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Unformatted text preview: 61 Problems for Section 3.6 &Periodic but Not Harmonic Excitation 31. A machine is loaded by a periodic &sawtooth¡shaped force, as depicted in Figure 3.50. The load is assumed to have existed for a very long time. Model this force using: (a) a oneterm Fourier series, (b) a threeterm Fourier series, and (c) a ¢veterm Fourier series. In each case solve for the response as a function of time. System properties are: m = 1 kg, k = 9 N/cm, & = 0 : 15 , T = 1 s, and A = 1 cm. Figure 3.50: Sawtooth loading. Solution: First of all, we realize that this problem can be solved exactly by solving the problem in discrete time intervals such that m & x 1 + c _ x 1 + kx 1 = A T t for & t & T m & x 2 + c _ x 2 + kx 2 = A T t ¡ A for T & t & 2 T . . . The displacements and velocities are continuous at t = nT for n = 1 ; 2 ; ¢¢¢ : Here, we use the Fourier expansion for the periodic forcing. The sawtooth function is given by At=T; & t & T: We ¢rst have to evaluate the Fourier coe£ cients as follows: a = 2 T Z T At T dt = A a p = 2 T Z T At T cos p! T tdt = 2 T 2 A cos p! T T + p! T (sin p! T T ) T ¡ 1 p 2 ! 2 T b p = 2 T Z T At T sin p! T tdt = ¡ 2 T 2 ( ¡ sin p! T T + p! T (cos p! T T ) T ) A p 2 ! 2 T : Note that the integration by part was used to evaluate a p and b p : For example, Z T t {z} u cos p! T tdt  {z } dv = t {z} u sin p! T t p! T  {z } v & & & & & & & & T ¡ Z T 1 dt {z} du sin p! T t p! T  {z } v : 62 CHAPTER 3 SDOF VIBRATION: WITH DAMPING Since ! T = 2 &=T; and p is an integer, cos p! T T = cos 2 &p = 1 ; and sin p! T T = 0 : Therefore, a p = 0 and b p = & A=p&: The Fourier series representation of the force is then F ( t ) = A 2 & 1 X p =1 A p& sin p 2 & T t: The one, three, and &veterm expansions are respectively, F 1 ( t...
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This note was uploaded on 02/26/2012 for the course 650 443 taught by Professor Benaroya during the Fall '11 term at Rutgers.
 Fall '11
 Benaroya

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