ME 440 Intermediate Vibrations Th, January 29, 2009 Section 1.11 © Dan Negrut, 2009 ME440, UW-Madison

Before we get started… Last Time: Discussed about periodic functions Covered the Fourier Series Expansion Went through one example Today: Covering material out of 1.11 Fourier Series Expansion, Complex Representation Three more examples HW Assigned: 1.68 and 2.35 out of the book NOTE: last time meeting in this room. We’ll to 3126ME starting on Feb. 3 (next lecture) 2

A Note on HW Problem One of the problems assigned last time: Problem 1.66 Doesn’t explicitly indicate what the expression of f(t) is Only shows a plot of it. Please work with following expression for f(t): 3

New Topic: Frequency Spectrum Recall the concept of fundamental frequency: By inspection of the terms in Fourier expansion, you notice that as n increases, the harmonic functions display gradually increasing oscillation frequencies. Frequency Spectrum: The plot of a n and b n versus n It shows how the amplitude of the harmonics entering the Fourier expansion changes as n increases Recall the discussion we had: as n increases (and therefore the oscillation frequency increases), I expect the frequency spectrum to be approaching the zero axis. Why? 4

Frequency Spectrum (Cntd) Example of Frequency Spectrum 5

Frequency Spectrum (Cntd) Note that Frequency Spectrum can be also provided in the form X n &  n, instead of a n and b n. Use transformation to move back and forth between these two representations 6 Example 1 Example 2

Example 2. Determine the Fourier expansion of the periodic function below Plot its frequency spectrum 7

New Topic: Fourier Expansion using Complex Notation Two important identities: 8 Therefore, Fourier expansion becomes Recall that  =2  /  and rewrite as

Fourier Expansion using Complex Notation (Cntd) Introduce notation: 9 Fourier expansion becomes Revisit computation of c n

Fourier Expansion using Complex Notation (Cntd) Similarly, 10 In conclusions, all the coefficient c n are computer as In particular, note that Legitimate question If you have c n, how do you compute a n and b n ?

Example Find the complex form of the Fourier series of the function whose definition in one period is 11

Example: Preamble The propeller and long steel shaft systems of large ships are susceptible to vibration problems. One source of axial excitation contributing to the vibration of such systems is the generation of pulse-type forces F(t) that result from a propeller blade passing the restricted area between the propeller and the hull of the ship. Consider that the axial forcing function F(t) can be represented by the rectangular pulse train shown on next slide in which and R is the rpm of the propeller at cruising speed. Use the a series expansion to determine whether a three- or four-bladed propeller should be used to avoid undesirable vibration if the following data pertain: R = 325 rpm d 0 = 20 in.(outside diameter of shaft) d i = 10 in.(inside diameter of shaft) L =175 ft(length of shaft) E = 30(10 6 ) psi (modulus of elasticity of steel) w = 490 lb/ft 3 (specific weight of steel) W = 22,500 lb(weight of either a three- or four-bladed propeller) 12

Example Determine the Fourier expansion for the following function: 13

Begin Chapter 2 Free Vibration of Single Degree of Freedom Systems 14