# Math 2250 Maple Lab 6, F2010. Mechanical Oscillations.
#
# NAME _______________________________________________ CLASSTIME ____
#
# There are three (3) problems in this project.
#
# Please answer the questions A, B, C , ... associated with
# each problem. The original worksheet "2250mapleL6-F2010.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 F2010 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. Assume m=4 and c=3.
#
# 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 above in item L6.1-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. Assume m=5 and k=3.5,
#
# 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.
#
# Assume m=4, k=41.
#
# 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. Look around
# the screen to see where maple printed the x,y-coordinates! 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*
>