Sample Mathematica code for the paper "Modeling a Diving Board"
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1 Sample Mathematica code for the paper "Modeling a Diving Board" Michael A. Karls and Brenda M. Skoczelas, Department of Mathematical Sciences, Ball State University, Muncie, IN Beam Data Original Data Remove headings from original data collected with World in Motion software. NOTE: At t = 1 second, World in Motion recorded values of (-2000, -2000) for the (x, y) data point at the 13th tape mark, at approximately x = 1.46 m. Looking at the nearby data, these clearly cannot be correct, so we interpolated the values from the corresponding points for x = 1.46 m at t = sec, which was recorded as ( , 0), and at t = sec, which was recorded as ( , ). These corrections are noted below in red. In[1]:= `, `, `, `, `, `, `,
2 2 DivingBoardSummer2009.nb `, `, `, `, `, `, `, `, `, `, `, `, `, `, `,
3 DivingBoardSummer2009.nb `, `, `, `, `, `, `, `, `, `, `, `, `, `, `,
4 4 DivingBoardSummer2009.nb `, `, `, `, `, `, `, Remove the first entry of each row, which corresponds to the time t at which the data was collected, for t 0 seconds. In[2]:= Beam DropBeam,, 1; Define the positions for the tape pieces along the beam. Check the length of this list - there should be fifteen points. In[3]:= xtape TableBeam1i, i, 1, LengthBeam1, 2 Out[3]= 0, , , , , , , , , , , , , , In[4]:= Lengthxtape Out[4]= 15 Steady-State Solution Solve the steady-state problem. In[5]:= In[6]:= ClearL, v, c, g DSolvec 2 V ''''x g, V0 0, V '0 0, V ''L 0, V '''L 0, Vx, x Out[6]= Vx 6 g L2 x 2 4 g L x 3 g x 4 24 c 2 In[7]:= vx : 6 g L2 x 2 4 g L x 3 g x 4 24 c 2 In[8]:= Out[8]= vx 6 g L 2 x 2 4 g L x 3 g x 4 24 c 2
5 DivingBoardSummer2009.nb 5 Initial Conditions Define initial conditions and model parameters. Length of beam is L meters. Parameters c, k, and g will be chosen dynamically below. The function f(x) is initial displacement of beam. Initial velocity g(x) is assumed to be zero. Define hx f x vx to be the extra amount by which the beam is bent initially. In[9]:= L 1.6; In[10]:= FitPartitionBeam1, 2, x, x 2, x Out[10]= x x 2 In[11]:= fx : a x b x 2 In[12]:= a ; b ; In[14]:= fx Out[14]= x x 2 In[15]:= In[16]:= hx : fx vx hx Out[16]= x x g x2 6.4 g x 3 g x 4 24 c 2 Construct Model Solve for the alpha values In[17]:= Clearalpha, Α alpha TableΑ. FindRootCosΑ L Α alpha; 1 CoshΑ L 2n 1Π, Α,, n, 0, 7; 2L Define position functions X n x, t In[20]:= Xx, n : CosΑn x CoshΑn x CoshΑn L CosΑn L SinΑn x SinhΑn x SinhΑn L SinΑn L
6 6 DivingBoardSummer2009.nb In[21]:= Define the A n Coefficients - for the Manipulate command to work, the coefficients A n have been found explicitly, which can be done with our choice of f(x)! ClearA, AN; An : SinL Αn SinhL Αn 8 g CosL Αn 8 c 2 L a b L Αn 4 g 8 L 4 Αn 4 CoshL Αn 8 g CosL Αn 8 c 2 L a b L Αn 4 g 8 L 4 Αn 4 8 c 2 Αn 2 a Αn SinL Αn a Αn 2 b SinL Αn SinhL Αn 2 c 2 Αn 4 L Cos2 L Αn Αn L Cosh2 L Αn Αn 6 CoshL Αn SinL Αn 3 CoshL Αn 2 Sin2 L Αn 6 CosL Αn 4 L Αn SinL Αn SinhL Αn 3 CosL Αn 2 Sinh2 L Αn AN TableAn, n, 1, 7; Define the B n Coefficients In[24]:= ClearΜ, B, BN; Μn : 1 2 4c2 Αn 4 k 2 ; Bn : k ANn 2Μn ; BN TableBn, n, 1, 7; Define time functions T n x, t In[28]:= Tt, n : ANnCosΜn t BNnSinΜn t 1 2 k t ; Define model yx, t vx X n x, tt n x, t In[29]:= yx, t : vx SumXx, ntt, n, n, 1, 7 Check to make sure our model is a function of c, k, and g: In[30]:= Out[30]= yx, t g x g x 3 g x 4 24 c 2 k t 2 1 c c g g c g Cos c 2 k 2 t 2 c c 2 k c g g c g k Sin c 2 k 2 t Cos x Cosh x Sin x Sinh x k t c 2 c g g c g
