# Math 2280 Maple Project 6, S2018. Tacoma Narrows. # # NAME _______________________ CLASSTIME ________ # # There are two (2) problems in this project. Please answer the questions # A, B, C , ... associated with each problem. The original worksheet # "2250mapleL6-narrows-S2018.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 S2018 revision of David Eyre's project # (original was year 2000). # __________L6.1. NONLINEAR MCKENNA MODELS # __________L6.2. MCKENNA NON-HOOKES LAW CABLE MODEL. # # L6-1. PROBLEM (NONLINEAR MCKENNA MODELS) # # There are three (3) parts L6-1A, L6-1B, L6-1C 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. # # Mouse copy into a maple worksheet. 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'); # # # 6-1A. 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. # # 6-1B. 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. # # 6-1C. 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? # # L6-2. PROBLEM ( MCKENNA'S NON-HOOKE'S LAW CABLE MODEL) # # There are three (3) parts L6-2A, L6-2B, L6-2C 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 for mouse copy. # 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. # # L6-2A. TORSIONAL OSCILLATION PLOT. Get the sample code above to produce # the plot of x(t) [that's what scene=[t,x] means]. # # L6-2B. 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. # # L6-2C. 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.