Explanation Given information: The weight of the bundle ( w b ) is 20 lb. The rotating weight including bundle weight ( W ) is 40 lb. The radius of the washer basket ( e ) is 9 inch. The spring constant ( k ) is 70 lb/ft. The damping ratio ( c c c ) is 0.05. The speed of the drum ( N ) is 250 rpm. The acceleration due to gravity ( g ) is 32.2 ft/s 2 . Calculation: Calculate the forced circular frequency ( ω f ) using the formula: ω f = 2 π N 60 Substitute 250 rpm for N . ω f = 2 π ( 250 ) 60 = 26.18 rad/s Calculate the mass of the rotating machine ( m ) using the formula: m = W g Substitute 40 lb for W and 32.2 ft/s 2 for g . m = 40 32.2 = 1.24224 lb ⋅ s 2 /ft Calculate the mass of the bundle ( w b ) using the formula: m b = w b g Substitute 20 lb for w b and 32.2 ft/s 2 for g . m b = 20 32.2 = 0.62112 lb ⋅ s 2 /ft Calculate the natural circular frequency ( ω n ) of the machine using the formula: ω n = k m Substitute 70 lb/ft for k and 1.24224 lb ⋅ s 2 /ft for m . ω n = 70 1.24224 lb ⋅ s 2 /ft = 7.5067 rad/s Calculate the critical damping coefficient ( c c ) using the formula: c c = 2 k m Substitute 70 lb/ft for k and 1.24224 lb ⋅ s 2 /ft for m . c c = 2 70 ( 1.24224 ) = 18.650 lb ⋅ s/ft Calculate the damping coefficient ( c ) from the damping ratio using the relation: Damping ratio = c c c Substitute 0.05 for damping ratio and 18.650 lb ⋅ s/ft for c c . 0.05 = c 18.650 c = 18.650 ( 0.05 ) c = 0.9325 lb ⋅ s/ft Calculate the unbalance force ( P m ) of the machine using the formula: P m = m b e ω f 2 Substitute 0.62112 lb ⋅ s 2 /ft for m b , 9 in. for e , and 26.18 rad/s for ω f . P m = 0.62112 × ( 9 in . × 1ft 12in . ) × ( 26.18 ) 2 = 319.28 lb Calculate the amplitude of vibration ( x m ) of the washing machine using the formula: x m = P m ( k − m ω f 2 ) 2 + ( c ω f ) 2 Substitute 319.28 lb for P m , 70 lb/ft for k , 1.24224 lb ⋅ s 2 /ft for m , 0

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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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 19.5, Problem 19.149P

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The amplitude of the motion (xm) and the magnitude of the force (Fm) transmitted to the sides of the washing machine.

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Chapter 19.5, Problem 19.148P

Chapter 19.5, Problem 19.150P

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A simplified model of a washing machine is shown. A bundle of wet clothes forms a weight wb of 20 lb in the machine and causes a rotating unbalance. The rotating weight is 40 lb (including wb) and the radius of the washer basket e is 9 in. Knowing the washer has an equivalent spring constant k = 70 lb/ft and damping ratio and during the spin cycle the drum rotates at 250 rpm, determine the amplitude of the motion and the magnitude of the force transmitted to the sides of the washing machine.

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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 amplitude of the motion ( x m ) and the magnitude of the force ( F m ) transmitted to the sides of the washing machine.

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The amplitude of the motion (xm) and the magnitude of the force (Fm) transmitted to the sides of the washing machine.

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