For an ensemble of attachments gears pulleys etc

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For an ensemble of attachments (gears, pulleys, etc), Rayleigh’s method for lumped masses gives Critical Speeds for Shafts where 𝑤 𝑖 is the weight of ith component and 𝑦 𝑖 is the deflection of ith component location.
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Critical Speeds for Shafts According to Dunkerley’s equation, the critical load is obtained by accounting for the contribution of all attachments as follows; where 𝜔 𝑖𝑖 is the critical speed of shaft due to its ith attachment, 𝜔 1 is the first critical speed of shaft.
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Dunkeley’s approach is more conservative than Rayleigh’s method resulting in lower critical speed. Therefore a critical speed of shaft and its attachments is obtained by: 1 𝜔 1 2 = 1 𝜔 𝑠 2 + ∑ 1 𝜔 𝑖𝑖 2
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A 25-mm diameter uniform steel shaft is 600 mm long between the two bearings. (a)Find the lowest critical speed of the shaft (b)If the goal is to double the critical speed, find the new diameter. (c)A half size model of the original shaft has what critical speed? Problem 2 From Table A-5 page 1015: 𝐸 = 207 GPa , and 𝛾 = 76 kN/m 3
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