Search results for: high-ratio differential speed rolling

Authors: Joon Ho Kim, Tae Kwon Ha

Abstract:

Effect of aging treatment on microstructural aspects of interfacial layers of the Cu/Al clad sheet produced by Differential Speed Rolling (DSR) process were studied by Electron Back Scattered Diffraction (EBSD). Clad sheet of Al/Cu has been fabricated by using DSR, which caused severe shear deformation between Al and Cu plate to easily bond to each other. Rolling was carried out at 100°C with speed ratio of 2, in which the total thickness reduction was 45%. Interface layers of clad sheet were analyzed by EBSD after subsequent annealing at 400°C for 30 to 120 min. With increasing annealing time, thickness of interface layer and fraction of high angle grain boundary were increased and average grain size was decreased.

Keywords: aluminium/copper clad sheet, differential speed rolling, interface layer, microstructure, annealing, electron back scattered diffraction

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Authors: Barenten Suciu

Abstract:

An electrical generator able to harness energy from the water waves and designed as a double-cone geared motor-generator (DCGMG), is proposed and theoretically investigated. Similar to a differential gear mechanism, used in the transmission system of the auto vehicle wheels, an angular speed differential is created between the cones rolling on two concentric circular rails. Water wave acting on the floating DCGMG produces and a gear-box amplifies the speed differential to gain sufficient torque for power generation. A model that allows computation of the speed differential, torque, and power of the DCGMG is suggested. Influence of various parameters, regarding the construction of the DCGMG, as well as the contact between the double-cone and rails, on the electro-mechanical output, is emphasized. Results obtained indicate that the generated electrical power can be increased by augmenting the mass of the double-cone, the span of the rails, the apex angle of the cones, the friction between cones and rails, the amplification factor of the gear-box, and the efficiency of the motor-generator. Such findings are useful to formulate a design methodology for the proposed wave-powered generator.

Keywords: amplification of angular speed differential, circular concentric rails, double-cone, wave-powered electrical generator

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Authors: H. B. Tibar, Z. Y. Jiang

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In this study, the influence of rolling process parameters such as the work roll cross angle and work roll shifting value on the strip shape and profile of aluminum have been investigated under dry conditions at a speed ratio of 1.3 using Hille 100 experimental mill. The strip profile was found to improve significantly with increase in work roll cross angle from 0^o to 1^o, with an associated decrease in rolling force. The effect of roll shifting (from 0 to 8mm) was not as significant as the roll cross angle. However, an increase in work roll shifting value achieved a similar decrease in rolling force as that of work roll cross angle. The effect of work roll shifting was also found to be maximum at an optimum roll speed of 0.0986 m/s for the desired thickness. Of all these parameters, the most significant effect of the strip shape profile was observed with variation of work roll cross angle. However, the rolling force can be a significantly reduced by either increasing the the work roll cross angle or work roll shifting.

Keywords: rolling speed ratio, strip shape, work roll cross angle, work roll shifting

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Authors: W. J. Kim

Abstract:

The behaviors of the recovery stress and strain of an ultrafine-grained Ni-50.2 at.% Ti alloy prepared by high-ratio differential speed rolling (HRDSR) were examined by a specially designed tensile-testing set up, and the factors that influence the recovery stress and strain were studied. After HRDSR, both the recovery stress and strain were enhanced compared to the initial condition. The constitutive equation showing that the maximum recovery stress is a sole function of the recovery strain was developed based on the experimental data. The recovery strain increased as the yield stress increased. The maximum recovery stress increased with an increase in yield stress. The residual recovery stress was affected by the yield stress as well as the austenite-to-martensite transformation temperature. As the yield stress increased and as the martensitic transformation temperature decreased, the residual recovery stress increased.

Keywords: high-ratio differential speed rolling, tensile testing, severe plastic deformation, shape memory alloys

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Authors: Z. Y. Jiang, H. B. Tibar, A. Aljabri

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Optimisation of the physical and mechanical properties of cold rolled thin strips is achieved by controlling the rolling parameters. In this paper, the factors affecting the asymmetrical cold rolling of thin low carbon steel strip have been studied at a speed ratio of 1.1 without lubricant applied. The effect of rolling parameters on the resulting microstructure was also investigated. It was found that under dry condition, work roll shifting and work roll cross angle can improve the strip profile, and the result is more significant with an increase of work roll cross angle rather than that of work roll shifting. However, there was no obvious change in microstructure. In addition, effects of rolling parameters on strip profile and microstructure have also been discussed.

Keywords: rolling speed ratio, microstructure, work roll cross angle, work roll shifting

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Authors: Vinod Kumar, Surjit Angra

Abstract:

The objective of this theoretical study is to develop simple design formulas for the prediction of minimum film thickness and maximum mean film temperature rise in lightly loaded high-speed rolling/sliding lubricated elliptical contacts incorporating starvation effect. Herein, the reported numerical analysis focuses on thermoelastohydrodynamically lubricated rolling/sliding elliptical contacts, considering the Newtonian rheology of lubricant for wide range of operating parameters, namely load characterized by Hertzian pressure (PH = 0.01 GPa to 0.10 GPa), rolling speed (>10 m/s), slip parameter (S varies up to 1.0), and ellipticity ratio (k = 1 to 5). Starvation is simulated by systematically reducing the inlet supply. This analysis reveals that influences of load, rolling speed, and level of starvation are significant on the minimum film thickness. However, the maximum mean film temperature rise is strongly influenced by slip in addition to load, rolling speed, and level of starvation. In the presence of starvation, reduction in minimum film thickness and increase in maximum mean film temperature are observed. Based on the results of this study, empirical relations are developed for the prediction of dimensionless minimum film thickness and dimensionless maximum mean film temperature rise at the contacts in terms of various operating parameters.

Keywords: starvation, lubrication, elliptical contact, traction, minimum film thickness

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Authors: J. Liu, Z. P. Yu, L. Xiong, Y. Feng, J. He

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According to the independence, accuracy and controllability of the driving/braking torque of the distributed drive electric vehicle, a control strategy of differential drive assisted steering was designed. Firstly, the assisted curve under different speed and steering wheel torque was developed and the differential torques were distributed to the right and left front wheels. Then the steering return ability assisted control algorithm was designed. At last, the joint simulation was conducted by CarSim/Simulink. The result indicated: the differential drive assisted steering algorithm could provide enough steering drive-assisted under low speed and improve the steering portability. Along with the increase of the speed, the provided steering drive-assisted decreased. With the control algorithm, the steering stiffness of the steering system increased along with the increase of the speed, which ensures the driver’s road feeling. The control algorithm of differential drive assisted steering could avoid the understeer under low speed effectively.

Keywords: differential assisted steering, control strategy, distributed drive electric vehicle, driving/braking torque

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Authors: M. Heydari Vini

Abstract:

There is a complicated relation between effective input parameters of cold rolling and output rolling force and exit thickness of strips.in many mathematical models, the effect of some rolling parameters have been ignored and the outputs have not a desirable accuracy. In the other hand, there is a special relation among input thickness of strips,the width of the strips,rolling speeds,mandrill tensions and the required exit thickness of strips with rolling force and the real exit thickness of the rolled strip. First of all, in this paper the effective parameters of cold rolling process modeled using an artificial neural network according to the optimum network achieved by using a written program in MATLAB,it has been shown that the prediction of rolling stand parameters with different properties and new dimensions attained from prior rolled strips by an artificial neural network is applicable.

Keywords: cold rolling, artificial neural networks, rolling force, real rolled thickness of strips

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Authors: Ali Moarrefzadeh

Abstract:

In metalworking, rolling is a metal forming process in which metal stock is passed through a pair of rolls. Rolling is classified according to the temperature of the metal rolled. If the temperature of the metal is above its recrystallization temperature, then the process is termed as hot rolling. If the temperature of the metal is below its recrystallization temperature, the process is termed as cold rolling. In terms of usage, hot rolling processes more tonnage than any other manufacturing process, and cold rolling processes the most tonnage out of all cold working processes. This article describes the use of advanced tubing inspection NDT methods for boiler and heat exchanger equipment in the petrochemical industry to supplement major turnaround inspections. The methods presented include remote field eddy current, magnetic flux leakage, internal rotary inspection system and eddy current.

Keywords: hot forming, cold forming, metal, rolling, simulation

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Authors: Alexander Pesin, Denis Pustovoytov, Mikhail Sverdlik

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Materials with ultrafine-grained structure and unique physical and mechanical properties can be obtained by methods of severe plastic deformation, which include processes of asymmetric rolling (AR). Asymmetric rolling is a very effective way to create ultrafine-grained structures of metals and alloys. Since the asymmetric rolling is a continuous process, it has great potential for industrial production of ultrafine-grained structure sheets. Basic principles of asymmetric rolling are described in detail in scientific literature. In this work finite element modeling of asymmetric rolling and metal forming processes in multiroll gauge was performed. Parameters of the processes which allow achieving significant values of shear strain were defined. The results of the study will be useful for the research of the evolution of ultra-fine metal structure in asymmetric rolling.

Keywords: asymmetric rolling, equivalent strain, FEM, multiroll gauge, profile, severe plastic deformation, shear strain, sheet

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Authors: Mohammadhadi Mirmohammadi, Reza Safian, Hossein Haddad

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This paper presents an innovative solution for complex multi-objective optimization problem which is a part of efforts toward maximizing rolling mill throughput and minimizing processing costs in tandem cold rolling. This computational intelligence based optimization has been applied to the rolling schedules of tandem cold rolling mill. This method involves the combination of two derivative-free optimization procedures in the form of nested loops. The first optimization loop is based on Improving Hit and Run method which focus on balance of power, force and reduction distribution in rolling schedules. The second loop is a real-coded genetic algorithm based optimization procedure which optimizes energy consumption and productivity. An experimental result of application to five stand tandem cold rolling mill is presented.

Keywords: derivative-free optimization, Improving Hit and Run method, real-coded genetic algorithm, rolling schedules of tandem cold rolling mill

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Authors: Cai-Wan Chang-Jian, Han-Ting Tsai

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In the processing industry of metal forming, rolling is the most used method of processing. In a cold rolling factory of stainless steel, there occurs a product defect on temper rolling process within cold rolling. It is called 'roll marks', which is a phenomenon of undesirable flatness problem. In this research, we performed a series of experimental measurements on the roll marks, and we used optical sensors to measure it and compared the vibration frequency of roll marks with the vibration frequency of key components in the skin pass mill. We found there is less correlation between the above mentioned data. Finally, we took measurement on the motor driver in rolling mill. We found that the undulation frequency of motor could match with the frequency of roll marks, and then we have confirmed that the motor’s undulation caused roll marks.

Keywords: roll mark, plane strain, rolling mill, stainless steel

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Authors: A. Pesin, D. Pustovoytov, M. Sverdlik

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During production of rolled steel strips the quality of the surface of finished strips influences steel consumption considerably. The most critical areas for crack formation during rolling are lateral sides of slabs. Deformation behaviors of the slab edge in roughing rolling process were analyzed by the finite element method with Deform-3D. In this study our focus is the analysis of the influence of edger’s form on the possibility to decrease surface cracking during roughing hot rolling.

Keywords: roughing hot rolling, FEM, crack, bulging

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Authors: Hyo-In Koh, Anders Nordborg, Alex Sievi, Chun-Kwon Park

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The noise source for the interior room of the high speed train is constituted by the rolling contact between the wheel and the rail, aerodynamic noise and structure-borne sound generated through the vibrations of bogie, connection points to the carbody. Air-borne sound is radiated through the panels and structures into the interior room of the trains. The high-speed lines are constructed with slab track systems and many tunnels. The interior noise level and the frequency characteristics vary according to types of the track structure and the infrastructure. In this paper the main sound sources and the transfer paths are studied to find out the contribution characteristics of the sources to the interior noise of a high-speed rail vehicle. For the identification of the acoustic power of each parts of the rolling noise sources a calculation model of wheel/rail noise is developed and used. For the analysis of the transmission of the sources to the interior noise noise and vibration are measured during the operation of the vehicle. According to operation speeds, the mainly contributed sources and the paths could be analyzed. Results of the calculations on the source generation and the results of the measurement with a high-speed train are shown and discussed.

Keywords: rail vehicle, high-speed, interior noise, noise source

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Authors: T. Vamsee Kiran, A. Gopi

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The undesired torque and flux ripple may occur in conventional direct torque control (DTC) induction motor drive. DTC can improve the system performance at low speeds by continuously tuning the regulator by adjusting the Kp, Ki values. In this differential evolution (DE) is proposed to adjust the parameters (Kp, Ki) of the speed controller in order to minimize torque ripple, flux ripple, and stator current distortion.The DE based PI controller has resulted is maintaining a constant speed of the motor irrespective of the load torque fluctuations.

