Horizontal washing machine are an important class of automatic washing machines. During spinning ... more Horizontal washing machine are an important class of automatic washing machines. During spinning extreme unbalance occurs producing forced vibrations excited by the rotating unbalance. The purpose of this paper is to investigate the effect of the various parameters of the washing machine on the vibration amplitude and velocity of the machine drum in terms of the spinning speed. This covers the effect of the isolators stiffness, damping coefficient and the drum mass for specific laundry capacity.
International Journal of Scientific & Technology Research, 2014
Robustness is one of the requirements used in controllers and compensators design. The designs pr... more Robustness is one of the requirements used in controllers and compensators design. The designs presented in the previous papers did not consider the robustness of the controller or compensator. Therefore, the objective of this paper is to investigate the robustness of I-PD, PD-PI and PI-PD controllers used to control second-order processes against uncertainty in the process parameters. A variation of ± 20 % in process parameters is considered through simulation to study its effect on the system performance parameters using the tuned controllers. With I-PD controller controlling an underdamped second order process, the variation in process natural frequency and damping ratio has almost no effect on the maximum percentage overshoot , maximum percentage undershoot , settling time and the phase margin of the control system. The variation of the process natural frequency produced a maximum change of 33 % in the system gain margin. The damping ratio change does not affect the gain margin ...
The problem of controlling an unstable delayed double integrating process using a feedforward fir... more The problem of controlling an unstable delayed double integrating process using a feedforward first-order lag-lead compensator is studied. The effect of time delay of the process in a range between 1 and 12 seconds is considered. The compensator is tuned using MATLAB optimization toolbox with five forms of the objective function in terms of the error between the step time response of the closed-loop control system and the response steady-state value. Using the proposed compensator with the delayed double integrating process indicates the robustness of the compensator in the time delay range used with superior time-based specifications compared with other techniques based on using PID controllers with the same process.
The problem of controlling an unstable delayed double integrating process with fractional delay u... more The problem of controlling an unstable delayed double integrating process with fractional delay using a feed forward first-order lag-lead compensator is studied. The effect of time delay of the process in a range between 0.1 and 0.9 seconds is considered. The compensator is tuned using MATLAB optimization toolbox with five forms of the objective function in terms of the error between the step time response of the closed-loop control system and the response steady-state value. Using the proposed compensator with the fractional delayed double integrating process indicates the robustness of the compensator in the time delay range used with superior time-based specifications compared with other technique based on PID controller .
Optimal design of shallow foundations used in supporting various machinery is of vital importance... more Optimal design of shallow foundations used in supporting various machinery is of vital importance. Badly designed foundations result in serious problems such as soil failure, large soil settlement, low vibration efficiency and resonance of structures near by the foundation. An optimal design approach based on using MATLAB optimization tool box is used to provide the optimal design of a machinery foundation on a silty soil. To minimize the foundation cost the foundation mass is used as an objective function. To control the high technical quality of the foundation from soil mechanics and vibrations point of view, 10 functional constraints are used. The soil isolation efficiency of the soil is responsible about the vibration transmitted to the surrounding. Therefore, the effect of the minimum isolation efficiency in the range of 71 – 81 % on the foundation optimal design is investigated in details. The performance of the foundation against the minimum isolation efficiency is outlined s...
In this paper a novel 2/2 feedforward compensator is proposed and applied to control a second-ord... more In this paper a novel 2/2 feedforward compensator is proposed and applied to control a second-order-like very slow process having 150 seconds settling time. The proposed compensator is capable of controlling the steady-state characteristics of the closed-loop control system and its dynamic characteristics. The MATLAB command “fsolve” is used to solve a set of nonlinear equations leading to tuning the proposed compensator. The resulting control system dynamics using the proposed compensator are completely satisfactory. The time response of the control system to a step input has no overshoot , the settling time is reduced to only 0.2 seconds and the steady-state error is as low as 0.01 for a unit step input.
This paper investigates the synthesis of 6 bar-3 sliders mechanism for motion generation having m... more This paper investigates the synthesis of 6 bar-3 sliders mechanism for motion generation having maximum time ratio and an assigned stroke. The mechanism output is a translation motion. The optimal dimensionless dimensions of the mechanism links are assigned using MATLAB optimization tool. The objective function to be maximized is the mechanisms time ratio subjected to four functional constraints controlling the performance of the mechanism. A desired dimensionless stroke between 3 and 8 is covers. In all the cases, the optimal dimensions are assigned and the transmission angel of the mechanism is kept between 45 and 135 degrees.
machines are extensively utilized in industrial life, since it represents a vital element in indu... more machines are extensively utilized in industrial life, since it represents a vital element in industrial processes. Therefore, early detection of faults of rotating machines is necessary to avoid the forced stopping for frequent maintenance in industrial processes. Various condition monitoring and detecting procedures are used to diagnose the rotating machinery faults based on vibration signature analysis, temperature monitoring, noise signature analysis, lubricant signature analysis, Artificial Intelligence (AI) techniques. many AI methods are in use for bearing defects diagnosis. For instance, Fuzzy Inference System (FIS); Artificial Neural Networks (ANNs) and Adaptive Neuro-Fuzzy Inference System (ANFIS). AI techniques are advanced technologically for classifying and detection various rotating machinery faults. This research work describes an inclusive method to some extent for detecting and classification of bearing faults using two methods of artificial intelligence which are AN...
This paper presents the 12 part of a series of research papers aiming at studying the development... more This paper presents the 12 part of a series of research papers aiming at studying the development of mechanical engineering in ancient Egypt. It presents the techniques and technology used by ancient Egyptians to cut hard stones and produce very complex artifacts. The ancient Egyptians used stone saws, drills and lathe turning and invented instrumentation to generate flat surfaces with accuracy better than 0.005 mm and measure dimensions from 1.3 mm to 10 km.
Volume 9, Issue 8, August 2020 Page 6 Abstract Highly oscillation in industrial processes is comp... more Volume 9, Issue 8, August 2020 Page 6 Abstract Highly oscillation in industrial processes is completely undesirable, and controller tuning has to solve this problem. PD-PI is a one of the second generation of the conventional PID controllers which is suggested to overcome this problem with improved performance regarding the spike characteristics associated with PID controllers. This work has proven that using the PD-PI controller is capable of solving the problems of the highly oscillated third order process. A highly oscillated third order process of 57% maximum overshoot and 75 seconds settling time is controlled using a PD-PI controller (through simulation). Different objective functions are tried in the tuning process of the controller assuming that the optimization problem is a constrained one. The overshoot and settling time are used to investigate the performance of the closed loop control system. The performance of the control system using the PD-PI controller is compared wi...
2019 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), 2019
The bearings are vital parts for any rotating machinery and work as a key role for any machinery ... more The bearings are vital parts for any rotating machinery and work as a key role for any machinery and dynamic system. For rotating machinery to work properly, the bearing should be healthy. This is necessary to ensure continuity and non-stop production. So, it is necessary to be supervised and monitor any rotating machinery for early detection of bearing faults. Different techniques are in use for bearing faults diagnosis such as envelope analysis, Wavelet analysis (WA), Short Time Fourier Transforms (STFT), Fuzzy Inference System (FIS) and Neural Networks (NN). This paper presents a different approach for detection and classification of bearing faults using Adaptive Neuro-Fuzzy Inference System (ANFIS). The proposed data were taken via using accelerometer which is mounted on the bearing housing in Machinery Fault Simulator (MFS). The data is measured in the X and Y directions at 2100 rev/min and 51.2 kHz rotating speed and sampling rate, respectively. Details of the design procedure...
Lag-lead compensators are well known in automatic control engineering. They have 4 parameters to ... more Lag-lead compensators are well known in automatic control engineering. They have 4 parameters to be adjusted (tuned) for proper operation. The frequency response of the control system or the root locus plot are traditionally used to tune the compensator in a lengthy procedure. A first order with an integrator process in a unity feedback loop of 67.3 % maximum overshoot and 12 seconds settling time is controlled using a lag-lead compensator (through simulation). The lag-lead compensator is tuned by minimizing the sum of absolute error of the control system using MATLAB. Four functional constrains are used to control the performance of the lag-lead compensated control system. The result was reducing the process oscillation to 2.438 % overshoot and an 0.648 seconds settling time. The performance of the lag-lead compensated system with the present tuning approach is compared with the classical tuning using the root locus technique. The comparison showed that the present tuning technique...
