II.1 Programming in MATLAB Getting Started Graphing

%% tgraph.m%% This m-file is a template for creating graphs of functions. It contains many of the% commands that are commonly used in graphing.%% In this template, the graphs of the sine function and the cosine function are created% in one figure window.%% This template shows:%% - several ways of plotting% - how to create a grid in a figure window% - how to label axes% - how to create a title for a figure window% - how to create a legend when more than one plot is drawn in a figure window%clear% Optional Feature. This sets all values to zero.for i = 1:361:1% This is the beginning of a loop. The index i increases% from 1 to 361 in increments of 1 each time through the% loop. More compact ways of doing loops is shown in% the template called tarray.%for i = 1:361 % The default increment is 1.theta(i) = (i-1)*2*pi/360; % theta is an array. theta varies from 0 to 2*pi radians.f(i) = sin(theta(i));% f is an array. The i-th entry of f is the sine function% evaluated at the i-th value of theta.p(i) = cos(theta(i));% p is an array. The i-th entry of p is the cosine% function evaluated at the i-th value of theta.end% This is the end of the loop.figuregrid onaxis([0 7 -1 1])hold%plot(theta,f)%plot(f)plot(theta,f,'r')plot(theta,p,'g')%plot(theta,p,'--')%plot(theta,p,':')%plot(theta,p,'r.')title('force and position')xlabel('t-axis')ylabel('force or positon')legend('force','position')% This opens a new figure window.% Optional Feature. This creates a grid on your graph.% Optional Feature. This specifies the axis limits% ([theta-min theta-max f-min f-max]).% Optional Feature. This holds the figure window for more% graphs.% This plots the array f versus theta (in radians).% This plots the array f versus the index (from 1 to% 361).% This plots the array f versus theta in red. The 'r'% indicates the color red.% This plots the array p versus theta in green. The 'g'% indicates the color green.% This plots the array p versus theta in large dashes.% This plots the array p versus theta in small dashes.% This plots the array p versus theta in red in small% red dots (zoom in to see them).% Optional Feature. This creates a title for the figure% window.% Optional Feature. This creates a label for the% horizontal axis.% Optional Feature. This creates a label for the vertical% axis.% Optional Feature. This creates a legend for the first% and second graphs in the figure window.Table 1: Template for Graphing3

ArraysNext, look over the second template shown in Table 2.This program determines the force reactions acting on the barshown in Fig. A2 – 2. The reactions are found by solving a setof three linear algebraic equations. The unknowns in thisproblem are⎛ F ⎞⎜ ⎟x =⎜N A⎟,⎝ N B ⎠where F is the friction force at A, N A is the normal force at A,and N B is the normal force at B. Create the M-file namedtarray.m and type in Table 2. Notice how the program defineselements of vectors and matrices (arrays). Also notice howthese arrays are manipulated using vector algebra. In thisprogram, a set of linear algebraic equations of the form Ax = bis solved by first calculating A -1 and then by calculating thesolution x = A -1 b.Next, save and close tarray.m, go to the commandwindow, typetarrayand hit return. The command window diplays b, A, and thesolution x. You’ll see that F = 31.6987 lb, N A = 68.3013 lb, andN B = 44.8288 lb.4

