This paper is the result of the need for dynamic analysis of certain elevating platforms. Project... more This paper is the result of the need for dynamic analysis of certain elevating platforms. Project engineers research institute of Faculty of Mechanical Engineering in Pristina. Platform supporting structure is specific because of its purpose to hole and to transit the loads, as well as the kinematics of lifting mechanism. This paper started with briefly review of the scissors-type stationary platform construction and transportation system. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
The paper presents guidelines for designer work in the field of elevator and offers some original... more The paper presents guidelines for designer work in the field of elevator and offers some original solutions in the field of automated calculation of modern elevator installation equipment, kinematics analysis of car door mechanism and car frame construction. Based on the analysis of previous researches in this field, considerable influence of characteristics of elevator loose material on operational velocity and capacity is noticed during vibratory elevating. Principles of loose material motion along the vibrating surface are briefly lamed in the paper. The influence of basic factors defining the operational velocity and capacity of the modern elevator is quoted. In order to determine necessary parameters, appropriate mechanical model defining the modern elevator and loose material is formed. Finally, recommendations for determination of basic parameters of modern elevator are given in order to achieve the maximal capacity that is the optimal elevating effects in general. Mathematic...
Composite light steel beams use back to back double C sections rather than hot rolled steel I bea... more Composite light steel beams use back to back double C sections rather than hot rolled steel I beams, but the general form of construction is similar to conventional composite construction. Importantly, welded shear connectors cannot be used for the relatively thin steel used in light steel construction, and therefore it has been necessary to develop alternative forms of shear connectors using powder actuated, or pneumatically driven pins. These shear connectors use profiled strip steel elements which are fixed by pins. To know the resistance and behavior of these innovative shear connectors some tests have been carried out determine the design resistance of these connectors. The aim of this project is to provide guidance on the design of composite beams using light steel sections, a carefully study has been carried out getting eventually design examples which illustrate the calculus method and can be adapted in a easy way to a design particular characteristic, and design tables to a...
The principle of in-process metrology is to obtain measurement data directly in the manufacturing... more The principle of in-process metrology is to obtain measurement data directly in the manufacturing process and to use this for effective process control and for reliable quality assurance. One prerequisite to significantly improve the manufacturing processes is the incorporation of traceable dimensional metrology directly on machine tools. In order to introduce a traceability chain into inprocess geometrical measurements, European project EMRP IND62 TIM was agreed between EC and European metrology association Euramet. One of the tasks of the project is to develop highly accurate robust temperature invariant measurement standards that can be used in the harsh environment of the production floor for verification and mapping of the measurement errors of machine tools. Laboratory for Production measurement (LTM) at the University of Maribor Faculty of Mechanical Engineering is taking part in the consortium of this joint research project. The main task of LTM is to develop, manufacture an...
Abstract— in this paper a theoretical and experimental research of composite beams steel connecti... more Abstract— in this paper a theoretical and experimental research of composite beams steel connections is presented. For every model (SP1, SP2, SP3, SP4, SP5, SP6, SP7 and SP8), a loading diagram was previously made, through adding the displacements. Eight types of specimens were being researched, composed of rigid and semi-rigid connections from which 4 connections are with IPE-profile and 4 connections with tube's section for the beam. From the numerical analysis of the researched models, and especially from the experimental research in the Laboratory for Structures in the Faculty of Mechanical Engineering – Pristina, specific conclusions were received that are ought to have theoretical and practical usage for researchers in this area of interest. Keywords— rigid and semi-rigid connections; beam connections; design; bending moment; rotation; plastic hinge.
Strojnícky časopis - Journal of Mechanical Engineering
In this paper deriving of dynamical model of the load-bearing structure in form of a Trusses of t... more In this paper deriving of dynamical model of the load-bearing structure in form of a Trusses of the excavator with rotor is elaborated. Finite Element Methods and real working conditions are used to derive the dynamical model and to perform the local linearization of the dynamic elastic line for the structure of the load-bearing of the excavator with rotor. As carriers with continuous masses were treated the upper bearing structure of the Dynamical model and the truss levels. Finally, in accordance with the requirements of the problem, the degrees of freedom and the reference nodes are selected. The second stage of reduction, represents the final formation of the dynamical model of the load-bearing infrastructure, which however must be more accurate during the dynamic behaviour of the excavator with the rotor. The exposed work with the formation of the dynamic model of the Truss bearing structure, enables the concrete solution of the oscillations of the excavator with rotor. In addi...
