Math Lab 9 3 November 2016 unid:

Transcription

1 Math Lab 9 3 November 2016 unid: Name: Instructions and due date: Due: 10 November 2016 at the start of lab. If extra paper is necessary, please staple it at the end of the packet. For full credit: Show all of your work, and simplify your final answers. Work together! However, your work should be your own (not copied from a group member).

2 Math Lab 9 - Page 2 of 7 1. Mass spring system Consider a mass spring dashpot system mx 00 (t)+cx 0 (t)+kx(t) =F 0 cos(!t), (1) where m =100kg, c =0N s/m, k =16N/m, andf 0 =300N. (Inthiscasewecan assume that the dashpot has been disconnected.) Assume the following initial conditions for x(t): x(0) = 0, x 0 (0) = 0, (2) i.e., the spring starts at rest, subject to a steady, periodic forcing F 0 cos(!t). (a) What is the natural angular frequency! 0 of this system (1),(2)? (Hint: the associated homogeneous equation.) Consider (b) Assume that! 6=! 0.Solvetheinitialvalueproblem(1),(2) by first computing the homogeneous and particular solutions (x H (t) andx P (t), respectively). (c) Using your answer from part (b), write down the special case when! =1/2. What is the period of the solution? (Hint: Consider the identity 2 sin((a B)/2 sin((a + B)/2) = cos B cos A.) Plot your solution with 0 apple t apple 125, and attach the plot. What phenomenon does this solution show?

3 Math Lab 9 - Page 3 of 7 (d) Now let! =! 0. Again solve the the initial value problem (1),(2). Plotyoursolution with 0 apple t apple 125, and attach the plot. What phenomenon does this solution show?

4 Math Lab 9 - Page 4 of 7 2. RLC circuits Consider the following series RLC circuit: R V L C Suppose a periodic voltage source v(t) =V 0 cos(!t) is applied to the circuit. Using Kircho s laws, the following di erential equation gives the charge q(t) ofthecapacitor. Take L q 00 (t)+r q 0 (t)+ 1 C q(t) =V 0 cos(!t) (3) L =0.25 V s A 1, R =0.4, C 1 =0.16 V C 1, V 0 =2V, and! =1s 1. (4) Note: For each equation listed in (4), theleft-handsidedenotesavariablefrom(3), while the right-hand side denotes an S.I. unit (for example V refers to volts). This gives us the inhomogeneous di erential equation: 0.25q 00 (t)+0.4q 0 (t)+0.16q(t) =2cos(t). (5) (a) Find the solution q H (t) ofthehomogeneousequationcorrespondingto(5). (b) Use the method of undetermined coe (5). cients to find a particular solution q P (t) to

5 Math Lab 9 - Page 5 of 7. (c) Write down the general solution to this equation (5). Identify the steady periodic and transient parts of this general solution. (d) Solve the initial value problem consisting of (5), together with the the initial conditions q(0) = 0, q 0 (0) = 0. You may use technology to do this if you do so, please print and attach (staple) your work. (e) On a single plot, graph the IVP solution found in part (d) as well as the steady periodic solution identified in part (c). (Please print and attach your plot.) Choose a time interval so that you can clearly see the convergence of the IVP solution to the steady periodic solution.

6 Math Lab 9 - Page 6 of 7 3. Suspension of a car Excited for the new series The Grand Tour, you start watching reruns of Top Gear, and end up in a mathematical modeling frenzy. Everyone is given an identical Mazda Miata, which has a sprung mass of 960 kg (the mass of the main body of the car that sits atop the suspension). They have to drive the car through a sinusoidal washboard-like road surface with an amplitude of 20 cm and a wavelength of 10 m. James May, a.k.a Captain Slow, states confidently that the best thing to do is to drive very slowly (30 kph). Hammond and Jeremy Clarkson say to drive as fast as the car can go (200 kph). The only one who takes his job seriously on the show is The Stig. The Stig assumes the car oscillates vertically as if it were a sprung mass m =960/4 kg =240kg on a single spring (with constant k = N/m) attached to a dashpot (with constant c =3000N sec/m). m ibu m x k c k c y Equilibrium position FIGURE model of a car. y In motion The unicycle y = 0 s (a) If the car s speed is given by s, determinetheheighty(t) ofthewashboardunder the car s wheel is at every time t. (b) Let x(t) betheheightofthesprungmassm, wherex =0whenthecarisstationary and not going over any bumps. The spring is compressed/stretched when the relative position (x y) ofthegroundandthemassmovefromtheneutralposition (x y =0),whilethedashpotexertsaforceproportionaltotherelativevelocity (x y) 0. Using The Stig s assumptions, write down a di erential equation expressing the position of the sprung mass x(t). Express the equation in the form Lx = f(t).

7 Math Lab 9 - Page 7 of 7 (c) At what car speed will practical resonance occur? (d) If the dashpot is disconnected, at what car speed will actual resonance start to occur? (e) Which driver, James May, or Jeremy Clarkson, will experience the rougher ride, as defined as the amplitude of the particular solution x p (t)? Compute each amplitude and decide. (f) Modify the damping coe cient c in the interval 0 apple c apple so that Clarkson has the roughest ride possible (maximum amplitude). State the value of c that achieves the maximum amplitude, and also the max amplitude. (Hint: 1) Express f(t) from part (b) as a function of c (which is unknown); and 2) use the amplitude equation on page 384, taking F 0 to be the amplitude of your new f(t).)