# Math 2250 Maple Project 7, F2008. Tacoma Narrows. # # NAME _______________________ CLASSTIME ________ VERSION A-K or L-Z # # Circle the version - see problem L7.1. There are three (3) problems in # this project. Please answer the questions A, B, C , ... associated # with # each problem. The original worksheet "2250mapleL7-F2008.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 F2008 revision of David Eyre's project # (original was year 2000). # __________L7.1. NONLINEAR MCKENNA MODELS # __________L7.2. MCKENNA NON-HOOKES LAW CABLE MODEL. > # L7.1. PROBLEM (NONLINEAR MCKENNA MODELS) # # There are three (3) parts L7.1A to L7.1C to complete. Mostly, this is # mouse copying. Retyping the maple code by hand is not recommended. # # NONLINEAR TORSIONAL MODEL WITH GEOMETRY INCLUDED. # Consider the nonlinear, forced, damped oscillator equation for # torsional # motion, with bridge geometry included, # # x'' + 0.05 x' + 2.4 sin(x)cos(x) = 0.06 cos (12 t/10) , # x(0) = x0, x'(0) = v0 # # and its corresponding linearized equation # # x'' + 0.05 x' + 2.4 x = 0.06 cos (12 t/10) , # x(0) = x0, x'(0) = v0. # # The spring-mass system parameters are m=1, c = 0.05, k = 2.4, w = 1.2 # , # F = 0.06. Maple code used to solve and plot the solutions appears # below. # > # # Use "copy as maple text" for maple 6+. > # x0:=0: a:=200: b:=300: # For part A. Change it for part B! > # v0:=0: m:=1: F := 0.06: w := 1.2: m:=1: c:= 0.05: k:= 2.4: > # with(DEtools): opts:=stepsize=0.1: > # deLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) + k*x(t) = > F*cos(w*t): > # IClinear:=[[x(0)=x0,D(x)(0)=v0]]: > # DEplot(deLinear,x(t),t=a..b,IClinear,opts,title='Linear'); > # deNonLinear:= m*diff(x(t),t,t) + c*diff(x(t),t) + > # k*sin(x(t))*cos(x(t)) = F*cos(w*t): > # ICnonlinear:=[[x(0)=x0,D(x)(0)=v0]]: > # > DEplot(deNonLinear,x(t),t=a..b,ICnonlinear,opts,title='NonLinear'); > # 7.1A. Let x0=0, v0=0. Plot the solutions of the linear and # nonlinear equations from t=200 to t=300. These plots # represent # the steady state solutions of the two equations. # # 7.1B. Let x0=1.2, v0=0. Plot the solutions of the linear and # nonlinear equations from t=220 to t=320. These plots # represent # the steady state solutions of the two equation, with new # starting value x0=1.2. [You must modify line 1 of the maple # code!] # # The two linear plots in A and B have to be identical to the # plot of xss(t). The reason is the superposition formula (see # E&P) x(t)=xh(t)+xss(t), even though the homogeneous solution # xh(t) is different for the two plots. This is because xh(t) # has limit zero at t=infinity. # # 7.1C. Determine the ratio of the apparent amplitudes (a number > 1) # for the nonlinear plots in A and B. Do "large sustained # oscillations" appear in the plot of the nonlinear # steady-state? > > #L7.1-A > #L7.1-B > #L7.1-C > # L7.2. PROBLEM ( MCKENNA'S NON-HOOKE'S LAW CABLE MODEL) # # There are three (3) parts L7.2A to L7.2C to complete. Mostly, this is # mouse copying. Retyping the maple code by hand is not recommended. # # The model of McKenna studies the bridge with a nonlinear, forced, # damped # oscillator equation for torsional motion that accounts for the # non-Hooke's law cables coupled to the equations for vertical motion. # The # equations in this case couple the torsional motion with the vertical # motion. The equations are: # # x'' + c x' - k G(x,y) = F sin wt, x(0) = x0, x'(0) = x1, # y'' + c y' + (k/3) H(x,y) = g , y(0) = y0, y'(0) = y1, # # where x(t) is the torsional motion and y(t) is the vertical motion. # The # functions G(x,y) and H(x,y) are the models of the force generated by # the # cable when it is contracted and stretched. Below is sample code for # writing the differential equations and for plotting the solutions. It # is # ready to copy with the mouse. > # > #with(DEtools): > #w := 1.3: F := 0.05: f(t) := F*sin(w*t): > #c := 0.01: k1 := 0.2: k2 := 0.4: g := 9.8: L := 6: > #STEP:=x->piecewise(x<0,0,1): > #fp(t) := y(t)+(L*sin(x(t))): > #fm(t) := y(t)-(L*sin(x(t))): > #Sm(t) := STEP(fm(t))*fm(t): > #Sp(t) := STEP(fp(t))*fp(t): > #sys := { > # diff(x(t),t,t) + c*diff(x(t),t) - > k1*cos(x(t))*(Sm(t)-Sp(t))=f(t), > # diff(y(t),t,t) + c*diff(y(t),t) + k2*(Sm(t)+Sp(t)) = g}: > #ic := [[x(0)=0, D(x)(0)=0, y(0)=27.25, D(y)(0)=0]]: > #vars:=[x(t),y(t)]: > #opts:=stepsize=0.1: > #DEplot(sys,vars,t=0..300,ic,opts,scene=[t,x]); > # The amazing thing that happens in this simulation is that the large # vertical oscillations take all the tension out of the springs and they # induce large torsional oscillations. # # L7.2A. TORSIONAL OSCILLATION PLOT. Get the sample code above to # produce # the plot of x(t) [that's what scene=[t,x] means]. # # L7.2B. ROADWAY TILT ANGLE. Estimate the number of degrees the roadway # tilts based on the plot. Recall that x in the plot is reported # in radians. Comment on the agreement of this result with # historical data and the video evidence in the film clip. # # Tip: Average the five largest amplitudes in the plot to find an # average maximum amplitude for t=0 to t=300. Convert to degrees # using Pi radians = 180 degrees. The film clip shows roadway # maximum tilt of 30 to 45 degrees, approximately. # # L7.2C. VERTICAL OSCILLATION PLOT. Modify the DEplot code to # scene=[t,y] # and plot the oscillation y(t) on t=0 to t=300. The plot is # supposed to show 30-foot vertical oscillations along the # roadway # that dampen to 7-foot vertical oscillations after 300 seconds. # # The agreement between these oscillation results and the # historical data for Tacoma Narrows, especially the visual data # present in the film clip of the bridge disaster, should be # clear # from the plots. This is your only answer check for the plot # results. > > #L7.2-A Torsional plot t-versus-x > #L7.2-b Roadway tilt angle estimate in degrees + comments. > #L7.2-C Vertical plot t-versus-y. >