Keywords: differential evolution, direct torque control, PI controller

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Authors: A. Pesin, D. Pustovoytov, M. Sverdlik

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Traditionally, the need in items represents a large body of rotation (e.g. shrouds of various process units: a converter, a mixer, a scrubber, a steel ladle and etc.) is satisfied by using them at engineering enterprises. At these enterprises large parts of bodies of rotation are made on stamping units or bending and forming machines. In Nosov Magnitogorsk State Technical University in alliance with JSC "Magnitogorsk Metal and Steel Works" there was suggested and implemented the technology for producing such items based on a combination of asymmetric rolling processes and plastic bending under conditions of the plate mill. In this paper, based on finite elemental mathematical simulation in technology of a combined process of asymmetric rolling and bending plastic has been improved. It is shown that for the same curvature along the entire length of the metal sheet it is necessary to introduce additional asymmetry speed when rolling front end and tape trailer. Production of large bodies of rotation at mill 4500 JSC "Magnitogorsk Metal and Steel Works" showed good convergence of theoretical and experimental values of the curvature of the metal. Economic effect obtained more than 1.0 million dollars.

Keywords: asymmetric rolling, plastic bending, combined process, FEM

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Authors: Riad Harouz, Brahim Mahfoud

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The threads and the circles on reinforced concrete are obtained by process of hot rolling with pebbles finishers in metal carbide which present a way of rolling around the outside diameter. Our observation is that this throat presents geometrical wear after the end of its cycle determined in tonnage. In our study, we have determined, in a first step, experimentally measurements of the wear in terms of thermo-mechanical parameters (Speed, Load, and Temperature) and the influence of these parameters on the wear. In the second stage, we have developed a mathematical model of lifetime useful for the prognostic of the wear and their changes.

Keywords: lifetime, metal carbides, modeling, thermo-mechanical, wear

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Authors: Abdulrahman Aljabri, Essam R. I. Mahmoud, Hamad Almohamedi, Zhengyi Jiang

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Cold rolled thin strip has received intensive attention through technological and theoretical progress in the rolling process, as well as researchers have focused on its control during rolling as an essential parameter for producing thinner strip with good shape and profile. An advanced approach has been proposed to analysis the thin strip profile in cold rolling of pair roll crossing and shifting mill using Finite Element Analysis (FEA) with an ALE technique. The ALE (Arbitrary Lagrangian-Eulerian) techniques to enable more flexibility of the ALE technique in the adjustment of the finite element mesh, which provides a significant tool for simulating the thin strip under realistic rolling process constraint and provide accurate model results. The FEA can provide theoretical basis for the 3D model of controlling the strip shape and profile in thin strip rolling, and deliver an optimal rolling process parameter, and suggest corrective changes during cold rolling of thin strip.

Keywords: pair roll crossing, work roll shifting, strip shape and profile, finite element modeling

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Authors: Seok Hong Min, Tae Kwon Ha

Abstract:

In the present study, hot-rolled sheets of Mg-xZn alloy s(x=6, 8, and 10 weight percent) were produced by employing casting, homogenization heat treatment, hot rolling, and annealing processes subsequently. Effect of Zn addition on the microstructure and mechanical properties of Mg-Zn alloys were also investigated in each process. Through calculation of phase equilibria of Mg-Zn alloys, solution treatment temperature was decided as temperatures from 350 oC, where supersaturated solid solution can be obtained. After solution treatment, hot rolling was successfully conducted by reduction of 60%. Compression and tension tests were carried out at room temperature on the samples as-cast, solution treated, hot-rolled and recrystallized after rolling.

Keywords: Mg-Zn alloy, heat treatment, microstructure, mechanical properties, hot rolling

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Authors: Ibtisam Masmali, Edwin Beggs

Abstract:

In this paper, we take example of differential calculi, on the finite group A4. Then, we apply methods of non-commutative of non-commutative differential geometry to this example, and see how similar the results are to those of classical differential geometry.

