- Email: [email protected]
Formal Method to Design a Macro-model of a Transport Vehicle Mechanical System Translational Motion
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 129 (2015) 362 – 368
International Conference on Industrial Engineering
Formal method to design a macro-model of a transport vehicle mechanical system translational motion Grinchenkov D.V., Mokhov V.A., Spiridonova I.A.* Platov South-Russian State Polytechnic University (NPI), 132, St. Prosvescheniya, Rostov region, Novocherkassk, 346428, Russian Federation
Abstract The article is devoted to the formalization [1] of the design of mathematical model for the computer simulation of complex mechanical systems. The simulation object is the mechanical system of the transport vehicle (railway or automotive). The complete system is split into subsystems (macro-elements), each of macro-element has one degree of freedom. The whole system of macro-elements can be considered as a translational moving system. A two-stage algorithm to construct a mathematical macro-model of the transport vehicle mechanical system - is presented. The results of the computer simulation are presented. This modeling method is beneficial to be used in the educational process to establish interdisciplinary connections, as well as to create e-learning resources for students of applied mathematics, information technology and engineering profiles. © 2015 2015The TheAuthors. Authors.Published Published Elsevier © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE(http://creativecommons.org/licenses/by-nc-nd/4.0/). 2015). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015) Keywords: Computer simulation; complex mechanical system; vehicle; formal method; object-oriented approach; e-leaning.
1. Introduction One of the significant areas of research in the field of solving the problems of modeling through the use of modern computer technologies is a computer simulation of complex mechanical systems. It should be noted that at present, particular significance is the formalization of mathematical models of systems and processes in the context of solving specific practical problems, due in no small measure to solving of the urgent problem of import substitution in the field of software systems CAD/CAM/CAE. A very important problem is the
* Corresponding author. Tel.: +7-928-121-61-77. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015)
doi:10.1016/j.proeng.2015.12.079
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
problem of teaching students to create new, including specialized software systems that support the functions of mathematical modeling complex mechanical systems [2-11]. The problem of modeling the dynamics of mechanical systems of vehicles [1,4-9] is a good example of the object of research in teaching students the principles of computer simulation [10, 11]. This task allows student to have the intuitive interpretation of system parameters. Intuitive interpretation also corresponds to results, which are presented in the form of dependency graphs from the time the motion. From a practical point of view, to the area of the considered mechanical systems are included trains and railway crews, and it includes means of road (automotive) transport, including vehicles of auto truck transport. 2. The ways to modeling of mechanical systems Due to the practical importance of a problem, currently the considerable number of works is devoted to creation of adequate mathematical model of mechanical systems of transport vehicles and their computer realization. In particular, for several years the design of the relevant modules of software system "Universal Mechanism" (UM) is developing under the supervising of prof. Pogorelov D.Y. [5, 6, 12, 13]. However, it should be noted that the study of models and methods [5-9, 12, 13] for implementation of mathematical software of software complex UM [14] is a separate task that requires a good mathematical training and deep knowledge in the subject area. This task is beyond the base curriculum for computer simulation training course for the bachelor studying, for which mechanical transport systems are not the main and only object to study. As a different approach to the construction of mathematical models of complex mechanical systems process can be used formal methods based on the principles of the existence of the Electro-mechanical analogy and on the using of "global" variables, objects, and elements of circuitry [15, 16]. An example of such approach is the use of simulation technology of mechanical systems by means of a formal language and a graphical representation of the modeled mechanical system using similar electronic circuits in accordance with international standard VHDL-AMS. However, in order to establish interdisciplinary connections in the learning process more rational approach may be the use of the method of generalized energy phase variables (GEPV) proposed in [10] for the design of mathematical models within CAD systems. The topology of the system in this method can be represented as in the form of the electrical circuit analog electronic devices and as in the classic form of a directed graph. The implementation steps of this method are based on mathematical and formal tools of the disciplines of applied mathematics. Objects for modeling by method GEPV can be discrete dynamical systems and sub-systems of different physical nature. It is worth noting that the GEPV analogy of phase variables, of topological and of component equations for different physical systems are consistent with the standard VHDL-AMS in which the corresponding phase variables appear with the names "across quantity" and "through quantity". The construction of mathematical models of complex mechanical systems can be carried out using classical methods of theoretical mechanics [2-4] with the use of modern technology implementation in an object-oriented approach also. Objectoriented approach is widely used now to solve different mathematical and specialized problems [17-20] by means of simulation by tools of object-oriented programming technology. Method GEPV and object-oriented approach at the correct introduction of the parameters of the system elements allows to construct a mathematical model which fully corresponds [1, 4] to the model obtained by mathematical tools of theoretical mechanics. It should be noted that when modeling systems consisting of the solids that perform various types of mechanical motion, the basic method GEPV [10] require to design of a complex system of circuits and the imposition of laws of interrelation variables based from the subject area. 3. Two-stage method to modeling of mechanical systems of transport vehicle As discussed in [1], for use in the educational process rationally [20] integrated application of methods of classical mechanics with GEPV method. When using the proposed technology the resulting system to describe the in relation to each other of movement of macro-elements in the direction of the main (target) movement can be seen as performing translational mechanical movement that corresponds to one of five basic types of physical subsystems, adequately simulated by the method GEPV [4]. Under the proposed approach, the macro system, shown in left part of figure 1, means to split the system into interconnected subsystems, each of which is characterized by separate
363
364
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
generalized coordinate. In right part of figure 1 shows a directed graph of the relationship of macro-elements, constructed to apply the method GEPV.