Compensators are used in place of classical PID controllers for possible achievement of better pe... more Compensators are used in place of classical PID controllers for possible achievement of better performance. Unsatisfactory dynamics of some industrial processes represent an engineering problem that hat to be solved. Highly oscillating processes and very slow processes are examples. In this paper a novel feedback PD compensator based is proposed and applied to control secondorder like processes having oscillating characteristics (damping ratio ≤ 0.8). The compensator proposed is capable of controlling the steady-state characteristics of the closed-loop control system and its dynamic characteristics. The advantage of the proposed compensator is its simple structure and tuning using an ITAE objective function. It was possible with the proposed compensator to satisfy a system performance with very low overshoot and settling time. The compensator tuning is reduced to simple relation with the process parameters with correlation coefficient as high as 0.9999.
Optimal synthesis of mechanisms is a successful approach for mechanism design to satisfy all the ... more Optimal synthesis of mechanisms is a successful approach for mechanism design to satisfy all the desired characteristics of the designed mechanism. The crank - slider mechanism has wide industrial applications and providing an offset feature provides a design with a stroke greater than its crank length and a time ratio greater than one. The optimal design problem in this case is a constrained multi-dimensional problem. Powell optimization technique is used to minimize a special objective function combining the mechanism stroke and time ratio. 2 functional constraint functions are used for the minimum and maximum transmission angle of the mechanism. The optimal results are fitted in a proper nonlinear model relating the normalized mechanism dimensions to the normalized stroke and time ratio. A comparison is conducted between the optimal design and the non-optimal one illustrating the advantages of the optimal approach.
International Journal of Innovation and Applied Studies, 2014
The objective of this paper is to investigate the dynamics of the barrel assembly-recoil mechanis... more The objective of this paper is to investigate the dynamics of the barrel assembly-recoil mechanism of military cannons when using air springs and a constant damping coefficient hydraulic damper in their recoil mechanisms. The elastic characteristics of the air spring is nonlinear and the recoil mechanism orientation introduces extra nonlinearity to the dynamic model of the system. An extremely nonlinear model of the barrel assembly is derived and solved using Runge-Kutta 4 method to provide the dynamic response of the barrel assembly upon firing. The simulation results using the data of a Howitzer M114 cannon are presented for recoil mechanism orientation ? 50 degrees. The performance of the recoil mechanism is evaluated through the minimum and maximum displacements of the barrel assembly and the settling time of its response upon firing. The effect of the number of air springs on the performance of recoil mechanism is investigated. The analysis shows that it is possible with air sp...
I. Introduction Dealing with process disturbance rejection is an important matter in control engi... more I. Introduction Dealing with process disturbance rejection is an important matter in control engineering. Not all types of controllers can succeed in achieving this objective. The aim of this series of research papers is to investigate controllers (or compensators) that are efficient in reducing the disturbance effect on the dynamics of the control system. Skogestad (2001) presented tuning rules for PID controllers for slow response with acceptance disturbance rejection. He considered pure time delay, integrating, integrating with delay and lag and double integrating processes [1]. Skogestad (2003) presented analytic tuning rules for PID controller. He modified the integral term to improve disturbance rejection for integrating processes. He considered first-order, second-order, pure time delay, integrating, integrating with lag and double integrating processes [2]. Arvanitis, Pasgianos and Kalogeropoulos (2005) presented simple methods for tuning three-term controllers for dead-time...
International Journal of Innovation and Applied Studies, 2014
Robustness is one of the requirements used in controllers and compensators design. This paper exa... more Robustness is one of the requirements used in controllers and compensators design. This paper examines the robustness of a Notch and a Sallen-Key compensator when used to control a highly oscillating second-order process. A variation of
This research paper aims at studying the generation of a crescent path using a 6 bar – 2 sliders ... more This research paper aims at studying the generation of a crescent path using a 6 bar – 2 sliders planar mechanism. This mechanism can generate crescent paths of different confutations depending on the location of the point on the mechanism coupler and the dimensions of the mechanism within a dimensions range assigned by an optimal synthesis of the mechanism. The x-span of the generated path equals the mechanism stroke. Normalized dimensions are used to provide wide application of the paths obtained. The generated crescent paths have a normalized y-span from 0.067 to 1.13 indicating the wide range of the generated crescent path.