This paper forms an integrated in order to predict the hysteretic moment-rotation curve and deter... more This paper forms an integrated in order to predict the hysteretic moment-rotation curve and determine the cyclic which can be used in the seismic design. In the first step the design method of EC3 and EC4 is studied. The standard design model has significant uncertainty, which underestimates the moment resistance, overestimates the initial stiffness and in certain cases the design failure mode does not follow the experimental one. An improvement of the EC3 resistance and stiffness model is proposed. An improvement of the EC3 resistance and stiffness model of the joints in large deformation regions, an extension of the model is proposed. In the second step of the analytical studies a semi-empirical method is proposed to predict the cyclic hysteretic behavior of the tested joints. The proposed method applies polygonal approximation of the hysteretic curve and coincides well with the experimental cyclic curve and shows a good agreement for absorbed energy during the increasing cycles. ...
This paper is the result of the need for optimization of the passenger lift in platforms by means... more This paper is the result of the need for optimization of the passenger lift in platforms by means of numerical method of the finite elements. The manner of modelling of supporting structures of the lift car frame is shown and identification of static behaviour of the car frame structure is made for the competent most critical lord case. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
Abstract: In this paper is presented mathematical modelling and analyzing of nominal conveying ab... more Abstract: In this paper is presented mathematical modelling and analyzing of nominal conveying ability of toothed belts with STD profile. For the designer and constructor of transmitters with toothed belts determination of nominal conveying ability for the respective belt profile has special practical importance. Under this power the constructor for the concrete conditions in which transmitter would worked, determine the conveying ability, respectively power which can carry the transmitter. From the mathematical expressions for each of the profiles STD belts through MathCAD program, are gained diagrams from that we can see change of nominal conveying ability, depending on the number of rotations of the driving pulley, and its kinematic diameter. Expressions for nominal conveying ability given in the paper are for the transmission ratio i = 1. When nominal conveying ability multiplied by factor, which takes into account, the impact of the belt width, then is obtained corrected conveying ability.
Abstract: System studied in this work is Luffing jib Crane used for materials handling. Cycle of ... more Abstract: System studied in this work is Luffing jib Crane used for materials handling. Cycle of work is jib luffing upwards. Research focuses on control of oscillations using modeling and simulations. Dynamic parameters investigated are: acceleration, angular velocity, forces and torque in main parts of crane, and influence of load swinging. Diagrams are presented as the solution results of the tested system. The aim is to design block diagram that represents model of crane and motion, analyze dynamics, look for optimal motion control functions and get conclusions about behavior of crane. Analysis will be done with computer application MapleSim.
Nowadays, the hydropower plant is considered as one of the more desirable sources of producing el... more Nowadays, the hydropower plant is considered as one of the more desirable sources of producing electrical energy due to its environmentally-friendly nature and extensive potential available throughout the world. On the other hand the hydropower plant allows the autonomous production of quantities of electrical energy capable of meeting the requirements of individual users starting from water resources which would otherwise be wasted. Based on the continuous requirement for renewable energy, a mini hydropower plant by using a propeller turbine is discussed due to its simple structure and easy production. The key parameters are studied for designing a hydropower plant, like water head, water flow-rate and turbine speed. The propeller turbine is considered for working under the best working conditions. During the detailed design of a hydropower plant some parameters are known and give us some indication about the geometry of the turbine and this is the starting point. The indication pa...
The purpose of this paper is to be experimental measurements and analysis of strain in the axis o... more The purpose of this paper is to be experimental measurements and analysis of strain in the axis of the rotor Excavator reductor the type SchRs 1300 (24 / 5), under the action of the rotor excavators loading during exploitation. With the development of the rotor excavators, problem of their dynamics, a phenomenon which must be taken into account in calculating such constructions. There fore, in this paper examined the measurements of dynamic analysis, under the action of the rotor excavator’s cargo. The basic source Excavator loads with resistance rotor is mine. The sharp variability of resistance to mining is due, above all, the specific geometry of the cutting, entry and exit periodical in which mine land. Resistance of mine in this paper will be simulated depending on the work regime of the Excavator. Measurements are made in real working conditions with maximum capacity during Excavator arm movement from right to left and the maximum depth in the lower level of the soil and the m...