& F
D
Fig.1. Left: sampling of the system and the interaction effects to method GEPV; Right: directed graph.
Directed graphs in a method GEPV allow to present conveniently and visually the topology of the system with different types of layout and interaction of the elements, as its are shown in figures 2, 3.
& F
Fig. 2. Left: second system; Right: directed graph to second system.
& F
Fig. 3. Left: third system; Right: directed graph to third system.
Proposed by the authors in [1] for use in the educational process the two-stage algorithm to construct mathematical macro-model of the mechanical system of the transport vehicle in the general form is presented below. Stage 1. First, it should make splitting (left parts of figures 1, 2, 3) of the system into subsystems (macroelements). Using the principles of the second kind Lagrange equations it should determine the inertial characteristics (reduced mass) of each object and the external action on the system. Stage 2. It should identify interaction effects of macro elements (subsystems) between themselves and with the external environment (constrains). It should build equivalent circuit and a directed graph (rights parts of figures 1, 2,
365
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
3) in accordance with the principles and algorithm of method application GEPV for the mechanical translational system. By formal tools should construct the mathematical model in the Cauchy problem form. Generalized information flow diagrams to implementation of the formalized steps of the proposed algorithm are showed in figure 4.
Sm j
S mi
K mi
Km j
I L ji
mj
L ji
Um j
U mi
mi
mGi
Mi
Mj
L
R
mGi
ioj
R ji
K L ji
ioj
K R ji
ª L*ji º ¬ ¼
U ji
ª R*ji º ¬ ¼
Fig. 4. Left: scheme of information flows of stage 1; Right: scheme of information flows of stage 2.
The most important aspect of drawing up of mathematical model by proposed methodology [1] is the correct description of the topology of the system: correct numbering of macro elements (subsystems) and indexation of the parameters of the energy elements in accordance with the following elements in the kinematic chain, i.e. a sequence of "transfer" to move from item to item. The reduced mass for the subsystem k that contains ‘ n ’ wheels pairs:
mk
mrm k
§ § § k ·2 · · ¨ m1 k 2n m2 k ¨ 1 ¨ i2 ¸¸ , Rk ¸ ¸ ¸ ¨ © ¨ ¹ © ¹¹ ©
(1)
where the notation means: m1 – mass of main element; m2 – mass of each wheel; R – radius of wheel; i – inertia radius of wheel. The force of external actions on the system causing its translational moving on the inclined surface consists of two parts:
F
r Fd r mC g sin D
r¦ j
Mj Rj
r mC g sin D .
(2)
This force [4] is directly applied to a driving macro-element, at the movement on an inclined surface it depends on the mass of whole system mC representing the total mass of all elements of system, but not their reduced mass analogs. At the movement up the inclined plane the component of influence of gravity has, naturally, the negative sign, and at the movement on a horizontal surface in general is absent. The traction force Fd is generated by the adhesion of the wheels with the support surface and depends on the moments acting on the axis of the driving wheels pairs with index ‘j ‘. At the stage of the braking system, the driving force is absent. It should be noted that only at this stage, the mathematical model must have components of friction
366
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
about "the earth" that are shown in graphs of right parts of figures 1, 2, 3 by dashed lines for subsystems (macroelements) that are including driving wheels pair. The resulting mathematical model [1, 4] when it presented in the form of the classical Cauchy problem contains the basis phase variables as absolute velocities U m for all macro-elements and as forces I L of elastic interaction. The interaction of subsystems « i o j » of any type (elastic L ji or friction R ji ) introduced in the mathematical model by dependence on the difference of velocities
U mi U m j , U j 0
U ji
Um j 0 Um j .
(3)
To automate the construction of a mathematical model by means of object-oriented programming is convenient to introduce a generalized representation of the components of the right side of the equations:
1 ; ª L*ji º mi ¬ ¼
K mi
1 L ji
where K L ji
K L ji U ji ; ª¬ R*ji º¼ 1 R ji
; K R ji
K R ji U ji ,
(4)
.
4. Resulting mathematical model
So, for the system presented in figure 1 the resulted mathematical macro-model beyond the stage of "braking" has the form: dI L21 dt dU m2 dt
ª¬ L*21 º¼ ;
dI L31 dt
ª¬ L*31 º¼ ;
dU m1
* º¼ 1 I L21 1 I L31 1 F ; K m1 1 ª¬ R21
dt
* º¼ 1 I L21 ; K m2 1 ª¬ R21
dU m3 dt
K m3 1 I L31 ;
dS mi dt
U mi , i 1,..,3 .
(5)
The resulting system of equations must be complemented by an unambiguous definition of the initial conditions of each stage of the movement. 5. Results of computer simulation
In figures 5, 6 are presented the results of computer simulation of the dynamics of the five macro elements with the help of a software product that is implemented using object-oriented programming based on the described algorithm for the construction of mathematical software.
Fig. 5. Left: graphs of displacements; Right: graphs of velocities.
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
Fig. 6. Left: graphs of accelerations; Right: graphs of elastic forces.
Object-oriented approach to the implementation of the technologies of constructing mathematical software allows simulate the behavior of the system in different modes: controlled motion, inertia and braking, which is clearly reflected in right part of figure 5. The definition and treatment of "exceptions" in the software product also allows you to track the permissible limits of relative movement of macro-elements. When reaching these values the characteristics of the elastic connecting elements change automatically. 6. Conclusions
Presented in this article, the technology of constructing mathematical models of complex mechanical systems is useful for use in the educational process in order to establish interdisciplinary links at example that is clear and intuitively understandable to the student. Development of electronic educational resources in this area allows you to combine educational material relating to the principles and foundations of mathematical and computer modeling, classical problems and methods of the subject area, use the principles of object-oriented approach to modeling, the characteristics and limitations of formal methods computer-aided design. When you set out the way to solve the described problem presents material [20] relating to such sections applied mathematics like graph theory, theory of algorithms, numerical methods of solution of problems of mathematical analysis and other sections of educational material related to the problem of modeling specific tasks programmatically. Additional elements which increase the quality of e-learning resources is also developing expert systems for selection of model parameters and their adjustment in accordance with the formal characteristics and analysis of the obtained results in order to improve the adequacy of the model. References [1] I.A. Spiridonova, D.V. Grinchenkov, On a formalization of a construction by two-stage of a mathematical macromodel to complex mechanical system, J University News. North-Caucasian Region. Technical Sciences Series. 6 (2013) 47–51. [2] D.C. Karnopp, D.L. Margolis, R.C. Rosenberg, System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems, fifth ed., Hoboken, New Jersey, 2012. [3] M.I. Bat, G.Yu. Dzhanelidze, A.S. Kelzon, Theoretical Mechanics at Examples and Problems, tenth ed., Lan, St. Petersburg, 2013. [4] I.A. Spiridonova, D.V. Grinchenkov, On a mathematical modeling of a complex mechanical system by formal and by classical methods, J. Russian Electromechanics. 5 (2013) 75–79. [5] D.Y. Pogorelov, Contemporary algorithms for computer synthesis of equations of motion of multibody systems, J. Computer and Systems Sciences International. 44(4) (2005) 503–51. [6] D.Y. Pogorelov, On numerical methods of modeling large multibody systems, J. Mechanism and Machine Theory. 