# 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* >