{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 256 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 257 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 258 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 23 62 "Math 2250 Maple Lab 3, Feb ruary 2006. Mechanical Oscillations." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "NAME _______________________ CLASSTIME ____ VERSION A-E, F-K, L-R, S-Z" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 67 "Circle the version - see problem L4.1. Th ere are three (3) problems" }}{PARA 0 "" 0 "" {TEXT 23 70 "in this pro ject. Please answer the questions A, B, C , ... associated " }}{PARA 0 "" 0 "" {TEXT 256 72 "with each problem. The original worksheet \"22 50mapleL2b-F2005.mws\" is a " }}{PARA 0 "" 0 "" {TEXT 257 71 "template for the solution; you must fill in the code and all comments. " }} {PARA 0 "" 0 "" {TEXT 258 72 "Sample code can be copied with the mouse . Use pencil freely to annotate " }}{PARA 0 "" 0 "" {TEXT 259 50 "the \+ worksheet and to clarify the code and figures." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 61 "The problem headers for t he Spring 2006 version of Mechanical" }}{PARA 0 "" 0 "" {TEXT 23 25 "O scillations maple lab 4." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT 23 48 " __________L4.1. UNDER-DAMPED FREE OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT 23 46 " __________L4.2. UNDAMPED FORCED OSCILL ATIONS." }}{PARA 0 "" 0 "" {TEXT 23 37 " __________L4.3. PRACTICAL RES ONANCE." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 46 "L4.1. PROBLEM (UNDER-DAMPED FREE OSCILLATIONS)" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "FREE OSCILLATIONS. Consid er the problem of free linear oscillations" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 32 " m x'' + c x' + k x=0 ," }}{PARA 0 "" 0 "" {TEXT 23 28 " x(0)=0, x'(0)=1." }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 66 "Here, m, \+ c and k are non-negative constants. The under-damped case" }}{PARA 0 " " 0 "" {TEXT 23 63 "is studied here, c^2 < 4km, as on page 327 in E&P. Depending on" }}{PARA 0 "" 0 "" {TEXT 23 43 "the first letter of your last name, assume:" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 " " {TEXT -1 115 " Version A-E: m=1, c=5 Version F-K: m=2, c =3\n Version L-R: m=4, c=6 Version S-Z: m=4, c=5\n\n" } {TEXT 23 68 " A. Display a Hooke's constant k > 0 so that the solut ion x(t) is" }}{PARA 0 "" 0 "" {TEXT 23 63 " under-damped. Check that x(t)=0 for infinitely many t>0." }}{PARA 0 "" 0 "" {TEXT 23 64 " Display the exact solution x(t) obtained by maple methods" }} {PARA 0 "" 0 "" {TEXT 23 31 " as in the example below." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 " B. Plot the exact symbolic solution x(t) on a suitable t-interval." }}{PARA 0 "" 0 "" {TEXT 23 62 " Check the graphic against Figure 5.4.9 page 3 28 of E&P." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 " C. Estimate from the graph the decimal value of the pseudo- period." }}{PARA 0 "" 0 "" {TEXT 23 70 " Display the graphical e stimate and also the exact pseudo-period" }}{PARA 0 "" 0 "" {TEXT 23 72 " 2Pi/w, where w is the natural frequency of the trigonometri c term" }}{PARA 0 "" 0 "" {TEXT 23 48 " in the solution x(t) fou nd in item 2.4.A." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 23 93 "EXAMPLE(Wrong parameters! Change it!) \+ # Use semicolons to see what you have done." }}{PARA 0 "" 0 " " {TEXT 23 84 "de:=3*diff(x(t),t,t)+1.5*diff(x(t),t)+4*x(t)=0: # Def ine the differential equation" }}{PARA 0 "" 0 "" {TEXT 23 86 "solve(3* r^2+1.5*r+4=0,r); # Solve the characteristic e quation." }}{PARA 0 "" 0 "" {TEXT 23 81 "ic:=x(0)=0,D(x)(0)= 1: \+ # Define the initial conditions" }}{PARA 0 "" 0 " " {TEXT 23 79 "p:=dsolve(\{de,ic\},x(t),method=laplace): # S ymbolically solve for x(t)" }}{PARA 0 "" 0 "" {TEXT 23 79 "X:=unapply( rhs(p),t): # Capture the dsolve symbolic" }}{PARA 0 "" 0 "" {TEXT 23 77 " \+ # answer as a function X(t)" }}{PARA 0 "" 0 "" {TEXT 23 69 "plot(X(t),t=0..5); # Plot the solution " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 23 66 "Maple tip: Click with the mouse on the graphic to print t he cursor" }}{PARA 0 "" 0 "" {TEXT 23 65 "location (left upper corner \+ of the maple window). The coordinates" }}{PARA 0 "" 0 "" {TEXT 23 66 " printed are of the form (x,y). From this coordinate information, a" }} {PARA 0 "" 0 "" {TEXT 23 40 "simple subtraction estimates the period. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#L4.1-A Define k, then solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 " # under-damped means mr^2+cr+k=0 has two conjugate complex roots." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#L4.1-B Plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "#L4.1-C Pseudo-period calculations. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 45 "L4.2. PROBLEM (UNDAMPED FORCED OSCILLATIONS )" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "FORCED LINEAR OSC ILLATIONS. Consider the undamped (c=0) forced problem" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 30 " mx'' + k x = \+ 5 cos(wt)," }}{PARA 0 "" 0 "" {TEXT 23 23 " x(0)=0, x'(0)=0," }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "where m, \+ k and w are non-negative constants. Depending on the first" }}{PARA 0 "" 0 "" {TEXT 23 33 "letter of your last name, assume:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 118 " Version A-E: m=1, \+ k=3.7 Version F-K: m=2, k=2.7\n Version L-R: m=3, k=4.8 \+ Version S-Z: m=4, k=4.6" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 72 " A. Choose the forcing angular freque ncy w to be 3 times larger than" }}{PARA 0 "" 0 "" {TEXT 23 72 " \+ the natural angular frequency w0, w0^2=k/m. Solve for x(t) using" }} {PARA 0 "" 0 "" {TEXT 23 72 " dsolve(). Plot the solution x(t) \+ on a suitable interval in order" }}{PARA 0 "" 0 "" {TEXT 23 68 " \+ to show the global behavior of the solution x(t). See Figure" }} {PARA 0 "" 0 "" {TEXT 23 24 " 5.6.2, page 350." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 " B. The solution x( t) is the sum of two functions, one of period" }}{PARA 0 "" 0 "" {TEXT 23 70 " 2Pi/w and the other of period 2Pi/w0. Display the \+ exact period," }}{PARA 0 "" 0 "" {TEXT 23 67 " as calculated fro m the solution formula for x(t) -- see page" }}{PARA 0 "" 0 "" {TEXT 23 23 " 350 for details." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 62 " C. Suggest a value for the forcing fr equency w so that the" }}{PARA 0 "" 0 "" {TEXT 23 67 " oscillati ons exhibit resonance. Show resonant behavior on a" }}{PARA 0 "" 0 " " {TEXT 23 51 " graph. Check against Figure 5.6.4, page 352." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "#L4.2-A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L4.2-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#L4.2-C" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 35 "L4.3. PROBLEM (PRACTICAL RESONANCE)" }}{PARA 0 "" 0 "" {TEXT 23 37 " Consider the damped forced problem" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 37 " mx'' + c x' + k x = 5 cos (w t)," }}{PARA 0 "" 0 "" {TEXT 23 23 " x(0)=0, x'(0)=0." }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 56 "Depending on the first letter of your last name, assume:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 111 " Version A-E: m=1, k=35 \+ Version F-K: m=2, k=38\n Version L-R: m=3, k=48 Vers ion S-Z: m=4, k=58" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 70 " A. Consider the damping constants c=2, c=1 and c=1/2. Compute the" }}{PARA 0 "" 0 "" {TEXT 23 68 " amplitude functio n C(w) [page 357] for these three equations," }}{PARA 0 "" 0 "" {TEXT 23 71 " then plot for w=0 to w=20 the three amplitude graphs on \+ a single" }}{PARA 0 "" 0 "" {TEXT 23 72 " set of axes. Compare a gainst Figure 5.6.9 page 357 of E&P (it has" }}{PARA 0 "" 0 "" {TEXT 23 38 " one curve, yours has 3 curves)." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 " B. For each case c=2, c=1, c=1/2, print the values w*, C* where" }}{PARA 0 "" 0 "" {TEXT 23 70 " C*=C(w*)=max \{C(w) : 0 <= w <= 20\}. The three data pai rs should" }}{PARA 0 "" 0 "" {TEXT 23 70 " show that C* becomes \+ larger as c tends to zero. SAVE YOUR MAPLE" }}{PARA 0 "" 0 "" {TEXT 23 22 " FILE FREQUENTLY" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 69 " Maple Hint: Use Maple's mouse interfa ce on the graphic of Part C." }}{PARA 0 "" 0 "" {TEXT 23 69 " Speci fically, click on a possible maximum (horizontal tangent) in" }}{PARA 0 "" 0 "" {TEXT 23 66 " the graph to display the values w*, C* on t he screen. Copy the" }}{PARA 0 "" 0 "" {TEXT 23 20 " values on pape r." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 30 "EXA MPLE(Beware! Wrong values!)" }}{PARA 0 "" 0 "" {TEXT 23 35 "F:=15: m:= 1: k:=25: c:='c': w:='w':" }}{PARA 0 "" 0 "" {TEXT 23 38 "C:=(w,c)->F/ sqrt((k-m*w*w)^2+(c*w)^2):" }}{PARA 0 "" 0 "" {TEXT 23 49 "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 "#L4.3-A Plot C(w), three graphics on one set of axes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#L4.3-B Table of six data values for w*, C*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "72 0 0" 5 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }