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Problem 4 (Resonance) Search for —Tacoma Narrows bridge collapse— on YouTube, and watch any of the top results. This bridge collapsed due to a phenomenon called resonance, in which a system is driven at its natural frequency, and the amplitude of the oscillation becomes extremely large. To use an everyday example, think of pushing a child on a swing every time she comes back to you, so that the child goes higher and higher. Clearly, resonance can be extremely dangerous, and engineers design structures to avoid resonance. In this problem, you will simulate the simple harmonic oscillator being driven at its natural frequency by solving the following initial-value problem: x? + ?²x = f? sin (?t), x(0) = 0, ?(0) = 0 (3) (a) (15 points) Using the definition of the Laplace transform, show that ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (4) Hint: This is very similar to finding the Laplace transforms of cos (?t) and sin (?t). (b) (5 points) Show that ? [(sin(?t) - ?t cos(?t)) / 2?³] = 1 / (s² + ?²)² (5) Hint: You can avoid using the definition of the Laplace transform by using the linearity of the Laplace transform and the following transforms that we have already derived: ? [sin(?t)] = ? / (s² + ?²), ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (6) (c) (15 points) Using Laplace transforms, solve (3) for x(t). Hint: You will use (5). (d) (5 points) Plot x(t) versus t for the following parameters: ? = 1 rad/s, f? = 1 m/s². Comment on your results. Do you observe resonance?

          Problem 4 (Resonance) Search for —Tacoma Narrows bridge collapse— on YouTube, and watch any of the top results. This bridge collapsed due to a phenomenon called resonance, in which a system is driven at its natural frequency, and the amplitude of the oscillation becomes extremely large. To use an everyday example, think of pushing a child on a swing every time she comes back to you, so that the child goes higher and higher. Clearly, resonance can be extremely dangerous, and engineers design structures to avoid resonance. In this problem, you will simulate the simple harmonic oscillator being driven at its natural frequency by solving the following initial-value problem: x? + ?²x = f? sin (?t), x(0) = 0, ?(0) = 0 (3) (a) (15 points) Using the definition of the Laplace transform, show that ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (4) Hint: This is very similar to finding the Laplace transforms of cos (?t) and sin (?t). (b) (5 points) Show that ? [(sin(?t) - ?t cos(?t)) / 2?³] = 1 / (s² + ?²)² (5) Hint: You can avoid using the definition of the Laplace transform by using the linearity of the Laplace transform and the following transforms that we have already derived: ? [sin(?t)] = ? / (s² + ?²), ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (6) (c) (15 points) Using Laplace transforms, solve (3) for x(t). Hint: You will use (5). (d) (5 points) Plot x(t) versus t for the following parameters: ? = 1 rad/s, f? = 1 m/s². Comment on your results. Do you observe resonance?

Problem 4 (Resonance) Search for —Tacoma Narrows bridge collapse— on YouTube, and watch any of the top results. This bridge collapsed due to a phenomenon called resonance, in which a system is driven at its natural frequency, and the amplitude of the oscillation becomes extremely large. To use an everyday example, think of pushing a child on a swing every time she comes back to you, so that the child goes higher and higher. Clearly, resonance can be extremely dangerous, and engineers design structures to avoid resonance. In this problem, you will simulate the simple harmonic oscillator being driven at its natural frequency by solving the following initial-value problem: x? + ?²x = f? sin (?t), x(0) = 0, ?(0) = 0 (3) (a) (15 points) Using the definition of the Laplace transform, show that ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (4) Hint: This is very similar to finding the Laplace transforms of cos (?t) and sin (?t). (b) (5 points) Show that ? [(sin(?t) - ?t cos(?t)) / 2?³] = 1 / (s² + ?²)² (5) Hint: You can avoid using the definition of the Laplace transform by using the linearity of the Laplace transform and the following transforms that we have already derived: ? [sin(?t)] = ? / (s² + ?²), ? [t cos(?t)] = (s² - ?²) / (s² + ?²)² (6) (c) (15 points) Using Laplace transforms, solve (3) for x(t). Hint: You will use (5). (d) (5 points) Plot x(t) versus t for the following parameters: ? = 1 rad/s, f? = 1 m/s². Comment on your results. Do you observe resonance?

Added by Mariano B.

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