Keywords: differential calculi, finite group A4, Christoffel symbols, covariant derivative, torsion compatible

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Authors: Vinayak Malaghan, Digvijay Pawar

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Speed is the independent parameter which plays a vital role in the highway design. Design consistency of the highways is checked based on the variation in the operating speed. Often the design consistency fails to meet the driver’s expectation which results in the difference between operating and design speed. Literature reviews have shown that significant crashes take place in horizontal curves due to lack of design consistency. The paper focuses on continuous speed profile study on tangent to curve transition for both day and night daytime. Data is collected using GPS device which gives continuous speed profile and other parameters such as acceleration, deceleration were analyzed along with Tangent to Curve Transition. In this present study, models were developed to predict operating speed on tangents and horizontal curves as well as model indicating the speed reduction from tangent to curve based on continuous speed profile data. It is observed from the study that vehicle tends to decelerate from approach tangent to between beginning of the curve and midpoint of the curve and then accelerates from curve to tangent transition. The models generated were compared for both day and night and can be used in the road safety improvement by evaluating the geometric design consistency.

Keywords: operating speed, design consistency, continuous speed profile data, day and night time

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Authors: Mamidi Ramakrishna Rao

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Doubly-fed induction generators (DFIG) due to their advantages like speed variation and four-quadrant operation, find its application in wind turbines. DFIG besides supplying power to the grid has to support reactive power (kvar) under grid voltage variations, should contribute minimum fault current during faults, have high efficiency, minimum weight, adequate rotor protection during crow-bar-operation from +20% to -20% of rated speed.  To achieve the optimum performance, a good electromagnetic design of DFIG is required. In this paper, a simple and heuristic global optimization – Differential Evolution has been used. Variables considered are lamination details such as slot dimensions, stack diameters, air gap length, and generator stator and rotor stack length. Two operating conditions have been considered - voltage and speed variations. Constraints included were reactive power supplied to the grid and limiting fault current and torque. The optimization has been executed separately for three objective functions - maximum efficiency, weight reduction, and grid fault stator currents. Subsequent calculations led to the conclusion that designs determined through differential evolution help in determining an optimum electrical design for each objective function.

Keywords: design optimization, performance, DFIG, differential evolution

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Authors: Henry Ogbemudia Omoregbee, Mabel Usunobun Olanipekun

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Debris fault diagnosis based on acoustic signals in rolling element bearing running at low speed and high radial loads are more of low amplitudes, particularly in the case of debris faults whose signals necessitate high sensitivity analyses. As the rollers in the bearing roll over debris trapped in grease used to lubricate the bearings, the envelope signal created by amplitude demodulation carries additional diagnostic information that is not available through ordinary spectrum analysis of the raw signal. The kurtosis value obtained for three different scenarios (debris induced, outer crack induced, and a normal good bearing) couldn't be used to easily identify whether the used bearings were defective or not. It was established in this work that the envelope spectrum analysis detected the fault signature and its harmonics induced in the debris bearings when bandpass filtering of the raw signal with the frequency band specified by kurtogram and spectral kurtosis was made.

Keywords: rolling bearings, rolling element bearing noise, bandpass filtering, harmonics, envelope spectrum analysis, spectral kurtosis

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Authors: Piyas Palit, Urbi Pal, Jitendra Mathur, Santanu Das

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Bearing is an important machinery part in the industry. When bearings fail to meet their expected life the consequences are increased downtime, loss of revenue and missed the delivery. This article describes the failure of a gearbox bearing in rolling contact fatigue. The investigation consists of visual observation, chemical analysis, characterization of microstructures using optical microscopes and hardness test. The present study also considers bearing life as well as the operational condition of bearings. Surface-initiated rolling contact fatigue, leading to a surface failure known as pitting, is a life-limiting failure mode in many modern machine elements, particularly rolling element bearings. Metallography analysis of crack propagation, crack morphology was also described. Indication of fatigue spalling in the ferrography test was also discussed. The analysis suggested the probable reasons for such kind of failure in operation. This type of spalling occurred due to (1) heavier external loading condition or (2) exceeds its service life.

Keywords: bearing, rolling contact fatigue, bearing life

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Authors: Jianrong Wu, Xuan Fu, Akihiro Higashi, Zhiming Tan

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CMOS image sensors with a rolling shutter are used broadly in the digital cameras embedded in mobile devices. The rolling shutter suffers the flicker artifacts from the fluorescent lamp, and it could be observed easily. In this paper, the characteristics of illumination flicker in motion case were analyzed, and two efficient detection methods based on matching fragment selection were proposed. According to the experimental results, our methods could achieve as high as 100% accuracy in static scene, and at least 97% in motion scene.