34(5) (1999) 791–800. [7] R. Sinha, C.J.J. Paredis, P.K. Khosla, Integration of mechanical CAD and behavioral modeling, Proc. IEEE/ACM. (2000) 31–36. [8] A.A. Zarifian, P.G. Kolpahchyan, Computer Modeling of Electric Locomotive as Controlled Electromechanical System, J Multibody System Dynamics. 22(4) (2009) 425–436. [9] R. Kovalev, N. Lysikov, G. Mikheev, D.Pogorelov, V. Simonov, V. Yazykov, S. Zakharov, I. Zharov, I. Goryacheva, S. Soshenkov, E. Torskaya, Freight Car Models and Their Computer-Aided Dynamic Analysis, J Multibody System Dynamics. 22(4) (2009) 399–423. [10] I.P. Norenkov, CAD Basics, fourth ed., BMSTU Publ., Moscow, 2009. [11] M.A. Bauer, M. Sarrafzadeh, F. Somezi, Fundamental CAD algorithms, IEEE Transaction on Computer-Aided Design of Integrated Circuits and Systems. 19(12) (2000) 1449–1475. [12] D.Y. Pogorelov, Jacobian Matrices Of the Motion Equations of a System of Bodies, J Computer and Systems Sciences International. 46(4) (2007) 563–577. [13] D.Y. Pogorelov, Simulation of Constraints by Compliant Joints, J Computer and Systems Sciences International. 50(1) (2011) 158–173. [14] Information on http://www.umlab.ru
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D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368 [15] M. Graziano, M. Ruo Roch, Automotive VHDL-AMS electro-mechanics simulations, New Trends and Developments in Automotive System Engineering. (2011) 541–566. [16] A.C.R. Silva, I.A. Grout, A methodology and a tool to design of mixed-signal technology, Proc. Electronic, Robotics and Automotive Mechanics Conference. (2007) 164–169. [17] A.N. Ivanchenko, A.A. Dorofeev, An object-oriented approach to computation of higher derivatives, J. University News. North-Caucasian Region. Technical Sciences Series. 1 (2014) 9–15. [18] A.N. Ivanchenko, A.A. Dorofeev, An object-oriented approach to construction of spline functions defined on linear spans, J University News. North-Caucasian Region. Technical Sciences Series. 1 (2015) 11–18. [19] I.V. Georgitca, S.S. Goncharov, V.A. Mokhov, Multiagent simulation of network attacks DDoS-type, E-J. Engineering Journal of Don. 26(3) (2013) 75–85. available at: http://ivdon.ru/uploads/article/pdf/IVD_79_georgitsa.pdf_1852.pdf. [20] D.V. Grinchenkov, D.N. Kushchiy, The methodological, technological and legal aspects of using the e-learning resources, J University News. North-Caucasian Region. Technical Sciences Series. 2 (2013) 118–123.
ScienceDirect Procedia Engineering 129 (2015) 362 – 368
International Conference on Industrial Engineering
Formal method to design a macro-model of a transport vehicle mechanical system translational motion Grinchenkov D.V., Mokhov V.A., Spiridonova I.A.* Platov South-Russian State Polytechnic University (NPI), 132, St. Prosvescheniya, Rostov region, Novocherkassk, 346428, Russian Federation
Abstract The article is devoted to the formalization [1] of the design of mathematical model for the computer simulation of complex mechanical systems. The simulation object is the mechanical system of the transport vehicle (railway or automotive). The complete system is split into subsystems (macro-elements), each of macro-element has one degree of freedom. The whole system of macro-elements can be considered as a translational moving system. A two-stage algorithm to construct a mathematical macro-model of the transport vehicle mechanical system - is presented. The results of the computer simulation are presented. This modeling method is beneficial to be used in the educational process to establish interdisciplinary connections, as well as to create e-learning resources for students of applied mathematics, information technology and engineering profiles. © 2015 2015The TheAuthors. Authors.Published Published Elsevier © by by Elsevier Ltd.Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE(http://creativecommons.org/licenses/by-nc-nd/4.0/). 2015). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015) Keywords: Computer simulation; complex mechanical system; vehicle; formal method; object-oriented approach; e-leaning.
1. Introduction One of the significant areas of research in the field of solving the problems of modeling through the use of modern computer technologies is a computer simulation of complex mechanical systems. It should be noted that at present, particular significance is the formalization of mathematical models of systems and processes in the context of solving specific practical problems, due in no small measure to solving of the urgent problem of import substitution in the field of software systems CAD/CAM/CAE. A very important problem is the
* Corresponding author. Tel.: +7-928-121-61-77. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the International Conference on Industrial Engineering (ICIE-2015)
doi:10.1016/j.proeng.2015.12.079
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
problem of teaching students to create new, including specialized software systems that support the functions of mathematical modeling complex mechanical systems [2-11]. The problem of modeling the dynamics of mechanical systems of vehicles [1,4-9] is a good example of the object of research in teaching students the principles of computer simulation [10, 11]. This task allows student to have the intuitive interpretation of system parameters. Intuitive interpretation also corresponds to results, which are presented in the form of dependency graphs from the time the motion. From a practical point of view, to the area of the considered mechanical systems are included trains and railway crews, and it includes means of road (automotive) transport, including vehicles of auto truck transport. 2. The ways to modeling of mechanical systems Due to the practical importance of a problem, currently the considerable number of works is devoted to creation of adequate mathematical model of mechanical systems of transport vehicles and their computer realization. In particular, for several years the design of the relevant modules of software system "Universal Mechanism" (UM) is developing under the supervising of prof. Pogorelov D.Y. [5, 6, 12, 13]. However, it should be noted that the study of models and methods [5-9, 12, 13] for implementation of mathematical software of software complex UM [14] is a separate task that requires a good mathematical training and deep knowledge in the subject area. This task is beyond the base curriculum for computer simulation training course for the bachelor studying, for which mechanical transport systems are not the main and only object to study. As a different approach to the construction of mathematical models of complex mechanical systems process can be used formal methods based on the principles of the existence of the Electro-mechanical analogy and on the using of "global" variables, objects, and elements of circuitry [15, 16]. An example of such approach is the use of simulation technology of mechanical systems by means of a formal language and a graphical representation of the modeled mechanical system using similar electronic circuits in accordance with international standard VHDL-AMS. However, in order to establish interdisciplinary connections in the learning process more rational approach may be the use of the method of generalized energy phase variables (GEPV) proposed in [10] for the design of mathematical models within CAD systems. The topology of the system in this method can be represented as in the form of the electrical circuit analog electronic devices and as in the classic form of a directed graph. The implementation steps of this method are based on mathematical and formal tools of the disciplines of applied mathematics. Objects for modeling by method GEPV can be discrete dynamical systems and sub-systems of different physical nature. It is worth noting that the GEPV analogy of phase variables, of topological and of component equations for different physical systems are consistent with the standard VHDL-AMS in which the corresponding phase variables appear with the names "across quantity" and "through quantity". The construction of mathematical models of complex mechanical systems can be carried out using classical methods of theoretical mechanics [2-4] with the use of modern technology implementation in an object-oriented approach also. Objectoriented approach is widely used now to solve different mathematical and specialized problems [17-20] by means of simulation by tools of object-oriented programming technology. Method GEPV and object-oriented approach at the correct introduction of the parameters of the system elements allows to construct a mathematical model which fully corresponds [1, 4] to the model obtained by mathematical tools of theoretical mechanics. It should be noted that when modeling systems consisting of the solids that perform various types of mechanical motion, the basic method GEPV [10] require to design of a complex system of circuits and the imposition of laws of interrelation variables based from the subject area. 3. Two-stage method to modeling of mechanical systems of transport vehicle As discussed in [1], for use in the educational process rationally [20] integrated application of methods of classical mechanics with GEPV method. When using the proposed technology the resulting system to describe the in relation to each other of movement of macro-elements in the direction of the main (target) movement can be seen as performing translational mechanical movement that corresponds to one of five basic types of physical subsystems, adequately simulated by the method GEPV [4]. Under the proposed approach, the macro system, shown in left part of figure 1, means to split the system into interconnected subsystems, each of which is characterized by separate
363
364
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
generalized coordinate. In right part of figure 1 shows a directed graph of the relationship of macro-elements, constructed to apply the method GEPV.
& F
D
Fig.1. Left: sampling of the system and the interaction effects to method GEPV; Right: directed graph.
Directed graphs in a method GEPV allow to present conveniently and visually the topology of the system with different types of layout and interaction of the elements, as its are shown in figures 2, 3.
& F
Fig. 2. Left: second system; Right: directed graph to second system.
