Explanation Given: Mass of motor is M . Equivalent spring constant is k . Unbalance rotating mass is m . Angular velocity of motor is ω f . Concept used: Write the expression for the motion of motor. M x ¨ + k x = P m sin ω f t Here, M is the mass of motor, k is the equivalent stiffness, ω f is the frequency of motion, t is the time and P m is the magnitude of unbalanced force. Divide the above expression by M . x ¨ + k x M = P m sin ω f t M ...... (1) Write the general equation of motion in differential form. x ¨ + ω n 2 x = F ...... (2) Here, ω n is the natural frequency of system. Compare equation (1) and equation (2). ω n 2 = k M k = M ω n 2 Write the expression for the amplitude of unbalanced forced due to rotating mass. P m = m r ω f 2 ...... (3) Here, m is the rotating mass and r is the distance of rotating mass from axis. Write the expression for amplitude of vibration. x m = P m k 1 − ( ω f 2 ω n 2 ) ...... (4) Here, x m is the amplitude of vibration. Substitute m r ω f 2 for P m , M ω n 2 for k in equation (4)

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

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…

Question

Chapter 19.4, Problem 19.113P

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Show that xm=r(m/M)(ωf/ωn)21−(ωf/ωn)2.

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Chapter 19.4, Problem 19.112P

Chapter 19.4, Problem 19.114P

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Show that x m = r ( m / M ) ( ω f / ω n ) 2 1 − ( ω f / ω n ) 2 .

Solution Summary: The author analyzes the amplitude of the motion of motor and divides it by M.

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Show that xm=r(m/M)(ωf/ωn)21−(ωf/ωn)2.

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Chapter 19 Solutions