Int J Adv Manuf Technol (2012) 58:421–429 DOI 10.1007/s00170-011-3420-5 ORIGINAL ARTICLE Performance enhancements of high-pressure die-casting die processed by biomimetic laser-remelting process Zhi-xin Jia & Ji-qiang Li & Li-Jun Liu & Hong Zhou Received: 16 December 2010 / Accepted: 30 May 2011 / Published online: June 2011 # Springer-Verlag London Limited 2011 Abstract Die service life improvement is an important problem in high-pressure die-casting industry Experiment results on die steel shows that biomimetic laser-remelting process provides a promising method to improve the service life of die-casting die A casting with uneven wall thickness was selected and problems existing in die-casting production were analyzed The corresponding die-casting die was processed by biomimetic laser-remelting process The application result indicates that the service life of the die processed by biomimetic laser-remelting process has been increased from 12,000 to 28,000 shots, which is more than twice that of no processed one under real high-pressure die-casting conditions The application of laser-remelting process provides desirable micro-structural changes in biomimetic units, which induces the intensified particles effect for improving the service life Keywords Die-casting die Thermal fatigue Laser-remelting process Introduction Die casting is a high-volume production process, which produces geometrically complex parts of nonferrous metals Z.-x Jia (*) : J.-q Li : L.-J Liu Ningbo Institute of Technology, Zhejiang University, 1# Qianhu Road, Ningbo 315100, People’s Republic of China e-mail: jzx@nit.zju.edu.cn H Zhou The Key Lab of Automobile Materials, The Ministry of Education, Jilin University, 5988# Renmin Road, Changchun 130025, People’s Republic of China with excellent surface finishes and low scrap rate The die castings are used extensively in automobile, motorcycle, computer, and consumer electronics These die castings are generally produced by using two steel die halves called the cover-die half and ejector-die half separately Each of the die halves usually contains a portion of the die cavity The process sequences are: (a) die closing, (b) cavity filling, (c) casting solidification, (d) parts ejection, and (e) lubrication The most important modes of failure in die-casting dies are thermal cracking, soldering, and corrosion Die wear and failure is a significant issue in diecasting industry, owing to the high cost of dies Nevertheless, owing to the harshness of service condition of the die-casting dies, the complexity of thermal fatigue processes, and the variety of factors affecting the process, die wear and failure has been a technical difficulty in die-casting industries for many years In order to prolong the service life of die-casting dies, many researchers have been engaged in the theoretical and experimental studies related Research group from The Ohio State University, USA, did a lot of work aiming at elucidating the life-limiting failure mechanisms in the die-casting die through experiments and CAE analysis [1–5] Venkatesan and Shivpuri [1, 2] carried out experiments under actual production conditions for a range of process and geometrical conditions with the accelerated erosive wear of core pins being used as a surrogate measure of die erosive wear Yu et al gave a study of corrosion of die materials and die coatings in aluminum die casting [3] They also studied effects of molten aluminum on H13 dies and coatings [4] Their experiments have shown that single-layer hard PVD and CVD coatings not protect the die steel surface from cracking Kulkarni et al investigated the thermal cracking behavior on nitrided die steels in liquid aluminum process- 422 ing [5] Srivastava et al developed a model to predict the thermal fatigue cracking using FEM software[6] Persson et al studied the simulation and evaluation of thermal fatigue cracking of hot work tool steels [7, 8] Domkin et al studied the soldering and did some work to tackle the problem of die life-time prediction based on a quantitative analysis of soldering in the framework of the full 3D simulations of the die casting process [9] Zheng et al established an evaluation system for the surface defect of casting and introduced artificial neural network to generalize the correlation between surface defects and die-casting parameters, such as mold temperature, pouring temperature, and injection velocity [10] Klobcar et al analyze the influence of aluminum alloy die-casting parameters, die material, and die geometry on in-service tool life by immersion testing and FEM method [11] With the development of laser technology, laser processing method is used to change the property of die material Grum et al reported results of CO2 laser repair surfacing of maraging steel with a Ni–Co–Mo alloy similar to the maraging steel [12] After laser surfacing of the DIN 1.2799 maraging steel a very favorable throughdepth residual-stress profile of the surfaced layer and the heat-affected zone is obtained Compressive residual stresses in the surface layer reduce the risk of formation and propagation of surface cracks Such a state of stress will considerably extend the tool life Persson et al studied the life-limiting failure mechanisms in dies aimed for brass die casting [13] They examined and evaluated cavity inserts and cores with respect to failure mechanisms after use in actual brass die casting They found that the dominating failure mechanism in the investigated tools was thermal fatigue cracking In order to improve the thermal conductivity of H13 die material, some studies that have been carried out to develop molds with higher thermal conductivity have concentrated on mixing copper and steel Beal et al manufacture the 3D structures from a mixture of H13 and copper powders by using a laser beam to sinter or melt the mixture of H13/Copper powder The method employed is based on the layer manufacturing technology [14] Khalid et al presents a novel approach to replace a conventional steel die by a bimetallic die made of Moldmax copper alloy coated with a protective layer of steel using laser cladding technology, direct metal deposition on the cavity surface for high-pressure die casting of aluminum alloys [15] Nature provides a whole host of superior multifunctional structures that can be used as inspirational systems for the design and synthesis of new, technologically important materials and devices Since the 1980s, Ren et al has been dedicating to the study of the cuticle morphologies and principles of soil animals They found Int J Adv Manuf Technol (2012) 58:421–429 that soil animals have “nonsmooth construction units,” which provide excellent anti-wear properties against soil [16] Recent works in the research group of Jilin University, China, also found that a considerable effect not only on wear resistance [17], but also on the thermal fatigue resistance [18, 19] when applied biomimetic principle on the die and