& F
Fig. 3. Left: third system; Right: directed graph to third system.
Proposed by the authors in [1] for use in the educational process the two-stage algorithm to construct mathematical macro-model of the mechanical system of the transport vehicle in the general form is presented below. Stage 1. First, it should make splitting (left parts of figures 1, 2, 3) of the system into subsystems (macroelements). Using the principles of the second kind Lagrange equations it should determine the inertial characteristics (reduced mass) of each object and the external action on the system. Stage 2. It should identify interaction effects of macro elements (subsystems) between themselves and with the external environment (constrains). It should build equivalent circuit and a directed graph (rights parts of figures 1, 2,
365
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
3) in accordance with the principles and algorithm of method application GEPV for the mechanical translational system. By formal tools should construct the mathematical model in the Cauchy problem form. Generalized information flow diagrams to implementation of the formalized steps of the proposed algorithm are showed in figure 4.
Sm j
S mi
K mi
Km j
I L ji
mj
L ji
Um j
U mi
mi
mGi
Mi
Mj
L
R
mGi
ioj
R ji
K L ji
ioj
K R ji
ª L*ji º ¬ ¼
U ji
ª R*ji º ¬ ¼
Fig. 4. Left: scheme of information flows of stage 1; Right: scheme of information flows of stage 2.
The most important aspect of drawing up of mathematical model by proposed methodology [1] is the correct description of the topology of the system: correct numbering of macro elements (subsystems) and indexation of the parameters of the energy elements in accordance with the following elements in the kinematic chain, i.e. a sequence of "transfer" to move from item to item. The reduced mass for the subsystem k that contains ‘ n ’ wheels pairs:
mk
mrm k
§ § § k ·2 · · ¨ m1 k 2n m2 k ¨ 1 ¨ i2 ¸¸ , Rk ¸ ¸ ¸ ¨ © ¨ ¹ © ¹¹ ©
(1)
where the notation means: m1 – mass of main element; m2 – mass of each wheel; R – radius of wheel; i – inertia radius of wheel. The force of external actions on the system causing its translational moving on the inclined surface consists of two parts:
F
r Fd r mC g sin D
r¦ j
Mj Rj
r mC g sin D .
(2)
This force [4] is directly applied to a driving macro-element, at the movement on an inclined surface it depends on the mass of whole system mC representing the total mass of all elements of system, but not their reduced mass analogs. At the movement up the inclined plane the component of influence of gravity has, naturally, the negative sign, and at the movement on a horizontal surface in general is absent. The traction force Fd is generated by the adhesion of the wheels with the support surface and depends on the moments acting on the axis of the driving wheels pairs with index ‘j ‘. At the stage of the braking system, the driving force is absent. It should be noted that only at this stage, the mathematical model must have components of friction
366
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
about "the earth" that are shown in graphs of right parts of figures 1, 2, 3 by dashed lines for subsystems (macroelements) that are including driving wheels pair. The resulting mathematical model [1, 4] when it presented in the form of the classical Cauchy problem contains the basis phase variables as absolute velocities U m for all macro-elements and as forces I L of elastic interaction. The interaction of subsystems « i o j » of any type (elastic L ji or friction R ji ) introduced in the mathematical model by dependence on the difference of velocities
U mi U m j , U j 0
U ji
Um j 0 Um j .
(3)
To automate the construction of a mathematical model by means of object-oriented programming is convenient to introduce a generalized representation of the components of the right side of the equations:
1 ; ª L*ji º mi ¬ ¼
K mi
1 L ji
where K L ji
K L ji U ji ; ª¬ R*ji º¼ 1 R ji
; K R ji
K R ji U ji ,
(4)
.
4. Resulting mathematical model
So, for the system presented in figure 1 the resulted mathematical macro-model beyond the stage of "braking" has the form: dI L21 dt dU m2 dt
ª¬ L*21 º¼ ;
dI L31 dt
ª¬ L*31 º¼ ;
dU m1
* º¼ 1 I L21 1 I L31 1 F ; K m1 1 ª¬ R21
dt
* º¼ 1 I L21 ; K m2 1 ª¬ R21
dU m3 dt
K m3 1 I L31 ;
dS mi dt
U mi , i 1,..,3 .
