# Math 2250 Maple Lab 6, August 2007. Mechanical Oscillations.
# 
# NAME _______________________  CLASSTIME ____  VERSION A-E, F-K, L-R, S-Z
# 
# Circle the version - see problem L6.1. There are three (3) problems in
# this project. Please answer the questions A, B, C , ... associated with
# each problem. The original worksheet "2250mapleL6-F2007.mws" is a
# template for the solution; you must fill in the code and all comments.
# Sample code can be copied with the mouse. Use pencil freely to annotate
# the worksheet and to clarify the code and figures.
# 
# The problem headers for the Fall 2007 version of Mechanical
# Oscillations maple lab 6.
# 
# __________L6.1. UNDER-DAMPED FREE OSCILLATIONS.
# __________L6.2. UNDAMPED FORCED OSCILLATIONS.
# __________L6.3. PRACTICAL RESONANCE.
> 
# L6.1. PROBLEM (UNDER-DAMPED FREE OSCILLATIONS)
# 
# FREE OSCILLATIONS. Consider the problem of free linear oscillations
# 
#           m x'' + c x' + k x=0,
#           x(0)=0,  x'(0)=1.
#                                                   
# Here, m, c and k are non-negative constants. The under-damped case is
# studied here, c^2 < 4km, as on page 327 in E&P. Depending on the first
# letter of your last name, assume:
# 
# Version A-E: m=1, c=5             Version F-K: m=2, c=4
# Version L-R: m=3, c=4             Version S-Z: m=4, c=5
# 
#    A. Display a Hooke's constant k > 0 so that the solution x(t) is
#       under-damped. Check that x(t)=0 for infinitely many t>0.
#       Display the exact solution x(t) obtained by maple methods
#       as in the example below.
# 
#    B. Plot the exact symbolic solution x(t) on a suitable t-interval.
#       Check the graphic against Figure 5.4.9 page 328 of E&P.
# 
#    C. Estimate from the graph the decimal value of the pseudo-period.
#       Display the graphical estimate and also the exact pseudo-period
#       2Pi/w, where w is the natural frequency of the trigonometric term
#       in the solution x(t) found in item 2.4.A.
# 
> # EXAMPLE(Wrong parameters! Change it!)             
> # Use semicolons to see what you have done.
> # Define the differential equation
> # de:=3*diff(x(t),t,t)+1.5*diff(x(t),t)+4*x(t)=0:   
> # Solve the characteristic equation.
> # solve(3*r^2+1.5*r+4=0,r);                         
> # Define the initial conditions
> # ic:=x(0)=0,D(x)(0)= 1:                            
> # Symbolically solve for x(t)
> # p:=dsolve({de,ic},x(t),method=laplace):           
> # Capture the dsolve symbolic solution as a function X(t)
> # X:=unapply(rhs(p),t):                             
> # Plot the solution
> # plot(X(t),t=0..5);                                
> 
# Maple tip: Click with the mouse on the graphic to print the cursor
# location (left upper corner of the maple window). The coordinates
# printed are of the form (x,y). From this coordinate information, a
# simple subtraction estimates the period.
> 
> #L6.1-A Define  k, then solve.
> # under-damped means mr^2+cr+k=0 has two conjugate complex roots.
> #L6.1-B Plot.
> #L6.1-C Pseudo-period calculations.
> 
# L6.2. PROBLEM (UNDAMPED FORCED OSCILLATIONS )
# 
# FORCED LINEAR OSCILLATIONS.  Consider the undamped (c=0) forced
# problem
# 
#      mx'' +  k x = 5 cos(wt),
#      x(0)=0,  x'(0)=0,
# 
# where m, k and w are non-negative constants. Depending on the first
# letter of your last name, assume:
# 
# Version A-E: m=1, k=3.5           Version F-K: m=2, k=2.5
# Version L-R: m=3, k=4.5            Version S-Z: m=4, k=4.5
# 
#    A.  Choose the forcing angular frequency w to be 3 times larger than
#        the natural angular frequency w0, w0^2=k/m. Solve for x(t) using
#        dsolve(). Plot the solution x(t) on a suitable interval in order
#        to show the global behavior of the solution x(t). See Figure
#        5.6.2, page 350.
# 
#    B. The solution x(t) is the sum of two functions, one of period
#       2Pi/w and the other of period 2Pi/w0. Display the exact period,
#       as calculated from the solution formula for x(t) -- see page
#       350 for details.
# 
#    C. Suggest a value for the forcing frequency w so that the
#       oscillations exhibit resonance.  Show resonant behavior on a
#       graph. Check against Figure 5.6.4, page 352.
# 
> 
> #L6.2-A
> #L6.2-B
> #L6.2-C
> 
# L6.3. PROBLEM (PRACTICAL RESONANCE)
#   Consider the damped forced problem
# 
#      mx'' + c x' + k x = 5 cos(w t),
#      x(0)=0,  x'(0)=0.
# 
# Depending on the first letter of your last name, assume:
# 
# Version A-E: m=1, k=30          Version F-K: m=2, k=36
# Version L-R: m=3, k=45          Version S-Z: m=4, k=55
# 
#    A. Consider the damping constants c=2, c=1 and c=1/2.  Compute the
#       amplitude function C(w) [page 357] for these three equations,
#       then plot for w=0 to w=20 the three amplitude graphs on a single
#       set of axes. Compare against Figure 5.6.9 page 357 of E&P 
#       (it has one curve, yours has 3 curves).
# 
#    B. For each case c=2, c=1, c=1/2, print the values w*, C* where
#       C*=C(w*)=max {C(w) : 0 <= w <= 20}. The three data pairs should
#       show that C* becomes larger as c tends to zero. SAVE YOUR MAPLE
#       FILE FREQUENTLY
# 
#    Maple Hint: Use Maple's mouse interface on the graphic of Part C.
#    Specifically, click on a possible maximum (horizontal tangent) in
#    the graph to display the values w*, C* on the screen. Copy the
#    values into your maple worksheet report.
# 
> #EXAMPLE(Beware! Wrong values!)
> #F:=15: m:=1: k:=25: c:='c': w:='w':
> #C:=(w,c)->F/sqrt((k-m*w*w)^2+(c*w)^2):
> #plot({C(w,4),C(w,3),C(w,2)},w=0..15,color=black);
> 
> #L6.3-A Plot C(w), three graphics on one set of axes
> #L6.3-B Table of six data values for w*, C*
>