tool surfaces to form a series of biomimetic units by laser Experiments were focused on the effects of laser input energy and biomimetic unit shape Zhang et al studied the size of units and investigated its effect on thermal fatigue behavior of 3Cr2W8V steel [19] They also studied the tensile property of H13 die steel with convex-shaped biomimetic surface [20] Shan et al did some experiments on injection molds by mimicking the injection conditions The results showed that the adhesion biomimetic molds have a beneficial effect on decreasing the adhesion to eject polymer parts [21] Studies [18–20] have shown that the biomimetic surface with units in varying shapes and distributions has an enhanced resistance not only to the thermal fatigue crack initiation but also to the crack propagation But so far these studies are merely experiment result in laboratory on specimen with simplified geometry by mimicking the real conditions Regarding the complex-shaped casting, especially the performance of die-casting die processed by biomimetic laser-remelting process under actual production conditions is still scarce In this paper, a set of die-casting die made of H13 is chosen to be processed by biomimetic laser-remelting process and its performance under actual production conditions is investigated The real die-casting conditions are supplied by our partner, Donghao Die-casting Co., Ltd The application result shows that service life of diecasting die processed by biomimetic laser-remelting process is prolonged from 12,000 to 28,000 shots The purpose of this study is to further reveal the effectiveness of thermal-fatigue-resistant mechanism of the units under actual production conditions, and finally to lay a foundation for the application of biomimetic laserremelting process in the design and manufacturing of die-casting dies in the future The rest of the paper is organized as follows Section gives the requirements and material of the selected casting, the die-casting parameters, the main problems, and the service life of the die in die-casting production Section shows the experiment parameters, method of biomimetic laser-remelting process, and the performance of the diecasting die under actual production conditions Section illustrates the microstructure of the unit Section describes the application of biomimetic laser-remelting process on the succeeded die-casting die Section gives conclusions Int J Adv Manuf Technol (2012) 58:421–429 423 The die casting and the die-casting die Due to the high cost of die-casting die, one die casting and the corresponding die were selected elaborately 2.1 The characteristics of the selected aluminum die casting The selected aluminum casting was produced by highpressure die casting, as shown in Fig 1, called cover, which is used in vehicles The material of the casting is ZL102 Though the geometry of the casting is not very complex, the dimension accuracy and the surface roughness are required strictly The inner surface of die casting is required to keep the original die-casting surface There are two platforms which have flatness checking requirements The outer surface of the casting is cleaned by shot blast and then sprayed with black paint, as depicted in Fig The average wall thickness of the die casting is about mm, which is thicker than general castings Moreover, the wall thickness is not even In the two-platform region, the max thickness reaches 18 mm, which create areas of high temperatures during solidification, the so called hot spots Furthermore, there are sharp angles or edges near the ribs on outer surface and platforms on inner surface, which are known to promote or increase the risk of soldering [9] and corner cracking [6] The strict requirements, uneven wall thickness, and corners in small radius lead to great difficulties in die-casting production and short service life of the die-casting die There are two reasons for this casting is selected One is the short service life of the die-casting die, which embarrassed our partner very much The other is our partner produces the casting in large quantities, about 10,000 pieces per month for the customer, which gives the great convenience to investigate change of the service life of die before and after being processed by biomimetic laser-remelting process 2.2 The die-casting process parameters The processing parameters for the selected die casting are listed below: Preheat temperature of the die, 200∼220°C Temperature of the aluminum molten liquid, 660°C Die cooling temperature, 250∼300°C Clamping force, 2,800 MPa Filling time, s Inlet temperature of circulating water, 25°C Outlet temperature of circulating water, 35°C One cycle time, 55 s 2.3 The defects regions on the casting In real die-casting production, defects on outer surface of the casting appear firstly on edges with small radius of the ribs, as shown in the red circle in Fig 2a While defects on inner surface of the die casting are concentrated on the boundary edges of the platforms, as shown in the red lines in Fig 2b 2.4 The service life of the die-casting die made by conventional process The die-casting die consists of two separate halves: the ejector die on the “bottom” side of the casting and the cover die on the “top” side of the casting The die halves are manufactured of hardened H13 die steel (0.36% C, 1.09% Si, 0.32% Mn, 5.12% Cr, 1.32% Mo, 0.80% V, and 1 is proposed Assume that the sensor is event driven, i.e., the plant output is sampled only when a new control input signal is received by the actuator; as depicted in Fig Then, at the outset of the k th sampling instant, the next sampling period is selected to be equal to the predicted time delay, i.e., Tkþ1 ¼ t pk Such an assumption becomes more realistic, when the delay 808 Int J Adv Manuf Technol (2012) 58:803–815 + − τ τ − τ τ − − − =τ τ =τ +τ − + − =τ + − + Fig Description of the variable sampling period scheme prediction error is small, i.e when t pk % t k For this case, Eq 13 can be simplified as: Z Tkþ1 ATkþ1 xðt þ Tkþ1 Þ ¼ e xðtÞ þ eAh Bdh uðt À Tk Þ; ð17Þ where t represents the time of the k th sampling instant Denoting xðt þ Tkþ1 Þ, x(t), and uðt À Tk Þ as xk + 1, xk, and uk − 1, respectively, Eq 17 can be re-written as xkþ1 ¼ 6kþ1 xk þ Γ kþ1 ukÀ1 ; ð18Þ ATkþ1 where and Γ kþ1 ¼ Γ ðTkþ1 Þ ¼ R Tkþ1 Ah6kþ1 ¼ 6ðTkþ1 Þ ¼ e e Bdh Clearly, Φk+ and Γk+ 1depend on Tk + 1; hence, the control input, uk−1, at the k th sampling instant, should be updated according to the state feedback law in an adaptive scheme To design a state feedback controller for such a delayed system, its model, described by Eq 18, can be represented in the augmented formà with the new augmented states  T Zk ¼ ZðkÞ ¼ xTk uTkÀ1 , i.e., " Zkþ1 ¼ xkþ1 uk # " ¼ 6kþ1 Γ kþ1 0mÂn 0mÂm #" xk ukÀ1 # " þ 0n;m Im;m # uk ¼ Akþ1 Zk þ B0 uk ð19Þ ! ! 0n;m Φkþ1 Γ kþ1 where Akþ1 ¼ , and B0 ¼ A state Im;m 0mÂn 0mÂm feedback controller, uk ¼ ÀKTkþ1 Zk , can be used to stabilize the new system represented by Eq 19 Therefore, the closed-loop equation can be re-written as Zkþ1 ¼ Akþ1 Zk þ B0 uk ¼ Akþ1 Zk À B0 KTkþ1 Zk ¼ ðAkþ1 À B0 KTkþ1 ÞZk ¼ Ψ kþ1 Zk ð20Þ where Ψ kþ1 ¼ Akþ1 À B0 KTkþ1 Linear quadratic regulator (LQR) or pole placement techniques can be used to design KTkþ1 Evidently, if the actual upcoming delay turns out to be smaller or larger than the predicted value, the performance degradation is expected However, simulation studies show the robustness of the proposed control method in the presence of delay prediction error For real-time implementation of such a method, the appropriate gain for the next sampling instant can be designed in the current step and therefore the computational delay in the upcoming instant can be neglected The proposed control scheme can be summarized as follows: Set the state feedback gain to zero Initialize the control vector with zero elements Calculation procedure for each sampling period, k, – – – – – – Estimate the next upcoming time delay, t pk , using the delay prediction algorithm proposed in Section Discretize the continuous system of Eq 12 by choosing the predicted delay as the upcoming sampling period (Tk+1) Consequently, the ZOH equivalents of the continuous system, namely, Φk+1 and Γk+1, are calculated using Eq 18 Calculate the augmented system of Eq 19 using Φk+1 and Γk+1 Design a stabilizing feedback gain for system, KTkþ1 , using the LQR or the pole placement technique Calculate the augmented state, Zk, using the control input uk−1 employed during the current step, when the signal xk is received by the controller Calculate the control input, uk ¼ ÀKTkþ1 Zk , and save it in the predefined input vector Transmit the calculated control input to the actuator through the network Apply the designed input uk during the (k+1)th sampling instant by a ZOH actuator and, simultaneously, trigger the sensors to send the new, time-stamped, plants data to the controller through the network The block diagram of the proposed control algorithm, together with the neural on-line time delay predictor, is summarized in Fig Stability analysis As described in Section 3, the variable sampling control scheme is proposed based on the idea of the event triggered sensors, i.e., the sensors send new plants data, when the new control inputs are received by the actuators Int J Adv Manuf Technol (2012) 58:803–815 809 KT τp k +1 k uk rk t k −1 tk − τ k − = t k −1 − t k − τ ′p′ τ k −1 [τ k − τ k −2 τ ′p k k τk−L] τ ′p + ek τp k k z ek τp e k = mean[e k − ek − ek − L ] [τ p k −1 τ pk − τ pk − L k −1 ] ek [t k xk ] Fig The proposed adaptive control algorithm with the specified delay predictor In practice, an excessively high sampling frequency may not necessarily guarantee a better performance, for, besides numerical problems, it may lead to excessive network traffic which causes longer delays and packet loss, which adversely affects the closed-loop stability and performance In order to ensure the closed-loop stability, Lemma proposed in the sequel, which is valid when, at each step, the sampling periods are selected as follows: Based on the expected network conditions, lower and upper bounds of the network delay are selected as Tmin and βTmin, respectively The possible network time delays are then decided at the outset of the design process The next sampling period is decided such that & Tkþ1 ¼ aTmin ; a ¼ ceilðt k =Tmin Þ ð21Þ a n d Tkþ1 Tss ¼ f Tmin 2Tmin Á Á Á bTmin g ¼ f T1 T2 Á Á Á Tb g, where Tss is a set of possible stable sampling periods for each instant Here, ceil(x) rounds x to the nearest integer towards infinity Remark For this purpose, the instantaneous network time delay, τk, must be calculated by comparing the send-time stamp attached to the last signal received by the ZOH, which is then triggered according to Tk+1 (Fig 2) To this 810 Int J Adv Manuf Technol (2012) 58:803–815 end, a modified ZOH is comprised near the plant, as depicted in Fig Remark The closed-loop system can be considered as a sampled data system with varying sampling rate; i.e., a hybrid system representation Consequently, changing the sampling period results in the controller switching among the corresponding controllers, already designed for various subsystems For sampling periods in the set Tss, the switched system can be represented as follows: Xb Zkþ1 ¼ q ðkÞAkþ1ji Zk þ B0 uk ð22Þ i¼1 i where qðkÞ ¼ f q1 ðkÞ q2 ðkÞ qb ðkÞ g is a switching signal vector specifying which subsystem will be activated at the instant k, where q i ðkÞ : f b g ! f g Pb and i¼1 q i ðkÞ ¼ for every positive integer k In particular, for 8i f b g, θi(k)=1 shows that i th subsystem is activated [33] Also, Akþ1ji is the systems matrix of i th subsystem at kth step, for i ¼ 1; ; b Since the model is assumed accurate, Akþ1ji depends only on an upcoming sampling period Ti, for every non-negative integer k Therefore, for the sake of simplicity, the index (k+1) can be deleted and it can be denoted by Ai Then, Eq 22 can be re-written as Xb Zkþ1 ¼ q ðkÞAi Zk þ B0 uk ð23Þ i¼1 i P Using the switching feedback law uk ¼ À bi¼1 qi ðkÞKTi Zk , the closed-loop equation can be written as Zkþ1 ¼ ¼ ¼ Xb Xb i¼1 qi ðkÞAi Zk À i¼1 qi ðkÞðAi À B0 KTi ÞZk i¼1 qi ðkÞΨ i Zk Xb Xb i¼1 ¼ ZkT Ψ Ti PΨ i Zk À ZkT PZk ¼ ZkT ðΨ Ti PΨ i À PÞZk ¼ ZkT ðÀQi ÞZk where Ψ Ti PΨ i À P ¼ ÀQi In order to have ΔVk < 0, Qi must be positive definite Therefore, it can be concluded from the theory of linear switched systems [34] and [35] that the switching closedloop control system will be stable provided that for all subsystems i f1 bg, corresponding to a discrete set of possible sampling periods Tss , there exists a common positive definite Matrix P such that Ψ Ti PΨ i À P ¼ ÀQi < Remark Lemma may be used to derive a set Tss for which the closed-loop stability is guaranteed Several methods for obtaining such a set can be considered For example, one may consider a positive definite matrix Q0 , and then obtain the solution P0 of the Lyapunov equation for a given value of T0, and check the negative definiteness of the expression Ψ T P0 Ψ À P0 This procedure can be repeated with other sampling periods with a pre-decided resolution Tmin, so that the widest valid interval f Tmin 2Tmin Á Á Á bTmin g can be obtained It is clear that such a set depends on any particular selected Q0 Simulation studies 5.1 Example qi ðkÞB0 KTi Zk ð24Þ where Ψ i ¼ Ai À B0 KTi and KTi are the stabilizing gain for i th subsystem In the sequel, the switched system theory is used to assess the stability of the proposed switching closed-loop system Lemma The system described by Eq 24 is asymptotically stable for all sampling periods Ti, satisfying Eq 21, provided that there exists a common positive definite matrix P such that Ψ i T PΨ i À P ¼ ÀQi ; T PZkþ1 À ZkT PZk ΔVk ¼ Vkþ1 À Vk ¼ Zkþ1 ð25Þ where Qi is positive definite for i ¼ 1; 2; ; b Proof For Eq 24, consider Vk ¼ ZkT PZk as a Lyapunov candidate function The asymptotic stability of such a system