(5)
The resulting system of equations must be complemented by an unambiguous definition of the initial conditions of each stage of the movement. 5. Results of computer simulation
In figures 5, 6 are presented the results of computer simulation of the dynamics of the five macro elements with the help of a software product that is implemented using object-oriented programming based on the described algorithm for the construction of mathematical software.
Fig. 5. Left: graphs of displacements; Right: graphs of velocities.
D.V. Grinchenkov et al. / Procedia Engineering 129 (2015) 362 – 368
Fig. 6. Left: graphs of accelerations; Right: graphs of elastic forces.
Object-oriented approach to the implementation of the technologies of constructing mathematical software allows simulate the behavior of the system in different modes: controlled motion, inertia and braking, which is clearly reflected in right part of figure 5. The definition and treatment of "exceptions" in the software product also allows you to track the permissible limits of relative movement of macro-elements. When reaching these values the characteristics of the elastic connecting elements change automatically. 6. Conclusions
Presented in this article, the technology of constructing mathematical models of complex mechanical systems is useful for use in the educational process in order to establish interdisciplinary links at example that is clear and intuitively understandable to the student. Development of electronic educational resources in this area allows you to combine educational material relating to the principles and foundations of mathematical and computer modeling, classical problems and methods of the subject area, use the principles of object-oriented approach to modeling, the characteristics and limitations of formal methods computer-aided design. When you set out the way to solve the described problem presents material [20] relating to such sections applied mathematics like graph theory, theory of algorithms, numerical methods of solution of problems of mathematical analysis and other sections of educational material related to the problem of modeling specific tasks programmatically. Additional elements which increase the quality of e-learning resources is also developing expert systems for selection of model parameters and their adjustment in accordance with the formal characteristics and analysis of the obtained results in order to improve the adequacy of the model. References [1] I.A. Spiridonova, D.V. Grinchenkov, On a formalization of a construction by two-stage of a mathematical macromodel to complex mechanical system, J University News. North-Caucasian Region. Technical Sciences Series. 6 (2013) 47–51. [2] D.C. Karnopp, D.L. Margolis, R.C. Rosenberg, System Dynamics: Modeling, Simulation, and Control of Mechatronic Systems, fifth ed., Hoboken, New Jersey, 2012. [3] M.I. Bat, G.Yu. Dzhanelidze, A.S. Kelzon, Theoretical Mechanics at Examples and Problems, tenth ed., Lan, St. Petersburg, 2013. [4] I.A. Spiridonova, D.V. Grinchenkov, On a mathematical modeling of a complex mechanical system by formal and by classical methods, J. Russian Electromechanics. 5 (2013) 75–79. [5] D.Y. Pogorelov, Contemporary algorithms for computer synthesis of equations of motion of multibody systems, J. Computer and Systems Sciences International. 44(4) (2005) 503–51. [6] D.Y. Pogorelov, On numerical methods of modeling large multibody systems, J. Mechanism and Machine Theory. 34(5) (1999) 791–800. [7] R. Sinha, C.J.J. Paredis, P.K. Khosla, Integration of mechanical CAD and behavioral modeling, Proc. IEEE/ACM. (2000) 31–36. [8] A.A. Zarifian, P.G. Kolpahchyan, Computer Modeling of Electric Locomotive as Controlled Electromechanical System, J Multibody System Dynamics. 22(4) (2009) 425–436. [9] R. Kovalev, N. Lysikov, G. Mikheev, D.Pogorelov, V. Simonov, V. Yazykov, S. Zakharov, I. Zharov, I. Goryacheva, S. Soshenkov, E. Torskaya, Freight Car Models and Their Computer-Aided Dynamic Analysis, J Multibody System Dynamics. 22(4) (2009) 399–423. [10] I.P. Norenkov, CAD Basics, fourth ed., BMSTU Publ., Moscow, 2009. [11] M.A. Bauer, M. Sarrafzadeh, F. Somezi, Fundamental CAD algorithms, IEEE Transaction on Computer-Aided Design of Integrated Circuits and Systems. 19(12) (2000) 1449–1475. [12] D.Y. Pogorelov, Jacobian Matrices Of the Motion Equations of a System of Bodies, J Computer and Systems Sciences International. 46(4) (2007) 563–577. [13] D.Y. Pogorelov, Simulation of Constraints by Compliant Joints, J Computer and Systems Sciences International. 50(1) (2011) 158–173. [14] Information on http://www.umlab.ru
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