In this paper, an example problem was used to demonstrate the application of Mathcad in solving the tasks of the dynamics of rigid bodies. The task was to determine angular acceleration for a given moment and the moment value at minimum acceleration, which makes this problem an optimization problem. Classical analytical solution of the problem is obtained using the D\u27Alembert principle. Mathcad code is given by which the solution is obtained. Functional dependence of angular acceleration to frictional moment is determined and final solution is calculated by the minimization function. Comparison of optimization solutions to classical solution confirmed the effectiveness of this method of problem solving. For this procedure Mathcad symbolic functions were used.U radu je na problemskom zadatku prikazana primjena MathCad-a u rješavanju zadataka iz dinamike krutih tijela. U zadatku je potrebno odrediti koliko je kutno ubrzanje za zadani moment i koliki je moment za minimalno ubrzanje, stoga ovaj problem predstavlja optimizacijski problem. Klasično analitičko rješenje zadatka je postavljeno koristeći D\u27Alembertov princip te je dana sintaksa MathCad koda kojim je dobiveno rješenje. U nastavku je za određivanje momenta definirana funkcija ovisnosti kutnog ubrzanja o momentu te je minimizacijom odredjen rezultantni moment. Usporedba rješenja optimizacije s klasičnim rješenjem potvrdila je učinkovitost ovog načina rješavanja zadatka. Za ovaj postupak su korištene MathCad simboličke funkcije

