ME 440 Intermediate Vibrations

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1 ME 440 Itermediate Vibratios Th, Jauary 29, 2009 Sectio 1.11 Da Negrut, 2009 ME440, UW-Madiso

2 Before we get started Last Time: Discussed about periodic fuctios Covered the Fourier Series Expasio Wet through oe example Today: Coverig material out of 1.11 Fourier Series Expasio, Complex Represetatio Three more examples HW Assiged: 1.68 ad 2.35 out of the book NOTE: last time meetig i this room. We ll to 3126ME startig o Feb. 3 (ext lecture) 2

3 A Note o HW Problem Oe of the problems assiged last time: Problem 1.66 Does t explicitly idicate what the expressio of f(t) is Oly shows a plot of it. Please work with followig expressio for f(t): 3

4 New Topic: Frequecy Spectrum Recall the cocept of fudametal frequecy: By ispectio of the terms i Fourier expasio, you otice that as icreases, the harmoic fuctios display gradually icreasig oscillatio frequecies. 4 Frequecy Spectrum: The plot of a ad b versus It shows how the amplitude of the harmoics eterig the Fourier expasio chages as icreases Recall the discussio we had: as icreases (ad therefore the oscillatio frequecy icreases), I expect the frequecy spectrum to be approachig the zero axis. Why?

5 Frequecy Spectrum (Ctd) Example of Frequecy Spectrum 5

6 Frequecy Spectrum (Ctd) Note that Frequecy Spectrum ca be also provided i the form X & q, istead of a ad b. Use trasformatio to move back ad forth betwee these two represetatios Example 1 Example 2 6

7 Example 2. Determie the Fourier expasio of the periodic fuctio below Plot its frequecy spectrum f(t) A -T 0 T 2T 3T 4T t 7

8 New Topic: Fourier Expasio usig Complex Notatio Two importat idetities: Therefore, Fourier expasio becomes Recall that w=2p/t ad rewrite as 8

9 Fourier Expasio usig Complex Notatio (Ctd) Itroduce otatio: Fourier expasio becomes Revisit computatio of c 9

10 Fourier Expasio usig Complex Notatio (Ctd) Similarly, I coclusios, all the coefficiet c are computer as Legitimate questio If you have c, how do you compute a ad b? I particular, ote that 10

11 Example Fid the complex form of the Fourier series of the fuctio whose defiitio i oe period is Periodic Fuctio exp(-t)

12 Example: Preamble The propeller ad log steel shaft systems of large ships are susceptible to vibratio problems. Oe source of axial excitatio cotributig to the vibratio of such systems is the geeratio of pulse-type forces F(t) that result from a propeller blade passig the restricted area betwee the propeller ad the hull of the ship. Cosider that the axial forcig fuctio F(t) ca be represeted by the rectagular pulse trai show o ext slide i which t = 60 R umber of propeller blades 12 ad R is the rpm of the propeller at cruisig speed. Use the a series expasio to determie whether a three- or four-bladed propeller should be used to avoid udesirable vibratio if the followig data pertai: R = 325 rpm d 0 = 20 i. (outside diameter of shaft) d i = 10 i. (iside diameter of shaft) L =175 ft (legth 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)

13 Example Determie the Fourier expasio for the followig fuctio: 13

14 Begi Chapter 2 Free Vibratio of Sigle Degree of Freedom Systems 14

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