User guide
Contents
1. Y To use an ODE for any model we must know the numer ical values of its solution These numerical values can only be obtained in some limited range which must be specified before solving the ODE In this case the range specified is the interval 0 20 3 3 Initial data Having specified the range press the Next button to arrive at the following W Simple Harmonic Motion ode Calcode _ o x File View Options Run Analyse Help A ose lesa ze wv Initial data 9 FIRST click and highlight a number i in the list on the left and THEN enter the initial or final Y value for that i in the edit box on the right v ES 1 x0 p For Help press F1 x Y 20 CHAPTER 3 USING THE WIZARD Before solving the ODE we also need to know the initial data That is we need to know the values of y and y2 at the beginning point of the range xo Here we have prescribed the initial data yi to 0 Y2 xo k This makes the solution y come out as the sin function Later you can of course change the initial data Press the scroll box to scroll to the value 2 and select it see the initial datum for y2 Press Next 3 3 3 4 Parameters There is only one parameter k and its value has been pre scribed to be 1 M Simple Harmonic Motion ode Calcode l olx File View Options Run Analyse Help Moca s eas a z lt w Parameter values For Help press F1 After the solution p2. z ccordinate of particle 1 etc Change the number of dimensions to 2 and the number of particles to 2 and click OK You should see the following twobody1 ode Calcode File View Options Run Analyse Help B ega For Help press F1 Only 1 ball is visible on the screen the second one sa lt L w What happened to 4 2 ORTHOHGRAPHIC PROJECTION 27 4 2 Orthohgraphic projection Right click to see the following shortcut menu Animate Reset Parameters v 3D view Orthographi v Show Trail Help Select orthographic projection CALCODE provides two basic types of 3d view using either parallel projection or orthographic projection Parallel projection is the default and gives a realistic view in which distant objects become smaller This realism is accentuated by realistic lighting ef fects However for this example we will use orthographic projection You should now see the following Bitwobodyi ode Calcode BRI File View Options Run Analyse Help IN oe a elas a eRe v For Help press F1 28 CHAPTER 4 3D ANIMATION 4 3 Setting parameters for 3d view The second ball is just visible at the edge of the screen To improve matters again right click and select Parameters from the short cut menu or use View 3d view Para meters You should see the following Witwobody1 ode Calcode olx File View Options Run Analyse Help fosuasamsa zeo
3. Parameters for 3D view 7 X Look At For Help press F1 X Ye As Under Clip planes at the bottom right change the Far value from 40 to 50 The second ball should now be clearly visible 7 twobody1 ode Calcode Tes Ple View Options Run Analyse Help Boe ea alas a Be ew For Help press F1 4 4 ANIMATION 29 4 4 Animation Again right click and select Animate from the short cut menu You should now see the balls in animated motion 4 5 Moving the z axis The z axis is not visible since its default position points per pendicularly out of the computer screen To make it visible left click and drag with the mouse You can do this even while the animation is going on a twobody1 ode Calcode File View Options Run Analyse Help Ao lt Sas Bj e v Y 2 For Help press F1 For a true 3d example try the Lorentz model ode with 3 dimensions and 1 ball 30 CHAPTER 4 3D ANIMATION M Lorentz Model ode Calcode File View Options Run Analyse Help Aosaesase D v For Help press F1 4 6 Reset To repeat the animation first right click and use reset from the short cut menu This should be enough to get you started Extensive help is available from the help menu For some further advanced features of CALCODE consult the user manual
