Explanation Given information: The mass of the uniform disk ( m ) is 8 kg . The radius of the disk ( r ) is 200 mm or 0 .2m . The rotating angle of disk ( θ ) is 3 ° . The magnitude of the static couple ( T ) is 50 N ⋅ m . The magnitude of the torsional couple ( T ) is T = T m sin ω f t . The magnitude of the couple ( T m ) is 60 N ⋅ m . Calculation: Calculate the mass moment of inertia ( I ¯ ) using the formula: I ¯ = 1 2 m r 2 Substitute 8 kg for m and 0 .2m for r . I ¯ = 1 2 × 8 × 0.2 2 = 0.16 kg ⋅ m 2 Calculate the torsional spring constant ( K ) using the formula: K = T θ Substitute 50 N ⋅ m for T and 3 ° for θ . K = 50 3 ° × π 180 = 50 0.05236 rad = 954.93 N ⋅ m/rad Calculate the natural circular frequency ( ω n ) using the formula: ω n = Κ I Substitute 954.93 N ⋅ m/rad for K and 0.16 kg ⋅ m 2 for I . ω n = 954.93 0.16 = 77

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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Free Damped Vibrations

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Chapter 19.4, Problem 19.104P

To determine

Find the range of values of (ωf) for which the amplitude of vibrations is less than the angle of rotation caused by a static couple of magnitude Tm.

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Chapter 19.4, Problem 19.103P

Chapter 19.4, Problem 19.105P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - Prob. 19.2P Ch. 19.1 - Prob. 19.3P Ch. 19.1 - Prob. 19.4P Ch. 19.1 - Prob. 19.5P Ch. 19.1 - Prob. 19.6P Ch. 19.1 - Prob. 19.7P Ch. 19.1 - Prob. 19.8P Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

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Find the range of values of ( ω f ) for which the amplitude of vibrations is less than the angle of rotation caused by a static couple of magnitude T m .

Solution Summary: The author calculates the mass moment of inertia using the formula: c

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Find the range of values of (ωf) for which the amplitude of vibrations is less than the angle of rotation caused by a static couple of magnitude Tm.

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Chapter 19 Solutions