Problem Set # 9 Math Methods Winter 2018 Due Date: Apr , 8.30am

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1 Problem Set # 9 Math Methods Winter 2018 Due Date: Apr , 8.30am The theme of this homework is solving ODEs. Remember, a good guess for a solution to an ODE is always Ae rt. Then figure out what values of r solve the ODE at ahnd. 1. Real and Imaginary Friends: In class, we tackled 2nd order ODE of the form ẍ + 2ζω o ẋ + ω 2 ox = 0; Note the 0 on the RHS means this equation is unforced. We saw that a guess of form x(t) = Ae rt, resulted in an underdamped (ζ < 1) solution of the form: x(t) = e ζωot ( A 1 e iω dt + A 2 e iω dt ) (1) where ω d = ω o 1 ζ 2 is the damped frequency of oscillation. Given the complex exponential form, it should come as no surprise that the constant A 1 and A 2 are complex numbers. In class, there was a hand-wavy (but correct!) claim that we can rewrite Eqn 1 in the following form, which more obviously reveals a damped, oscillation: where D 1 and D 2 are real numbers. x(t) = e ζωot (D 1 cos ω d t + D 2 sin ω d t) (2) (a) Show that this claim is in fact true provided that A 1 = A 2 i.e., A 1 = a 1 + ib 1 and A 2 = a 2 + ib 2 = a 1 ib 1 are complex conjugates. (b) Furthermore show that it must be true that D 1 = 2a 2 and D 2 = 2b 2. (c) Lastly, we know that D 1 and D 2 are solved by using initial conditions. Typically, these are given as: x(0) = x o and ẋ(0) = v o. Thus, show that D 1 = x o and D 2 = vo+ζωoxo ω d. 2. A real shock to the system Mountain Bikers. A daredevil mountain biker, who goes by the name Hotdogger, launches himself off a cliff that is 2 m high, as shown in the figure below. Then his friend, known in the mountain bike community as Smallfry, does the same. Hotdogger has a mass of 100 kg. Smallfry tips the scales at only 50 kg. All mountain bikes come equipped with a modern suspension system (spring plus adjustable damping). The system can be modeled as shown in Figure 4. The mass m represent the mass of the rider plus the bike (m bike = 10 kg). The spring is quite stiff and has spring constant k = 250 kn/m. The damper constant c can be set by the rider by adjusting a turn-screw on the bike. Let s assume both Hotdogger and Smallfry are riding the same bike with damping constant set to c = 1500 Ns/m. Finally, recall that the equation of motion for this system is given by: m z + cż + kz = 0 where we are taking z(t) = x(t) δx o to be the oscillations about the static equilibrium position (the spring pre-compressed by the mass by an amount δx o = mg/k). Note also that the total mass m = m rider + m bike. 1

2 Figure 1: Mountain biker hucks off a cliff. (a) Develop an expression for and compute the initial velocity v o of the rider just at the moment s/he begins to impact the ground. Hint: This is Intro Physics conservation of energy not fundamentally a math problem just yet. (b) Make a table for both Hotdogger and Smallfry that computes the following: ω o = k/m ζ = c 2mω o. (c) Determine if the mechanical response is overdamped, critically damped, or underdamped. (d) Make a detailed plot for each rider showing the ensuing vertical displacement z(t). Take t = 0 to be the instant just when the tire(s) begin to contact the ground (therefore, you have information for the initial displacement z(0) = δx o = mg/k). Plotting with a software package is recommended. (e) Compare and contrast the ensuing motion Hotdogger and Smallfry will experience. Annotating plots from 2dto highlight interesting/salient features of the motion is highly recommended. (f) A mountain biker wants to have a some cushion when landing a big jump, but doesn t want to oscillate too much. Given your results from part 2d, what adjustments, if any, should Hotdogger and Smallfry make to their suspension systems. Be sure to clearly explain your rationale and argue your case quantitatively. 3. Pedal Faster, I Hear Banjos Hotdogger and Smallfry are two superstar mountain bikers who are vying for a world championship time at the Red Bull games. They are still hucking off cliffs. And they are going to pedal their little hearts out over the entire race course going for the cup. Now we have a forced 2nd order ODE of the form: mẍ + cẋ + kx = F o cos ωt where the RHS term represents a forcing on the mountain bike suspension to the pumping of legs during pedaling; the rider pedals at a rate ω = 10 rad/s with an force of F o = 500 N. (a) Assume the riders are on a straight away with no cliffs, no bumps otherwise, such that the only oscillation in the system is due to the forcing term (i.e., x(0) = 0 and ẋ(0) = 0. 2