4. by double clicking on the desktop icon or using Start Programs CALCODE You should see the following screen 1 2 CHAPTER 1 INSTALLATION W Untitled Calcode Bis File View Options Run Analyse Help Bcoeasgiasua tee License Missing Cancel Get License For Help press F1 Press the Get License button You should now see the following screen W Untitled Calcode File View Options Run Analyse Help No s a METT x A Files created successfully 7 File names are C CalcFile1 txt and C CalcFile2 txt To get your Calcode licence please send these two files as attachments to dr_c_k_raju hotmail com Do NOT edit these files in any way E For Help press F1 Press OK You may open the files CalcFilel txt and CalcFile2 txt in any text editor to view the information you are sending But do NOT edit these files or save them after viewing that would make these files invalid The file 1 2 OBTAINING THE LICENSE FILE 3 CalcFilel txt contains basic information about your sys tem processor s memory operating system etc The sec ond file CalcFile2 txt contains a code to check that the first file has not been changed in the minutest way Why is this information needed This information is needed since the CALCODE license is specific to a given com puter This process may change in future versions Send these two files CalcFilel
5. relate to the movement of mass points in 3 dimensions CALCODE visualizes this using little coloured balls in place of those mass points As an example open the file 2bodyl ode and run GS twobody1 ode Calcode File View Options Run Analyse Help ei a 5 0 000000 0 005000 0 010000 0 015000 0 020000 0 025000 0 030000 0 035000 0 040000 0 045000 0 050000 0 055000 0 060000 0 065000 n a7annn For Help press F1 0 000000 0 099996 0 199969 0 299886 0 399711 0 499397 0 598882 0 698086 0 796912 0 895247 0 992964 1 089917 1 185943 1 280865 1 274447 26 CHAPTER 4 3D ANIMATION 4 1 Starting 3d view Now click the 3d view button or use the Menu item View 3d view View 3d You should see the following 7 twobody1 ode Calcode File View Options Run Analyse Help 10 0 050000 For Help press F1 no M ana 0 0 000000 0 00000 1 0 005000 0 09999 f 2 0 010000 0 19996 S i l 3 0 015000 0 29988 No of dimensions 1 2 or fl 4 0 020000 0 39971 No of particles max 7 fr 5 0 025000 0 49939 _ 6 0 030000 0 59888 It is assumed that the y values are so numbered 7 0 035000 0 69808 ha r 8 0 040000 0 79691 ERE 9 0 045000 0 89524 inthe 2dcase 11 0 055000 1 08991 12 0 060000 1 185943 3 131619 iH 1 13 0 065000 1 280865 3 213583 i nn7nnnn 1 274447 2 Rss H U gt c lt H K y3 x coordinate of particle 2 0 99296 while in the 3 d case y3
6. shown by the legend the blue curve is our original solution Y1 and the red curve is the graph of the function sin x 0 1 Looking at the two graphs we can easily see how close the solution Y1 is to the function sin x 0 1 Why 0 1 This is just a small offset In this case we know that the solution Y1 is the function sin x so if we had simply entered sin x for comparison the two graphs would have coincided 2 6 Zeros Maxima and Minima CALCODE can easily calculate the sections or zeros of the solution or of a function of the solution This feature can 13 2 6 ZEROS MAXIMA AND MINIMA be used for example to calculate the time period of the You should see the pendulum Use Analyse Sections 3 following W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help 2 lt D v Moga a For Help press F1 Press OK to see the zeros listed out W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help He 3 Aos ala Calcode Output x 3 141591 F 0 000002 type 1 yfi x 6 283184 F 0 000002 type 1 sn Y x 0 000002 y 2 1 000000 For Help press F1 Note that the first zero is the same as the value of 7 The maxima and minima of the solution can be similarly calculated using Analyse Extrema 14 CHAPTER 2 GETTING STARTED 2 7 Example 2 the simple pendulum Let us pause for a moment t
7. solves the equations and displays the following screen 2 Simple Harmonic Motion ode Calcode View Options Run Analyse t File A FIRST EXAMPLE wan nauk WN ka Drag the bar separating the two views so that you can D g 8 For Help press F1 ecaooocoo0ooceceoecececeo BH H s H 57 see the numbers more clearly Ea Simple Harmonic Motion ode Calcode File View Options Run Analyse Help T Aloe 0 000000 0 050000 0 100000 0 150000 0 200000 0 250000 0 300000 0 350000 0 400000 0 450000 0 500000 0 550000 0 600000 0 650000 0 700000 0 750000 0 800000 0 850000 0 900000 0 950000 1 000000 1 050000 1 100000 1 150000 9 TAn For Help press Fi What you are seeing is the solution of the differential 0 000000 0 049979 0 099833 0 149438 0 198669 0 247404 0 295520 0 342898 0 389418 0 434966 0 479426 0 522687 0 564642 0 605186 0 644218 0 681639 0 717356 0 751280 0 783327 0 813415 0 841471 0 867423 0 891207 0 912764 n anna l 1 000000 0 998750 0 995004 0 988771 0 980067 0 968912 0 955336 0 939373 0 921061 0 900447 0 877583 0 852525 0 825336 0 796084 0 764842 0 731689 0 696707 0 659983 0 621610 0 581683 0 540302 0 497571 0 453596 0 408487 n T a e K 8 CHAPTER 2 GETTING STARTED equation 2 1 The left panel shows the numerical values of the solution under the column Y 1 The right panel shows the same so
