Question

Seems long but it's not too bad.
the book used in class is:
Advanced Engineering Mathematics by Erwin Kreyszig
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Thank you

Problem 3 (Frequency Response Testing) A common application for Fourier in mechanical design relates to the concept of resonance that we discussed earlier in the semester. Recall that resonance occurs when you drive a mechanical or electrical system at or near its natural frequency, whích then causes large amplitude oscillations. For relatively simple systems like the mass spring damper, the natural frequencies can be determined from the governing differential equation. But for complex mechanical parts like the one below, computing the natural frequency is not easy: For these situations, a modal analysis if often performed to determine the natural frequencies. There are many ways to perform a modal analysis, but one way is to strike the system with an impulsive forcing and observe the mechanical vibrations. A Fourier Series expansion is then performed on the response, and the resulting coefficients are presented as a power spectrum shown below: Power Spectrum for Frequency Response Test 30 25 20 15 10 0 2 46 8 10 12 14 16 18 20 Frequency (Hz) Each frequency on the x-axis is known as a mode of the response. In this problem, we are going to go backwards frop our usual Fourier methodology. We are going to take the power spectrum (i.e. Fourier Series coefficients and their respective frequencies) and reconstruct the time-domain mechanical response. Answer the following questions: a. From the power spectrum above, identify the three modes with the largest b. Using the three modes you identified in part (a), reconstruct the mechanical response. What are the amplitudes of these frequencies? vibrations to the modal test, f (t). You may assume that the power spectrum given above is for the odd coefficients, bn. f(t) b sin(ot)+ b sin(o2t) + bsin(w t) c. Plot your f(t) from o to 5s

Answer 1

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