Keywords: illumination flicker, embedded camera, rolling shutter, detection

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Authors: Taher El-Bitar, Maha El-Meligy, Eman El-Shenawy, Almosilhy Almosilhy, Nader Dawood

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The steel contains 0.3% C and 0.004% B, beside Mn, Cr, Mo, and Ni. The alloy was processed by using 20-ton capacity electric arc furnace (EAF), and then refined by ladle furnace (LF). Liquid steel was cast as rectangular ingots. Dilatation test showed the critical transformation temperatures Ac₁, Ac₃, M_s and M_f as 716, 835, 356, and 218 °C. The ingots were austenitized and soaked and then rough rolled to thin slabs with 80 mm thickness. The thin slabs were then reheated and soaked for finish rolling to 6.0 mm thickness plates. During the rough rolling, the roll force increases as a result of rolling at temperatures less than recrystallization temperature. However, during finish rolling, the steel reflects initially continuous static recrystallization after which it shows strain hardening due to fall of temperature. It was concluded that, the steel plates were successfully heat treated by quenching-tempering at 250 ºC for 20 min.

Keywords: armor steel, austenitizing, critical transformation temperatures (CTTs), dilatation curve, martensite, quenching, rough and finish rolling processes, soaking, tempering, thermo-mechanical processing

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Authors: Mohammed Tahir, Jonas Lagergren

Abstract:

Vancron 40, a nitrided powder metallurgical tool Steel, is used in cold work applications where the predominant failure mechanisms are adhesive wear or galling. Typical applications of Vancron 40 are among others fine blanking, cold extrusion, deep drawing and cold work rolls for cluster mills. Vancron 40 positive results for cold work rolls for cluster mills and as a tool for some severe metal forming process makes it competitive compared to other type of work rolls that require higher precision, among others in cold rolling of thin stainless steel, which required high surface finish quality. In this project, three roll materials for cold rolling of stainless steel strip was examined, Vancron 40, Narva 12B (a high-carbon, high-chromium tool steel alloyed with tungsten) and Supra 3 (a Chromium-molybdenum tungsten-vanadium alloyed high speed steel). The purpose of this project was to study the depth profiles of the ironed stainless steel strips, emergence of galling and to study the lubrication performance used by steel industries. Laboratory experiments were conducted to examine scratch of the strip, galling and surface roughness of the roll materials under severe tribological conditions. The critical sliding length for onset of galling was estimated for stainless steel with four different lubricants. Laboratory experiments result of performance evaluation of resistance capability of rolls toward adhesive wear under severe conditions for low and high reductions. Vancron 40 in combination with cold rolling lubricant gave good surface quality, prevents galling of metal surfaces and good bearing capacity.

Keywords: Vancron 40, cold rolling, adhesive wear, galling, surface finish, lubricant, stainless steel

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Authors: Mohamed Abdel-Monem, Gamal Sowilam, Omar Hegazy

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This paper presents a technical assessment of an electric vehicle with two independent rear-wheel motor and an improved traction control system. The electric differential and the control strategy have been implemented to assure that in a straight trajectory, the two rear-wheels run exactly at the same speed, considering the same/different road conditions under the left and right side of the wheels. In case of turning to right/left, the difference between the two rear-wheels speeds assures a vehicle trajectory without sliding, thanks to a harmony between the electric differential and the control strategy. The present article demonstrates a complete model and analysis of a traction control system, considering four different traction scenarios, for two independent rear-wheels motors for electric vehicles. Furthermore, the vehicle model, including wheel dynamics, load forces, electric differential, and control strategy, is designed and verified by using MATLAB/Simulink environment.

Keywords: electric vehicle, energy saving, multi-motor, electric differential, simulation and control

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Authors: Martin K. Steiger, Lukas Heisler, Hans-Georg Brachtendorf

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Deep neural networks and their variants form the backbone of many AI applications. Based on the so-called residual networks, a continuous formulation of such models as ordinary differential equations (ODEs) has proven advantageous since different techniques may be applied that significantly increase the learning speed and enable controlled trade-offs with the resulting error at the same time. For the evaluation of such models, high-performance numerical differential equation solvers are used, which also provide the gradients required for training. However, whether classical gradient-based methods are even applicable or which one yields the best results has not been discussed yet. This paper aims to redeem this situation by providing empirical results for different applications.

Keywords: deep neural networks, gradient-based learning, image processing, ordinary differential equation networks

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Authors: Fuziyah Ishak, Siti Norazura Ahmad

Abstract:

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

Keywords: accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations

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