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"Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 203 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle1" -1 200 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle1" -1 204 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle2" -1 201 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle2" -1 205 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle3 " -1 202 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 53 "Math 2250 Maple Lab 6, \+ F2009 Mechanical Oscillations." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 72 "NAME ________________ _______ CLASSTIME ____ VERSION A-E, F-K, L-R, S-Z" }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "Ci rcle the version - see problem L6.1. There are three (3) problems in" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "this project. Please answer the questions A, B, C , ... associated with" }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 65 "each problem. The original worksheet \+ \"2250mapleL6-F2009.mws\" is a" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "template for the solution; you must fill in the code and all comments." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "Samp le code can be copied with the mouse. Use pencil freely to annotate" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 50 "the worksheet and to \+ clarify the code and figures." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 55 "The problem headers f or the F2009 version of Mechanical" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 25 "Oscillations maple lab 6." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 47 "_________ _L6.1. UNDER-DAMPED FREE OSCILLATIONS." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 45 "__________L6.2. UNDAMPED FORCED OSCILLATIONS." } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 36 "__________L6.3. PRACT ICAL RESONANCE." }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 46 "L6.1. PR OBLEM (UNDER-DAMPED FREE OSCILLATIONS)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 "FREE OSCILLATIO NS. Consider the problem of free linear oscillations" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 31 " \+ m x'' + c x' + k x=0," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 27 " x(0)=0, x'(0)=1." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 50 " \+ " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "Here, m, c a nd k are non-negative constants. The under-damped case is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "studied here, c^2 < 4km, as on \+ page 327 in E&P. Depending on the first" }{TEXT 204 0 "" }}{PARA 200 " " 0 "" {TEXT 204 33 "letter of your last name, assume:" }{TEXT 204 0 " " }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 55 "Version A-E: m=1, c=5 Version F-K: m=2, c=4" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 55 "Version L-R: m=3, c=4 \+ Version S-Z: m=4, c=5" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 " A. Display a Hooke's con stant k > 0 so that the solution x(t) is" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 62 " under-damped. Check that x(t)=0 for infini tely many t>0." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 " \+ Display the exact solution x(t) obtained by maple methods" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 30 " as in the example bel ow." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " B. Plot the exact symbolic solution x(t) on a s uitable t-interval." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 61 " Check the graphic against Figure 5.4.9 page 328 of E&P." } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " C. Estimate from the graph the decimal value of the p seudo-period." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " \+ Display the graphical estimate and also the exact pseudo-period" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " 2Pi/w, where w \+ is the natural frequency of the trigonometric term" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 47 " in the solution x(t) found in it em 2.4.A." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 205 52 "# EXAMPLE(Wrong parameters! Change it !) " }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 44 "\n# Use semi colons to see what you have done." }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 34 "# Define the differential equat ion" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# de:=3*diff(x(t),t,t) +1.5*diff(x(t),t)+4*x(t)=0: " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 36 "# Solve the characteristic equation." }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# solve(3*r^2+1.5*r+4=0,r) ; " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 205 31 "# Define the initial conditions" } {MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# ic:=x(0)=0,D(x)(0)= 1: \+ " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> \+ " 0 "" {MPLTEXT 1 205 29 "# Symbolically solve for x(t)" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# p:=dsolve(\{de,ic\},x(t),method=lapla ce): " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 57 "# Capture the dsolve symbolic solution as a functio n X(t)" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# X:=unapply(rhs(p) ,t): " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 19 "# Plot the solution" }{MPLTEXT 1 205 0 "" }{MPLTEXT 1 205 53 "\n# plot(X(t),t=0..5); \+ " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 66 "Maple ti p: Click with the mouse on the graphic to print the cursor" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 "location (left upper corner of the maple window). The coordinates" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "printed are of the form (x,y). From this coordinat e information, a" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 40 "si mple subtraction estimates the period." }{TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 30 "#L6.1-A Define k, then solve." }{MPLTEXT 1 205 0 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 65 "# under-damped mean s mr^2+cr+k=0 has two conjugate complex roots." }{MPLTEXT 1 205 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 13 "#L6.1-B Plot." } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 35 "# L6.1-C Pseudo-period calculations." }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 45 "L6.2. PROBLEM (UNDAMPED FORCED OSCILLATIONS )" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 "FORCED LINEAR OSCILLATIONS. Consider the undamped (c=0) force d problem" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 29 " mx'' + k x = 5 cos(wt)," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 22 " x(0)=0, x'(0)=0," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 "where m, k and w are non-negative constants. Depending on the \+ first" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 33 "letter of you r last name, assume:" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 57 "Version A-E: m=1, k=3.5 \+ Version F-K: m=2, k=2.5" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 58 "Version L-R: m=3, k=4.5 Version S-Z: m=4, k=4.5" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " A. Choose the forcing angular frequency w to be 3 ti mes larger than" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " \+ the natural angular frequency w0, w0^2=k/m. Solve for x(t) using" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 71 " dsolve(). Plo t the solution x(t) on a suitable interval in order" }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 67 " to show the global behavior of the solution x(t). See Figure" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 23 " 5.6.2, page 350." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " B. The soluti on x(t) is the sum of two functions, one of period" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 69 " 2Pi/w and the other of period 2P i/w0. Display the exact period," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " as calculated from the solution formula for x(t) - - see page" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 22 " 35 0 for details." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 61 " C. Suggest a value for the forcing \+ frequency w so that the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " oscillations exhibit resonance. Show resonant behavior \+ on a" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 50 " graph. C heck against Figure 5.6.4, page 352." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 7 "#L6.2-A" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 7 "#L6.2-B" } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 7 "#L 6.2-C" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 35 "L6.3. PROBLEM (PRAC TICAL RESONANCE)" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 36 " \+ Consider the damped forced problem" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 36 " mx'' + c x' + \+ k x = 5 cos(w t)," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 22 " \+ x(0)=0, x'(0)=0." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 56 "Depending on the first letter o f your last name, assume:" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 54 "Version A-E: m=1, k=30 \+ Version F-K: m=2, k=36" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 54 "Version L-R: m=3, k=45 Version S-Z: m=4, k=55" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " A. Consider the damping constants c=2, c=1 and c=1/2 . Compute the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 " \+ amplitude function C(w) [page 357] for these three equations," } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 " then plot for w =0 to w=20 the three amplitude graphs on a single" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 71 " set of axes. Compare against Fig ure 5.6.9 page 357 of E&P (it has" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 37 " one curve, yours has 3 curves)." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 " \+ B. For each case c=2, c=1, c=1/2, print the values w*, C* where" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 " C*=C(w*)=max \{ C(w) : 0 <= w <= 20\}. The three data pairs should" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 69 " show that C* becomes larger as c tends to zero. SAVE YOUR MAPLE" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 21 " FILE FREQUENTLY" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 " Maple Hint: Use Maple's mouse interface on the graphic of Part C." }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 68 " Specifically, click on a possible m aximum (horizontal tangent) in" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 " the graph to display the values w*, C* on the screen. Copy the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 43 " values into your maple worksheet report." }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 31 "#EXA MPLE(Beware! Wrong values!)" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 36 "#F:=15: m:=1: k:=25: c:='c': w:='w':" } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 39 "# C:=(w,c)->F/sqrt((k-m*w*w)^2+(c*w)^2):" }{MPLTEXT 1 205 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 50 "#plot(\{C(w,4),C(w,3),C( w,2)\},w=0..15,color=black);" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 52 "#L6.3-A Plot C(w), three graphics on one set of axe s" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 43 "#L6.3-B Table of six data values for w*, C*" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{PARA 202 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }