Explanation Given information: The mass of the motor ( M ) is 180 kg. The unbalance mass of the rotor ( m ) is 85 g. The distance between the center of rotor to axis of rotation ( r ) is 150 mm. The equivalent spring constant ( k ) is 7.5 kN/m. The maximum acceleration of the motor ( x m ) is 9 mm/s 2 . The speed of the motor ( N ) is 100 rpm. The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Calculate the frequency of the periodic force ( ω f ) using the relation: ω f = 2 π N 60 Substitute 100 rpm for N . ω f = 2 π ( 100 rpm ) 60 = 10.472 rad / s Calculate the magnitude periodic force ( P m ) using the formula: P m = m r ω f 2 Substitute 85 g for m , 150 mm for r , and 10.472 rad / s for ω f . P m = ( 85 g × kg 1,000 g ) × ( 150 mm × 1 1 , 000 m mm ) × 10.472 2 = 1.39820 N Calculate the natural circular frequency ( ω n ) using the formula: ω n = k M Substitute 180 kg for M and 7.5 kN/m for k . ω n = 7.5 × 10 3 180 = 6.455 rad/s Calculate the damping factor ( c c c ) of the system using the formula: ω f 2 x m = P m k ( 1 − ( ω f ω n ) 2 ) 2 + ( 2 ( c c c ) ω f ω n ) 2 x m = ω f 2 P m k ( 1 − ( ω f ω n ) 2 ) 2 + ( 2 ( c c c ) ω f ω n ) 2 Substitute 1

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

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the init…

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Chapter 19.5, Problem 19.146P

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The damping factor (ccc).

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Chapter 19.5, Problem 19.145P

Chapter 19.5, Problem 19.147P

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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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The damping factor ( c c c ) .

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The damping factor (ccc).

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Chapter 19 Solutions