hw3_spring_09_soln

# hw3_spring_09_soln - EML 6267 Assignment#3 Spring 2009...

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EML 6267 Assignment #3 Spring 2009 Please remember to put your name on your assignment. Due: Monday, 2/16 1. (60 pts.) Complete problem 1 from chapter 3 of the text. For part f), use plot limits of zero to 60000 rpm and zero to 10 mm. a) Calculate the directional orientation factor. Using this value, compute and plot the real and imaginary parts (in m/N) of the oriented frequency response function vs. frequency (in Hz). The directional orientation factor is found by first projecting the force, F , into the u direction, and then projecting that result into the x direction. () ( ) ( ) ( ) 671 . 0 35 cos 35 70 cos cos cos = = = α β µ The oriented frequency response function is the product of the directional orientation factor and the single degree of freedom frequency response for this system. It is written as: ( ) + = 2 2 2 2 2 1 2 1 r r r i r k FRF orient ζ , where n r ω = and m k n = . The result is shown in Fig. s.3.1.b. This figure was produced using e_3_1_a.m. 0 200 400 600 800 1000 -2 0 2 x 10 -7 Real (m/N) 0 200 400 600 800 1000 -4 -2 0 x 10 -7 Frequency (Hz) Imag (m/N) Figure s.3.1.b: Oriented frequency response function.

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b) Determine the minimum value of the real part of the oriented frequency response function and the corresponding chatter frequency. Calculate b lim,crit . For a single degree of freedom system, we can calculate the minimum value of the real part of the frequency response function using: () ζ + = 1 4 1 Re min k FRF . For this example, the minimum value of the real part of the oriented frequency response function is then: 4 10 6 . 1 1 4 Re min × = + = µ k FRF orient mm/N, where we’ve converted from m/N to mm/N. The corresponding chatter frequency is approximated as: ( ) 5 . 528 1 = + = n c f f Hz. This result is confirmed in e_3_1_a.m using the M ATLAB ® min command. The critical stability limit is determined using: [] 1 . 2 10 6 . 1 1500 2 1 Re min 2 1 4 crit lim, = × = = orient s FRF K b mm.
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hw3_spring_09_soln - EML 6267 Assignment#3 Spring 2009...

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