8. CALCODE 3 3 Contents 1 Installation 1 1 1 Basic installation 1 1 2 Obtaining the license fle 1 2 Getting started 5 2 1 Afirstexample 6 2 2 Switching VIEWS 6 seek we we ww AE hee 8 De IEEE 05 5 A ode he ass eae cl o gt ec Bibi Ge ee we E 10 2 4 Analysing the solution 10 2 5 2 6 2 COMPAN gadane Zeros Maxima and Minima Example 2 the simple pendulum Using the Wizard 3 1 3 2 3 3 3 4 3 9 3 6 3 0 3 8 Equations and Parameters RADo ugna fan Sh eck a te ad Oe R Initial data lt 0 4 Yale BAe ER ee eS Parameters ace de Great ge ch te te oro de sn 8 Error COMO seraa dee ee oS RE Graphics display LEH A ieee E O ah et ee Repeating the wizard 3d animation 4 1 4 2 4 3 4 4 4 5 4 6 Starting sd VIEW aa E K ara aoei oh e e i Orthohgraphic projection Setting parameters for 3d view AMA One oS Td othe EO 0 Moving the z axis 0 0 TROSC tick St N vos i Seg et es Rw a a 11 i 14 15 15 18 19 20 21 2l 22 22 Chapter 1 Installation 1 1 Basic installation If you obtained CALCODE install files as a zip file simply open the file and press the Install button Otherwise double click on Setup exe Select the directory in which you want to install Calcode 1 2 Obtaining the license file After installation start CALCODE
9. arameter values and initial data can be quickly varied using the Repeat function in the Analyse menu Press next 3 5 ERROR CONTROL 21 3 5 Error control There are only two values used for error control Wsimple Harmonic Motion ode Calcode Bis File View Options Run Analyse Help Bosusgasn ze w Error Control For Help press F1 One is the permissible error also known as absolute er ror The other is the maximum step size The numerical solution proceeds in steps and too large or too small a step can increase the overall error For most common purposes the default values should be OK 3 6 Graphics display In this screen you can elect to see fewer or more graphs Generally too many graphs tend to clutter the screen You don t have to think much about this choice since you can always change it later You can also elect to see symbols on the graphs useful for black and white printing 22 CHAPTER 3 USING THE WIZARD 3 7 Output The final screen of the Wizard is the output screen W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help A D e 9 e s FOL 6 wl S Enter the number of equally spaced points in the range at which Calcode should numerical values of the nek thins T Store output data to file Output file name tose For Help press F1 K CALCODE outputs the solution at a number of equally spaced points You can choose to inc
10. e Harmonic Motion ode Calcode File View Options Run Analyse Help IN os a 6las a2 F574 No X Yia LYS WA 284 14 200000 0 998027 0 062797 f 285 14 250000 0 993641 0 11259 286 14 300000 0 986772 0 16211 287 14 350000 0 977436 0 21123 288 14 400000 0 965658 0 259817 289 14 450000 0 951465 0 30775 290 14 500000 0 934895 0 35492 291 14 550000 0 915988 0 40120 292 14 600000 0 894791 0 44648 293 14 650000 0 871358 0 49064 294 14 700000 0 845747 0 53358 295 14 750000 0 818022 0 575187 296 14 800000 0 788252 0 615357 297 14 850000 0 756512 0 653981 298 14 900000 0 722881 0 690971 299 14 950000 0 687444 0 72623 300 15 000000 0 650288 0 75968f 301 15 050000 0 611507 0 79123 302 15 100000 0 571197 0 82081 303 15 150000 0 529459 0 84833 304 15 200000 0 486399 0 873733 305 15 250000 0 442122 0 89695 306 15 300000 0 396741 0 917931 307 15 350000 0 350367 0 936617 aagos one 1E Annnnn n n110 4 For Help press F1 Identify the buttons for split view graph view and report view by positioning the mouse cursor over them If the cursor is kept stationary for a second a little help message will be shown By clicking on these buttons you can switch between a view which shows only numbers report or only graph or both For example here is the graph view simple Harmonic Motion ode Calcode File View Options Run Analyse Help Aos ala s e z Je S
