Explanation Given Information: The mass ( m ) of the advertising panel is 48 kg. The length ( 2a ) of the advertising panel is 2.4 m. The width ( 2b ) of the advertising panel is 1.6 m. The period ( τ 1 ) for panel to complete a full revolution is 6s. The period ( τ 2 ) for frame to complete a full revolution is 12s. Calculation: Show the principal axes as in Figure (1). Let the principal axes as x ' , y ' and z ' . Express the centroidal moment of inertia ( I ¯ x ' ) of about the principal axes x ' : I ¯ x ' = 1 3 m ( a 2 + b 2 ) Express the centroidal moment of inertia ( I ¯ y ' ) of about the principal axes y ' : I ¯ y ' = 1 3 m a 2 Express the centroidal moment of inertia ( I ¯ z ' ) of about the principal axes z ' : I ¯ z ' = 1 3 m b 2 Express the term θ ˙ as below: θ ˙ = ω 1 Express the angular velocity about axes x ' : ω x ' = ω 2 sin θ Differentiate the above equation with respect to θ ω ˙ x ' = ω 2 cos θ θ ˙ Substitute ω 1 for θ ˙ . ω ˙ x ' = ω 1 ω 2 cos θ Express the angular velocity about axes y ' : ω y ' = ω 2 cos θ Differentiate the above equation with respect to θ ω ˙ y ' = − ω 2 sin θ θ ˙ Substitute ω 1 for θ ˙ . ω ˙ y ' = − ω 1 ω 2 sin θ Express the angular velocity about axes z ' : ω z ' = ω 1 Differentiate the above equation with respect to θ . ω z ' = 0 Since the acceleration of the mass center is zero the resultant force acting on the column CD is zero. Express the moment about axes x ' : ∑ M x ' = I ¯ x ' ω ˙ x ' − ( I ¯ y ' − I ¯ z ' ) ω y ' ω z ' Substitute 1 3 m ( a 2 + b 2 ) for I ¯ x ' , 1 3 m a 2 for I ¯ y ' , 1 3 m b 2 for I ¯ z ' , ω 1 ω 2 cos θ for ω ˙ x ' , ω 1 for ω z ' , and ω 2 cos θ for ω y ' . ∑ M x ' = [ 1 3 m ( a 2 + b 2 ) × ω 1 ω 2 cos θ − ( 1 3 m a 2 − 1 3 m b 2 ) × ω 2 cos θ × ω 1 ] = 1 3 m a 2 ω 1 ω 2 cos θ + 1 3 m b 2 ω 1 ω 2 cos θ − 1 3 m a 2 ω 1 ω 2 cos θ + 1 3 m b 2 ω 1 ω 2 cos θ = 2 3 m b 2 ω 1 ω 2 cos θ Express the moment about axes y ' : ∑ M y ' = I ¯ y ' ω ˙ y ' − ( I ¯ z ' − I ¯ x ' ) ω x ' ω z ' Substitute 1 3 m ( a 2 + b 2 ) for I ¯ x ' , 1 3 m a 2 for I ¯ y ' , 1 3 m b 2 for I ¯ z ' , − ω 1 ω 2 sin θ for ω ˙ y ' , ω 1 for ω z ' , and ω 2 sin θ for ω x ' . ∑ M y ' = ( 1 3 m a 2 × − ω 1 ω 2 sin θ ) − ( 1 3 m b 2 − 1 3 m ( a 2 + b 2 ) ) × ω 2 sin θ × ω 1 = − 1 3 m a 2 ω 1 ω 2 sin θ − 1 3 m b 2 ω 1 ω 2 sin θ + 1 3 m a 2 ω 1 ω 2 sin θ + 1 3 m b 2 ω 1 ω 2 sin θ = 0 Express the moment about axes z ' : ∑ M z ' = I ¯ z ' ω ˙ z ' − ( I ¯ x ' − I ¯ y ' ) ω x ' ω y ' Substitute 1 3 m ( a 2 + b 2 ) for I ¯ x ' , 1 3 m a 2 for I ¯ y ' , 1 3 m b 2 for I ¯ z ' , 0 for ω ˙ z ' , ω 2 sin θ for ω x ' and ω 2 cos θ for ω y '

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 18, Problem 18.154RP

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Express the dynamic reaction (MD) exerted on column CD by its support at D.

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Chapter 18, Problem 18.153RP

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Express the dynamic reaction ( M D ) exerted on column CD by its support at D .

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Express the dynamic reaction (MD) exerted on column CD by its support at D.

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Chapter 18 Solutions