Katarina Pisačić

Marko Horvat

Vinko Višnjić

English

Hrčak - Portal of scientific journals of Croatia

Pisačić K., Horvat M., Višnjić V.                                                                           MathCad za inženjere: optimizacija rješenja problemskog zadatka dinamike krutih tijela  ISSN 1846-6168 UDK 621:004    MATHCAD ZA INŽENJERE: OPTIMIZACIJA RJEŠENJA PROBLEMSKOG  ZADATKA IZ DINAMIKE KRUTIH TIJELA   MATHCAD FOR ENGINEERS: OPTIMIZATION OF RIGID BODY DYNAMICS  EXAMPLE PROBLEM   Katarina Pisačić, Marko Horvat, Vinko Višnjić  Stručni članak Sažetak: U radu je na problemskom zadatku prikazana primjena MathCad-a u rješavanju zadataka iz dinamike krutih tijela. U zadatku je potrebno odrediti koliko je kutno ubrzanje za zadani moment i koliki je moment za minimalno ubrzanje, stoga ovaj problem predstavlja optimizacijski problem. Klasično analitičko rješenje zadatka je postavljeno koristeći D'Alembertov princip te je dana sintaksa MathCad koda kojim je dobiveno rješenje. U nastavku je za određivanje momenta definirana funkcija ovisnosti kutnog ubrzanja o momentu te je minimizacijom odredjen rezultantni moment. Usporedba rješenja optimizacije s klasičnim rješenjem potvrdila je učinkovitost ovog načina rješavanja zadatka. Za ovaj postupak su korištene MathCad simboličke funkcije.  Ključne riječi: MathCad, optimizacija, dinamika krutih tijela, symbolics  Professional paper Abstract: In this paper, an example problem was used to demonstrate the application of Mathcad in solving the tasks of the dynamics of rigid bodies. The task was to determine angular acceleration for a given moment and the moment value at minimum acceleration, which makes this problem an optimization problem. Classical analytical solution of the problem is obtained using the D'Alembert principle. Mathcad code is given by which the solution is obtained. Functional dependence of angular acceleration to frictional moment is determined and final solution is calculated by the minimization function. Comparison of optimization solutions to classical solution confirmed the effectiveness of this method of problem solving. For this procedure Mathcad symbolic functions were used.  Key words: MathCad, optimization, rigid body dynamics, symbolics   1. INTRODUCTION  In the recent studies of rigid body dynamics various programming languages and software are been used. MathCad is equation solver and graphing software used by scientists for solving various scientific problems [1,2]. This article presents a concise example of rigid body dynamics problems solved in MathCad and presents simple optimization problem. D'Alemberts principle is used to create artificial equilibrium state by balancing active forces with inertial force which equals to m a  . This principle enables solving dynamics problems similarly to statics and is well known in studies of vector and engineering mechanics [3,4].  Furthermore when acceleration equals 0 dynamic problem becomes static for 0m a    and D'Alembert's force becomes 0. Analytical solution for this case is easily obtained and it is used to validate optimized dynamic solution. 2. DYNAMIC EXAMPLE PROBLEM  Problem solved for this presentation is modified textbook example [4].   2.1. Sample problem  Due to rotation frictional moment 2.5 NmOM   appears at pivot O (see Figure 1.). Drum mass is D 6 kgm   and radius of gyration 225 mmsi  while weights masses are 1 12 kgm   and 2 7 kgm  , drum radiuses are 1 200 mmr   and 2 300 mmr  .  1. Determine angular acceleration of the grooved drum.  2. Determine maximum frictional moment when weights are still moving       Tehnički glasnik 8, 4(2014), 327-331 327Pisačić K., Horvat M., Višnjić V.                                                                           MathCad za inženjere: optimizacija rješenja problemskog zadatka dinamike krutih tijela     To obtain solution following will be done: 1) Free body diagram  2) Dynamic equilibrium equations 3) Static equilibrium equations  4) Solving equations using MathCad  2.2. Free body diagram  Identifying correct relations between the bodies is necessary to draw corresponding free body diagram.   Free body diagram is shown in Figure 2.     2.3. Dynamic equilibrium equations  It is requested to determine angular acceleration and analytical solution will be obtained by the use of D'Alembert's principle. Vector equations are following: For weight A:  1 1 1 10   0yF m g m a S                            (1)  For weight B:  2 2 2 20   0yF m g m a S                           (2)    For drum:  1 1 2 20   0O O OM S r S r I M                  (3) Kinematic relationships are:  1 1a r                                                                     (4)  2 2a r                                                                    (5)  2.4. Static equilibrium equations  Second problem statement requests the calculation maximum value of frictional moment OM . If frictional moment is large enough, thy system will become still and the problem becomes static. Therefore static equilibrium equations will be used to obtain solution.  For weight A:  1 10   0yF m g S                (6)  For weight B:  2 20   0yF m g S                (7)  For drum:  1 1 2 20   0O OM S r S r M                 (8)   3. MATHCAD SOLUTION    One way of solving system of linear equations in MathCad is using given-find block [5].   For successful calculation four steps should be made: 1) Define variables  2) Define initial values 3) Given-Find solve block  4) Store solutions   3.1. Mathcad solution of dynamic equilibrium  To enable proper calculation all constants must be defined [6, 7], local variable definitions are used for easier manipulation. Unlike global definitions, local definitions are valid from the place where are written to the right and below.   3.1.1. Defining variables  Green wavy line indicates that some previous definitions are overridden. MathCad recognizes dozens physical constants and units. If variable name is underlined then is recommended to rename it (Figure 3.).  1m2mA B O 2r1r2m gB2 2m a2S1m g1 1m a1S  A O 1S  2SOxFOyFOm gOMFigure 1. Rigid body dynamics problem Figure 2. Free body diagram 328 Technical journal 8, 4(2014), 327-331Pisačić K., Horvat M., Višnjić V.                                                                           MathCad za inženjere: optimizacija rješenja problemskog zadatka dinamike krutih tijela    3.1.2. Initial values  For all unknowns initial values and units should be defined (Figure 4.). Similar as defining constants local variable definition is used. For nonlinear systems it is more effective to define 1 rather than 0, in some cases 0 fails performance of Given-Find block.     3.1.3. Solving system   Given-Find solve block is numerical solve block where the number of unknown variables must be same as number of equations in the block. Solution of given system is shown in Figure 5.            3.1.4. Storing solution  Solve function Find is used to obtain solution. In this example vector is defined with same number of elements as given equations, containing variable names in same order as in Find function and thus solutions are stored. This is optional, but usually done for further calculation needs. If solutions have various physical quantities, numerical values are separately recalled (Figure 6.).      3.2. Mathcad solution of static equilibrium  For static equilibrium solution similar procedure is done. Constants are same as in dynamic equilibrium and this step is unnecessary.   3.2.1. Initial values  Although mostly same constants are used, number of equations is changed and different initial values are defined using appropriate physical units, Mathcad syntax for this procedure is shown in Figure 7.     Figure 7. Defining initial values for second solution  3.2.2. Solving system and storing solutions  In this step solve block Given-Find is used. Solutions are stored and numerical values are recalled separately because different physical units are used. Different units can be recalled inside same matrix only when symbolical evaluation is used (arrow).                      g 9.81ms2IO mD is2 0.304m2kgMO 2.5N mr2 300mmr1 200mm is 225mmmD 6kgm1 12kg m2 7kg 0.3131s2S1 116.968 N S2 69.328 NS1S2a1a2Find S1 S2 a1 a2  S1 r1 S2 r2 IO  MO 0m1 g m1 a1 S1 0Givenm2 a2 m2 g S2 0a1  r1a2  r2 0rads2a1 0ms2 a2 0ms2S2 0kNS1 0kNS1 0kN S2 0kN MO 1N mS1 r1 S2 r2 MO 0m1 g S1 0m2 g S2 0S1S2MOFind S1 S2 MO 117.72kg ms268.67kg ms22943.0kg m mms2MO 2.943JFigure 3. MathCad code defining variables Figure 4. Defining initial values Figure 5. Solution block Given-Find Figure 6. Recalling solutions Figure 8. Second solution block Tehnički glasnik 8, 4(2014), 327-331 329Pisačić K., Horvat M., Višnjić V.                                                                           MathCad za inženjere: optimizacija rješenja problemskog zadatka dinamike krutih tijela  4. OPTIMIZATION OF DYNAMIC SOLUTION  Another approach is possible to obtain solution for maximum frictional moment OM . When frictional moment reaches high value weights decelerate, subsequent angular acceleration reaches minimum value. Solution of problem is finding frictional moment for minimum angular acceleration. Although many known optimization procedures are known [8, 9], for this purpose built in MathCad functions will be used.  4.1. Defining solution function  In this step function for obtaining range of solutions is defined. System of equations solved in chapter 3.2.2 is optimized using symbolical calculation in MathCad.   4.1.1. Defining variables  When running symbolic calculations use of units requires some additional calculation steps and in this calculation physical units are omitted for faster calculation (Figure 9.).            