PalmIII system dynamics 2nd solman

Solutions Manual c to accompany System Dynamics, Second Edition by William J Palm III University of Rhode Island Solutions to Problems in Chapter One Solutions Manual Copyright 2010 The McGraw-Hill Companies All rights reserved No part of this manual may be displayed, reproduced, or distributed in any form or by any means without the written permission of the publisher or used beyond the limited distribution to teachers or educators permitted by McGraw-Hill for their individual course preparation Any other reproduction or c translation of this work is unlawful www.elsolucionario.net www.elsolucionario.net 1.1 W = mg = 3(32.2) = 96.6 lb 1.2 m = W/g = 100/9.81 = 10.19 kg W = 100(0.2248) = 22.48 lb m = 10.19(0.06852) = 0.698 slug 1.3 d = (50 + 5/12)(0.3048) = 15.37 m 1.4 n = 1/[60(1.341 × 10−3 )] = 12.43, or approximately 12 bulbs 1.5 5(70 − 32)/9 = 21.1◦ C 1.6 ω = 3000(2π)/60 = 314.16 rad/sec Period P = 2π/ω = 60/3000 − 1/50 sec 1.7 ω = rad/sec Period P = 2π/ω = 2π/5 = 1.257 sec Frequency f = 1/P = 5/2π = 0.796 Hz c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.8 Physical considerations require the model to pass through the origin, so we seek a model of the form f = kx A plot of the data shows that a good line drawn by eye is given by f = 0.2x So we estimate k to be 0.2 lb/in Skipping ahead to Section 1.5, we can solve this problem using the least squares method, based on equation (1.5.3) The script file is x = f = num den k = [4.7,7.2,10.6,12.9]-4.7; [0,0.47,1.15,1.64]; = sum(x.*f); = sum(x.^2); num/den The result is k = 0.1977 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.9 The script file is x = [0:0.01:1]; subplot(2,2,1) plot(x,sin(x),x,x),xlabel( x (radians) ),ylabel( x and sin(x) ), gtext( x ),gtext( sin(x) ) subplot(2,2,2) plot(x,sin(x)-x),xlabel( x (radians) ),ylabel( Error: sin(x) - x ) subplot(2,2,3) plot(x,100*(sin(x)-x)./sin(x)),xlabel( x (radians) ), ylabel( Percent Error ),grid The plots are shown in the figure x and sin(x) 0.6 Error: sin(x) − x x 0.8 sin(x) 0.4 0.2 0 0.5 x (radians) 0.5 x (radians) −0.05 −0.1 −0.15 −0.2 0.5 x (radians) Percent Error −5 −10 −15 −20 Figure : for Problem 1.9 From the third plot we can see that the approximation sin x ≈ x is accurate to within 5% if |x| ≤ 0.5 radians c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.10 For θ near π/4, π π + cos 4 θ− π 3π 3π + cos 4 θ− 3π f (θ) ≈ sin For θ near 3π/4, f (θ) ≈ sin c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.11 For θ near π/3, π π − sin 3 θ− π 2π 2π − sin 3 θ− 2π f (θ) ≈ cos For θ near 2π/3, f (θ) ≈ cos c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.12 For h near 25, √ f (h) ≈ 1 25 + √ (h − 25) = + (h − 25) 10 25 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.13 For r near 5, f (r) ≈ 52 + 2(5)(r − 5) = 25 + 10(r − 5) For r near 10, f (r) ≈ 102 + 2(10)(r − 10) = 100 + 20(r − 10) c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net 1.14 For h near 16, √ f (h) ≈ 1 16 + √ (h − 16) = + (h − 16) 16 f (h) ≥ if h > −16 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.5 The closed-form solution is y(t) = 65 −3t 4t + e e 7 After five steps, t = 0.05 and the exact solution is y(0.05) = 8.8647 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.6 The closed-form solution is y(t) = 6e−5t After five steps, t = 0.1 and the exact solution is y(0.1) = 3.6392 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.7 The closed-form solution is y(t) = + sin t After five steps, t = 1.5 and the exact solution is y(1.5) = 6.9975 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.8 The closed-form solution is y(t) = 12 − cos 3t After five steps, t = 0.5 and the exact solution is y(0.5) = 11.8585 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.9 The closed-form solution is y(t) = 3.25 − 12.5e−4t After five steps, t = 0.125 and the exact solution is y(0.125) = −4.3316 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.10 The closed-form solution is y(t) = 65 −3t 4t + e e 7 After five steps, t = 0.05 and the exact solution is y(0.05) = 8.8647 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.11 The equations of motion are given by equations (3) and (4) in Example 2.2.4 These equations are programmed in the following m file function ydot = mast(t,y) Q=sqrt(2020+1650*cos(1.33+y(1))); ydot(1) = y(2); ydot(2) = (1/25400)*(-17500*cos(y(1))+(626000./Q).*sin(1.33+y(1))); ydot = [ydot(1);ydot(2)]; This file is called as follows Since we not know how long it will take for the mast to reach 90◦, we must experiment with the stop time Since the geometry on which the equations are based breaks down if θ ≥ 90◦, we start with a low value of the stop time, say seconds, and gradually increase the stop time uintil the plot shows that 90◦ has been reached To two decimal places, the answer is 8.62 seconds [t,y] = ode45(@mast,[0, 8.62],[pi/6,0]); plot(t,(180/pi)*y(:,1)) c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.12 The closed-form solution is y(t) = 6e−5t After five steps, t = 0.1 and the exact solution is y(0.1) = 3.6392 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.13 The closed-form solution is y(t) = + sin t After five steps, t = 1.5 and the exact solution is y(1.5) = 6.9975 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.14 The closed-form solution is y(t) = 3.25 − 12.5e−4t After five steps, t = 0.125 and the exact solution is y(0.125) = −4.3316 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.15 The closed-form solution is y(t) = 6e−5t After five steps, t = 0.1 and the exact solution is y(0.1) = 3.6392 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.16 The closed-form solution is y(t) = + sin t After five steps, t = 1.5 and the exact solution is y(1.5) = 6.9975 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.17 The closed-form solution is y(t) = 12 − cos 3t After five steps, t = 0.5 and the exact solution is y(0.5) = 11.8585 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.18 The closed-form solution is y(t) = + sin t After five steps, t = 1.5 and the exact solution is y(1.5) = 6.9975 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net D.19 The closed-form solution is y(t) = 12 − cos 3t After five steps, t = 0.5 and the exact solution is y(0.5) = 11.8585 c 2010 McGraw-Hill This work is only for non-profit use by instructors in courses for which the textbook has been adopted Any other use without publisher’s consent is unlawful www.elsolucionario.net ... without publisher’s consent is unlawful www.elsolucionario.net Solutions Manual c to accompany System Dynamics, Second Edition by William J Palm III University of Rhode Island Solutions to Problems... horizontal and vertical velocity components are vx = v0x sin θ v0y = v0 cos θ (2) Establish a coordinate system at the point where the mass leaves the surface, with x positive to the right and y positive