11. imple Harmonic Motion For Help press F1 You can also use the View menu to switch between views 10 CHAPTER 2 GETTING STARTED 2 3 Options Various aspect of the view such as fonts titles colors etc can all be customized using the Options menu 2 4 Analysing the solution Now click on Analyse Phase plots You should see the following screen YW Simple Harmonic Motion ode Calcode File View Options Run Analyse Help B eg ax lt Be v Simple Harmonic Motion Phase plots HD y axis For Help press F1 Enter 1 for the number of plots followed by the values 1 and 2 Press OK to see a phase plot plot of Y1 vs Y2 2 5 COMPARING 11 M Simple Harmonic Motion ode Calcode File View Options Run Analyse Help Aosus lase ze Be amp For Help press F1 If you wanted to see a plot of just Y1 you could have entered the values 0 and 1 to plot YO vs Y1 The convention here is that the independent variable x is called YO 2 5 Comparing Next select Compare from the Analyse menu You should see the following W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help Aosua ase zjev For Help press F1 12 CHAPTER 2 GETTING STARTED Enter sin x 0 1 in the text box and press OK You should see the following Simple Harmonic Motion ode Calcode File View Options Run Analyse Help Mose 6 las2 zee For Help press Fi As
12. lution as a graph with the legend Y1 coloured in blue Thee other column Y 2 gives the numerical values of the derivative a and these are likewise shown as a graph with the legend Y2 coloured in red 2 2 Switching views Note that that two views are connected If you click on some numerical value the row is highlighted and the correspond ing point is also highlighted on the graph Simple Harmonic Motion ode Calcode File View Options Run Analyse Help B eg Sas L Flt tw No X YU Y 0 0 000000 0 000000 1 000000 1 0 050000 0 049979 0 998750 2 0 100000 0 099833 0 995004 3 0 150000 0 149438 0 988771 4 0 200000 0 198669 0 980067 5 0 250000 0 247404 0 968912 6 0 300000 0 295520 0 955336 7 0 350000 0 342898 0 939373 8 0 400000 0 389418 0 921061 9 0 450000 0 434966 0 900447 10 0 500000 0 479426 0 877583 11 0 550000 0 522687 0 852525 12 0 600000 0 564642 0 825336 13 0 650000 0 605186 0 796084 14 0 700000 0 644218 0 764842 15 0 750000 0 681639 0 731689 16 0 800000 0 717356 0 696707 17 0 850000 0 751280 0 659983 18 0 900000 0 783327 0 621610 19 0 950000 0 813415 0 581683 20 1 000000 0 841471 0 540302 21 1 050000 0 867423 0 497571 22 1 100000 0 891207 0 453596 23 1 150000 0 912764 7 he N 94 TAn NA ADNA n ILEO 4 gt For Help press F1 Similarly if you click on some point in the graph the corre sponding numerical value scrolls into view and is highlighted 2 2 SWITCHING VIEWS 9 Ea Simpl
13. o take stock What exactly has been achieved by any of the above These features of simple harmonic motion are well known So what was the need for CALCODE This is made clearer by repeating the above steps but opening the example file Jacobian Elliptic Functions ode in place of Simple Harmonic Motion ode In this case the solution is the non elementary function sn a This makes for a slight but noticeable difference between simple harmonic motion and the simple pendulum The difference is so slight that the two are often confused with each other But it is a noticeable and over a period of time the difference can grow very large For example if the simple pendulum is used to make a clock such a clock would not be accurate enough to navigate by However the easy example of simple harmonic motion is a good way to learn the capabilities of CALCODE Chapter 3 Using the Wizard One of the advantages of using CALCODE is that the differ ential equations to be solved can be entered symbolically To see how this is done exit Calcode and restart At the opening screen press the Wizard button first button on the left or use File Wizard 3 1 Equations and Parameters Press next at the welcome screen to arrive at the following 15 16 CHAPTER 3 USING THE WIZARD f Untitled Calcode File View Options Run Analyse Help 3 lass 7 e l Number of equations x Write your equations in standard form then enter
14. rease or decrease this number Generally a larger number of points leads to a smoother plot but a slower solution You can also choose whether to save the numerical output to a file for later use or for use with some other software Press the Finish button to see the solution once again 3 8 Repeating the wizard Any time you wish to make changes to what you entered in the Wizard you can rerun the Wizard Alternatively you can choose File New Edit to arrive at a property sheet or a tabbed dialog box 3 8 REPEATING THE WIZARD 23 W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help ii For Help press F1 D W amp S d 2 Kl xv 1 0 050000 0 049979 oe eee eet ee al 2 0 100000 0 099833 3 0 150000 0 149438 Write your equations in standard form then 4 0 200000 0 198669 Fioni loam mate about te standerd iom of 5 0 250000 0 247404 Load from file 6 0 300000 0 295520 T 7 0 350000 0 342398 Number of Equations 3 0 400000 0 389418 Number of Parameters h 9 0 450000 0 434966 10 0 500000 0 479426 11 0 550000 0 522687 ok Taa Hem 12 0 600000 0 564642 IR 5 e j 13 0 650000 0 605186 0 796084 s d n TANANAN n E E eA K n 7AA Sa 3 8 Here you can quickly change any values you want without having to go through the entire Wizard Chapter 4 3d animation Many ODEs arise in physics That is how Newtonian physics began and that is where it ends These ODEs
