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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 23 64 "Math 2250 Maple Lab 2b,  N
ovember 2005. Mechanical Oscillations." }}{PARA 0 "" 0 "" {TEXT 23 0 "
" }}{PARA 0 "" 0 "" {TEXT 23 72 "NAME _______________________  CLASSTI
ME ____  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. Th
ere are three (3) problems" }}{PARA 0 "" 0 "" {TEXT 23 58 "in this pro
ject. 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-F2005.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 2005 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 -1 115 " Version A-E: m=1, c=5             Versi
on 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 tha
t the solution x(t) is" }}{PARA 0 "" 0 "" {TEXT 23 63 "       under-da
mped. 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 328 of E&P." }}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 
"" {TEXT 23 70 "    C. Estimate from the graph the decimal value of th
e pseudo-period." }}{PARA 0 "" 0 "" {TEXT 23 70 "       Display the gr
aphical estimate and also the exact pseudo-period" }}{PARA 0 "" 0 "" 
{TEXT 23 72 "       2Pi/w, where w is the natural frequency of the tri
gonometric term" }}{PARA 0 "" 0 "" {TEXT 23 48 "       in the solution
 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 i
t!)             # 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 characte
ristic 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 dsolve sy
mbolic" }}{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 the 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 infor
mation, 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 
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 conjugate co
mplex 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-period c
alculations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT 23 44 "2.5. PROBLEM (UNDAMPED FORCED OSCILLATIONS )" }
}{PARA 0 "" 0 "" {TEXT 23 0 "" }}{PARA 0 "" 0 "" {TEXT 23 71 "FORCED L
INEAR 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 firs
t" }}{PARA 0 "" 0 "" {TEXT 23 33 "letter of your last name, assume:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 23 118 " Versio
n 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 angu
lar frequency 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 solu
tion 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 Figu
re" }}{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 solu
tion 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 th
e exact period," }}{PARA 0 "" 0 "" {TEXT 23 67 "       as calculated f
rom 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 \+
frequency w so that the" }}{PARA 0 "" 0 "" {TEXT 23 67 "       oscilla
tions 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 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 "   Co
nsider 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          Version 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 function C(w) [page 357
] for these three equations," }}{PARA 0 "" 0 "" {TEXT 23 71 "       th
en 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 "0 0 0" 54 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }