Explanation Given: The weight of the block, W = 12 lb . The magnitude of the damping force or damping coefficient, c = 0.7 lb ⋅ s / ft . The spring stiffness, k = 53 lb / ft . Displacement of blcok from the equilibrium position, y = 0.62 ft . Conclusion: Express the natural frequency. ω n = k W / g (I) Here, acceleration due to gravity is g . Substitute 12 lb for W , 32 .2 ft / s 2 for g and 53 lb / ft for k in Equation (I). ω n = 53 12 / 32.2 = 11.925 rad / s Express the critical damping factor. c c = 2 W g ω n (II) Here, natural frequency is ω n . Substitute 12 lb for W , 32 .2 ft / s 2 for g and 11.925 rad / s for ω n in Equation (II). c c = 2 [ 12 32.2 ] ( 11.925 ) = 8.889 lb ⋅ s / ft Write the condition for system to be underdamped. c < c c Express damping natural frequency. ω d = ω n 1 − ( c / c c ) 2 (III) Substitute 11.925 rad / s for ω n , 0.7 lb ⋅ s / ft for c and 8.889 lb ⋅ s / ft for c c in Equation (III). ω d = 11.925 1 − ( 0.7 / 8.889 ) 2 = 11.888 rad / s Express the mass of the block. m = W g (IV) Substitute 12 lb for W and 32 .2 ft / s 2 for g in Equation (IV). m = 12 32.2 = 0.3727 slug Write the general condition for the underdamped system. y = D [ e − [ c 2 m ] t sin ( ω d t + ϕ ) ] (V) Here, phase angle is ϕ , time is t and mass of block is m . Take the time derivative of Equation (V) to get, y ˙ = v = D [ e − [ c 2 m ] t ω d cos ( ω d t + ϕ ) + [ − c 2 m ] e − [ c 2 m ] t sin ( ω d t + ϕ ) ] = D e − [ c 2 m ] t [ ω d cos ( ω d t + ϕ ) − c 2 m sin ( ω d t + ϕ ) ] (VI) Apply the initial condition: At t = 0 Substitute 0 for t and 0

Engineering Mechanics: Dynamics (14th Edition)

14th Edition

ISBN: 9780133915389

Author: Russell C. Hibbeler

Publisher: PEARSON

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Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

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Chapter 22.6, Problem 72P

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The position of the block as a function of time.

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Chapter 22.6, Problem 71P

Chapter 22.6, Problem 73P

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Chapter 22 Solutions

Engineering Mechanics: Dynamics (14th Edition)

Ch. 22.1 - A spring is stretched 175 mm by an 8-kg block. If...Ch. 22.1 - Prob. 2P Ch. 22.1 - A spring is stretched 200 mm by a 15-kg block. If...Ch. 22.1 - When a 20-lb weight is suspended from a spring,...Ch. 22.1 - Prob. 5P Ch. 22.1 - Prob. 6P Ch. 22.1 - Prob. 7P Ch. 22.1 - Prob. 8P Ch. 22.1 - A 3-kg block is suspended from a spring having a...Ch. 22.1 - Prob. 10P

Ch. 22.1 - Prob. 11P Ch. 22.1 - 22-12. Determine the natural period of vibration...Ch. 22.1 - The body of arbitrary shape has a mass m, mass...Ch. 22.1 - Determine the torsional stiffness k, measured in...Ch. 22.1 - Prob. 15P Ch. 22.1 - Prob. 16P Ch. 22.1 - If the natural periods of oscillation of the...Ch. 22.1 - Prob. 18P Ch. 22.1 - Prob. 19P Ch. 22.1 - A uniform board is supported on two wheels which...Ch. 22.1 - If the wire AB is subjected to a tension of 20 lb,...Ch. 22.1 - The bar has a length l and mass m. It is supported...Ch. 22.1 - The 20-kg disk, is pinned at its mass center O and...Ch. 22.1 - Prob. 24P Ch. 22.1 - If the disk in Prob. 22-24 has a mass of 10 kg,...Ch. 22.1 - Prob. 26P Ch. 22.1 - Prob. 27P Ch. 22.1 - Prob. 28P Ch. 22.1 - Prob. 29P Ch. 22.2 - Determine the differential equation of motion of...Ch. 22.2 - Determine the natural period of vibration of the...Ch. 22.2 - Determine the natural period of vibration of the...Ch. 22.2 - Prob. 33P Ch. 22.2 - Determine the differential equation of motion of...Ch. 22.2 - Prob. 35P Ch. 22.2 - Prob. 36P Ch. 22.2 - Prob. 37P Ch. 22.2 - Prob. 38P Ch. 22.2 - Prob. 39P Ch. 22.2 - If the slender rod has a weight of 5 lb, determine...Ch. 22.6 - If the block-and-spring model is subjected to the...Ch. 22.6 - Prob. 42P Ch. 22.6 - A 4-lb weight is attached to a spring having a...Ch. 22.6 - Prob. 44P Ch. 22.6 - Prob. 45P Ch. 22.6 - Prob. 46P Ch. 22.6 - Prob. 47P Ch. 22.6 - Prob. 48P Ch. 22.6 - Prob. 49P Ch. 22.6 - Prob. 50P Ch. 22.6 - The 40-kg block is attached to a spring having a...Ch. 22.6 - The 5kg circular disk is mounted off center on a...Ch. 22.6 - Prob. 53P Ch. 22.6 - Prob. 54P Ch. 22.6 - Prob. 55P Ch. 22.6 - Prob. 56P Ch. 22.6 - Prob. 57P Ch. 22.6 - Prob. 58P Ch. 22.6 - Prob. 59P Ch. 22.6 - The 450-kg trailer is pulled with a constant speed...Ch. 22.6 - Prob. 61P Ch. 22.6 - Prob. 62P Ch. 22.6 - Prob. 63P Ch. 22.6 - The spring system is connected to a crosshead that...Ch. 22.6 - Prob. 65P Ch. 22.6 - Prob. 66P Ch. 22.6 - Prob. 67P Ch. 22.6 - The 200-lb electric motor is fastened to the...Ch. 22.6 - Prob. 69P Ch. 22.6 - If two of these maximum displacements can be...Ch. 22.6 - Prob. 71P Ch. 22.6 - Prob. 72P Ch. 22.6 - Prob. 73P Ch. 22.6 - Prob. 74P Ch. 22.6 - Prob. 75P Ch. 22.6 - Prob. 76P Ch. 22.6 - Prob. 77P Ch. 22.6 - Prob. 78P Ch. 22.6 - Prob. 79P

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