Horizontal washing machine are an important class of automatic washing machines. During spinning ... more Horizontal washing machine are an important class of automatic washing machines. During spinning extreme unbalance occurs producing forced vibrations excited by the rotating unbalance. The purpose of this paper is to investigate the effect of the various parameters of the washing machine on the vibration amplitude and velocity of the machine drum in terms of the spinning speed. This covers the effect of the isolators stiffness, damping coefficient and the drum mass for specific laundry capacity.
International Journal of Scientific & Technology Research, 2014
Robustness is one of the requirements used in controllers and compensators design. The designs pr... more Robustness is one of the requirements used in controllers and compensators design. The designs presented in the previous papers did not consider the robustness of the controller or compensator. Therefore, the objective of this paper is to investigate the robustness of I-PD, PD-PI and PI-PD controllers used to control second-order processes against uncertainty in the process parameters. A variation of ± 20 % in process parameters is considered through simulation to study its effect on the system performance parameters using the tuned controllers. With I-PD controller controlling an underdamped second order process, the variation in process natural frequency and damping ratio has almost no effect on the maximum percentage overshoot , maximum percentage undershoot , settling time and the phase margin of the control system. The variation of the process natural frequency produced a maximum change of 33 % in the system gain margin. The damping ratio change does not affect the gain margin ...
The problem of controlling an unstable delayed double integrating process using a feedforward fir... more The problem of controlling an unstable delayed double integrating process using a feedforward first-order lag-lead compensator is studied. The effect of time delay of the process in a range between 1 and 12 seconds is considered. The compensator is tuned using MATLAB optimization toolbox with five forms of the objective function in terms of the error between the step time response of the closed-loop control system and the response steady-state value. Using the proposed compensator with the delayed double integrating process indicates the robustness of the compensator in the time delay range used with superior time-based specifications compared with other techniques based on using PID controllers with the same process.
The problem of controlling an unstable delayed double integrating process with fractional delay u... more The problem of controlling an unstable delayed double integrating process with fractional delay using a feed forward first-order lag-lead compensator is studied. The effect of time delay of the process in a range between 0.1 and 0.9 seconds is considered. The compensator is tuned using MATLAB optimization toolbox with five forms of the objective function in terms of the error between the step time response of the closed-loop control system and the response steady-state value. Using the proposed compensator with the fractional delayed double integrating process indicates the robustness of the compensator in the time delay range used with superior time-based specifications compared with other technique based on PID controller .
Optimal design of shallow foundations used in supporting various machinery is of vital importance... more Optimal design of shallow foundations used in supporting various machinery is of vital importance. Badly designed foundations result in serious problems such as soil failure, large soil settlement, low vibration efficiency and resonance of structures near by the foundation. An optimal design approach based on using MATLAB optimization tool box is used to provide the optimal design of a machinery foundation on a silty soil. To minimize the foundation cost the foundation mass is used as an objective function. To control the high technical quality of the foundation from soil mechanics and vibrations point of view, 10 functional constraints are used. The soil isolation efficiency of the soil is responsible about the vibration transmitted to the surrounding. Therefore, the effect of the minimum isolation efficiency in the range of 71 – 81 % on the foundation optimal design is investigated in details. The performance of the foundation against the minimum isolation efficiency is outlined s...
In this paper a novel 2/2 feedforward compensator is proposed and applied to control a second-ord... more In this paper a novel 2/2 feedforward compensator is proposed and applied to control a second-order-like very slow process having 150 seconds settling time. The proposed compensator is capable of controlling the steady-state characteristics of the closed-loop control system and its dynamic characteristics. The MATLAB command “fsolve” is used to solve a set of nonlinear equations leading to tuning the proposed compensator. The resulting control system dynamics using the proposed compensator are completely satisfactory. The time response of the control system to a step input has no overshoot , the settling time is reduced to only 0.2 seconds and the steady-state error is as low as 0.01 for a unit step input.