15. s screen shows the first of the two equations 3 1 But there are two further changes of notation First y is being denoted as Y 1 which could also be written as plain Y1 or yl We use this because in a printed text y is easier to read but on a computer Y 1 or Y1 is easier to enter Second primes are being used to denote derivatives with respect to x This again is easier to indicate on the screen than the fraction SL With these notational changes the equations 18 CHAPTER 3 USING THE WIZARD for simple harmonic motion are yi y2 y 2 k kxyl 3 2 To see the second equation press the scroll boxes and when the number 2 comes into view SELECT it M Simple Harmonic Motion ode Calcode ol x File View Options Run Analyse Help Aos a 6a sae te w RHS of equations P For Help press F1 Note that the operation of multiplication is explicitly in dicated by writing k x k x yl It is also correct to write k 2 y1 but NOT ky1 3 2 Range Press the Next button to arrive at the following 3 3 INITIAL DATA 19 W simple Harmonic Motion ode Calcode _ o x File View Options Run Analyse Help A os 68 sa ze e RHS of equations P FIRST click and highlight a number i in the list on the left and THEN enter the tight hand side of the equation i in the edit box on the right Press the Help button to learn more about symbolic equation La l kky For Help press F1 X
16. the following information Press the help button to leam more about the standard form of Load from file Number of Equations fl Number of Parameters jo sa Hee For Help press F1 X Press the button Load from File again select simple harmonic motion ode and this is what you should see W Simple Harmonic Motion ode Calcode File View Options Run Analyse Help Doe Ww alas Fle l Number of equations x Br Write your equations in standard form then enter the following information Press the help button to learn more about the standard form of Number of Equations E Number of Parameters fi Back Next gt Cancel Help For Help press F1 X The equation being solved is the same equation 2 1 So why does the above screen say that the number of equations to be solved is 2 The reason is that the equation 2 1 has been re written 3 1 EQUATIONS AND PARAMETERS 17 as the following two equations dyi EE Y2 dy2 7 ST e Uj 3 1 E y 3 1 Our original y is now being called y1 So there are 2 equations The only parameter is k to this we will assign a number whose value we might later want to vary without changing the rest of the equation Press the Next button to arrive at the following W Simple Harmonic Motion ode Calcode Biel File View Options Run Analyse Help BS osus ase ze w RHS of equations P For Help press F1 Thi
17. txt and CalcFile2 txt as attachments to ckr ckraju net You should receive back a file CalcLicense txt Simply copy this file to your root directory C Once again you can view the contents of this file but do not edit it in any way else the license will become invalid Chapter 2 Getting started After you have obtained the license file CALCODE should start up and you should see an opening screen like this Untitled Calcode File View Options Run Analyse Help A osusgmasa tev For Help press F1 D CHAPTER 2 GETTING STARTED 2 1 A first example As a first example of CALCODE s capabilities use File Open and select Simple Harmonic Motion ode as the file to open Untitled Calcode File View Options Run Analyse Help D Z Open HEI Jacobian Elliptic Functions ode Simple Harmonic Motion ode Ballistics with Air Resistance ode Ballistics with Air Resistance Polar Coordi Lorentz Model ode twobody1 ode Amea File name Files of type Calcode Files ode Cancel For Help press F1 X This loads the ordinary differential equations for simple harmonic motion usually written as Be ST T 7 1 together with the initial data y 0 0 4 0 1 The range over which the equation is to be solved is specified as 0 20 Now click on the the Solve equation button the one next to the print button Meer alas CALCODE immediately
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