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Modelling and Optimization of Vibratory Finishing Process
Modelling and Optimization of Vibratory Finishing Process F. Hashinioto, Timken Research, The Timken Company, Canton, OH,USA Submitted by D. 6.DeBra (1) Received on January 9,1996
ABSTRACT Vibratory finishing has been employed for final finishing of products, because of the capability of the finishing consistency with lower cost. However, it has taken the trial and error method to find a proper process set-up due to the fact that the fundamcntals of vibratory finishing have not been established yet. This paper describes the fundamental principles of vibratory finishing and proposes the mathematical modelling which makes the prediction of surface roughness and stock removal possible. The validity of the modelling is discussed with experimental results, and an algorithm to design an optimum process of vibratory finishing is proposed.
Kelwords: Vibratory finishing, Barrel, Modelling
1. INTRODUCTION
Vibratory finishing has been widely employed for final surface finishing of products, because of the capability of finishing consistency with considerably lowcr manufacturing cost. So far, many advantages of vibratory finishing in terms of product quality and cost reduction have been reported [1]-[5], however, the systematical fundamentals of vibratory finishing have not been established yet. So, it has taken the trial and error method to find a proper process set-up, even if only surface roughness of incoming parts to be finished is changed. Furthermore, there is no common way to evaluate the vibratory finishing system, including the process set-up conditions, equipment, media, solution, ctc. In order to apply the vibratory finishing technology to the final finishing process for specialty products, it is essential to develop the fundamentals for the control of process parameters such as surface roughness and stock removal.
machines with working capacities of 0.4 m 3 and 1.0 m 3 were ernploycd with various hnds of media. The frequency and amplitude wcre 21 Hz and 5 mm, respectively. The cylindrical specimens, with diameter rangc of 6 mm to 250 mm. made of carburized steels with hardness of 62 HRC wcre tested. Fig.1 shows SEM pictures of finished surface with various process time. The initial surface with roughness of 0.28 pmRa before the operation had grinding grooves with directional marks as a typical ground surfacc[Fig.I(a)]. The grooves were eliminated gradually from the top surfacc[Fig.l(b)] and, after 45 minutcs operation, the grooves disappeared complctcly and the finished surface cnded with an inherent surface texture of vibratory finishing, which had no directional machining marks [Fig.l(c)]. The inherent surface had a constant roughness namcd “roughness limitation Dr” of the vibratory finishing. The roughness limitation Dr
This paper dcscribes the basic rules of the vibratory finishing process and dcrivcs the fundamental formulas as the mathematical modelling of the process that makes the prediction of process paramcters possible. Then, a formula to determine an optimum process time is proposcd. The validity of the modelling is discussed with experimental results, and an algorithm to design an optimum process of vibratory finishing is proposed. 2. BASIC RULES OF VIBRATORY FINISHING In order to understand the characteristics of the vibratory finishing process. the number of experimental tests was carried out. Two bawl-shaped tumbling
Annals of the ClRP Vol. 45/7/1996
( ~ 1 4 5min. 0.06 umRa
(d)180 min. 0.06 umRa
Fig. 1 Surface changes during vibratory finishing
303
depends on system paramctcrs of the vibratory finishing which are the types of equipment, media and compound, work material, vibration modc and amplitude, ctc. Once the finished surface reached thc inhcrcnt surfacc texture, the texture as well as the roughness were maintained throughout the rest of thc proccss. Therefore, after the process time of 1 5 minutes. the operation should be stopped to save timc and reduce cost. These are very typical features and can be summarized as a basic rule of vibratory finishing.
basic rulc rcgarding stock removal can be described as follons: [Rule 31 The vibralory finishing has a constant stock removal rate m in thc stcady statc process.
.
L
Em
3 .c
14 12 10
[Rule 11The finishing surface of components to be treated by vibratory finishing has an inherent surface texture with a constant roughness named "roughness limitation Dr". Fig.2 shows the
surface roughness changes of workpieccs with various initial roughness Ir during the vibratory finishing. It is clearly shown that there is the roughness limitation Dr in the vibratory finishing system. In this case, Dr is about 0.064 pmRa. Where the initial roughness Ir was high, the roughness was rapidly reduced and the change rate was higher than the case of lower initial roughness. The higher roughness during the finishing process also had the higher rate of roughness change, and the rate was gradually reduced as the roughness approached the roughness limitation Dr. The feature is also a very typical characteristic [1].[4], and can be summarized as a basic rulc of the vibratory finishing process. [Rule 21 The greatcr the difference between roughness of finished surface and the roughness limitation Dr is, the faster the rate of roughness change is.
.90 120 150 180 Time min. Fig.3 Stock removal during vibratory finishing (Experimental) 30
0
60
3. MODELLING OF VIBRATORY FINISHING 3.1 Surface roughness
Based on Rules 1 and 2 , assume that the rate of roughness change dRa(t)/dt is proponional to the roughness dflcrcncc betwcn Ra(t) and the roughness limitation Dr, where Ra(0 is the surface roughncss d c r the process time 1. Thercforc. the Rule 2 can be described by the following differential equation.
Solving for the average roughness function Ra(t) from Eq.( 1) givcs 0.6
0.3
9
Since Ha(0) = Ir at t = 0 and R a ( 4 = Dr at t the constant CI and A are obtained from Eq.(2).
0.2 0.1
2 -
60 90 120 150 180 Time min. Fig.2 Roughness changes during vibratory finishing(Experimenta1)
0
30
Fig.3 shows the stock removal in workpiecc diameter with respect to the proccss time. The stock removal pattern consists of transient and steady statc periods. In the transient period the stock removal rate was high, and after that the stock removal rate became a constant in the steady state period. Furthermore, the rougher the initial roughness, the greater was the stock removal rate. It is clear that the transient stock removal rate depends on the initial roughness Ir, but the stock removal rate in the steady state period is maintained at a constant, no matter what the initial roughness Ir is. Therefore, the 304
C1= ( I r - D r )
7-
q
(3)
In Eq.(l), the constant -4 reprcscnts the rate of roughness changc but has thc unit of inverse of timc. So, it is more convenient and rcalistic to represcnt thc constant A by thc following equation: A=-- 1
T
where T is defined as the time constant of the vibratory finishing system and represents the degree of the response time of the system. Now, substituting Eq.(3) and Eq.(5) to Eq.(2), the average roughness hnction
Ra(0 during finishing operation can bc given by the following equation.
--1 R a ( f )= (Ir-Dr).e
--t ~ a ( t= )Re(0) - R a ( f )= ( I f - I h X
Substituting Eq.( 12) into Eq.( 11) gives +Dr
(6) I
This means that the roughness of finishing components can be predicted by Eq.(6) under the known system parameters (time constant T and roughness limitation Dr), if the initial roughness Ir is measured before the operation. The time constant T can be measured by the process time t when the surface roughness becomcs the value of R a m given by the following equation.
Sr(t) = 2 x 4 x ( I r - Dr)(l- e-?)
Also, substituting Eq.(lO) and Eq.(13) into Eq.(8), the total stock removal S f ) in diameter can be wrinen as follows:
--f S ( t )= m . t + 2 x J x ( I r - D r ) ( l - e
Ru(T) = (Ir - Dr)e-l
+ Dr
T)(14)
(7) 3-3. Optimum Process Time
3-2. Stock Removal There are two types of stock removal mechanism in the vibratory finishing process. One is based on the microcutting action with a constant stock removal rate m and is represented by Sm. The other is based on the change of roughness given by Eq46) and is represented by Sr. Therefore, the total stock removal S(0 of the vibratory finishing process can be described as:
The aim of employing vibratory finishing is to obtain the inherent surface texture and to achieve the aimed roughness with the minimum process time and cost. The finishing process should be stopped when the surface roughness reaches the aimed roughness Ar. The process time is defined as the optimum process time Topt. In order to find the optimum time Topr out. substitute Ra(0 = A r and t = Topt into Eq.(6).
--rapt The first term Sm(0 represents the steady slate process, and the second term Sr(0 does the transient process.
Ar=(Ir-Dr)(l-e
T ) + D r (15)
Solving for Top[ from Eq.(21) gives
The Rule 3 can be described as: (9) Solving for Sm(0 from Eq.(9) by using Sm(O)=O at t=O gives
where m is the constant stock removal per unit process time. While the surface roughness Ra(t) is changing until the roughness reaches the roughness limitation Dr, the stock removal Sr(0 will be close to the change of the maximum peak-to-valley height Rt on the surface profile to be finished. The maximum roughness height Rt can be approximately written as Rf ;r 4 Ru . So, thc stock removal Sr(0 in diameter is :
-
where dRt and dRa are the change of maximum roughness height and average roughness, respectively. From Eq.(6), the change of average roughncss dRa(0 is:
Topf = -T .log e
(Ar - Dr)
( I f - Dr)
(16)
where Topt>O. Eq.(16) reveals the minimum time of the vibratory finishing process to achicvc the aimed roughness with the minimum cost. 4. VERIFICATION
OF MODELLING
The system parameters such as the time constant T, the roughness limitation Dr and the constant stock removal rate m can be measured from the experimental results shown in Figs.2 and 3, and thc system parameters measured are: - Tinie constant : T = 15 minutes - Roughness limitation : Dr = 0.064 pmRa - Constant stock removal rate : m = 0.05 pdmin. Fig.4 shows the mathematical model simulations calculated by Eq.(6) with the above system parameters and is corresponding to the test results of Fig.2. The results of theoretical simulations for roughness changes during the vibratory finishing process accurately coincide with the experimental results.
305
tcsturc was crcrltcd as shown in Fig.l(c). The final roughncss and stock removal predicted by thc modclling are 0.076 pmRa and 3.8 pm in diamctcr. rcspcctivcly.
0.8
2E
0.6
The modelling proposed hcrc not only makes the accurate prcdiction of process parameters possible, but also mathematically describes the characteristics of vibratory finishing summarized as basic rules. Furthermore, the modelling has a capability to design the optimum process time.
3
0.4 E
c
0.2 LY
0 90 120 150 180 Time min. Fig.4 Simulations of roughness changes (Theoretical)
0
30
60
( I ) The basic rules of vibratory finishing are clarified. and the fundamentals are established. (2) The mathematical modelling developed here makes the accurate prediction of roughness and stock removal of finished products possible. (3) The optimum finishing process is defined in terms of minimum process time and cost to achieve the aim. and the formula to design the optimum process is newly proposed.
10
.E
5. CONCLUSIONS
6. NOMENCLATURE
90 120 150 180 Time min. Fig.5 Simulations of stock removal process (Theoretical)
0
30
60
: constant : constant : aimed averagc roughncss in Ra : roughncss limitation in Ra : initial roughness in Ra : constant stock removal rate in diamcter : average roughness function in Ra : average roughncss changc in Ra : maximum roughness height in Rt : masimum roughness height change in Rt : stock removal function : stock rcmovcd in stcady statc process : stock removal due to surface roughness change : finishing process time : time constant of vibratory finishing system : optimum proccss timc to achicvc the aimcd average roughness Ar
'g
80 70 I60 50 i= VI 40 v)
g
80 30 h 20
k 0" E .=
10
0
0.25 0.5 0.75 1 1.25 1.5 Initial Roughness Ir umRa Fig.6 Diagram for optimum process time
0
Fig5 also shows the theoretical results of simulations for the stock removal tcst results, and Eq.(14) with the above system parameters is used for the simulations. The theoretical results agree with thc expcrimcntal results shown in Fig.3 vcry wcll.
7. REFERENCES
Fig.6 is the diagram to determine the optimum proccss time according to the initial roughncss Ir of incoming workpieces. In this example, the aimed roughness Ar of 0.076 pmRa is used in Eq.(16). From this diagram, in the case of Fig.1, it is easily found that the optimum process time Topr is 1 3 minutes where the initial roughness I r of incoming workpieces is 0.28 pmRa. The optimum process time corresponds to the time when the ground marks disappeared and the inhercnt surfacc
[3] Gillespie,L.K., 1975, A Quantitative Approach to Vibratory Deburring Effectiveness,SME, MRR75-
306
[ 11 Babichcv,A.P.,1964, The Vibratory Polishing of Turbine Blades, Machine & Tooling, 35: 41-44 [ 2 ] Dargis,R.G., 1983, Chemical Acceleratcd-Vibratory
Deburring of Stcel Parts, SME,MR83-913 11
of Centrifugal Barrcl Machines for Polishing of Stcel Parts, Kikai Gijyutsu (Japanese),Vol.36-9:;17-51 [j] Safrenek.W.H., Secrest, A.C.. Tum.J.C.,1976, Chemical Accelerators for Vibratory Finishing, Manufacturing Engineering Nov. :32-33 [J] 0hn0,I.. Trend of Development
ABSTRACT Vibratory finishing has been employed for final finishing of products, because of the capability of the finishing consistency with lower cost. However, it has taken the trial and error method to find a proper process set-up due to the fact that the fundamcntals of vibratory finishing have not been established yet. This paper describes the fundamental principles of vibratory finishing and proposes the mathematical modelling which makes the prediction of surface roughness and stock removal possible. The validity of the modelling is discussed with experimental results, and an algorithm to design an optimum process of vibratory finishing is proposed.
Kelwords: Vibratory finishing, Barrel, Modelling
1. INTRODUCTION
Vibratory finishing has been widely employed for final surface finishing of products, because of the capability of finishing consistency with considerably lowcr manufacturing cost. So far, many advantages of vibratory finishing in terms of product quality and cost reduction have been reported [1]-[5], however, the systematical fundamentals of vibratory finishing have not been established yet. So, it has taken the trial and error method to find a proper process set-up, even if only surface roughness of incoming parts to be finished is changed. Furthermore, there is no common way to evaluate the vibratory finishing system, including the process set-up conditions, equipment, media, solution, ctc. In order to apply the vibratory finishing technology to the final finishing process for specialty products, it is essential to develop the fundamentals for the control of process parameters such as surface roughness and stock removal.
machines with working capacities of 0.4 m 3 and 1.0 m 3 were ernploycd with various hnds of media. The frequency and amplitude wcre 21 Hz and 5 mm, respectively. The cylindrical specimens, with diameter rangc of 6 mm to 250 mm. made of carburized steels with hardness of 62 HRC wcre tested. Fig.1 shows SEM pictures of finished surface with various process time. The initial surface with roughness of 0.28 pmRa before the operation had grinding grooves with directional marks as a typical ground surfacc[Fig.I(a)]. The grooves were eliminated gradually from the top surfacc[Fig.l(b)] and, after 45 minutcs operation, the grooves disappeared complctcly and the finished surface cnded with an inherent surface texture of vibratory finishing, which had no directional machining marks [Fig.l(c)]. The inherent surface had a constant roughness namcd “roughness limitation Dr” of the vibratory finishing. The roughness limitation Dr
This paper dcscribes the basic rules of the vibratory finishing process and dcrivcs the fundamental formulas as the mathematical modelling of the process that makes the prediction of process paramcters possible. Then, a formula to determine an optimum process time is proposcd. The validity of the modelling is discussed with experimental results, and an algorithm to design an optimum process of vibratory finishing is proposed. 2. BASIC RULES OF VIBRATORY FINISHING In order to understand the characteristics of the vibratory finishing process. the number of experimental tests was carried out. Two bawl-shaped tumbling
Annals of the ClRP Vol. 45/7/1996
( ~ 1 4 5min. 0.06 umRa
(d)180 min. 0.06 umRa
Fig. 1 Surface changes during vibratory finishing
303
depends on system paramctcrs of the vibratory finishing which are the types of equipment, media and compound, work material, vibration modc and amplitude, ctc. Once the finished surface reached thc inhcrcnt surfacc texture, the texture as well as the roughness were maintained throughout the rest of thc proccss. Therefore, after the process time of 1 5 minutes. the operation should be stopped to save timc and reduce cost. These are very typical features and can be summarized as a basic rule of vibratory finishing.
basic rulc rcgarding stock removal can be described as follons: [Rule 31 The vibralory finishing has a constant stock removal rate m in thc stcady statc process.
.
L
Em
3 .c
14 12 10
[Rule 11The finishing surface of components to be treated by vibratory finishing has an inherent surface texture with a constant roughness named "roughness limitation Dr". Fig.2 shows the
surface roughness changes of workpieccs with various initial roughness Ir during the vibratory finishing. It is clearly shown that there is the roughness limitation Dr in the vibratory finishing system. In this case, Dr is about 0.064 pmRa. Where the initial roughness Ir was high, the roughness was rapidly reduced and the change rate was higher than the case of lower initial roughness. The higher roughness during the finishing process also had the higher rate of roughness change, and the rate was gradually reduced as the roughness approached the roughness limitation Dr. The feature is also a very typical characteristic [1].[4], and can be summarized as a basic rulc of the vibratory finishing process. [Rule 21 The greatcr the difference between roughness of finished surface and the roughness limitation Dr is, the faster the rate of roughness change is.
.90 120 150 180 Time min. Fig.3 Stock removal during vibratory finishing (Experimental) 30
0
60
3. MODELLING OF VIBRATORY FINISHING 3.1 Surface roughness
Based on Rules 1 and 2 , assume that the rate of roughness change dRa(t)/dt is proponional to the roughness dflcrcncc betwcn Ra(t) and the roughness limitation Dr, where Ra(0 is the surface roughncss d c r the process time 1. Thercforc. the Rule 2 can be described by the following differential equation.
Solving for the average roughness function Ra(t) from Eq.( 1) givcs 0.6
0.3
9
Since Ha(0) = Ir at t = 0 and R a ( 4 = Dr at t the constant CI and A are obtained from Eq.(2).
0.2 0.1
2 -
60 90 120 150 180 Time min. Fig.2 Roughness changes during vibratory finishing(Experimenta1)
0
30
Fig.3 shows the stock removal in workpiecc diameter with respect to the proccss time. The stock removal pattern consists of transient and steady statc periods. In the transient period the stock removal rate was high, and after that the stock removal rate became a constant in the steady state period. Furthermore, the rougher the initial roughness, the greater was the stock removal rate. It is clear that the transient stock removal rate depends on the initial roughness Ir, but the stock removal rate in the steady state period is maintained at a constant, no matter what the initial roughness Ir is. Therefore, the 304
C1= ( I r - D r )
7-
q
(3)
In Eq.(l), the constant -4 reprcscnts the rate of roughness changc but has thc unit of inverse of timc. So, it is more convenient and rcalistic to represcnt thc constant A by thc following equation: A=-- 1
T
where T is defined as the time constant of the vibratory finishing system and represents the degree of the response time of the system. Now, substituting Eq.(3) and Eq.(5) to Eq.(2), the average roughness hnction
Ra(0 during finishing operation can bc given by the following equation.
--1 R a ( f )= (Ir-Dr).e
--t ~ a ( t= )Re(0) - R a ( f )= ( I f - I h X
Substituting Eq.( 12) into Eq.( 11) gives +Dr
(6) I
This means that the roughness of finishing components can be predicted by Eq.(6) under the known system parameters (time constant T and roughness limitation Dr), if the initial roughness Ir is measured before the operation. The time constant T can be measured by the process time t when the surface roughness becomcs the value of R a m given by the following equation.
Sr(t) = 2 x 4 x ( I r - Dr)(l- e-?)
Also, substituting Eq.(lO) and Eq.(13) into Eq.(8), the total stock removal S f ) in diameter can be wrinen as follows:
--f S ( t )= m . t + 2 x J x ( I r - D r ) ( l - e
Ru(T) = (Ir - Dr)e-l
+ Dr
T)(14)
(7) 3-3. Optimum Process Time
3-2. Stock Removal There are two types of stock removal mechanism in the vibratory finishing process. One is based on the microcutting action with a constant stock removal rate m and is represented by Sm. The other is based on the change of roughness given by Eq46) and is represented by Sr. Therefore, the total stock removal S(0 of the vibratory finishing process can be described as:
The aim of employing vibratory finishing is to obtain the inherent surface texture and to achieve the aimed roughness with the minimum process time and cost. The finishing process should be stopped when the surface roughness reaches the aimed roughness Ar. The process time is defined as the optimum process time Topt. In order to find the optimum time Topr out. substitute Ra(0 = A r and t = Topt into Eq.(6).
--rapt The first term Sm(0 represents the steady slate process, and the second term Sr(0 does the transient process.
Ar=(Ir-Dr)(l-e
T ) + D r (15)
Solving for Top[ from Eq.(21) gives
The Rule 3 can be described as: (9) Solving for Sm(0 from Eq.(9) by using Sm(O)=O at t=O gives
where m is the constant stock removal per unit process time. While the surface roughness Ra(t) is changing until the roughness reaches the roughness limitation Dr, the stock removal Sr(0 will be close to the change of the maximum peak-to-valley height Rt on the surface profile to be finished. The maximum roughness height Rt can be approximately written as Rf ;r 4 Ru . So, thc stock removal Sr(0 in diameter is :
-
where dRt and dRa are the change of maximum roughness height and average roughness, respectively. From Eq.(6), the change of average roughncss dRa(0 is:
Topf = -T .log e
(Ar - Dr)
( I f - Dr)
(16)
where Topt>O. Eq.(16) reveals the minimum time of the vibratory finishing process to achicvc the aimed roughness with the minimum cost. 4. VERIFICATION
OF MODELLING
The system parameters such as the time constant T, the roughness limitation Dr and the constant stock removal rate m can be measured from the experimental results shown in Figs.2 and 3, and thc system parameters measured are: - Tinie constant : T = 15 minutes - Roughness limitation : Dr = 0.064 pmRa - Constant stock removal rate : m = 0.05 pdmin. Fig.4 shows the mathematical model simulations calculated by Eq.(6) with the above system parameters and is corresponding to the test results of Fig.2. The results of theoretical simulations for roughness changes during the vibratory finishing process accurately coincide with the experimental results.
305
tcsturc was crcrltcd as shown in Fig.l(c). The final roughncss and stock removal predicted by thc modclling are 0.076 pmRa and 3.8 pm in diamctcr. rcspcctivcly.
0.8
2E
0.6
The modelling proposed hcrc not only makes the accurate prcdiction of process parameters possible, but also mathematically describes the characteristics of vibratory finishing summarized as basic rules. Furthermore, the modelling has a capability to design the optimum process time.
3
0.4 E
c
0.2 LY
0 90 120 150 180 Time min. Fig.4 Simulations of roughness changes (Theoretical)
0
30
60
( I ) The basic rules of vibratory finishing are clarified. and the fundamentals are established. (2) The mathematical modelling developed here makes the accurate prediction of roughness and stock removal of finished products possible. (3) The optimum finishing process is defined in terms of minimum process time and cost to achieve the aim. and the formula to design the optimum process is newly proposed.
10
.E
5. CONCLUSIONS
6. NOMENCLATURE
90 120 150 180 Time min. Fig.5 Simulations of stock removal process (Theoretical)
0
30
60
: constant : constant : aimed averagc roughncss in Ra : roughncss limitation in Ra : initial roughness in Ra : constant stock removal rate in diamcter : average roughness function in Ra : average roughncss changc in Ra : maximum roughness height in Rt : masimum roughness height change in Rt : stock removal function : stock rcmovcd in stcady statc process : stock removal due to surface roughness change : finishing process time : time constant of vibratory finishing system : optimum proccss timc to achicvc the aimcd average roughness Ar
'g
80 70 I60 50 i= VI 40 v)
g
80 30 h 20
k 0" E .=
10
0
0.25 0.5 0.75 1 1.25 1.5 Initial Roughness Ir umRa Fig.6 Diagram for optimum process time
0
Fig5 also shows the theoretical results of simulations for the stock removal tcst results, and Eq.(14) with the above system parameters is used for the simulations. The theoretical results agree with thc expcrimcntal results shown in Fig.3 vcry wcll.
7. REFERENCES
Fig.6 is the diagram to determine the optimum proccss time according to the initial roughncss Ir of incoming workpieces. In this example, the aimed roughness Ar of 0.076 pmRa is used in Eq.(16). From this diagram, in the case of Fig.1, it is easily found that the optimum process time Topr is 1 3 minutes where the initial roughness I r of incoming workpieces is 0.28 pmRa. The optimum process time corresponds to the time when the ground marks disappeared and the inhercnt surfacc
[3] Gillespie,L.K., 1975, A Quantitative Approach to Vibratory Deburring Effectiveness,SME, MRR75-
306
[ 11 Babichcv,A.P.,1964, The Vibratory Polishing of Turbine Blades, Machine & Tooling, 35: 41-44 [ 2 ] Dargis,R.G., 1983, Chemical Acceleratcd-Vibratory
Deburring of Stcel Parts, SME,MR83-913 11
of Centrifugal Barrcl Machines for Polishing of Stcel Parts, Kikai Gijyutsu (Japanese),Vol.36-9:;17-51 [j] Safrenek.W.H., Secrest, A.C.. Tum.J.C.,1976, Chemical Accelerators for Vibratory Finishing, Manufacturing Engineering Nov. :32-33 [J] 0hn0,I.. Trend of Development