4.1.2. Defining initial values  For effective use of symbolic calculation initial values for unknown variables are defined to override any stored numerical value (Figure 10.).         When full symbolic solution is wanted, all used physical quantities are to be defined this way, regardless if they are already defined as constants.  4.1.3. Defining solution function  Following is function formed of symbolic solution of dynamic equilibrium equations and function argument is frictional moment.  For obtaining solution solve function is used combined with simplify and float keywords for optimizing result. Solution of system is five elements function vector (Figure 11.).     4.2. Optimizing solution function  Angular acceleration as stated in solve function is fifth element. Default starting index (MathCad built in variable ORIGIN) is 0 so the function alfa(x) representing angular acceleration dependent on momentum is defined. This function is used to extract single solution value from solution vector (Figure 12.).       4.2.1. Solution plot  Range variable is defined for plotting solution an linear plot of solution is made.        mome 0 0.1 300.10.20.30.40.50.60.70.80.911.11.2...0 1 2 31123momentangular accelerationrj MO S1 r1 S2 r2 IO  MO 0m1 g m1 a1 S1 0m2 a2 m2 g S2 0a1  r1a2  r2solveS1S2a1a2simplifyfloat1.698 MO 112.724 1.485 MO 73.042 0.141 MO0.416 0.212 MO 0.625 0.707 MO 2.082 MO MO a1 a1 a2 a2  S1 S1 S2 S2m1 12 m2 7r1 0.200 r2 0.300 is 0.225mD 6g 9.81IO mD is2 0.304alfa M( ) rj M( )0 4Figure 9. Defined variables with omitted units Figure 10. Initial values for symbolic calculation Figure 11. Defining whole solution function Figure 12. Defining single solution function Figure 13. Solution plot 330 Technical journal 8, 4(2014), 327-331Pisačić K., Horvat M., Višnjić V.                                                                           MathCad za inženjere: optimizacija rješenja problemskog zadatka dinamike krutih tijela  Mathematical solution of problem shows linear (Figure 13.) dependence between moment and angular acceleration, thus solution plot is represented with line and minimum of such function in this case would be  .   4.2.2. Minimizing function  For obtaining physically correct solution, minimum value of angular acceleration equals zero and maximum frictional moment equals root of function alfa(M).  Root function in this form requires defining initial value for solution and syntax is as follows in Figure 14.        5. CONCLUSION  Short theoretical problem is used to show one approach to solving optimization problems by use of symbolic calculations in MathCad engineering software. Physics laws demand that real conditions are considered while solutions are validated. Use of mathematical software shortens the process of solution verification and enables devising more theoretical problems and obtaining reliable solutions in shorter time even so without the use of graphical software.    6. REFERENCES  [1] Domnisoru, C.: Using MathCad in Teaching Power Engineering, IEEE Transactions on education, VOL. 48, NO. 1, February 2005 [2] Ochkov, V. F.; Bogomolova, E. P.: Teaching Mathematics with Mathematical Software, Journal of Humanistic Mathematics, 5(1), 265-285. DOI: 10.5642/jhummath.201501.15, January 2015 [3] Matejiček, F.: Kinetika sa zbirkom zadataka, prvo izdanje, Strojarski fakultet, Slavonski Brod, 2011. [4] Meriam, J. L.; Kraige, L. G.: Engineering Mechanics Volume 2 Dynamics 6th edition SI version, John Wiley & Sons, Inc, 2007. [5] MathCad 14 User's Guide, Parametric Technology Corporation, 2007. [6] Maxfield, B.: Essental MathCad for Engineerng, Science, and math ISE, Academic Press, 2012. [7] Ochkov, V.F.; Lehenkyi, V.I.; Minaeva E.A.: Physicyl quantities dimensions and units in mathematica packages, Математичні машини і системи 1, 2009. [8] Scitovski, R., Truhar, N.,  Tomljanović, Z.:  Metode optimizacije, Sveučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, 2014. [9] Matijević, D.; Sabo, K.; Scitovski, R.: Globalna optimizacija Lipschitz- neprekidnih funkcija, Sveučilište J.J. Strossmayera u Osijeku, Odjel za matematiku Kontakt autora:  Katarina Pisačić, dipl. ing. Sveučilište Sjever Sveučilišni centar Varaždin 104. brigade 3 42000 Varaždin e-mail: katarina.pisacic@unin.hr  Marko Horvat, dipl. ing. Sveučilište Sjever Sveučilišni centar Varaždin 104. brigade 3 42000 Varaždin e-mail: marko.horvat@unin.hr  Izv. prof. dr. sc. Vinko Višnjić. Sveučilište Sjever Sveučilišni centar Varaždin 104. brigade 3 42000 Varaždin e-mail: vinko.visnjic@unin.hr    M0 root alfa M0  M0  2.943M0 3Figure 14. Root function solution Tehnički glasnik 8, 4(2014), 327-331 331

MathCad za inženjere: optimizacija rješenja problemskog zadatka iz dinamike krutih tijela

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