This paper is the result of the need for optimization of the passenger lift in platforms by means... more This paper is the result of the need for optimization of the passenger lift in platforms by means of numerical method of the finite elements. The manner of modelling of supporting structures of the lift car frame is shown and identification of static behaviour of the car frame structure is made for the competent most critical lord case. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
This paper forms an integrated approach of optimization model for vehicle and model of minimizati... more This paper forms an integrated approach of optimization model for vehicle and model of minimization of referred mass of rotational elements. Optimization model with the criterion of minimal consumption vehicle include kinematics and dynamics parameters, technical regulations set before hoisting system and recommendations for the performing of some partial calculations. Minimization model of referred mass of rotational elements take into account referred mass of motor rotor, sheaves, drums, and gear transmission. Restriction of rotation elements minimization model are based on prescribed rope angling and disposition of excavation with rotor on ground four levels. Flywheel moments for motor rotor and referred mass of elements are listed in the relevant data sheets or catalogues. We can use the approximate empirical formulas given in specialized bibliography to
In this paper is the appointment of external loads caused by the resistance scrabble, which will ... more In this paper is the appointment of external loads caused by the resistance scrabble, which will enable the establishment of mathematical model of excavations with rotors, which are incorporated constructive and relevant parameters. Appointment of external loads enables solid solutions to design, reliability to excavations with rotors. Technique using computer programs and package will classical methods of reviewing and analyzing the loads of excavations with rotors, which appear in real conditions.
Crawler cranes belong to the family of mobile cranes that have complex structure with big dimensi... more Crawler cranes belong to the family of mobile cranes that have complex structure with big dimensions and mechanisms and high security requests. Main cycles of the work of Crawler Cranes can be divided into: lifting and lowering the working load, boom lifting and lowering, rotation (slewing) around its vertical axes in 360o, and translational movement forward/backwards. In this workshop, we are going to simulate the work of crawler crane while rotating around its vertical axes when fully engaged with load, which hangs in some height. The aim is to see the effects of dynamic forces (or moments) in the crane’s construction during this work cycle, particularly at the start and end of the rotation when working load swings. Reasons for doing this study are occasions of crane instability, while the crane rotates and load swings. We will research the effects of angular speed and workload swaying (swinging) in overall stability of crawler crane by looking for results of swaying angles, dynam...
In the paper is presented experimental analysis of steel space truss structure. The space truss s... more In the paper is presented experimental analysis of steel space truss structure. The space truss structure was loaded by spatial set of forces that simulate real condition of the structure with eight balanced member forces (4 chord and 4 diagonal members), up to structure failure. It was realized in a specially designed test facility. Tested space truss structure samples were made in real scale, according to the model originated after finite element analysis and optimization. Basic idea was to construct and test a space truss structure that can be made in average technology conditions, without special tools and requirements. Besides, results of a stress-strain finite element analysis are presented and comparison of the two analyses is given for the most critical regions of the space truss structure. Values of measured and calculated strains across model samples and characteristic measuring points are presented.
The paper presents guidelines for designer work in the field of elevator and offers some original... more The paper presents guidelines for designer work in the field of elevator and offers some original solutions in the field of automated calculation of modern elevator installation equipment, kinematics analysis of car door mechanism and car frame construction. Based on the analysis of previous researches in this field, considerable influence of characteristics of elevator loose material on operational velocity and capacity is noticed during vibratory elevating. Principles of loose material motion along the vibrating surface are briefly lamed in the paper. The influence of basic factors defining the operational velocity and capacity of the modern elevator is quoted. In order to determine necessary parameters, appropriate mechanical model defining the modern elevator and loose material is formed. Finally, recommendations for determination of basic parameters of modern elevator are given in order to achieve the maximal capacity that is the optimal elevating effects in general. Mathematic...
This paper is the result of the need for dynamic analysis of certain elevating platforms. Project... more This paper is the result of the need for dynamic analysis of certain elevating platforms. Project engineers research institute of Faculty of Mechanical Engineering in Pristina. Platform supporting structure is specific because of its purpose to hole and to transit the loads, as well as the kinematics of lifting mechanism. This paper started with briefly review of the scissors-type stationary platform construction and transportation system. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
The paper presents guidelines for designer work in the field of elevator and offers some original... more The paper presents guidelines for designer work in the field of elevator and offers some original solutions in the field of automated calculation of modern elevator installation equipment, kinematics analysis of car door mechanism and car frame construction. Based on the analysis of previous researches in this field, considerable influence of characteristics of elevator loose material on operational velocity and capacity is noticed during vibratory elevating. Principles of loose material motion along the vibrating surface are briefly lamed in the paper. The influence of basic factors defining the operational velocity and capacity of the modern elevator is quoted. In order to determine necessary parameters, appropriate mechanical model defining the modern elevator and loose material is formed. Finally, recommendations for determination of basic parameters of modern elevator are given in order to achieve the maximal capacity that is the optimal elevating effects in general. Mathematic...
Composite light steel beams use back to back double C sections rather than hot rolled steel I bea... more Composite light steel beams use back to back double C sections rather than hot rolled steel I beams, but the general form of construction is similar to conventional composite construction. Importantly, welded shear connectors cannot be used for the relatively thin steel used in light steel construction, and therefore it has been necessary to develop alternative forms of shear connectors using powder actuated, or pneumatically driven pins. These shear connectors use profiled strip steel elements which are fixed by pins. To know the resistance and behavior of these innovative shear connectors some tests have been carried out determine the design resistance of these connectors. The aim of this project is to provide guidance on the design of composite beams using light steel sections, a carefully study has been carried out getting eventually design examples which illustrate the calculus method and can be adapted in a easy way to a design particular characteristic, and design tables to a...
The principle of in-process metrology is to obtain measurement data directly in the manufacturing... more The principle of in-process metrology is to obtain measurement data directly in the manufacturing process and to use this for effective process control and for reliable quality assurance. One prerequisite to significantly improve the manufacturing processes is the incorporation of traceable dimensional metrology directly on machine tools. In order to introduce a traceability chain into inprocess geometrical measurements, European project EMRP IND62 TIM was agreed between EC and European metrology association Euramet. One of the tasks of the project is to develop highly accurate robust temperature invariant measurement standards that can be used in the harsh environment of the production floor for verification and mapping of the measurement errors of machine tools. Laboratory for Production measurement (LTM) at the University of Maribor Faculty of Mechanical Engineering is taking part in the consortium of this joint research project. The main task of LTM is to develop, manufacture an...
Abstract— in this paper a theoretical and experimental research of composite beams steel connecti... more Abstract— in this paper a theoretical and experimental research of composite beams steel connections is presented. For every model (SP1, SP2, SP3, SP4, SP5, SP6, SP7 and SP8), a loading diagram was previously made, through adding the displacements. Eight types of specimens were being researched, composed of rigid and semi-rigid connections from which 4 connections are with IPE-profile and 4 connections with tube's section for the beam. From the numerical analysis of the researched models, and especially from the experimental research in the Laboratory for Structures in the Faculty of Mechanical Engineering – Pristina, specific conclusions were received that are ought to have theoretical and practical usage for researchers in this area of interest. Keywords— rigid and semi-rigid connections; beam connections; design; bending moment; rotation; plastic hinge.
Strojnícky časopis - Journal of Mechanical Engineering
In this paper deriving of dynamical model of the load-bearing structure in form of a Trusses of t... more In this paper deriving of dynamical model of the load-bearing structure in form of a Trusses of the excavator with rotor is elaborated. Finite Element Methods and real working conditions are used to derive the dynamical model and to perform the local linearization of the dynamic elastic line for the structure of the load-bearing of the excavator with rotor. As carriers with continuous masses were treated the upper bearing structure of the Dynamical model and the truss levels. Finally, in accordance with the requirements of the problem, the degrees of freedom and the reference nodes are selected. The second stage of reduction, represents the final formation of the dynamical model of the load-bearing infrastructure, which however must be more accurate during the dynamic behaviour of the excavator with the rotor. The exposed work with the formation of the dynamic model of the Truss bearing structure, enables the concrete solution of the oscillations of the excavator with rotor. In addi...