is guaranteed as long as ΔVk < Now, In order to assess the effectiveness of the proposed delay prediction algorithm, a typical delay history is captured through the Internet In this simulation, the PING command was used to obtain the round trip time (RTT) between a PC in our laboratory (at the Iran University of Science and Technology) as the controlled plant and the server of www google.com as the central controller RTT can be regarded as the total time needed to transmit a signal over a network up to a destination and, i.e., the overall transmission, receive, and service times,2 or in fact the total delay in the control loop The time history of the observed delays is shown in Fig The presented algorithm is then used to determine the order of the delay as a Markov process As shown in Fig 5, it is evident that the order of the process can be estimated as three Afterwards, the proposed neural model is trained using a part of the captured delay vector and is tested with the rest The estimation error history is shown in Service time is the time required for network components such as routers to direct information to various paths Int J Adv Manuf Technol (2012) 58:803–815 Fig The delay of a typical communicational network vs data packet number Fig 6, with a mean of about 1.9% and a maximum of just less than 8.8% 5.2 Example 811 Fig Error of the delay prediction algorithm in seconds using the MLP neural model with modifier vs data packet number 0 0 7 6 515:38 À18:18 0 7 7 Á 7u 7x þ 0 0 x¼6 7 7 7 60 0 À17:86 517:07 In this example, the networked control of a two axis example of a three-axis milling machine tool is considered Each axis moves on a linear slide which is driven through a ball screw by a DC motor with a tachometer and a linear encoder, i.e., all the states are directly measured Denoting the position and velocity of the ith axes by Ri and Vi, respectively, the state vector would be as x ¼ ½R1 ; V1 ; R2 ; V2 T The state equations for a typical system is reported in [36] and [37] as The eigenvalues of A are ½0; À18:18; 0; À17:86, and the continuous-time system is unstable The response of the continuously controlled system, assuming an ideal zerodelay network, and under an initial condition of ½2 1T is shown in Fig Several attempts for Fig Estimation of the Markov order of network delay process Fig Closed-loop response with a fixed control gain through an ideal zero-delay network under an initial condition of ½ T yðtÞ ¼ xðtÞ 812 networked control of this system have been reported in the literature Lian et al [36] proposed a method to stabilize this system for τk ≤ 0.03 s, and Zhang and Yu [37] guaranteed the stability for τk ≤0.5 s [37] In order to verify the effectiveness of the method proposed in this paper, a network with a uniformly distributed random sequence of delays, in the range of 0.7–1.6 s, with a mean value of about 1.1 s and a standard deviation of 0.3, is assumed Since the proposed control method is a variable sampling scheme, the closed-loop stability must be checked using Lemma This can be done by finding an interval for time varying sampling periods, wherein the NCS with the proposed switched control law remains stable For this purpose, a nominal sampling period of 1.18 s is selected and using the MATLAB/LMI Toolbox, the discrete Lyapunov equation is solved for a positive definite matrix P Then, by assuming Tmin =0.1 s and by checking the positive definiteness of various Qk for different sampling periods in the range addressed by Eq 21, the stability interval is obtained Figure shows the minimum eigenvalue of Qk as a function of the sampling times Obviously, the proposed method guarantees the stability of the control system for all sampling periods in the interval of 0.7–1.6 s Closed-loop response of the networked system, illustrated in Fig 9, demonstrates the exceptional effectiveness of the proposed algorithm, compared to the response of the system controlled using an analog controller, shown in Fig To discuss the robustness of the proposed method with respect to the delay prediction error, new simulation studies for cases with 25% and 75% prediction errors are conducted The results shown in Figs 10 and 11, respectively, reveal the stability of the closed-loop system, even under prediction errors Fig Minimum eigenvalue of Q as a function of sampling period Int J Adv Manuf Technol (2012) 58:803–815 Fig Closed-loop response with the proposed variable gain method through a network under an initial condition of ½ T , with the assumption of perfect delay prediction 5.3 Example The results obtained in Section are applied to a DC motor driving a roller for transportation of paper sheets Friction in the motor and transmission, and the slippage between the roller and the paper sheet are neglected The CT motor-roller dynamics is obtained as " Á x¼ 0 qrR 7u xþ6 J m þ n2 J R # yðtÞ ¼ xðtÞ Fig 10 Closed-loop response with the proposed variable gain method through a network under an initial condition of ½ T , and with a 25% error in predicted delay Int J Adv Manuf Technol (2012) 58:803–815 813 Fig 11 Closed-loop response with the proposed variable gain method through a network under an initial condition of ½ T , and with a 75% error in predicted delay Fig 13 Closed-loop response with the proposed variable gain method through a network under an initial condition of ½ T where JM ¼ 1:95  10À5 kg m2 , JR ¼ 6:5  10À5 kg m2 , rR ¼ 14  10À3 m, and q=0.2 denote motor and roller mass moment of inertia, roller radius, and the transmission ratio between motor and roller, respectively [38] For the purpose of networked control of this system, Cloosterman et al [38] proposed a method to stabilize this system for t k 0:002 s Here, the network delay is assumed to be a uniformly distributed random sequence in the range of 0.2–2.2 s, with a mean value of 1.8 s and a standard deviation of 0.54 The closed-loop stability is studied using Lemma 2, with a similar procedure as in Example Figure 12 shows the minimum eigenvalue of Qk as a function of time varying sampling periods The closed-loop response of the networked control system, with guaranteed stability, is shown in Fig 13 To discuss the robustness of the proposed method with respect to the delay prediction error, new simulation study for case with 55% prediction errors is conducted (Fig 14) The result reveals the stability of the closed-loop system, even under prediction errors Fig 12 Minimum eigenvalue of Q as a function of sampling period Conclusion In this paper, a method for real-time control of networked control systems is proposed The proposed technique involves two new contributions compared to the previously existing methods: (1) a new algorithm for real-time prediction of the network delay and (2) a new variable discrete-time state feedback control based on the predicted Fig 14 Closed-loop response with the proposed variable gain method through a network under an initial condition of ½ T , and with