This paper investigates the synthesis of 6 bar-3 sliders mechanism for motion generation having m... more This paper investigates the synthesis of 6 bar-3 sliders mechanism for motion generation having maximum time ratio and an assigned stroke. The mechanism output is a translation motion. The optimal dimensionless dimensions of the mechanism links are assigned using MATLAB optimization tool. The objective function to be maximized is the mechanisms time ratio subjected to four functional constraints controlling the performance of the mechanism. A desired dimensionless stroke between 3 and 8 is covers. In all the cases, the optimal dimensions are assigned and the transmission angel of the mechanism is kept between 45 and 135 degrees.
machines are extensively utilized in industrial life, since it represents a vital element in indu... more machines are extensively utilized in industrial life, since it represents a vital element in industrial processes. Therefore, early detection of faults of rotating machines is necessary to avoid the forced stopping for frequent maintenance in industrial processes. Various condition monitoring and detecting procedures are used to diagnose the rotating machinery faults based on vibration signature analysis, temperature monitoring, noise signature analysis, lubricant signature analysis, Artificial Intelligence (AI) techniques. many AI methods are in use for bearing defects diagnosis. For instance, Fuzzy Inference System (FIS); Artificial Neural Networks (ANNs) and Adaptive Neuro-Fuzzy Inference System (ANFIS). AI techniques are advanced technologically for classifying and detection various rotating machinery faults. This research work describes an inclusive method to some extent for detecting and classification of bearing faults using two methods of artificial intelligence which are AN...
This paper presents the 12 part of a series of research papers aiming at studying the development... more This paper presents the 12 part of a series of research papers aiming at studying the development of mechanical engineering in ancient Egypt. It presents the techniques and technology used by ancient Egyptians to cut hard stones and produce very complex artifacts. The ancient Egyptians used stone saws, drills and lathe turning and invented instrumentation to generate flat surfaces with accuracy better than 0.005 mm and measure dimensions from 1.3 mm to 10 km.
Volume 9, Issue 8, August 2020 Page 6 Abstract Highly oscillation in industrial processes is comp... more Volume 9, Issue 8, August 2020 Page 6 Abstract Highly oscillation in industrial processes is completely undesirable, and controller tuning has to solve this problem. PD-PI is a one of the second generation of the conventional PID controllers which is suggested to overcome this problem with improved performance regarding the spike characteristics associated with PID controllers. This work has proven that using the PD-PI controller is capable of solving the problems of the highly oscillated third order process. A highly oscillated third order process of 57% maximum overshoot and 75 seconds settling time is controlled using a PD-PI controller (through simulation). Different objective functions are tried in the tuning process of the controller assuming that the optimization problem is a constrained one. The overshoot and settling time are used to investigate the performance of the closed loop control system. The performance of the control system using the PD-PI controller is compared wi...
2019 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), 2019
The bearings are vital parts for any rotating machinery and work as a key role for any machinery ... more The bearings are vital parts for any rotating machinery and work as a key role for any machinery and dynamic system. For rotating machinery to work properly, the bearing should be healthy. This is necessary to ensure continuity and non-stop production. So, it is necessary to be supervised and monitor any rotating machinery for early detection of bearing faults. Different techniques are in use for bearing faults diagnosis such as envelope analysis, Wavelet analysis (WA), Short Time Fourier Transforms (STFT), Fuzzy Inference System (FIS) and Neural Networks (NN). This paper presents a different approach for detection and classification of bearing faults using Adaptive Neuro-Fuzzy Inference System (ANFIS). The proposed data were taken via using accelerometer which is mounted on the bearing housing in Machinery Fault Simulator (MFS). The data is measured in the X and Y directions at 2100 rev/min and 51.2 kHz rotating speed and sampling rate, respectively. Details of the design procedure...
Lag-lead compensators are well known in automatic control engineering. They have 4 parameters to ... more Lag-lead compensators are well known in automatic control engineering. They have 4 parameters to be adjusted (tuned) for proper operation. The frequency response of the control system or the root locus plot are traditionally used to tune the compensator in a lengthy procedure. A first order with an integrator process in a unity feedback loop of 67.3 % maximum overshoot and 12 seconds settling time is controlled using a lag-lead compensator (through simulation). The lag-lead compensator is tuned by minimizing the sum of absolute error of the control system using MATLAB. Four functional constrains are used to control the performance of the lag-lead compensated control system. The result was reducing the process oscillation to 2.438 % overshoot and an 0.648 seconds settling time. The performance of the lag-lead compensated system with the present tuning approach is compared with the classical tuning using the root locus technique. The comparison showed that the present tuning technique...