3 Determine the forced response of the system x(t) in this case. Hint: We ve already solved this problem before! Look back at the Batmobile problem from when we studied complex exponentials. (b) Later in the course, there s a monster 5 m cliff. The riders continue to pedal as before. Determine the total response of the mountain biker. Plot your solution using a software package, and annotate the parts of the plot where the unforced response dominates, and where the forced response dominates. 4. C u L8R: Oscillations in an RLC cirucit Consider the electrical circuit in Fig. 2. It consists of a resistor, inductor, and capacitor in series wiht a voltage source V s. This tidy little model represents every transmission line in the real world, including antennas. Electrical circuits can oscillate too! Want to receive a signal on the FM dial cruising up or down I-81? Want to send or receive a signal from outerspace? Or just your WiFi network? Yes, yes, and yes. So it is definitely worthwhile to study it a bit further here we go! Figure 2: RLC circuit. Parameters values are: R = 50Ω; C = 100 pf; L = 1µH. Image credit: The governing equation for this system is given by: L q + R q + 1 C q = V s(t) (3) where q is the charge on the capacitor. This equation derives from using the definition of current equals charge per unit time (i(t) = dq/dt) and applying KVL around a loop (sum of voltage gains and drops are equal). (a) First let s assume there is no forcing, such that V s (t) = 0. Given Eqn 3, make an educated guess for q(t) = Ae rt and show that your educated guess is a very good one by solving the characteristic equation. It would very well be useful to recast Eqn 3 in the canonical form before plunging ahead: q + 2ζω o q + ω 2 oq = 0 (b) Given the parameter values in the caption of Figure 2 is the system under-, over-, or critically damped? (c) Assume the initial conditions are such that: q(0)/c = V c (0) = 5 V, and q(0) = i c (0) = 0. Plot the resulting response of the system for the voltage across the capacitor V c (t). (d) What is the period of damped oscillation T d? (As a helpful hint: Recall the factor of 2π difference between frequency and angular frequency ω d, and of course period is 1 over the frequency.) What is the exponential decay time τ? Plot your solution over an appropriate time domain to visualize. 3

4 (e) Now let s turn on the voltage source V s (t) = V in cos ωt to force the system. Let s set the forcing frequency ω = ω o, and V in = 0.5 V. Use the same initial conditions are given previously. Find the total response of the system (forced + unforced). Carefully plot your solution to visualize the action of the unforced solution and the forced part of the solution. You can gain some intuition and deepen your understanding of this phenomenon by viewing the following animation: 5. The Vertical Pendulum: Oscillations or No? Consider the vertical pendulum arrangement shown in Fig. 3. It consists of a thin bar with mass M = 2 kg and length L = 1 m. Two spring (spring constant k = 50 N/m) are attached on both sides of the bar, both a distance a from the pivot point. Figure 3: Inverted pendulum. Is it stable? That depends! The equation of motion for this system is given by: J θ + ( 2ka 2 MgL/2 ) θ = 0 where J = 1 3 ML2 is the mass moment of inertia. The term in parentheses expresses the springs trying to push the pendulum back to vertical, while gravity tries to pull it down. (a) Write the equation of motion in the form: θ + λθ = 0. (b) Is λ always positive? For what values of a will will λ > 0? What is the critical value a crit which yields λ = 0? (c) Now, try out a guess for θ(t) = Ae rt. Write the characteristic equation which allows us to solve for the valid values of r. Be very careful to note whether r is real or imaginary; this has profound consequences for the solution! (d) Solve the differential equation for a = 0.75 m. Assume the initial conditions θ(0) = θ o and θ(0) = 0. Graph your solution for θ(t). Be sure to clearly label the exact value of the period of oscillation, T. (e) Now assume that a = 0.25 m. Is λ positive or negative? Solve the differential equation again from part a). Assume the same IC s as in part b. Graph your solution for θ(t). (f) Explain the result from parts b and c. Does the bar always oscillate about the vertical (equilibrium) position? Provide an intuitive explanation of your mathematical result i.e., why does the exact position of the spring-bar contacts matter? 4

5 6. Leaving on a Jet Plane We re leaving this problem set on a jet plane...or maybe just solving a model of one. The system shown in Figure 4(a), is a model of a strut that supports an airplane wing. (Or perhaps it is a good model of a shoulder joint with arm extended. Try flapping your wings! Start with your arm fully extended, raise it slightly, then try to steady it back at the horizontal position.) The mass m represents a small engine, the spring constant k represents the springiness of the wing/beam. You decide to add a damper c to control the nature of the vibrations encountered during flight. (a) System parameters: m = 100 kg, k = 100 N/m, a = 1/3 m ; l = 1 m. (b) Single strut supports airplane wing Figure 4: Vibrational model of airplane wing and the real deal. (a) The second order ODE that describes the motion of the system can be written as: J θ + ca 2 θ + ka 2 θ = 0 where J = ml 2 is the mass moment of inertia, and θ(t) measures the rotation of the wing. (b) Assume that c = 100 N-s/m. Is the system over-, under-, or critically damped? Plot θ(t) as a function of time. Assume initial conditions of θ(0) = 10 and θ(0) = π rad/sec. (c) Repeat part b, but this time using c = 2000 N-s/m. Plot θ(t) on the same axes you used above, so that you can easily compare and contrast the two systems. (d) Finally, repeat one last time using a value of c = 600 N-s/m. (e) Which of the three damper values you would choose for your airplane wing? Justify. 5