Math Lab 9 23 March 2017 unid:

Transcription

1 Math Lab 9 23 March 2017 unid: Name: Instructions and due date: Due: 30 March 2017 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 (t) + cx (t) + kx(t) = F 0 cos(ωt), (1) where m = 100 kg, c = 0 N s/m, k = 16 N/m, and F 0 = 300 N. (In this case we can assume that the dashpot has been disconnected.) Assume the following initial conditions for x(t): x(0) = 0, x (0) = 0, (2) i.e., the spring starts at rest, subject to a steady, periodic forcing F 0 cos(ωt). (a) What is the natural angular frequency ω 0 of this system (1),(2)? (Hint: the associated homogeneous equation.) Consider (b) Assume that ω ω 0. Solve the initial value problem (1),(2) by first computing the homogeneous and particular solutions (x H (t) and x P (t), respectively). (c) Using your answer from part (b), write down the special case when ω = 1/2. Notice that it is a superposition of two sinusoidal functions, one with the natural angular frequency and one with the forcing angular frequency of.5. What is the period of the solution? (Hint: Find the period of each term, and then find the least common period, which will be a period for the sum. Plot your solution with 0 t 125, and attach the plot. What phenomenon does this solution show? (Hint: consult the text or the class notes for Friday.)

3 Math Lab 9 - Page 3 of 7 (d) Now let ω = ω 0. Again solve the the initial value problem (1),(2). Plot your solution with 0 t 125, and attach the plot. (Hint: This is case 2 of undetermined coefficients, as we previewed in class on Wednesday.) 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(ωt) is applied to the circuit. Using Kirchoff s laws, the following differential equation gives the charge q(t) of the capacitor. L q (t) + R q (t) + 1 C q(t) = V 0 cos(ωt) (3) Notice that the is the same mathematical differential equation as for forced mass-spring configurations, except that the letters have changed meanings. Take L = 0.25 V s A 1, R = 0.4 Ω, C 1 = 0.16 V C 1, V 0 = 2 V, and ω = 1 s 1. (4) Note: For each equation listed in (4), the left-hand side denotes a variable from (3), while the right-hand side denotes an S.I. unit (for example V refers to volts). This gives us the inhomogeneous differential equation: 0.25q (t) + 0.4q (t) q(t) = 2 cos(t). (5) (a) Find the solution q H (t) of the homogeneous equation corresponding to (5). (b) Use the method of undetermined coefficients to find a particular solution q P (t) to (5).

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. 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 Thus the natural frequency of horizontal back-and-fo disk is 2k/3m, which is 2/ times th k/m of a mass on a spring that is sliding without fric out sliding. It is interesting (and perhaps surprising) tha not depend on the radius of the disk. It could be eithe radius of one meter (but of the same mass). Example 5 Suppose that a car oscillates vertically as if it were a m 3. Suspension of a car spring (with constant k = N/m), attached to a c = 3000 N s/m). Suppose that this car with the da Excited for the new series The Grand Tour, you start watching along a washboard reruns of roadtop surface Gear, with an amplitude o and end up in a mathematical modeling frenzy. EveryoneL is= given 10 m (Fig. an identical 5.6.6). At what Mazda car speed will resonan Miata, which has a sprung mass of 960 kg (the mass of the main body of the car that sits Solution We think of the car a unicycle, as pictured in Fig atop the suspension). They have to drive the car through a sinusoidal washboard-like upward displacement of the mass m from its equilibr road surface with an amplitude of 20 cm and a wavelength 2π s of 10 m. James May, a.k.a y = a cos force of gravity, because it merely displaces the equilib Captain Slow, states confidently that the best thing L Surface to do is 9ofSection to drive very 5.4. slowly We write(30 thekph). equation of the road sur s Hammond and Jeremy Clarkson say to drive as fast as the car can go (200 kph). s = 0 y = a cos 2πs (a = 0.05 m, L = The only one who takes his job seriously on the show is The Stig. The Stig assumes L FIGURE The washboard the car oscillates vertically as if road itsurface were a sprung mass m = 960/4 kg = 240 kg on a single spring (with constant k = of Example 5. When the car is in motion, the spring is stretched by th N/m) attached to second a dashpot law, F = (with ma, gives constant c = 3000 N sec/m). mx = k(x y); k m Equilibrium position arson Learning Solu Not For Resale Or Distribution c FIGURE model of a car. x y k m y In motion The unicycle c y = 0 s that is, mx + kx = ky. If the velocity of the car is v, then s = vt in Eq. (15), s mx + kx = ka cos 2πvt L This is the differential equation that governs the vertic comparing it with Eq. (4), we see that we have forc frequency ω = 2πv/L. Resonance will occur when ω numerical data to find the speed of the car at resonance v = L k 2π m = (a) If the car s speed is given by s, determine the height y(t) of the washboard under 14 2π 800 the car s wheel is at every time t. that is, about 33.3 mi/h (using the conversion factor of (b) Let x(t) be the height of the sprung mass m, where x = 0 when the car is stationary and not going over any bumps. The spring is compressed/stretched when the relative position (x y) of the ground and the mass move from the neutral position (x y = 0), while the dashpot exerts a force proportional to the relative velocity (x y). Using The Stig s assumptions, write down a differential 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 coefficient c in the interval 0 c 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).)