Compensators are used in place of classical PID controllers for possible achievement of better pe... more Compensators are used in place of classical PID controllers for possible achievement of better performance. Unsatisfactory dynamics of some industrial processes represent an engineering problem that hat to be solved. Highly oscillating processes and very slow processes are examples. In this paper a novel feedback PD compensator based is proposed and applied to control secondorder like processes having oscillating characteristics (damping ratio ≤ 0.8). The compensator proposed is capable of controlling the steady-state characteristics of the closed-loop control system and its dynamic characteristics. The advantage of the proposed compensator is its simple structure and tuning using an ITAE objective function. It was possible with the proposed compensator to satisfy a system performance with very low overshoot and settling time. The compensator tuning is reduced to simple relation with the process parameters with correlation coefficient as high as 0.9999.
Optimal synthesis of mechanisms is a successful approach for mechanism design to satisfy all the ... more Optimal synthesis of mechanisms is a successful approach for mechanism design to satisfy all the desired characteristics of the designed mechanism. The crank - slider mechanism has wide industrial applications and providing an offset feature provides a design with a stroke greater than its crank length and a time ratio greater than one. The optimal design problem in this case is a constrained multi-dimensional problem. Powell optimization technique is used to minimize a special objective function combining the mechanism stroke and time ratio. 2 functional constraint functions are used for the minimum and maximum transmission angle of the mechanism. The optimal results are fitted in a proper nonlinear model relating the normalized mechanism dimensions to the normalized stroke and time ratio. A comparison is conducted between the optimal design and the non-optimal one illustrating the advantages of the optimal approach.
International Journal of Innovation and Applied Studies, 2014
The objective of this paper is to investigate the dynamics of the barrel assembly-recoil mechanis... more The objective of this paper is to investigate the dynamics of the barrel assembly-recoil mechanism of military cannons when using air springs and a constant damping coefficient hydraulic damper in their recoil mechanisms. The elastic characteristics of the air spring is nonlinear and the recoil mechanism orientation introduces extra nonlinearity to the dynamic model of the system. An extremely nonlinear model of the barrel assembly is derived and solved using Runge-Kutta 4 method to provide the dynamic response of the barrel assembly upon firing. The simulation results using the data of a Howitzer M114 cannon are presented for recoil mechanism orientation ? 50 degrees. The performance of the recoil mechanism is evaluated through the minimum and maximum displacements of the barrel assembly and the settling time of its response upon firing. The effect of the number of air springs on the performance of recoil mechanism is investigated. The analysis shows that it is possible with air sp...
I. Introduction Dealing with process disturbance rejection is an important matter in control engi... more I. Introduction Dealing with process disturbance rejection is an important matter in control engineering. Not all types of controllers can succeed in achieving this objective. The aim of this series of research papers is to investigate controllers (or compensators) that are efficient in reducing the disturbance effect on the dynamics of the control system. Skogestad (2001) presented tuning rules for PID controllers for slow response with acceptance disturbance rejection. He considered pure time delay, integrating, integrating with delay and lag and double integrating processes [1]. Skogestad (2003) presented analytic tuning rules for PID controller. He modified the integral term to improve disturbance rejection for integrating processes. He considered first-order, second-order, pure time delay, integrating, integrating with lag and double integrating processes [2]. Arvanitis, Pasgianos and Kalogeropoulos (2005) presented simple methods for tuning three-term controllers for dead-time...
International Journal of Innovation and Applied Studies, 2014
Robustness is one of the requirements used in controllers and compensators design. This paper exa... more Robustness is one of the requirements used in controllers and compensators design. This paper examines the robustness of a Notch and a Sallen-Key compensator when used to control a highly oscillating second-order process. A variation of
This research paper aims at studying the generation of a crescent path using a 6 bar – 2 sliders ... more This research paper aims at studying the generation of a crescent path using a 6 bar – 2 sliders planar mechanism. This mechanism can generate crescent paths of different confutations depending on the location of the point on the mechanism coupler and the dimensions of the mechanism within a dimensions range assigned by an optimal synthesis of the mechanism. The x-span of the generated path equals the mechanism stroke. Normalized dimensions are used to provide wide application of the paths obtained. The generated crescent paths have a normalized y-span from 0.067 to 1.13 indicating the wide range of the generated crescent path.
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Papers by Galal Hassaan