Mathematical Functions

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	Mathematical Functions in AutoCAD Example: The catenary function by Héctor Monroy, BSCE, MSCE, MSEC CAD Manager. CSA group, Inc.
	Sometimes engineers lose one of their most powerful tools when they use one of their most powerful tools -- the mathematical tool that's right inside the software. AutoCAD is very good drafting software for making arcs and lines, but sometimes this simple view of things is not sufficient to represent typical engineering graphics. For example: if we have a cable supported by two towers, we need to define catenary curve to represent the cable, yet AutoCAD doesn't have a "catenary" command. Also, if you need to analyze a beam deflection there isn't an easy way to do it. The real CAD tool power is the possibility of using CAD with a geometric calculator, not only for measuring areas and distances, but to graph complex functions. This article brings you a small AutoLISP routine to demonstrate how to graph mathematical functions and parametric equations. If you know AutoLISP, skip the next two paragraphs. If you don't know AutoLISP, you'll need to understand how to write mathematical expressions in AutoLISP. First rule: All AutoLISP expressions are written between parentheses. Second rule: all mathematical expressions are written in prefix notation. Third rule: mathematic operations between real numbers produce real numbers, between integers produce integers and between integers, and real produce real. Examples: 1 + 3 is (+ 1 3) in AutoLISP 1 + 2 / 4 is (+ 1 (/ 2 4)) in AutoLISP (+ 1 3) = 4 (+ 1.0 3) = 4.0 (/ 5 3) = 1 (/ 5.0 3) =1.66667 To understand in depth, see your AutoLISP manual. EVALUATING FUNCTIONS IN THE XY PLANE Typically, AutoCAD users work in the XY plane, X values increasing left to right and Y values increasing bottom to top. In the graphfx.lsp file, the GraphFx AutoLISP function evaluates a mathematical expression in the form Y=F(X) and returns a sorting list with the evaluated points. The AutoLISP function has four arguments: the mathematical function Y=F(X) in AutoLISP terms, the first X value, the last X value and the increasing value to X. The AutoLISP function Printpntlst graphs a 3DPOLYLINE using a list with sorted points. For example to generate a list with evaluated points using the trigonometric function sin(x) between 0 and 10 increasing x in 0.1, you need to write the following expression in the AutoCAD command line: (graphFx '(sin x) 0 10 0.1) To print the points, write (Printpntlst (graphFx '(sin x) 0 10 0.1)) sin(x) graphic Also, you can generate more complex and practical graphs. For example, the following equation defines the deflection curve for a propped cantilever beam with a uniform load (q): to express the last equation in AutoLISP you need to write the following line: (* (/ (* q x x) (* 48 E I)) (+ (* 3 L L) (* -5 L x) (* 2 x x))) If the length (L) of the beam is 10m and simplifying and scaling the expression only for demonstration we have: (printpntlst (graphFx '(* x x -0.0005 (+ 300 (* -50 x) (* 2 x x))) 0 10 0.1)) Deflection curve for a propped cantilever beam EVALUATING PARAMETRIC FUNCIONS IN THE XYZ SPACE If we have a graph described by three functions, one for X coordinate, one for Y coordinate, one for Z coordinate and a common parameter we can use the GraphPar AutoLISP function (graphfx.lsp). This AutoLISP function has six arguments: three mathematical functions X=F(q), Y=F(q) and Z=F(q) in AutoLISP terms, the first q parameter value, the last q parameter value and the increasing value of q. For example, we can describe such parabolic motion with parametric equations in the XY plane using a time parameter q measured in time units from the initial projection point: Expressing the last equations in AutoLISP we have: (+ X0 (* Vx0 q)) (+ Y0 (* Vy0 q) (* 0.5 g q q)) 0 If q start in 0 and end in 200, X0=0, Y0=0 and simplifying and scaling the parametric function we have: (printpntlst (graphpar '(* 100 q) '(- (* 100 q) (* q q)) '0 0 200 1)) Parabolic motion THE CABLE PROBLEM One of the most typical problems in engineering is to fix a cable between two points. The function to describe the shape is a catenary. The mathematical expression is: Where w is the weight per unit length of the cable, t is the tension developed in the cable, and y0 is the minimum height of the cable. AutoLISP don't have the hyperbolic cosine, but that is not a problem since we can define it with the following function (defun cosh(angles / angles) (/ (/ (+ (exp angles)(exp (* angles -1))) 2)) ) In the catenary.lsp file we have a routine to print a cable between two points defined the points in AutoCAD, the weight per unit length of the cable and the tension. The file has three additional functions: hyperbolic cosine (cosh), hyperbolic sine (sinh), and arc hyperbolic sine (arcsinh) Below we can see the graphFx AutoLISP use in the c:catenary function: (printpntlst (graphFx '(+ (* global_c (- (cosh (/ (- x global_xv) global_c)) 1)) global_yv) x1 x2 1.0)) Cable between two points CONCLUSION With the GraphFx and GraphPar AutoLISP routines we can graphic mathematical functions, also we can include the routines inside others routines within we like to have only the kernel of a specific problem (i.e. The cable problem).   [an error occurred while processing this directive]