Explanation Given information: The spring constant is 7.5 kN / m , the maximum acceleration of the motor is 9 mm / s 2 , the rotational speed of motor is 100 rpm , the mass of the motor is 180 kg , the equivalent mass of the rotor is 85 g , the location of equivalent mass is from axis of rotation is 150 mm . The figure illustrates the forces. Figure-(1) Write the expression of angular speed of motor. ω = 2 π N 60 ..... (I) Here, the angular speed of the motor is ω , the rotational speed of the motor is N . Write the expression of maximum amplitude of steady state response of motor. X m = P m k [ 1 − ( ω ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω ω n ) ] 2 ...... (II) Here, the angular frequency is ω , the natural angular frequency is ω n , the damping coefficient is c , the critical damping coefficient is c c , the amplitude of applied force is P m , the spring constant is k , the maximum amplitude is X m . Write the expression of amplitude of force as equivalent to net rotating unbalanced mass. P m = m u e ω 2 ...... (III) Here, the unbalanced mass is m u , the distance of unbalance mass from the axis of rotation is e . Write the expression of natural frequency of vibration. ω n = k m ...... (IV) Here, the mass of the rotor is m , the natural frequency of vibration is ω n . Write the expression of maximum acceleration. a max = ( X m ) ω 2 ...... (V) Here, the maximum acceleration is a max . Substitute P m k [ 1 − ( ω ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω ω n ) ] 2 for X m in Equation (V). a max = ( P m k [ 1 − ( ω ω n ) 2 ] 2 + [ 2 ( c c c ) ( ω ω n ) ] 2 ) ω 2 ...... (VI) Calculation: Substitute 100 rpm for N in Equation (I). ω = 2 π ( 100 rpm ) 60 = 200 π rpm 60 = 10.471 rad / s Substitute 85 g for m u and 150 mm for e , 10

Vector Mechanics for Engineers: Dynamics

11th Edition

ISBN: 9780077687342

Author: Ferdinand P. Beer, E. Russell Johnston Jr., 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

To determine

The damping factor.

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Chapter 19.5, Problem 19.145P

Chapter 19.5, Problem 19.147P

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Vector Mechanics for Engineers: Dynamics

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The damping factor.

Solution Summary: The author explains the damping factor and the amplitude of steady state response of the motor.

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The damping factor.

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Chapter 19 Solutions