The Implementation of Optimum MRR on Digital PC-Based Lathe System
Tian-Syung Lan1 and Kuei-Shu Hsu2 1
Department of Mechanical Engineering, De Lin Institute of Technology, Tuchen, Taiwan
236, R.O.C. 2
Department of Mechanical and Automation Engineering, Kao Yuan University. Lu-Chu Hsiang, Kaohsiung 821, Taiwan, R.O.C. E-mail:
[email protected]
Abstract Optimizing the profit of an individual cutting tool is crucial to the computer numerical controlled (CNC) machining industry. In this paper, the mathematical modeling, the dynamic solution, and the decision criteria through Calculus of Variations are introduced to achieve the optimal material removal rate (MRR) control of a cutting tool under the fixed tool life. To realize the optimum MRR, a commercialized lathe system with a DSP (Digital Processor Controller) and a man-machine interface is developed. Additionally, the implementation of dynamic MRR control for a real-world industrial case is experimentally performed on our proposed digital PC-based lathe system. It is found that the surface roughness of all machined work-pieces not only stabilizes as the tool consumed, but also accomplishes the recognized standard for finish turning. In this study, the adaptability of the dynamic control of optimum MRR as well as the realization of the digital PC-based lathe system are absolutely guaranteed.
Keywords: material removal rate; digital lathe control, man-machine interface; surface roughness *
Corresponding author: Kuei-Shu Hsu
Mailing address: Department of Mechanical and Automation Engineering, Kao Yuan
University, Lu-Chu Hsiang, Kaohsiung 821, Taiwan, R.O.C. E-mail:
[email protected]
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1. List of Symbols
a = average volume of material machined per unit part.
B = upper speed limit of MRR. bM ′(t ) = marginal operation cost at the material removal rate M ′(t ) ; where b is a constant. bM ′ 2 (t ) = operational cost at time t. c = overall holding cost of unit chip per unit time, including chip holding and finished
part holding costs. M (t ) = cumulated volume of material machined during time interval [0, t ] . M ′(t ) = MRR at time t. P = revenue per unit part machined. T = tool life for the dynamic model, which denotes the cumulated machining time before a tool replacement. ~ t = time when the optimum MRR reaches the upper limit B .
2. Introduction
The material removal rate (MRR) is an important control factor of machining operation [1, 2], and the control of machining rate is also crucial for production planners. Since the modern computer numerical controlled (CNC) machines are extensively used to perform various tasks, e.g., from job shops to flexible manufacturing systems (FMS), the problem of optimal machining control of CNC has received considerable attention [3].
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Therefore, the necessity of finding the optimal MRR control for maximum profit as well as the implementation of dynamic optimization for an individual tool in the CNC machine cannot be overlooked. The mathematical modeling and the optimal solution of MRR were presented in [1]. This paper is aimed at developing a PC-based machining system to achieve dynamic MRR control by manipulating the feed-rate in accordance with constant depth of cut. Fuh et al. [4] designed a variable structure system (VSS) controller for CNC turning machines. Rober and Shin [5] developed a PC-based open architecture controller that can override the pre-programmed (???) feed rate of the CNC milling machines. In [6], Kim and Kim proposed an adaptive cutting force control approach for machining centers based on direct cutting force measurement. In addition, in [7, 8, 9, 10], the idea of adaptive control has been applied to help select the optimal feed-rate. Generally, the basic objective of adaptive control is to maintain consistent performance of a system in the presence of uncertainty or unknown variation in plant parameters. However, these existing on-line control schemes are all considerably expensive. Moreover, none of them is guaranteed to achieve the maximum profit. On the other hand, many researchers employed the direct-drive method to rotate the ball-screw shaft, in which continuous adjustment of feed-rate can be guaranteed. However, it has been reported that the performance of the direct-drive method is hampered by some practical limitations. The CNC has an feed-rate adjustment function that can be used to adjust the value of desired federate in percentage, which is often done manually using a setting dial on the control panel.
In order to incorporate the digital
PC-based system into commercialized CNC machine tools, this study develops the interface between the digital PC-based control method and the CNC machines. Recently, the profit optimization of an individual cutting tool in machining
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operation has become an important issue for CNC machining industry. In this paper, the mathematical modeling and dynamic MRR control to achieve the optimal machining profit for a cutting tool during a fixed life time is introduced. To realize the dynamic solution, the control scheme on a commercialized lathe system with DSP (Digital Processor Controller) and a Man-machine Interface is developed. Additionally, a real-world industrial machining task is performed on our proposed digital PC-based lathe system.
3. Theoretical Background
In the previous research [1], the cutting process is regarded as a continuous single-tool turning operation without breakdown. The data for both the fixed tool life time and the upper MRR limit are obtained from the maximal machining conditions suggested in the machining handbook. In general, if the machine is operated within the tool life time, it will not break even with the highest machining rate. The machining costs are divided into two different categories [1, 2], the operational cost and the holding cost. While the marginal operation cost is a linear increasing function of the production rate [11], the operational cost of the machine is directly proportional to the square of the MRR [1, 2]. In most manufacturing cases, all chips from cutting and the finished products are usually held and stored at the machine until tool replacement. All parts are paid at a given price after machining operation [1, 2]. In [1], P
M (T ) is used to describe the contribution of one tool under machining a
operation with a fixed life time T.
∫
T 0
bM ' 2 (t )dt and
∫
T 0
cM (t )dt represent the
operational cost and the overall holding cost during time interval [0, T ] , respectively. 4
Therefore, the mathematical model and its constraints [1] are listed as below.
⎧ ⎪Max ⎪ ⎪ ⎪ s.t. ⎪⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩
⎧ M (T ) ⎨P a ⎩
∫ [bM ' T
2
0
]
⎫ (t ) + cM (t ) dt ⎬ ⎭
M (0) = 0 M (T ) is free 0 ≤ M ' (t ) ≤ B ∀ t ∈ [0, T ]
′ Let M * (t ) be the optimal solution of the model, and assume that time interval
(0, ~t )
is the maximal subinterval of [0,T] to satisfy Euler Equation [11, 12]. There are
two feasible cases to be discussed.
′ 3.1 Case 1: M * (t ) will not touch B before T.
In this case, it is assumed that M ' (t ) will never reach the upper speed limit B before tool life time T . From Euler Equation [11, 12], the transversality condition of salvage value for free M (T ) [11, 12], and the boundary conditions, one can obtain the optimal solution [1] for Case 1 as ′ c 1 P M * (t ) = t + ( − cT ) 2b 2b a M * (t ) =
(1)
c 2 1 P t + ( − cT )t 4b 2b a
(2)
Before solving the other case, one property is proposed [1, 2] and discussed as follows:
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′ PROPERTY: If M * (t ) touches the upper MRR limit B , it will stay to be B from the
touch time ~ t to the end of the event time T. Proof:
′ From Eq.(1), M * (t ) is a strictly increasing linear function of t. And it holds ′ ′ for the subinterval of [0, ~ t ) subject to 0 ≤ M * (t ) ≤ B . Since M * (t )
cannot contradict Euler Equation [11, 12] to be a decreasing function of t, the property is verified.
′ 3.2 Case 2: M * (t ) will touch B before T.
In this case, we assume that M ' (t ) will reach the upper limit B at time ~ t . From the transversality condition of salvage value for free end value M (t~ ) [11, 12], and the PROPERTY, one will have 1 P ~ t = T − ( − 2bB) c a
(3)
Therefore, the optimal solution [1] for Case 2 is shown as follows: 1 P ⎧c t t] + ( − cT ) , if t ∈ [0, ~ ⎪ 2b 2b a M (t ) = ⎨ ⎪ t ,T ] , if t ∈ (~ ⎩B
(4)
⎧c 2 1 P ~ ⎪ 4b t + 2b ( a − cT )t , if t ∈ [0, t ] M (t ) = ⎨ ⎪ ~ ~ , if t ∈ (~ t ,T ] ⎩M ( t ) + B(t − t )
(5)
*′
*′
3.3 Decision Criteria From Eq.(3), two decision criteria are classified. 1.
When
P ≤ 2bB , it means that ~ t ≥ T . This contradicts the assumption of Case 2. a
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That is, the optimal control of MRR will not reach the upper speed limit within the tool life time. The optimal solution is Case 1. 2.
When
P > 2bB , it means that ~ t < T . That is, the optimal control of MRR will a
reach the upper speed limit within the tool life time. The optimal solution is Case 2.
4. The Implementation A schematic diagram of the experimental set-up is illustrated in Figure 1. The overall experimental set-up consists of a commercialized lathe system, a DSP processor controller, and a Man-machine Interface system. The characteristics of each component are given as below:
4.1 The Digital Lathe System The layout of the digital lathe system via PC-based control for realization of the optimum MRR control through the Man-machine interface controller is illustrated in Figure 1, in which the digital lathe system consists of four components: the user interface, networking, simulation, and Lathe control scheme.
PMC32-6000
Pulse Input
Motion Control
PC
Human Machine Interface
Encoder Output
Feed AC Servo Motor
PCI Bus
Frequency Transformer 232/485
Frequency Input Rotation Rate
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Spindle induced Motor
Figure 1: Layout of the digital lathe system.
The characteristics of each component are described as below:
1. The user interface provides programs with the following functions: ● Generation of the desired reference input and calculation of the feedback
information generated from the commanded input for the feed actuator servo systems. ● Providing the parameter setting of spindle motor system based on the theoretical
development.
2. The networking provides communication protocol programs with the following functions: Modbus, the industrial communication network equipment, generally consists of one master side and many slave sides maneuvered by a specific address. In the process of communication operation, the master side delivers a serial command signal to the slave side and waits to receive feedback signals from the slave side after the subsequent command is provided. The slave side obtains the request signal for a start to verify the assigned number with its address number, and the master side demands to deliver the data signal with the assigned number correctly. In other words, if there is error message in the communication process, the slave side does not deliver any returned pass corresponding data to the master side and send a message error to check the mistakes instead. Generally, the Modbus has two parts: RTU and ASCII modules. RTU module supports the binary data format and 16 bit unsigned integer and it is used to cope with large data material by big-endian. The CRC (Cyclical Redundancy Check) is adopted
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to examine error in communication format. In this paper, the ASCII module is used in less quantity with character format communication and LRC (Longitudinal Redundancy Check) in error examination. 3. The digital Lathe control scheme for maneuvering the cutting has the following characteristics: A servo motor with screw gears for driving the cutting tool base in the feed directions, and the frequency transformer with induced motor manipulated for the spindle rate via 232-485 converter by the PC-based are included in the proposed lathe re-equip system. The PMC32-6000 motion control card supports the pulse output. There is only one output signal wired for each channel, and the card also supports quadrature encoder which encodes the position of the spindle shaft of the motors for encoding, pulse/direction counting or up/down counting.
4.2 The Man-Machine Interface
The PMC32-6000 DSP-Based motion control card was designed to provide a powerful computation process. The code composer by TI is used to develop and debug the DSP hardware, and the Labview by NI is selected as the man-machine interface. The MATLAB is utilized to plot the experimental results. The DSP processor on the mother board is a TMS320C31 with 1 Mbyte of SRAM. The mother board has eight independent analog-to-digital converters (each with 12-bit resolution) with a programmable sampling frequency up to 50 KHz. The unique architecture is designed so that at the instant that the sampling interrupt occurred, all eight input values will be converted and can be read simultaneously by all DSPs. The PMC32-6000 has six 12-bit independent analog output
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ports in total. In this manner, the computation and communication power can be tailored to individual applications. A man-machine programming interface provides the operator to set machining parameters of the turning process by spindle rate as well as the feedforward speed is shown in Figure 2.
Figure 2: The man-machine programming interface
The BORLAND C++ BUILDER is utilized as the human-machine interface, and the programmed interface is shown as Figure 3.
Figure 3: Window of the programmed human-machine interface
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5. The Experiment
A numerical example of a single-tool finish-turning operation for specific S45C steel fixture plates from AirTAC Corporation in Taiwan is referenced for the experiment. All data compiled are transformed into SI units as well as US dollars and listed as follows: P = 5.45 dollars , a = 17355 mm 3 , b = 1.7 × 10 −8 (dollars - min)/mm 6 ,
c = 6.625 × 10 −8 dollars /(min − mm 3 ) , B = 16470 mm 3 / min , and T = 40 min . The desk-top lathe control system is proposed. The feedrate is generated by the servo-drive which controls the servo-motor, and the spindle speed is controlled by the inverter. The φ 25mm × 180mm S45C work-pieces are selected for the experiment, and the length of turn for each work-piece is 150mm . The spindle speed ranges from 1328 rpm to 1340 rpm , and the feed rate is selected as 2 mm sec .
6. Results and Discussion
The surface roughness of all finished work-pieces is measured at the front, middle, and rear regions. The surface roughness of all thirty-three finished work-pieces under the digital lathe system is plotted as Figure 4.
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Figure 4: Plot of the measured surface roughness
It can be seen from Figure 4 that the surface roughness tends to increase as the tool life is consumed. However, the growing rate of surface roughness decelerates and stabilizes as the tool life consumed. Besides, it is also found that the machined work-pieces are all within a surface roughness of 2.4 μ m which satisfactorily matches the recognized standard for finish turning. With the experimental result, the adaptability and applicability of the dynamic machining model are guaranteed.
7. Concluding Remarks
In recent years, a significant progress in CNC machine tools has been made as high productivity and machine tool lift as important role factors in manufacturing processes. The MRR is an important factor of machining operation, and the control of machining speed is crucial for production planners. This paper not only introduces the idea of dynamic machining optimization for an individual tool during a fixed life time, but also 12
contributes a reliable and applicable technique in approaching dynamic MRR control for maximum profit. A digital controlled PC-based lathe system is used to realize the dynamic MRR control. It is shown that the machined parts are all within a surface roughness of 2.4 μm . The experimental results demonstrate that the adaptability and applicability of the dynamic machining model are promised.
Acknowledgement This research was partially supported by the National Science Council in Taiwan through Grant NSC 92-2213-E-244-004.
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