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2. Laplace Transforms: Damped Oscillator A damped simple harmonic oscillator follows the differential equation where wo k/m is the undamped angular frequency of the oscillator, and < Ethe damping ratio. Suppose such system is perturbed with the initial condition I(t =0) =1 and it( =0) =0 (A) Take the Laplace transform and show that the transformed displacement X(s) can be written 25Wo t X(s) (B) Split the expression for X(s) into the sum of two fractions. the first with numerator (s- -Ewa) and the second with numerator (Ewo). Show that the first fraction corresponds to the solution or(t) (C) Find the inverse Laplace transform of the second fraction from part (B), and state the full solution I(()) for this problem. (D) What is the behaviour of the solution as we reduce the damping ratio to zero? In other words, what is the form of a(() in the limit that E 0? 1 3. Fourier Series: Approximating a Saw Tooth Function The teeth on typical handsaw for cutting wood have the shape shown in the figure below (from blackburntools.com) 60° Triangular saw file cross-section GO° 60° Gullet Point line Tooth Point 60° 60 Baseline Saw plate Handle Suppose we want to find a Fourier Series representation for such sawtooth profile. Firstly, we can idealize the profile of single tooth according to the following figure, where the lengths are in arbitrary units that can be scaled to different saw sizes. A 378 1/12 1/2 (A) Show that this setup approximately results in the desired angle: (B) Find the equations of the two straight line segments that make up the saw tooth 2 (C) The Fourier Sine series (see equation 2.2 in the notes) that represents the sawtooth profile can be written in the form f(x) "sin(2nzx) where b, Show that this is indeed the correct form for bre and find the correct values for two positive rational fractions A and B. In other words. solve the problem and write your answer in the form above. HINT: You will need to make use of the following integrals: 1/12 4n2%² cos(nm/6)-cos(nz) 2nm cos(nx/6) sin(nz/6) sin(nm) sin(2nzx)da 24nx 4n2%² (D) Generate plot of the Fourier representation of the saw tooth by taking taking the first 5 terms in the Fourier sine series you have just derived. You may use any software to achieve this, please include the plot and any code in your solution. HINT: Something like MATLAB would be ideal for this, but it even be done in a spreadsheet. To use spreadsheet try the following: (i) create column of x values from 0 to 0.5 in steps of 0.01 (ii) create column for n = 1 by copying down the formula (iii) create the other columns for n=2ton=5, you should be able to copy across the n column if you have properly set it up (iv) create column of y values by summing the n through 72 5 columns (v) plot versus y using scatterplot Here is the plot obtained using spreadsheet: 1.4 0.35 0.3 0.25 0.2 0.15 11 0.05 n 01 nz 13 04

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