a 55% error in predicted delay 814 upcoming network time delay A sufficient condition for closed-loop asymptotic stability is derived using the switched linear systems theory To illustrate the effectiveness of the proposed delay prediction algorithm, we used a realistic delay history obtained through a typical internet connection between two distant nodes Simulation studies reveal that the delay prediction algorithm approximately predicts the network occurring time delays Further simulation studies of two manufacturing processes, i.e., a DC motor driving a roller for transportation of paper sheets and a milling machine, show the effectiveness of the proposed variable state feedback control in the real time The proposed algorithm provides a stable and exceptionally desirable response compared to the ideal continuous-time control, i.e., when the network time delay is zero The robustness of the proposed method with respect to the delay prediction error is also shown through simulation studies The result 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time discounting has been developed for varying demand pattern over a fixed planning horizon Optimal solutions with complete backlogging and without backlogging are established and has proven that the total variable cost is convex The main contribution of this paper is based on the assumption of the demand which is the linear function of the instantaneous stock level I(t) In this paper, a new type of demand has been considered which will help us to obtain qualitative insights without much analytical complexity Optimal solutions of the proposed models are derived and effects of deterioration on the inventory replenishment policies are studied with the help of numerical examples Keywords Economic order quantity (EOQ) · Deteriorating items · Backlogging · Time discounting · Varying demand Introduction Deterministic inventory models have been developed in the literature based on the demand rate which is to R Uthayakumar Gandhigram Rural Institute-Deemed University, Gandhigram, 624 302, Dindigul District, Tamil Nadu, India e-mail: uthayagri@gmail.com M Rameswari (B) R.V.S College of Engineering and Technology, Dindigul District, Tamil Nadu, India e-mail: sivarameswari1977@gmail.com be either constant, time dependent, or stock dependent The no-shortage economic order quantity (EOQ) model was solved analytically by Donaldson [1], where the demand is assumed to be a linearly increasing function of time Two types of time varying demand patterns are linear (positive or negative) demand and exponential (increasing or decreasing) demand It is also experienced that for consumer goods, sales increase with the increase of the inventory Basically, a large number of people buy more inventory by being attracted towards large inventory This requires consideration of the demand to be a function of onhand inventory As a result, many papers were focused on inventory models using some form of functional dependencies between the demand rate and on-hand inventory Recently, considerable attention has been received by inventory theory research which involves situations in which the demand rate is dependent on the level of inventory A basic model in which the demand rate of an item is a function of the instantaneous inventory level is investigated by Bakker and Urban [2] The demand rate of an item is dependent on the instantaneous inventory level until a given inventory level is achieved after which it becomes the constant demand rate which was focused by Datta and Pal [3] The optimal solution may not be provided by an inventory system that possesses an inventorylevel-dependent demand It may be desirable to order larger quantities, resulting in stock remaining at the end of cycle, due to potential profits resulting from the increase demand This phenomenon was discussed by Bakker and Urban [2] Inventory models in which the demand rate depends on the inventory level are based on the common real-life observation that greater product availability tends to stimulate more sales In this 818 regard, both marketing and environmental stimuli was emphasized in this paper, by considering a stock-leveldependent demand rate, with the objective of profit maximization The items that incur a gradual loss in quality or quantity over time while in inventory are usually called deteriorating items There are items such as highly volatile substances, radio active materials, etc., in which the rate of deterioration is very large Therefore, loss from deterioration should not be ignored The certain commodities were observed to shrink with time by a proportion which can be approximated by a negative exponential function of time The first attempt to derive optimal policies for deteriorating items was made by Ghare and Schrader [4] who derived a revised form of the EOQ model assuming exponential decay Ting and Chung [5] studied the inventory replenishment model for deteriorating items with a linear trend in demand considering shortages Dave and Patel [6] discussed an inventory model for deteriorating items without shortages and the-time dependent demand patterns, mainly, used in the literature were, (1) linearly time dependent and (2) exponentially time dependent Yang et al [7] provided various inventory models with time-varying demand patterns under inflation and investigated replenishment models with shortages Lately, Goyal and Giri [8] reviewed the contributions on the literature in modeling of deteriorating inventory By using two numerical examples, they suggested that their policy outperforms the traditional approach Dohi et al [9] discussed inventory systems with and without backlogging allowed under the condition that the demand follows a Wiener process for an infinite time span, taking into account time value from a viewpoint different from that of Trippi and Lewin [10] Literature review In 1997, Teng et al [11] established various inventory replenishment policies to solve the problem of determining the time and the number of replenishment, analytically compared various models, and identified the best alternative based on minimizing total relevant cost In the real market, the demand rate of any product is always in a dynamic state Demand of a product may vary with time or with price or instantaneous level of stock displayed in a retail shop Much attention has so far been paid to inventory modeling with timedependent demand A heuristic approach to determine the EOQ in the general case of a deterministic timevarying demand pattern is started with the work of Silver and Meal [12] An important area of inven- Int J Adv Manuf Technol (2012) 58:817–840 tory studies had developed production-recycling system now-a-days, due to growing environmental concern and environmental regulations in industry For this reason, inventory models with stock-dependent demand and the optimal production for an inventory control system of deteriorating multi-items where items