(a) Explanation Given: The differential equation for the deflection y ( x ) of the shaft is E I d 4 y d x 4 − ρ ω 2 y = 0 such that y ( 0 ) = 0 , y ″ ( 0 ) = 0 , y ( L ) = 0 , y ″ ( L ) = 0 and α 4 = ρ ω 2 E I . Calculation: The differential equation for the deflection y ( x ) of the shaft is given as follows: E I d 4 y d x 4 − ρ ω 2 y = 0 … … ( 1 ) The boundary conditions are as follows: y ( 0 ) = 0 , y ″ ( 0 ) = 0 , y ( L ) = 0 and y ″ ( L ) = 0 The auxiliary equation for equation ( 1 ) is as follows: ( E I ) m 4 − ρ ω 2 = 0 m 4 − ρ ω 2 E I = 0 m 4 − α 4 = 0 ( α = ρ ω 2 E I ) ( m 2 + α 2 ) ( m 2 − α 2 ) = 0 The roots for the above equation are as follows: m = ± i α m = ± α The general solution of equation ( 1 ) is as follows: y ( x ) = c 1 cos ( α x ) + c 2 sin ( α x ) + c 3 cos h ( α x ) + c 4 sinh ( α x ) … … ( 2 ) Substitute 0 for x in equation ( 2 ) . y ( 0 ) = c 1 cos ( 0 ) + c 2 sin ( 0 ) + c 3 cos h ( 0 ) + c 4 sinh ( 0 ) c 1 + c 3 = 0 ( y ( 0 ) = 0 ) Label the above equation as follows: c 1 + c 3 = 0 … … ( 3 ) Differentiate equation ( 2 ) with respect to x . y ′ ( x ) = − α c 1 sin ( α x ) + α c 2 cos ( α x ) + α c 3 sinh ( α x ) + α c 4 cos h ( α x ) Differentiate the above equation with respect to x . y ″ ( x ) = − α 2 c 1 cos ( α x ) − α 2 c 2 sin ( α x ) + α 2 c 3 cosh ( α x ) + α 2 c 4 sin h ( α x ) Substitute 0 for x in the above equation. y ″ ( 0 ) = − α 2 c 1 cos ( 0 ) − α 2 c 2 sin ( 0 ) + α 2 c 3 cosh ( 0 ) + α 2 c 4 sin h ( 0 ) − α 2 c 1 + α 2 c 3 = 0 c 3 − c 1 = 0 Label the above equation as follows: c 3 − c 1 = 0 … … ( 4 ) Add equation ( 3 ) and equation ( 4 ) . c 1 + c 3 + c 3 − c 1 = 0 2 c 3 = 0 c 3 = 0 Substitute the value of c 3 in equation ( 4 ) . 0 − c 1 = 0 c 1 = 0 Substitute the value of c 1 and c 3 in equation ( 2 ) (b) To determine The angular speed ω n and the fundamental critical speeds ω 1 of the shaft.

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.9, Problem 31E

(a)

To determine

The eigenvalues and the eigenfunctions for the differential equation EId4ydx4−ρω2y=0 such that y(0)=0, y″(0)=0, y(L)=0 and y″(L)=0.

(b)

To determine

The angular speed ωn and the fundamental critical speeds ω1 of the shaft.

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Chapter 3.9, Problem 30E

Chapter 3.9, Problem 32E

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The eigenvalues and the eigenfunctions for the differential equation E I d 4 y d x 4 − ρ ω 2 y = 0 such that y ( 0 ) = 0 , y ″ ( 0 ) = 0 , y ( L ) = 0 and y ″ ( L ) = 0 .

Solution Summary: The author explains the differential equation for the deflection y(x) of the shaft.