This paper forms an integrated in order to predict the hysteretic moment-rotation curve and deter... more This paper forms an integrated in order to predict the hysteretic moment-rotation curve and determine the cyclic which can be used in the seismic design. In the first step the design method of EC3 and EC4 is studied. The standard design model has significant uncertainty, which underestimates the moment resistance, overestimates the initial stiffness and in certain cases the design failure mode does not follow the experimental one. An improvement of the EC3 resistance and stiffness model is proposed. An improvement of the EC3 resistance and stiffness model of the joints in large deformation regions, an extension of the model is proposed. In the second step of the analytical studies a semi-empirical method is proposed to predict the cyclic hysteretic behavior of the tested joints. The proposed method applies polygonal approximation of the hysteretic curve and coincides well with the experimental cyclic curve and shows a good agreement for absorbed energy during the increasing cycles. ...
This paper is the result of the need for optimization of the passenger lift in platforms by means... more This paper is the result of the need for optimization of the passenger lift in platforms by means of numerical method of the finite elements. The manner of modelling of supporting structures of the lift car frame is shown and identification of static behaviour of the car frame structure is made for the competent most critical lord case. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
Abstract: In this paper is presented mathematical modelling and analyzing of nominal conveying ab... more Abstract: In this paper is presented mathematical modelling and analyzing of nominal conveying ability of toothed belts with STD profile. For the designer and constructor of transmitters with toothed belts determination of nominal conveying ability for the respective belt profile has special practical importance. Under this power the constructor for the concrete conditions in which transmitter would worked, determine the conveying ability, respectively power which can carry the transmitter. From the mathematical expressions for each of the profiles STD belts through MathCAD program, are gained diagrams from that we can see change of nominal conveying ability, depending on the number of rotations of the driving pulley, and its kinematic diameter. Expressions for nominal conveying ability given in the paper are for the transmission ratio i = 1. When nominal conveying ability multiplied by factor, which takes into account, the impact of the belt width, then is obtained corrected conveying ability.
Abstract: System studied in this work is Luffing jib Crane used for materials handling. Cycle of ... more Abstract: System studied in this work is Luffing jib Crane used for materials handling. Cycle of work is jib luffing upwards. Research focuses on control of oscillations using modeling and simulations. Dynamic parameters investigated are: acceleration, angular velocity, forces and torque in main parts of crane, and influence of load swinging. Diagrams are presented as the solution results of the tested system. The aim is to design block diagram that represents model of crane and motion, analyze dynamics, look for optimal motion control functions and get conclusions about behavior of crane. Analysis will be done with computer application MapleSim.
Nowadays, the hydropower plant is considered as one of the more desirable sources of producing el... more Nowadays, the hydropower plant is considered as one of the more desirable sources of producing electrical energy due to its environmentally-friendly nature and extensive potential available throughout the world. On the other hand the hydropower plant allows the autonomous production of quantities of electrical energy capable of meeting the requirements of individual users starting from water resources which would otherwise be wasted. Based on the continuous requirement for renewable energy, a mini hydropower plant by using a propeller turbine is discussed due to its simple structure and easy production. The key parameters are studied for designing a hydropower plant, like water head, water flow-rate and turbine speed. The propeller turbine is considered for working under the best working conditions. During the detailed design of a hydropower plant some parameters are known and give us some indication about the geometry of the turbine and this is the starting point. The indication pa...
The purpose of this paper is to be experimental measurements and analysis of strain in the axis o... more The purpose of this paper is to be experimental measurements and analysis of strain in the axis of the rotor Excavator reductor the type SchRs 1300 (24 / 5), under the action of the rotor excavators loading during exploitation. With the development of the rotor excavators, problem of their dynamics, a phenomenon which must be taken into account in calculating such constructions. There fore, in this paper examined the measurements of dynamic analysis, under the action of the rotor excavator’s cargo. The basic source Excavator loads with resistance rotor is mine. The sharp variability of resistance to mining is due, above all, the specific geometry of the cutting, entry and exit periodical in which mine land. Resistance of mine in this paper will be simulated depending on the work regime of the Excavator. Measurements are made in real working conditions with maximum capacity during Excavator arm movement from right to left and the maximum depth in the lower level of the soil and the m...