7 DivingBoardSummer2009.nb 7
8 8 DivingBoardSummer2009.nb Finding c, k, and g via the Manipulate command - Figures 3, 4, 5, and 6 in the paper are found this way! Since beam isn't moving until the fourth set of data points, drop the first four sets of data points. In[31]:= In[32]:= In[33]:= Beam DropBeam, 1, 4; data TablePartitionBeami, 2, i, 1, LengthBeam; Clearc, k, g
9 DivingBoardSummer2009.nb 9 In[34]:= ManipulateShowListPlotTable j 1, dataj, 15, 2, j, 1, LengthBeam, PlotStyle 30 PointSize0.02, RGBColor0, 0, 1, PlotRange 0.3, 0.3, AxesLabel "t", "y", ListPlotTable j 1 j 1, Evaluateydataj, 15, 1,, j, 1, LengthBeam. c Χ1, k Χ2, g Χ3, PlotStyle RGBColor1, 0, 0, PlotRange 0.3`, 0.3`, Joined True, Χ1, 1, "c", 1, 15, Χ2, 0, "k", 0, 4, Χ3, 0, "g", 0, 10, TrackedSymbols Χ1, Χ2, Χ3, SaveDefinitions True c k 0.94 g 9.8 Out[34]=
10 10 DivingBoardSummer2009.nb Here's how to plot Figure 6 without using the Manipulate command In[35]:= ShowListPlotTable j 1, dataj, 15, 2, j, 1, LengthBeam, 30 PlotStyle PointSize0.02, RGBColor0, 0, 1, PlotRange 0.3, 0.3, AxesLabel "t", "y", ListPlotTable j 1 j 1, Evaluateydataj, 15, 1,, j, 1, LengthBeam c 12.87, k 0.94, g 9.8, PlotStyle RGBColor1, 0, 0, PlotRange 0.3`, 0.3`, Joined True Out[35]= Find error with a choice of c, k, and g via a Mean Sum of the Squares for Error Calculation Find the error with the choices of c, k, and g (entered as c1, k1, and g1 so that the Manipulate command will work correctly) that give a reasonable fit graphically. The function all data points. Hc, k, g computes the mean of the sum of the squares for error (MSSE) over In[36]:= Hc, k, g : EvaluateSum dataj, i, 2 ydataj, i, 1, j 1 30 LengthBeam 15 2, i, 1, 15, j, 1, LengthBeam In[37]:= H12.87, 0.94, 9.8 Out[37]= Find values of c, k, and g that minimize MSSE via the FindMinimum command, starting with c = c1, k = k1, and fixing g = 9.8 m sec 2. Then compute error with the resulting c and k! In[38]:= c ; k1 0.94; In[40]:= Out[40]= FindMinimumHc, k, 9.8, c, c1, k, k , c , k
11 DivingBoardSummer2009.nb 11 In[41]:= H12.84, 1.01, 9.8 Out[41]= Plot model vs. measured data at all points with our new choices for c and k with g fixed at 9.8 m sec 2 - this is Figure 7 in the paper: In[42]:= g1a ShowTableGraphics3DRGBColor0, 0, 1, PointSize0.011, PointTablextapej, i 1, BeamAll, 2 ji, i, 1, LengthBeamAll, 1, 30 j, 1, 15, Axes True, True, True, AxesLabel "x", "t", "y", BoxRatios 3, 2, 1, ImageSize 600; In[43]:= g2a Show TableParametricPlot3Dxtapei, j 30, yxtapei, j. c 12.84, k 1.01, g 9.8, 30 j, 0, 32, Axes True, True, True, AxesLabel "x", "t", "y", Ticks xtape, Automatic, Automatic, BoxRatios 3, 2, 1, PlotRange 0, L, 0, LengthBeam 30, 0.25, 0.25, ImageSize 600, PlotStyle Thick, Red, i, 1, 15;
12 12 DivingBoardSummer2009.nb In[44]:= Showg2a, g1a, Boxed False, ViewPoint 0.8, 1.5, Out[44]= y x t
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