Please show all work for Exercise 2 please! The general solution to Equation 3, unless w = √ and B = 0, isX(t) = G₁e¬Bt/(2M) cos (t√/1² – (B/(2M)}²)+ C₂e-B²/(2M) sin t√√23 - (B/(2M))²)NW(5)Bw- cos (wt -atan (M(W² 525)))/M² (S2² - w²)² + B²w²whereand atan(x) = arctan(x), the inverse tangent of x. It is usefulto have a measure of how a certain system (for this lab, the bridge), amplifies orreduces forces (resulting from walking humans) applied to it. The ratio of theamplitude of the input (i.e., the right hand side of the equation) to the long-term amplitude of the output (i.e., the solution to the differential equation) asa function of input frequency is called gain (we explore this in class in §4.6 ofBrannan & Boyce). The gain function for Equation 3 is1(6)G(w) =./M² (S2²3 - w²)² + B²w²There are a number of important observations to make about G(w): (1) be-cause it's a ratio of output to input amplitudes, it doesn't depend on the inputamplitude (for our bridge, the number of people or their mass); (2) it gets largeras the damping B decreases; and (3) in general, it gets larger as the forcingfrequency w approaches the value 2o-though the maximum amplitude will notoccur exactly at w = No. (Be sure you can see how these are reflected in theexpression above.)Exercise 2: The primary utility of the gain function is to tell us how thesolution to an oscillator responds to a forcing term. (1) We said above that G(w) does not depend on the number of people ortheir mass, and that it increases with smaller B; why are these state-ments true?(2) Which terms in Equation 5 persist for large times? For large times, whathappens to the dependence of the solution on the initial conditions?(3) What is the amplitude of the equilibrium oscillation in Equation 5? Howdoes the gain function relate the amplitude of the equilibrium oscillationto the amplitude of a forcing term in the form of Equation 3? Inparticular, what is the maximum displacement of the bridge given again function G(w), a walking frequency w, a number of people N, andthe lateral force per person W?

Answered: Image /qna-images/answer/d2dfe426-248a-49e8-9c65-f92214e64774.jpg

We said above that G(w) does not depend on the number of people or their mass, and that it increases with smaller B; why are these state- ments true? Which terms in Equation 5 persist for large times? For large times, what happens to the dependence of the solution on the initial conditions? What is the amplitude of the equilibrium oscillation in Equation 5? How does the gain function relate the amplitude of the equilibrium oscillation to the amplitude of a forcing term in the form of Equation 3? In particular, what is the maximum displacement of the bridge given a gain function G(w), a walking frequency w, a number of people N, and the lateral force per person W?

We said above that G(w) does not depend on the number of people or their mass, and that it increases with smaller B; why are these state- ments true? Which terms in Equation 5 persist for large times? For large times, what happens to the dependence of the solution on the initial conditions? What is the amplitude of the equilibrium oscillation in Equation 5? How does the gain function relate the amplitude of the equilibrium oscillation to the amplitude of a forcing term in the form of Equation 3? In particular, what is the maximum displacement of the bridge given a gain function G(w), a walking frequency w, a number of people N, and the lateral force per person W?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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