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Homework10asol.fall11

# Homework10asol.fall11 - ME 352 Machine Design I Fall...

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ME 352 - Machine Design I Name of Student ______________________________ Fall Semester 2011 Lab Section Number ___________________________ Homework No. 10 (30 points). Due at the beginning of lecture on Wednesday, November 16th. Problem 1. Solve Problem 15.26, see page 746, using: (i) The Exact Solution; (ii) The Dunkerley approximation; and (iii) The Rayleigh-Ritz approximation. Problem 2. Solve Problem 15.27, see page 747, using the Dunkerley approximation.

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2 Solution to Problem 1. A steel shaft is simply supported by two rolling element bearings at A and B as shown in Figure 1. The length of the shaft is 1.45 m and two flywheels with weight 1 2 W W 300 N = = are rigidly attached to the shaft at the locations shown. One flywheel is 0.35 m to the right of the left bearing at A and the other flywheel is 0.35 m to the left of the right bearing at B. The weight of the shaft can be neglected. The influence coefficients are 5 11 a 12.600 x10 cm/N = and 5 21 a 9.250 x10 cm/N. = (i) Determine the first and second critical speeds of the shaft using the exact solution. (ii) Determine the first critical speed using the Dunkerley approximation and Rayleigh-Ritz equation. Figure 1. A Steel Shaft Simply Supported by Two Rolling Element Bearings. The exact solutions for the first and second critical speeds of the shaft can be written as 2 11 1 22 2 11 1 22 2 11 22 12 21 1 2 2 2 1 2 ( a m a m ) ( a m a m ) 4 ( a a a a ) m m 1 1 , ω ω 2 + ± + = (1) From the symmetry of the loading, the influence coefficients 5 11 22 a a 12.600 x10 cm/N = = (2) From Maxwell's reciprocity theorem, the influence coefficients 5 12 21 a a 9.250 x10 cm/N = = (3)
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Homework10asol.fall11 - ME 352 Machine Design I Fall...

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