were either complementary and/or substitute were formulated with a resource constraint which was presented by Maity and Maiti [13] An extended product inspection policy is considered by the work of Wang [14], where product inspections are performed in the middle of the production run and after the inspection, all products until the end of the production run are fully reworked Yeh et al [15] was extended by Yeh and Chen [16] to consider allowable shortages for the imperfect production processes The deterministic EOQ with partial backordering was discussed by Pentico et al [17] His objective was to develop a set of equations that are both simpler to use and have a more understandable and intuitive form that closely resemble to the basic EOQ and the EOQ with full backordering A model for the economic production quantity (EPQ) that simultaneously considered the effect of production defects and machine breakdowns was developed by Chakraborty et al [18] in their model The model including stockout and backorder costs was extended by Dye [19] The demand, production and deterioration rates to vary overtime was allowed by Goyal and Giri [20] An imperfect production process, inflation and multiple deliveries in their situational scenario was included by Lo et al [21] Perishable inventory, a constant backordering, demand that decreases linearly with the decreasing inventory level, a finite production rate, inflation and a finite planning horizon for items with a stock-dependent demand rate was included by Jolai et al [22] A perishable item that follows a two-parameter Weibull distribution was considered by him and shortage was allowed and partially backlogged at a fixed rate The EPQ model for multiple products with planned backorders was discussed by Li et al [23], but they assume that all the demand not originally satisfied from stock is willing to wait for the units to be delivered upon production By comparing the optimal total average costs of the two systems, they evaluated the impact of postponement on the manufacturer under four circumstances and also found that the key factors in postponement decisions were the variance of the machine utilization rates and the variance of the backorder costs The composite EOQ model which he extended to incorporate lost sales was developed by Sharma [24], i.e., when a fraction of shortage quantity is not backordered Lately, multiproduct environment in which one item has a shelflife constraint was considered by him [25] In reference Int J Adv Manuf Technol (2012) 58:817–840 [26], cyclic production in which the annual demand was satisfied by the optimal cycle time was developed by him, and a certain quantity of the item was produced in each cycle Shelf-life and manufacturing capacity constraints [27] were imposed by Sharma Each item in the family had a certain shelf life A capacity constraint was also imposed on the system Procurement of input items corresponding to the finished products was modeled to facilitate smaller order quantities A production environment in which multiple items are produced in a cycle was considered by Sharma [28] The optimal inventory replenishment models for deteriorating items taking into account time discounting with constant demand was developed by Kun et al [29] Discounted cash flow (DCF) approach to determine the optimal number of replenishment and the corresponding cycle length, consisting of positive and negative inventory periods is applied in this paper In addition, it is proved that the total variable cost functions are convex Finally, sensitivity of the optimal solution with respect to various parameters of the system has been discussed Assumptions The following assumptions are made throughout this paper: Replenishment rate is infinite and lead time is zero A single item is considered over a prescribed period of H units of time When inventory system allows shortages, m + replenishment are made during the entire time horizon H The last replenishment is made at time t = H, just to replenish any shortages generated in the last cycle The constant rate of deterioration θ, is only applied to on-hand inventory Two models are analyzed: model I in which backlogging is not permitted and model II in which complete backlogging is permitted with a finite shortage cost C2 per unit time 819 The function D(t) is given by D(t) = α + β I(t) I(t) > =α 10 I(t) ≤ where α is the consumption rate, α > 0, β is the stock-dependent consumption rate parameter, ≤ β ≤ m denotes the number of replenishment periods during the time horizon H C is the purchasing cost per unit C1 is the inventory holding cost per unit, per unit time A is the ordering cost per order C2 is the shortage cost per unit, per unit time A0 is the ordering cost per order at time zero C0 is the purchasing cost per order at time zero C10 is the inventory holding cost per order at time zero C20 is the shortage cost per order at time zero Assuming continuous compounding of inflation, the ordering cost, purchasing cost of the item, outof-pocket inventory holding cost and shortage cost at any time t are A(t) = A0 eγ t , C(t) = C0 eγ t , C1 (t) = C10 eγ t and C2 (t) = C20 eγ t where ≤ γ ≤ is a constant inflation rate 11 R represents the discount rate of net inflation 12 θ is a constant deterioration rate 13 γ is a constant inflation rate 14 T j is the total time which is elapsed up to and including the jth replenishment cycle (j = 1,2, m) where Tm = H and T0 = 15 t j is the time at which the inventory level in the jth replenishment cycle drops to zero (j = 1,2, m) 16 Q is the optimal order quantity 17 BQ is the optimal backorder quantity 18 TC is the total variable cost of the system Model I: no shortages permitted Notations For developing the proposed models, the following notations are used throughout this paper The demand rate function D(t), is deterministic and is a known function of instantaneous stock level I(t): 5.1 Model formulation The total time horizon H has been divided into m equal parts of length T so that T = H/m Hence, the reorder times over the planning horizon H are T j = jT (j = 0,1, m − 1) To start with, the inventory level I(t) during the first replenishment cycle is considered This 820 Int J Adv Manuf Technol (2012) 58:817–840 inventory level is depleted by the effects of demand and deterioration So, the variation of instantaneous stock level I(t), with respect to t is governed by the following differential equation: dI(t) = −α − (θ + β)I(t), dt 0≤t≤T α e(θ +β)(T−t) − , (θ + β) 0≤t≤T C10 α (γ −R)H (θ+β)H e m −e m (θ + β)(θ + β + R − γ ) × (1) with the boundary condition I(T) = The solution of Eq can be represented by I(t) = CH = − e(γ −R)H C10 α + − e(γ −R)H/m (θ + β)(γ − R) × − e(γ −R)H/m (2) TC(m) = CR + CP + CH So, TC(m) = A0 + m−1 e(γ −R)T j × j=0 = A0 − e(γ −R)H − e(γ −R)H/m (θ+β)H C0 α e m −1 (θ + β) − e(γ −R)H C10 α + (γ −R)H/m 1−e (θ + β)(θ + β + R − γ ) (3) × e (θ+β)H m − e(γ −R)H/m × The total purchasing cost (CP) is