{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Math 2250 Maple Lab 6, F20 08 Mechanical Oscillations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "NAME _______________________ CLASSTIME ____ V ERSION A-E, F-K, L-R, S-Z" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 70 "Circle the version - see problem L6.1. There are t hree (3) problems in" }}{PARA 0 "" 0 "" {TEXT -1 71 "this project. Ple ase answer the questions A, B, C , ... associated with" }}{PARA 0 "" 0 "" {TEXT -1 65 "each problem. The original worksheet \"2250mapleL6-F 2008.mws\" is a" }}{PARA 0 "" 0 "" {TEXT -1 70 "template for the solut ion; you must fill in the code and all comments." }}{PARA 0 "" 0 "" {TEXT -1 71 "Sample code can be copied with the mouse. Use pencil free ly to annotate" }}{PARA 0 "" 0 "" {TEXT -1 50 "the worksheet and to cl arify the code and figures." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The problem headers for the F2008 version of Me chanical" }}{PARA 0 "" 0 "" {TEXT -1 25 "Oscillations maple lab 6." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "_________ _L6.1. UNDER-DAMPED FREE OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT -1 45 "__________L6.2. UNDAMPED FORCED OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT -1 36 "__________L6.3. PRACTICAL RESONANCE." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "L6.1 . PROBLEM (UNDER-DAMPED FREE OSCILLATIONS)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "FREE OSCILLATIONS. Consider the problem of free linear oscillations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " m x'' + c x' + k x=0," }} {PARA 0 "" 0 "" {TEXT -1 27 " x(0)=0, x'(0)=1." }}{PARA 0 " " 0 "" {TEXT -1 50 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 69 "Here, m, c and k are non-negative con stants. The under-damped case is" }}{PARA 0 "" 0 "" {TEXT -1 70 "studi ed here, c^2 < 4km, as on page 327 in E&P. Depending on the first" }} {PARA 0 "" 0 "" {TEXT -1 33 "letter of your last name, assume:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Version A -E: m=1, c=5 Version F-K: m=2, c=4" }}{PARA 0 "" 0 "" {TEXT -1 55 "Version L-R: m=3, c=4 Version S-Z: m=4, c=5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 " A. D isplay a Hooke's constant k > 0 so that the solution x(t) is" }}{PARA 0 "" 0 "" {TEXT -1 62 " under-damped. Check that x(t)=0 for infin itely many t>0." }}{PARA 0 "" 0 "" {TEXT -1 63 " Display the exac t solution x(t) obtained by maple methods" }}{PARA 0 "" 0 "" {TEXT -1 30 " as in the example below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 " B. Plot the exact symbolic solution x( t) on a suitable t-interval." }}{PARA 0 "" 0 "" {TEXT -1 61 " Che ck the graphic against Figure 5.4.9 page 328 of E&P." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " C. Estimate from the graph the decimal value of the pseudo-period." }}{PARA 0 "" 0 "" {TEXT -1 69 " Display the graphical estimate and also the exact p seudo-period" }}{PARA 0 "" 0 "" {TEXT -1 71 " 2Pi/w, where w is t he natural frequency of the trigonometric term" }}{PARA 0 "" 0 "" {TEXT -1 47 " in the solution x(t) found in item 2.4.A." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "# EXAMPLE (Wrong parameters! Change it!) \n# Use semicolons to see w hat you have done." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "# Def ine the differential equation\n# de:=3*diff(x(t),t,t)+1.5*diff(x(t),t) +4*x(t)=0: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "# Solve th e characteristic equation.\n# solve(3*r^2+1.5*r+4=0,r); \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "# Define the \+ initial conditions\n# ic:=x(0)=0,D(x)(0)= 1: \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "# Symbolically solve \+ for x(t)\n# p:=dsolve(\{de,ic\},x(t),method=laplace): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "# Capture the dsolve symbol ic solution as a function X(t)\n# X:=unapply(rhs(p),t): \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# Plot th e solution\n# plot(X(t),t=0..5); " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Maple tip: Click with the mouse on the graphic to print t he cursor" }}{PARA 0 "" 0 "" {TEXT -1 65 "location (left upper corner \+ of the maple window). The coordinates" }}{PARA 0 "" 0 "" {TEXT -1 66 " printed are of the form (x,y). From this coordinate information, a" }} {PARA 0 "" 0 "" {TEXT -1 40 "simple subtraction estimates the period. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "#L6.1-A Define k, then solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "# under-damped means mr^2+cr+k=0 ha s two conjugate complex roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#L6.1-B Plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "#L 6.1-C Pseudo-period calculations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "L6.2. PROBLEM (UNDAMPED FORCED OSCILLATIONS )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "FORCED LINEAR OSCILLATIONS. Consider the undamped (c=0) forced problem" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 " mx'' + k x = 5 cos(wt)," }}{PARA 0 "" 0 "" {TEXT -1 22 " x(0)=0, x'(0)=0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "where m, k and w are non-negati ve constants. Depending on the first" }}{PARA 0 "" 0 "" {TEXT -1 33 "l etter of your last name, assume:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 57 "Version A-E: m=1, k=3.5 Version F-K: m=2, k=2.5" }}{PARA 0 "" 0 "" {TEXT -1 58 "Version L-R: m=3, k=4 .5 Version S-Z: m=4, k=4.5" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 " A. Choose the forcing angular fre quency w to be 3 times larger than" }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ the natural angular frequency w0, w0^2=k/m. Solve for x(t) using" }}{PARA 0 "" 0 "" {TEXT -1 71 " dsolve(). Plot the solution x(t) on a suitable interval in order" }}{PARA 0 "" 0 "" {TEXT -1 67 " \+ to show the global behavior of the solution x(t). See Figure" }} {PARA 0 "" 0 "" {TEXT -1 23 " 5.6.2, page 350." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 " B. The solution x(t) is the sum of two functions, one of period" }}{PARA 0 "" 0 "" {TEXT -1 69 " 2Pi/w and the other of period 2Pi/w0. Display the exact p eriod," }}{PARA 0 "" 0 "" {TEXT -1 66 " as calculated from the so lution formula for x(t) -- see page" }}{PARA 0 "" 0 "" {TEXT -1 22 " \+ 350 for details." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 61 " C. Suggest a value for the forcing frequency w so th at the" }}{PARA 0 "" 0 "" {TEXT -1 66 " oscillations exhibit reso nance. Show resonant behavior on a" }}{PARA 0 "" 0 "" {TEXT -1 50 " \+ graph. Check against Figure 5.6.4, page 352." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L6.2-A" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "#L6.2-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L6.2-C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "L6.3. PROBLEM (PRACTICAL RESONANCE)" }} {PARA 0 "" 0 "" {TEXT -1 36 " Consider the damped forced problem" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 " mx'' + c x' + k x = 5 cos(w t)," }}{PARA 0 "" 0 "" {TEXT -1 22 " x(0)= 0, x'(0)=0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Depending on the first letter of your last name, assume:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "Version A -E: m=1, k=30 Version F-K: m=2, k=36" }}{PARA 0 "" 0 "" {TEXT -1 54 "Version L-R: m=3, k=45 Version S-Z: m=4, k=55" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " A. Co nsider the damping constants c=2, c=1 and c=1/2. Compute the" }} {PARA 0 "" 0 "" {TEXT -1 67 " amplitude function C(w) [page 357] \+ for these three equations," }}{PARA 0 "" 0 "" {TEXT -1 70 " then \+ plot for w=0 to w=20 the three amplitude graphs on a single" }}{PARA 0 "" 0 "" {TEXT -1 71 " set of axes. Compare against Figure 5.6.9 page 357 of E&P (it has" }}{PARA 0 "" 0 "" {TEXT -1 37 " one cur ve, yours has 3 curves)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 66 " B. For each case c=2, c=1, c=1/2, print the val ues w*, C* where" }}{PARA 0 "" 0 "" {TEXT -1 69 " C*=C(w*)=max \{ C(w) : 0 <= w <= 20\}. The three data pairs should" }}{PARA 0 "" 0 "" {TEXT -1 69 " show that C* becomes larger as c tends to zero. SAV E YOUR MAPLE" }}{PARA 0 "" 0 "" {TEXT -1 21 " FILE FREQUENTLY" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 " Maple \+ Hint: Use Maple's mouse interface on the graphic of Part C." }}{PARA 0 "" 0 "" {TEXT -1 68 " Specifically, click on a possible maximum (h orizontal tangent) in" }}{PARA 0 "" 0 "" {TEXT -1 65 " the graph to \+ display the values w*, C* on the screen. Copy the" }}{PARA 0 "" 0 "" {TEXT -1 43 " values into your maple worksheet report." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "#EXA MPLE(Beware! Wrong values!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#F:=15: m:=1: k:=25: c:='c': w:='w':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "#C:=(w,c)->F/sqrt((k-m*w*w)^2+(c*w)^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#plot(\{C(w,4),C(w,3),C(w,2)\},w=0. .15,color=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "#L6.3-A Plot C(w), three gra phics on one set of axes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#L6.3-B Table of six data values for w*, C*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 6 0" 55 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }