]

[3] [4] [5] [6] (b)maxim force Fig.9 Curves of slider acceleration from simulation and experiment [7] [8] [9]

As shown in Fig.9a, the simulation result and experiment result are compared, and the change tendencies of the slider acceleration under the idle spinning condition are basically same. However, the experiment curve possesses more fluctuation trace than the simulation, due to the neglecting to the elastic deformation in the simulation model. As shown in Fig.9b, the simulation result and experiment result are compared, and it can be found that the simulation result is bigger than the experiment result in the same tendency. In the ranges from 110° to 180°, the slider acceleration has little change of vibration, while in the ranges from 310° to 330°, the slider acceleration has acute change of vibration. IV. CONCLUSIONS A continuous contact model of high speed press with clearance has been presented above. The dynamic function for high speed press is developed on the basis of the kinetics universal function. By computer simulation, clearance and its size as well as the spinning speed of crank are further analyzed concerning the impacts on both the output error and the kinetic responses of high speed press, with the conclusions drawn as follows: A) When the clearance size changes, the peak values of the dynamic error of the mechanism change correspondingly. In idle spinning, the greater the clearance, the greater the corresponding peak error value. But when the press operates with its top power, the greater the clearance is, the less the corresponding peak error value, with less acute change as well. B) At various spinning speeds of the crank, when the press is in idle spinning, the greater the crank speed, the bigger the peak values of the slider displacement error, speed error and acceleration error. Furthermore, as the press operates with its top power, the greater the crank speed, the greater the peak error change. And slight changes are observed between 110°to 180° and 310° to 330°. REFERENCES

[1] Dhande S G, Chakraborty J, "Analysis and Synthesis of Mechanical Error in Linkages – A Stochastic Approach, " ASME Journal of Engineering for Industry, 1973,95:672–676

Choi J H, Lee S J , Choi D H, "Tolerance Optimization for Mechanisms with Lubricated Joints, " Multibody System Dynamics 2,1998: 145–168 Meng Xianju, Zhang Ce, Zhan Meijing. "Analytical Model of Accuracy Probability for Linkages with Clearances, " Journal of Machine Design.2004,21(9) :35-37 Song Li,Yang Jian,Cao Weiqing. "The Motion Error Synthesis of Planar Linkages Through Monte-Carlo Method, " Mechanical Science And Technology.1997, 16(3):479-482 Song Li,Yang Jian. "The Emulating Experimental Study of Motion Error of Planar Linkages with Joint Clearance, " Journal of Machine Design. 1998, 7(7):29-31 Dong Xia,Wang Kedian. "A Linkage Model with Revolute and Its Application to Analyzing the Precision of A Complex Linkage, " Mechanical Science And Technology .2005, 24(4):479-483 P Ravn. "A Continuous Analysis Method for Planar Multibody Systems with Joint Clearance" Multibody System Dynamics2., 1998: 1–24 P Flores, J Ambrosio. "Revolute joints with clearance in multibody systems ". Computers and Structures 82 (2004): 1359–1369. P Flores, J Ambrosio, J C P Claro, et al. "A study on dynamics of mechanical systems including joints with clearance and lubrication ". Mechanism and Machine Theory 41 (2006) :247–261.

Jia Fang, became a associate professor in College of Mechanical Engineering in Southeast University in 2003, and a principle investigator for several sponsored projects, such as "the Development of 65T servo Press", and "Development NC-controlled Press with high speed" which sponsored by the Department of Science and Technology of Jiangsu Province.

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