m + I(0)e(γ −R)T j−1 CP = C0 j=1 m j=1 CP = × α e(θ+β)T − e(γ −R)T j−1 (θ + β) = C0 − e(γ −R)H − e(γ −R)H/m (4) The holding cost during the first replenishment cycle is T I(t)e(γ −R)t dt T = C10 H1 = α e(θ+β)(T−t) − e(γ −R)t dt (θ + β) (γ −R)H (θ+β)H C10 α e m −e m (θ + β)(θ + β + R − γ ) + C10 α − e(γ −R)H/m (θ + β)(γ − R) Hence, the present values of the total holding cost (CH) during the H is given as m H1 e(γ −R)T j−1 CH = − e(γ −R)H − e(γ −R)H/m C10 α × − e(γ −R)H/m (θ + β)(γ − R) − e(γ −R)H − e(γ −R)H/m (6) 5.2 Optimal solution procedure (θ+β)H C0 α e m −1 (θ + β) H1 = C10 (5) Consequently, the total variable cost of the system during the entire time period H is given by Since there are m replenishment in the H, the total replenishment cost (CR) is given by CR = A0 − e(γ −R)H − e(γ −R)H/m j=1 m e(γ −R)T j−1 = H1 j=1 If the variable m is treated as a continuous variable, the second-order derivative d TC(m) is positive is shown dm2 Theorem in the Appendix Consequently, TC(m) is convex with respect to m Therefore the optimal number m* of replenishment of TC(m) is the smallest positive integer m such that TC(m + 1) ≥ TC(m) Using the optimal solution procedure was described above, the optimal order quantity can be found as Q = I(0) = (θ+β)H α e m −1 θ +β (by Eq 2) Numerical example Data considered to illustrate model I are as follows: α = 900 units, θ = 0.05, A0 = $400.00, C10 = $25 per unit per year, C0 = $50 per unit, R = 0.15, H = years, β = 0.25 and γ = 0.05 By using the solution procedure is developed, the optimal values of replenishment number, order quantity and total variable cost are given by m* = 2, Q* = 1,376.4, and TC(m) = $12,522, respectively Int J Adv Manuf Technol (2012) 58:817–840 821 decision making will be of great help by the sensitivity analysis Using the numerical example given in the preceding section, the sensitivity analysis of various parameters has been done In fact, there is no change in the ordering quantity and total cost for different values of H The measures of system sensitivity considered here are Q’/Q and TC’/TC, where Q’, TC’ are based on the estimated values of parameters and Q, TC are based on the original values of the parameters, respectively (Fig 1) Results of the sensitivity analysis are exhibited in Tables 1, 2, and Fig Pictorial representation of the inventory cycles 7.1 Comment on the sensitivity analysis Sensitivity analysis The main conclusions can be drawn from the sensitivity analysis and are, thus, given as: In any decision-making situation, change in the values of parameters due to uncertainties can take place In order to examine the implications of these changes, When the consumption rate (α) decreases or increases, the ordering quantity (Q) and the present value of total cost (TC) will also decrease or increase Table Sensitivity analysis with respect to various parameters on ordering quantity and total system cost for the model without shortages Parameters α θ Percentage of under estimation and over estimation of parameters H=3 m Q Q /Q TC TC /TC H=4 m Q Q /Q TC TC /TC H = 30 m Q Q /Q TC TC /TC H=3 m Q Q /Q TC TC /TC H=4 m Q Q /Q TC TC /TC H = 30 m Q Q /Q TC TC /TC −50 −40 −30 −20 −10 +10 +20 +30 +40 +50 2.0786 685.9 0.4984 62,970 0.5029 2.7715 685.9 0.4983 80,100 0.5029 20.7864 685.9 0.4983 230,860 0.5029 1.1117 2,600.2 1.8891 131,010 1.0462 1.4823 2,600.1 1.8891 166,640 1.0462 11.1171 2,600.2 1.8891 480,300 1.0462 2.0765 824.0 0.5987 75,420 0.6023 2.7686 824.0 0.5987 95,930 0.6023 20.7646 824.0 0.5987 276,500 0.6023 1.2672 2,275.6 1.6533 129,540 1.0345 1.6897 2,275.5 1.6532 164,780 1.0345 12.6725 2,275.5 1.6532 474,930 1.0346 2.0749 962.1 0.6990 87,870 0.7017 2.7665 962.1 0.6990 111,770 0.7017 20.7491 962.1 0.6990 322,140 0.7017 1.4387 1,999.1 1.4524 128,250 1.0242 1.9183 1,999.1 1.4524 163,140 1.0242 14.3869 1,999.2 1.4525 470,200 1.0242 2.0737 1,100.2 0.7994 10,032 0.8011 2.7650 1,100.2 0.7993 127,600 0.8011 20.7374 1,100.2 0.7993 367,780 0.8011 1.6282 1,761.7 1.2799 127,110 1.0151 2.1709 1,761.7 1.2799 161,690 1.0151 16.2820 1,761.7 1.2799 466,020 1.0151 2.0728 1,238.3 0.8997 11,277 0.9005 2.7638 1,238.3 0.8997 143,440 0.9005 20.7284 1,238.3 0.8997 413,420 0.9006 1.8383 1,555.9 1.1304 126,100 1.0071 2.4511 1,555.9 1.1304 160,400 1.0071 18.3833 1,555.9 1.1304 462,320 1.0071 2.0721 1,376.4 1.0000 12,522 1.0000 2.7628 1,376.4 1.0000 159,280 1.0000 20.7212 1,376.4 1.0000 459,070 1.0000 2.0721 1,376.4 1.0000 125,220 1.0000 2.7628 1,376.4 1.0000 159,280 1.0000 20.7212 1,376.4 1.0000 459,070 1.0000 2.0715 1,514.5 1.1003 13,767 1.0994 2.7620 1,514.5 1.1003 175,110 1.0994 20.7152 1,514.5 1.1003 504,710 1.0994 2.3333 1,218.8 0.8855 124,440 0.9938 3.1110 1,218.8 0.8855 158,290 0.9938 23.3327 1,218.8 0.8855 456,220 0.9938 2.0710 1,652.6 1.2007 15,012 1.1988 2.7614 1,652.6 1.2006 190,950 1.1988 20.7103 1,652.6 1.2007 550,350 1.1988 2.6263 1,079.6 0.7844 123,770 0.9884 3.5017 1,079.6 0.7844 157,430 0.9884 26.2629 1,079.6 0.7844 453,760 0.9884 2.0706 1,790.7 1.3010 16,256 1.2982 2.7608 1,790.7 1.3010 206,780 1.2982 20.7062 1,790.7 1.3010 595,990 1.2983 2.9568 956.1 0.6947 123,200 0.9839 3.9424 956.1 0.6947 156,710 0.9839 29.5677 956.2 0.6947 451,680 0.9839 2.0703 1,928.7 1.4013 17,501 1.3976 2.7603 1,928.8 1.4013 222,620 1.3976 20.7026 1,928.8 1.4013 641,640 1.3977 3.3317 846.1 0.6147 122,740 0.9802 4.4422 846.1 0.6147 156,130 0.9802 33.3166 846.1 0.6147 449,990 0.9802 2.0699 20,669 1.5017 18,746 1.4971 2.7599 2,066.9 1.5017 238,450 1.4971 20.6995 2,066.9 1.5016 687,280 1.4971 3.7598 747.6 0.5431 122,390 0.9774 5.0131 747.5 0.5431 155,680 0.9774 37.5981 747.5 0.5431 448,700 0.9774 822 Int J Adv Manuf Technol (2012) 58:817–840 Table Sensitivity analysis with respect to various parameters on ordering quantity and total system cost for the model without shortages Parameters C C1 Percentage of under estimation and over estimation of parameters H=3 m Q Q /Q TC TC /TC H=4 m Q Q /Q TC TC /TC H = 30 m Q Q /Q TC TC /TC H=3 m Q Q /Q TC TC /TC H=4 m Q Q /Q TC TC /TC H = 30 m Q Q /Q TC TC /TC −50 −40 −30 −20 −10 +10 +20 +30 +40 +50 1.2410 2,385.4 1.7331 64,330 0.5137 1.6546 2,385.5 1.7331 81,830 0.5137 12.4096 2,385.5 1.7331 235,850 0.5138 5.0823 543.2 0.2441 121,070 0.9001 6.7764 543.2 0.2441 154,000 0.9001 50.8230 543.2 0.2441 443,860 0.9000 1.3887 2,110.6 1.5334 76,920 0.6143 1.8516 2,110.6 1.5334 97,840 0.6143 13.8868 2,110.7 1.5335 282,000 0.6143 3.7637 739.3 0.3322 121,930 0.9065 5.0183 739.3 0.3322 155,100 0.9065 37.6374 739.2 0.3322 447,030 0.9065 1.5445 1,881.9 1.3673 89,250 0.7128 2.0593 1,881.9 1.3673 113,530 0.7128 15.4448 1,881.9 1.3673 327,220 0.7128 3.0510 918.4 0.4127 122,860 0.9134 4.0679 918.4 0.4127 156,280 0.9134 30.5095 918.4 0.4127 450,430 0.9134 1.7095 1,688.1 1.2264 101,390 0.8097 2.2793 1,688.1 1.2265 128,960 0.8097 17.0950 1,688.1 1.2264 371,700 0.8097 2.6037 1083.1 0.4867 123,730 0.9199 3.4716 1,083.1 0.4867 157,390 0.9199 26.0370 1,083.1 0.4867 453,620 0.9198 1.8849 1,521.4 1.1054 113,360 