This paper is the result of the need for optimization of the passenger lift in platforms by means... more This paper is the result of the need for optimization of the passenger lift in platforms by means of numerical method of the finite elements. The manner of modelling of supporting structures of the lift car frame is shown and identification of static behaviour of the car frame structure is made for the competent most critical lord case. After that, the procedure for creating the model for the leverage calculation is exposed. Strength, and the scarring capacity of construction, is proving by the structure analysis by Finite Element Method.
This paper forms an integrated approach of optimization model for vehicle and model of minimizati... more This paper forms an integrated approach of optimization model for vehicle and model of minimization of referred mass of rotational elements. Optimization model with the criterion of minimal consumption vehicle include kinematics and dynamics parameters, technical regulations set before hoisting system and recommendations for the performing of some partial calculations. Minimization model of referred mass of rotational elements take into account referred mass of motor rotor, sheaves, drums, and gear transmission. Restriction of rotation elements minimization model are based on prescribed rope angling and disposition of excavation with rotor on ground four levels. Flywheel moments for motor rotor and referred mass of elements are listed in the relevant data sheets or catalogues. We can use the approximate empirical formulas given in specialized bibliography to
In this paper is the appointment of external loads caused by the resistance scrabble, which will ... more In this paper is the appointment of external loads caused by the resistance scrabble, which will enable the establishment of mathematical model of excavations with rotors, which are incorporated constructive and relevant parameters. Appointment of external loads enables solid solutions to design, reliability to excavations with rotors. Technique using computer programs and package will classical methods of reviewing and analyzing the loads of excavations with rotors, which appear in real conditions.
Crawler cranes belong to the family of mobile cranes that have complex structure with big dimensi... more Crawler cranes belong to the family of mobile cranes that have complex structure with big dimensions and mechanisms and high security requests. Main cycles of the work of Crawler Cranes can be divided into: lifting and lowering the working load, boom lifting and lowering, rotation (slewing) around its vertical axes in 360o, and translational movement forward/backwards. In this workshop, we are going to simulate the work of crawler crane while rotating around its vertical axes when fully engaged with load, which hangs in some height. The aim is to see the effects of dynamic forces (or moments) in the crane’s construction during this work cycle, particularly at the start and end of the rotation when working load swings. Reasons for doing this study are occasions of crane instability, while the crane rotates and load swings. We will research the effects of angular speed and workload swaying (swinging) in overall stability of crawler crane by looking for results of swaying angles, dynam...
In the paper is presented experimental analysis of steel space truss structure. The space truss s... more In the paper is presented experimental analysis of steel space truss structure. The space truss structure was loaded by spatial set of forces that simulate real condition of the structure with eight balanced member forces (4 chord and 4 diagonal members), up to structure failure. It was realized in a specially designed test facility. Tested space truss structure samples were made in real scale, according to the model originated after finite element analysis and optimization. Basic idea was to construct and test a space truss structure that can be made in average technology conditions, without special tools and requirements. Besides, results of a stress-strain finite element analysis are presented and comparison of the two analyses is given for the most critical regions of the space truss structure. Values of measured and calculated strains across model samples and characteristic measuring points are presented.
The paper presents guidelines for designer work in the field of elevator and offers some original... more The paper presents guidelines for designer work in the field of elevator and offers some original solutions in the field of automated calculation of modern elevator installation equipment, kinematics analysis of car door mechanism and car frame construction. Based on the analysis of previous researches in this field, considerable influence of characteristics of elevator loose material on operational velocity and capacity is noticed during vibratory elevating. Principles of loose material motion along the vibrating surface are briefly lamed in the paper. The influence of basic factors defining the operational velocity and capacity of the modern elevator is quoted. In order to determine necessary parameters, appropriate mechanical model defining the modern elevator and loose material is formed. Finally, recommendations for determination of basic parameters of modern elevator are given in order to achieve the maximal capacity that is the optimal elevating effects in general. Mathematic...
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