%% tarray.m%% This m-file is a template for creating arrays. It contains many of the% commands that are commonly used for arrays.%% In this template, a set of linear algebraic equations is solved.%% This template shows:%% - several ways to input a vector% - how to transpose a vector and a matrix% - how to invert a matrix% - how to swap a column in a vector with another column vector% - how to multiply vectors and matrices% - how to perform a for-end loop more efficiently%clearW = 100;L = 2;theta = pi/4;beta = pi/6;b = [W,0,W*L/2*cos(beta)]'% An array is defined by enclosing the elements in% brackets, separating them by commas.% This creates a row vector. The prime following the% right bracket transposes the row vector, making it a% column vector.%% b = [W;0;W*L/2*cos(beta)] % This is another way to define a column vector. Each row% (the rows are one element long) is separated by a% semicolon.A = [0,1,cos(theta);1,0,-sin(theta);0,0,L*cos(theta-beta)] %A matrix is defined by% listing its elements row by row. Each% row is separated by a semicolon.ainv = inv(A);% inv inverts a matrix.% detofA = det(A) % det takes the determinant of a matrix.% Atranspose = A' % a prime after a matrix transposes the matrix.% A(:,2) = [10,10,10]' % Using the colon, the second row of A is replaced with% the column vector [10,10,10]'.x = ainv*b%time1 = [0:1:2*pi]% multiplication/addition of vectors and matrices uses% the same syntax that is used for scalars.% Instead of using a for-end loop. The following syntax%can be used. This instruction creates a row vector of% values starting from 0, stepping in increments of% 1,ending just before 2pi. The loop is created using the% brackets.%time2 = [0:2*pi] % The default increment is 1.%c = cos(time1) % Creates a row vector called c. The array is created% using the parentheses, inside which the array time1% appears.Table 2: Template for Arrays5

To try out a few other vector operations, type incleara = [1, 2, 3]b = [2, 3, 4]and then type in each of the following quantities and hit enter:a*b’ a’*b dot(a,b)dot(a,b’) cross(a,b) cross(a’,b’)Notice that the dot function does not distinguish betweencolumn vectors and row vectors but that the other operationsdo.FunctionsThe third and final template shows you how to createfunctions. This will help you become more efficient inprogramming. Create the M-file called tfunction.m and type inTable 3. The M-file tfunction.m turns the force reactions in thesecond template into a function of beta and theta. In thecommand window, type in[F,NA]=tfunction(pi/6,pi/4)and hit enter. The reactions F and N A are displayed.6

function [F,NA] = tfunction(beta,theta)%% tfunction.m%% This m-file is a template for a function.%% In this template, the function tfunction finds the friction force F and the normal% force NA.% The inputs to the function are the variables beta (the angle of the bar) and theta% (the angle of the incline). These two inputs are in radians.% The outputs to the function are F and NA.% Notice the syntax for the function. The output variables are surrounded by brackets,% the input variables are surrounded by parentheses, and the name of the function is% the same as the name of the file.%% This template shows:%% - the syntax for a function%W = 100;L = 2;b = [W,0,W*L/2*cos(beta)]';A = [0,1,cos(theta);1,0,-sin(theta);0,0,L*cos(theta-beta)];ainv = inv(A);x = ainv*b;F = x(1);NA = x(2);Table 3: Template for FunctionsThe M-file tfunction.m can also be called from withinanother program. So can the M-files test1 and test2. Toillustrate this, create the M-file named test2.m and type inTable 4. This program determines the critical angle beta of thebar for which the bar remains in static equilibrium but belowwhich the bar slides (See Fig. A2 – 2). The program test2.mcalls the function tfunction.m in a loop, sweeping over a rangeof values of beta between 0 and theta. Each time the function iscalled, the program compares the value of the friction forcewith the maximum allowable level of friction, F max = µN A , tosee if it’s been exceeded. To run test2.m, go to the commandwindow, type intest2and hit enter. The command window displaysbetacritical = 26.55, which is the critical angle betain degrees, and a figure window appears showing two graphs.One graph is of the friction force F versus beta and the secondgraph is of the maximum friction force (above which the barslides) versus beta.Figure A2 – 2: A bar rests on a rough surface at oneend and on a smooth inclined surface at the other end.7

clearmu = 0.5;theta = pi/4;error = 100;betamax = 0;for i = 1:101beta = (i-1)*theta/100;[F,NA]=tfunction(beta,theta);x(i) = beta*180/pi;y(i) = F;FMAX(i)=mu*NA;er = abs(F-FMAX(i));% The function tfunction is called.% Notice that x in test2.m is% different than x in tfunction.m.% Quantities are not passed% between programs.endif er

Key Commandsclc, clear, edit, what, whoReview Questions1. What is the MATLAB desktop?2. Describe the purpose of a semicolon placed at the end of aninstruction.3. What command erases what’s in the command window?4. What command clears all of the variables in the workspace?5. Describe the purpose of the commands who and what.ProblemsA2 – 1A W = 50 lb block rests on a rough surface when the frictionforce F acting on the block is less than µN, in which N denotesthe normal force acting on the block by the surface and µ = 0.5is the static friction coefficient. When F exceeds 0.5N, theblock begins to slide. The friction force F and the normal forceN depend on the incline angle θ by F = 50cosθ and N = 50sinθ.Plot the functions F(θ) and µN(θ) on the same graph letting θrange between 0 and 90 ○ . Determine from the graph the largestangle θ max for which the block remains at rest.Answer: θ max = 26.6 ○ (Note that θ max can also be foundanalytically.)A2 – 2A triangular truss is acted upon by a P = 500 lb load, as shown.The load causes a tensile force of F AB = P/sinθ in member ABand a compressive force of F AC = Pcosθ /sinθ in member AC.Plot F AB + F AC versus θ. For what angle θ is F AB + F ACminimal?Answer: θ min = 60 ○ (Note that θ min can also be foundanalytically.)9

A2 – 3A L = 15 foot inextensible cord is tensioned by a spring havinga spring constant of k = 1000 pounds per foot and anunstretched length of a = 0.25 feet. A W = 20 pound weighthangs from point A, as shown. After some analysis, the tensionT in the cord (in pounds), the angle θ of the spring, and thestretch u of the spring were found to be related to each other bythe three equationsa = ( a + u) cosθ,W = ku sinθ,T = ku cosθ.It was determined from these equations that the angle θ of thespring is constrained by the equation0 = f(θ) = a – [a + W/(ksinθ)]cosθTo find the angle θ, plot f(θ) over the range of θ between 0 and90 ○ and locate from the graph where f(θ) = 0. Then, find thetension T in the cord and the stretch u in the spring.Answer: θ = 30.321 ○ , T = 34.2 pounds, u = 0.475 inches.A2 – 5A bridge is supported by cables that are suspended from acircular arch, as shown. The number of cables is 2n + 1. Thecables are tensioned to equal tensions T. After some analysis, inorder to support a 100 kilo-pound weight, it has beendetermined that the tension in the cables isT =1+2n∑Wsin kθ2k = 1 [cos kθ−1/(n + 1)] +sin2( kθ)πwhere θ = . Graph the tension T versus n for a range of2 ( n + 1)n between 1 and 10.Answer:T = 34.3, 23.1, 17.7, 14.4, 12.2 ,10.5, 9.27, 8.29, 7.50, 6.84 kilo - pounA2 – 4After analyzing the triangular truss given in problem A2 – 2, itwas determined that the unknown internal forces F AB and F ACand the unknown reactions at B and C, denoted by R Bx , R By , andR Cx , are related to each other by the five equations−FFFABAB− F− Fcosθ+ FAC= 0,sinθ= 500cosθ+ RBx= 0,sinθ+ RBy= 0,+ R = 0ABABACCxin which θ = 60 ○ . Placing the unknowns in the column vector x= [F AB F AC R Bx R By R Cx ] T , write these equations in matrix-vectorform as Ax = b. Then write a short program, find A -1 and then x= A -1 b.Answer: F AB = 577 lb, F AC = 289 lb, R Bx = -289 lb, R By = 500 lb,R Cx = 289 lb.10

A2 – 6A stone ornamental bridge being considered as a gateway to aEuropean city combines Greek and gothic elements. The cablesare not just ornamental. They also pull together beam A andpillars B. The cables are tensioned to the same tension to pulldown on the beam with a force of 2,500 pounds. The number ofcables is 2n. After some analysis, it has been determined thatthe tension in the cables isT =2Hn∑250022 2k = 1 L [2 − k /( n + 1)] + H [ k /( n +kwhere L = 7 feet and H = 25 feet. Graph the tension T versus nfor a range of n between 5 and 10.Answer: T = 361, 303, 261, 229, 205, 185 pounds.1)]211