MEE-3011: Homework #4 Solution of coupled systems of di¤erential equations This is due Monday December 11, the last day of class . Name (print): TUID: i h g f e d c b a use the above values (a,b,c, etc) when a quantified (numerical) answer is needed. #— — — — — — — — — — — — the …elds below will be completed by Dr. Peridier— — — — — — — — — — — — — — # 1. Problem 1 score [5 pt] /comments: ... ... ... ... ... ... 2. Problem 2 score [5 pt] /comments ... ... ... ... ... ... 3. Problem 3 score [10 pt] /comments. ... ... ... ... ... MEE-3011: Homework #1 Assemble submission for this problem in the following order: (1) (2) (3) Problem #1 this page on top, followed by your handwritten derivations (in order) your matlab code and plots Name (print): Write your solutions on this page, and attach additional pages to explain your answer, including your matlab code and plots. [1] The intention of this problem is to con…rm that you can compute the motion of this commercial scale using four of the methods we have seen in this course. At time t = 0 an object of mass m [kg] is placed on the scale pan. The scale has spring constant k and damping coe¢ cient b: (The governing equation has "mg" on the right-hand side because x = 0 accounts for the static displacement of the pan but not of the object to be wieghed.) Use these values: symbol m k b units [kg] [N/m] [N/m sec] YOUR VALUE max([b,c,d])*10 round(mean([a,b,c,d]*1000) k=20 ... and here is the governing equation: m d2 x dx +b + kx = mg (for t dt2 dt XXXXX 0); x(0) = dx dt (0) =0 Solve this problem 4 ways: a. Derive the solution by hand and use this solution to estimate how long (t99:5% ) it takes the scale to come within 1/2 % of its resting position. Report this time here: t99:5% =________________ b. Compute and plot (for 0 t t99:5% ) x(t) in [cm] using the 2nd -order recipe. Provide hand-written derivation as needed. t99:5% ) x(t) in [cm] using the generalization of the 1st -order recipe that calls uses expm(). ! ! ! ! dZ x You will need to set up the problem as = [A] Z + Rhs(t); where Z = : dx dt dt c. Compute and plot (for 0 t What are the eigenvalues of [A]; and does this make sense to you? Provide hand-written derivation/discussion. d. Compute and plot (for 0 t t99:5% ) x(t) in [cm] using ode45(). Provide hand-written derivation as needed. MEE-3011: Homework #4 Problem #2 Assemble submission for this problem in the following order: (1) (2) (3) this page on top, followed by your handwritten derivations (in order) your matlab code and plots Name (print): Attach additional pages as needed to explain your code, and also your matlab code and plots. [2] It turns out that, for spring-loaded scales, dry friction is a complicating factor. Background. Again, say that at time t = 0 an object of mass m [kg] is placed on the scale pan. The scale again has a spring constant k but instead the motion is damped out by frictional e¤ects in the mechanism. The maximum magnitude of the dry dx friction force is Ff sign( ): dt However, this maximum friction value is exerted only if the net restoring force of the spring exceeds the friction drag. If the spring force is smaller thav friction force (i.e. if (k jx mg k j) < Ff ) the two forces (restoring force and friction) will be in equilibrium and the net force on the mass is zero.. Say at time t = 0 we place a mass m on the scale The governing equation is: d2 x m 2 = dt ( k (x mg k ) Ff sign( 0; dx ); dt (if x mg k > Ff k (otherwise) ) ) ; x(0) = 0; dx (0) = 0; dt The above equation is nonlinearly dependent on the dependent variable x(t), so there is no closed-form solution. Furthermore, since the equation is nonlinear we cannot use our 1st or 2nd order recipes. Instead, we must integrate the above nonlinear 2nd -order ODE as two coupled 1st -order di¤erential equations using ode45() in matlab.. Try integrating the above system to steady state and plot x(t) versus t . Remark on any features of the plot that distinguishes it from conventional damped oscillation (such as problem 1). Note especially where the mass comes to rest. Use these values: symbol m k Ff units [kg] [N/m] [N] (dimensionless) friction coe¢ cient YOUR VALUE-please enter it here max([b,c,d])*10 round(mean([a,b,c,d]*1000) mg mean([a; b; c; d; e]) sum([a; b; c; d; e; f; g; h]) XXXXX MEE-3011: Homework #4 Assemble submission for this problem in the following order: (1) (2) (3) (4) Problem #3 this page on top, followed by your handwritten derivations (in order) your matlab code your plots Name (print): We are going to consider the impact on a vehicle’s traction using proposed "inerter" device discussed in class. [3] Consider the using a quarter-car model below. The schematics below use the following notation mc mt mi ks kt b MI chassis mass tire mass inerter-casing mass shock-absorber spring constant tire spring constant system damping inerter inertance value [kg] [kg] [kg] [kg/sec2 ] [kg/sec2 ] [kg/sec] [kg] 10(a+b+c+d) [kg] 12 [kg] 2 [kg] 2(e+f+g) [kN/m] 100 [kN/m] =1% [sec] (proportional damping of [K] matrix assumed) 10 max(a,b,c,d) [kg] (a) Use the Lagrange method to obtain the coupled governing equations assuming no damping for each case below. i. No inerter (you should get two coupled di¤erential equations in C(t) and T (t) ) ii. The inerter in parallel with the shock as shown above (two coupled di¤erential equations in C(t) and T (t) ) iii. The inerter in series with the shock as shown above (three coupled equations in C(t); I(t) and T (t) ) (b) Set each case up in matrix vector form: ! ! ! [M] Z• (t) + [B] Z_ (t) + [K] Z (t) ! = rhs y(t) ! ! Here, Z (t) = [C(t) T (t)]0 for cases (i) and (ii), and Z (t) = [C(t) I(t) T (t)]0 for case (iii). Note also that [B] = [K] so that we will be able to use our "proportional damping" modal-coordinate eigenvalue method. (c) The goal of this calculation is to the plot variation in the tire’s e¤ective traction during and after it goes over a speed bump. Assume the speed bump is 1/4 meter high, that it takes 1/2 second to traverse the speed bump, and that the tire encounters the speed bump at t = 0; i.e. y(t) = 1 4 [m] 0 sin(2 t) for 0 [sec] for 12 [sec] t t 1 2 [sec] Use our eigenvalue/proportional damping method to plot the change in e¤ective traction , three seconds, where Ftraction =kt ( y(t) T (t) ) Ftraction ; for the …rst (d) Brie‡y discuss your three plots, and explain why you think the racing industry was interested in inerter technology.