0.9053 2.5133 1,521.4 1.1053 144,200 0.9053 18.8494 1,521.4 1.1053 415,620 0.9053 2.2965 1235.2 0.5550 124,520 0.9257 3.0619 1,235.3 0.5551 158,390 0.9257 22.9646 1,235.2 0.5550 456,510 0.9257 2.0721 1,376.4 1.0000 125,220 1.0000 2.7628 1,376.4 1.0000 159,280 1.0000 20.7212 1,376.4 1.0000 459,070 1.0000 1.3225 2225.5 1.0000 134,510 1.0000 1.7633 2,225.6 1.0000 171,100 1.0000 13.2249 2,225.6 1.0000 493,160 1.0000 2.2725 1,248.9 0.9074 136,970 1.0938 3.0300 1,248.9 0.9074 174,220 1.0938 22.7248 1,248.9 0.9074 502,140 1.0938 1.4060 2082.5 0.9358 132,110 0.9822 1.8747 2,082.5 0.9357 168,040 0.9821 14.0600 2,082.5 0.9358 484,340 0.9821 2.4877 1,135.9 0.8253 148,630 1.1870 3.3169 1,135.9 0.8253 189,060 1.1870 24.8770 1,135.9 0.8253 544,910 1.1870 1.4920 1953.2 0.8777 129,870 0.9655 1.9894 1,953.2 0.8776 165,200 0.9655 14.9204 1,953.2 0.8776 476,140 0.9655 2.7197 1,035.0 0.7519 160,230 1.2796 3.6262 1,035.0 0.7520 203,810 1.2796 27.1966 1,035.0 0.7520 587,440 1.2796 1.5808 1835.5 0.8248 127,780 0.9499 2.1078 1,835.5 0.8247 162,530 0.9499 15.8082 1,835.5 0.8248 468,450 0.9499 2.9706 944.2 0.6860 171,770 1.3718 3.9608 944.2 0.6860 218,490 1.3718 29.7057 944.2 0.6860 629,750 1.3718 1.6725 1728.0 0.7764 125,810 0.9353 2.2300 1,728.0 0.7764 160,020 0.9353 16.7252 1,728.0 0.7764 461,230 0.9352 3.2430 862.1 0.6264 183,270 1.4636 4.3240 862.1 0.6264 233,120 1.4636 32.4300 862.1 0.6264 671,900 1.4636 1.7673 1629.3 0.7321 123,940 0.9214 2.3565 1,629.2 0.7321 157,660 0.9214 17.6735 1,629.2 0.7321 454,400 0.9214 That is, changes in (α) will lead to the positive changes in Q and TC The change in deterioration rate (θ) leads to a negative change on the Q and TC That is, Q and TC decreases with the increase of (θ) The change in the stock-dependent consumption rate (β) leads to a negative change on the Q and TC That is, Q and TC decrease with the increase of (β) and the holding cost (C1 ) leads to a positive change in the Q and the present value of the TC The change in discount rate of net inflation rate (R) leads to a positive change in the Q and the present value of TC The change in the ordering cost (A) and the purchase unit cost (C) leads to a negative change in the Q and a positive change in the present value of the TC That is, Q decreases with the increase of A and C whereas TC increases with the increase of A and C The TC is more sensitive to the consumption rate (α), the C, and the deterioration rate (θ) as compared with other parameters These results are illustrated in Fig 2a–h 7.2 Special case In this section, the important particular case where R = and H = 1.0 is studied, and the differences with classical EOQ model is compared When R = and H = 1.0, then the total variable cost is given by, TC(T) = A0 + C0 α e(θ +β)T − (θ + β) + C10 α (θ + β) × − eγ − eγ T e(θ +β)T − eγ T (θ + β − γ ) × + − eγ − eγ T C10 α(1 − eγ T ) (θ + β) (7) [...]... of the cylinder in 2-mm-thick layers, maintaining the position of the tool and the chip inside the hole (see below) The aim of this analysis was to evaluate the chip formation and the interface tool/chip/piece and to investigate the cause of the elevated packing factor of the chip Figure 8 illustrates the procedure and shows the top surface of the layer cut from approximately 1 .5 mm below the top of. .. characterization of the integrity can be performed by the evaluation of the alterations of the structure under the surface, as measurements of plastic deformations, microhardness, among others Plastic deformations consist in the deformation and change of orientation of the grains near the surface of the material after the cutting The measured values correspond to the vertical distance from the surface to the point... complete the workpiece Thus, at the end of the second cycle, the same distance between holes of 1 .5 times the diameter of the tool was obtained This strategy was applied in the dry tests to avoid thermal influences that could compromise the results of the experiment 2.2 Tools The tools used in the experiments were coated carbide drills, DIN 653 7 K, provided by Walter AG Company The diameter of the tools... from the top, a higher packing factor was observed This region is the same as that where the microchipping of the margins occurred Figure 7a shows schematically the mechanics of cutting and shear zones, for a better understanding of the results presented in Fig 7b, which shows the transverse section of the chip on the depth of 3 mm below the top of the drill Figure 7b shows the disordered packing of the. .. greater values of the roughness on the machined surface due to the higher friction on the interface tool/chip/workpiece caused by the absence of the coolant and lubricant functions performed by the cutting fluid The MQL application condition resulted in the lowest roughness values in both analyzed regions of the holes Near the Region near the beginning of the 1200th hole Region near the bottom of the 1200th... the passage of the tool and the consequent greater roughness The microhardness measurements on the surface of the last holes, in the region near the bottom, resulted in values approximately twice as large as the bulk material hardness, which corroborates the hypothesis of the occurrence of microwelding, because applying high thermal and mechanical loads to the welded chip causes microhardening of the. .. r(ã) is the trajectory of the origin of , A(ã) is the orthogonal matrix and is the cutting tool fixed in the tool coordinate system The swept volume SVk of the tool between kth and (k+1)th NC lines is expressed as the union of the instantaneous tool volumes TV within the normalized time interval [0, 1] (Eq 12) Then the machined workpiece Wo is to be calculated either by the subtraction of the inital... due to the loss of these functions and how they affect the tool wear and the quality of the machined surface Therefore, in view of the complexity and extent of difficulties and different conditions in this type of change process, this work presents a study of drilling of AISI P20 steel with carbide tools, with different conditions for the application of cutting fluid The main goal was to evaluate the. .. involved by the material of the piece When the chip flow is compromised, it leads to packing and clogging of the chip and can cause the collapse of the tool [2023] It is also known that these problems are caused by the loss of the primary functions of the cutting fluids, which are lubrication, cooling, and transport of chips [6, 18, 19] However, it is not clear what are the changes in the interface... Figure 18c shows the part contour in sectional view which is reconstructed from the TDM model Figure 19 shows the real and virtual workpiece of a tested sample The NC program is composed of 10,600 lines and the part size is 70ì118ì44 mm3 The zero point of the workpiece is located in the middle of the workpiece on the top The cutting tool undergoes the three-axis movement The detail of the virtual workpiece