{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 }{CSTYLE "" -1 23 "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 65 "Math 2250 Maple Lab 2b, S eptember 2004. Mechanical Oscillations." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 "NAME _______________________ CLASST IME ____ VERSION A-E, F-K, L-R, S-Z" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 66 "Circle the version - see problem 2.4. T here are three (3) problems" }}{PARA 0 "" 0 "" {TEXT 23 58 "in this pr oject. Please answer the questions A, B, C , ..." }}{PARA 0 "" 0 "" {TEXT 23 52 "associated with each problem. The original worksheet" }} {PARA 0 "" 0 "" {TEXT 23 65 "\"2250mapleL2b-F2004.mws\" is a template \+ for the solution; you must" }}{PARA 0 "" 0 "" {TEXT 23 65 "fill in the code and all comments. Sample code can be copied with" }}{PARA 0 "" 0 "" {TEXT 23 61 "the mouse. Use pencil freely to annotate the workshe et and to" }}{PARA 0 "" 0 "" {TEXT 23 29 "clarify the code and figures ." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 59 "The \+ problem headers for the Fall 2004 version of Mechanical" }}{PARA 0 "" 0 "" {TEXT 23 26 "Oscillations maple lab 2b." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 47 " __________2.4. UNDER-DAMPED FR EE OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT 23 45 " __________2.5. UNDAM PED FORCED OSCILLATIONS." }}{PARA 0 "" 0 "" {TEXT 23 36 " __________2. 6. PRACTICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 45 "2.4. PROBLEM (UNDER-DAMPED FREE OSCILLATIONS)" }} {PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 67 "FREE OSCI LLATIONS. Consider 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 3 27 in E&P. Depending on" }}{PARA 0 "" 0 "" {TEXT 23 43 "the first lett er of your last name, assume:" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 56 " Version A-E: m=1, c=4 Versio n F-K: m=2, c=4" }}{PARA 0 "" 0 "" {TEXT 23 56 " Version L-R: m=3, c=5 Version S-Z: m=4, c=5" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }} {PARA 0 "" 0 "" {TEXT 23 68 " A. Display a Hooke's constant k > 0 s o that the solution x(t) is" }}{PARA 0 "" 0 "" {TEXT 23 63 " und er-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 map le methods" }}{PARA 0 "" 0 "" {TEXT 23 31 " as in the example be low." }}{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 Figu re 5.4.9 page 328 of E&P." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT 23 70 " C. Estimate from the graph the decimal value o f the pseudo-period." }}{PARA 0 "" 0 "" {TEXT 23 70 " Display th e graphical estimate and also the exact pseudo-period" }}{PARA 0 "" 0 "" {TEXT 23 72 " 2Pi/w, where w is the natural frequency of the \+ trigonometric term" }}{PARA 0 "" 0 "" {TEXT 23 48 " in the solut ion x(t) found 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: # Define the differential equation" }}{PARA 0 "" 0 "" {TEXT 23 86 "solve(3*r^2+1.5*r+4=0,r); # Solve the c haracteristic equation." }}{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) : # Symbolically solve for x(t)" }}{PARA 0 "" 0 "" {TEXT 23 79 "X:=unapply(rhs(p),t): # Capture the ds olve 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); # Pl ot the solution" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 23 66 "Maple tip: Click with the mouse on the gr aphic to print the cursor" }}{PARA 0 "" 0 "" {TEXT 23 65 "location (le ft upper corner of the maple window). The coordinates" }}{PARA 0 "" 0 "" {TEXT 23 66 "printed are of the form (x,y). From this coordinate in formation, a" }}{PARA 0 "" 0 "" {TEXT 23 40 "simple subtraction estima tes the period." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "#2.4-A Define k, then solve." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "# under-damped means mr^2+cr+k=0 has two conju gate complex roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "#2.4 -B Plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#2.4-C Pseudo-p eriod calculations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 23 44 "2.5. PROBLEM (UNDAMPED FORCED OSCILLATION S )" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "FO RCED LINEAR OSCILLATIONS. 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 f irst" }}{PARA 0 "" 0 "" {TEXT 23 33 "letter of your last name, assume: " }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 58 " Vers ion A-E: m=1, k=3.5 Version F-K: m=2, k=2.5" }}{PARA 0 "" 0 "" {TEXT 23 59 " Version L-R: m=3, k=4.5 Version S-Z: m=4, \+ k=4.5" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 72 " A. Choose the forcing angular frequency w to be 3 times larger th an" }}{PARA 0 "" 0 "" {TEXT 23 72 " the natural angular frequen cy 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 ord er" }}{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, o ne of period" }}{PARA 0 "" 0 "" {TEXT 23 70 " 2Pi/w and the othe r of period 2Pi/w0. Display the exact period," }}{PARA 0 "" 0 "" {TEXT 23 67 " as calculated from 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. Su ggest a value for the forcing frequency w so that the" }}{PARA 0 "" 0 "" {TEXT 23 67 " oscillations exhibit resonance. Show resonant \+ behavior on a" }}{PARA 0 "" 0 "" {TEXT 23 51 " graph. Check agai nst Figure 5.6.4, page 352." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#2.5-A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#2.5-B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#2.5-C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 23 34 "2.6. 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 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 55 " Vers ion A-E: m=1, k=30 Version F-K: m=2, k=36" }}{PARA 0 "" 0 "" {TEXT 23 55 " Version L-R: m=3, k=45 Version S-Z: m=4, k=55" }}{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 function C(w) [page 357] for these three equations," }}{PARA 0 "" 0 "" {TEXT 23 71 " the n plot for w=0 to w=20 the three amplitude graphs on a single" }} {PARA 0 "" 0 "" {TEXT 23 72 " set of axes. Compare against Figur e 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, pri nt the values w*, C* where" }}{PARA 0 "" 0 "" {TEXT 23 70 " C*=C (w*)=max \{C(w) : 0 <= w <= 20\}. The three data pairs should" }} {PARA 0 "" 0 "" {TEXT 23 70 " show that C* becomes larger as c t ends to zero. SAVE YOUR MAPLE" }}{PARA 0 "" 0 "" {TEXT 23 22 " F ILE FREQUENTLY" }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 69 " Maple Hint: Use Maple's mouse interface on the graphi c of Part C." }}{PARA 0 "" 0 "" {TEXT 23 69 " Specifically, click o n a possible maximum (horizontal tangent) in" }}{PARA 0 "" 0 "" {TEXT 23 66 " the graph to display the values w*, C* on the screen. Copy \+ the" }}{PARA 0 "" 0 "" {TEXT 23 20 " values on paper." }}{PARA 0 " " 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 30 "EXAMPLE(Beware! Wr ong 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 51 "#2.6-A Plot C(w), three graphics on one set of axes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "#2.6-B Table of six data values for w*, C*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "75 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }