(a) Explanation Given information: The weight of the car ( W ) is 30000 lb. The acceleration due to gravity ( g ) is 32 .2 ft / s 2 . Calculation: Calculate the mass of the railroad car ( m ) using the formula: m = W g Substitute 30,000 lb for W and 32 .2 ft / s 2 for g . m = 30000 lb 32 .2 ft / s 2 = 931.67 lb ⋅ s 2 / ft Refer the time displacement graph from the textbook. Take the time between two peak points or the time period of oscillation as 0.41 sec. Calculate the frequency of damped vibration ( ω d ) using the formula: ω d = 2 π t d Here, t d the time period of oscillation. Substitute 0.41 sec for t d . ω d = 2 π 0.41 s = 15.325 rad / s The expression for the amplitude of forced vibration for light damping as follows: x = x 0 e − ( c / 2 m ) t sin ( ω d t + ϕ ) (1) Here, x is the amplitude of forced vibration, x 0 is the initial amplitude, c is the damping coefficient, ω d is the frequency of damped vibration, t is the time period, and ϕ is the phase difference. Consider at first peak point. Take displacement at first peak point as x 1 at the time t 1 . At peak point, the displacement is maximum and the value of sin ( ω d t + ϕ ) is one. Substitute x 1 for x , t 1 for t , and 1 for sin ( ω d t + ϕ ) in equation (1). x = x 0 e − ( c / 2 m ) t sin ( ω d t + ϕ ) x 1 = x 0 e − ( c / 2 m ) ( t 1 ) ( 1 ) x 1 = x 0 e − ( c / 2 m ) t 1 (2) Consider at second peak point. Take displacement at second peak point as x 2 at the time t 2 . At peak point, the displacement is maximum and the value of sin ( ω d t + ϕ ) is one. The time at second peak point is the sum of time at first peak point and time period of oscillation. Calculate the time at second peak point ( t 2 ) using the relation: t 2 = t 1 + t d Substitute x 2 for x , ( t 1 + t d ) for t , and 1 for sin ( ω d t + ϕ ) in equation (1). x = x 0 e − ( c / 2 m ) t sin ( ω d t + ϕ ) x 2 = x 0 e − ( c / 2 m ) ( t 1 + t d ) ( 1 ) x 2 = x 0 e − ( c / 2 m ) ( t 1 + t d ) (3) Divide the equation (3) by equation (2) (b) To determine Find the spring constant ( k ).

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 19.132P

(a)

To determine

Find the damping constant (c).

(b)

To determine

Find the spring constant (k).

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Chapter 19.5, Problem 19.131P

Chapter 19.5, Problem 19.133P

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Problem 2. A projectile of mass m = 10 kg travelling with a velocity of 50m/s strikes and becomes embedded in a massless board supported by a spring of stiffness k-6.4x 104 N/m and a dashpot with damping coefficient, c. a) Determine the damping coefficient, c, for the fastest decay of the vibrations. b) For the value of c calculated above, determine the time required for the board to reach the maximum displacement. What is the value of the maximum displacement? (Hint: Velocity will be zero at the point of maximum displacement) c) [Bonus] What percentage of energy has been dissipated at the point of maximum displacement?

1. A system consists of a mass, a spring, and a dashpot. The mass weighs 19.3 lb. The spring constant of the spring is 45 lb/in. The damping constant of the dashpot is .06 lb*s/in. The amplitude of the first cycle is 3 inches. Find the amplitude 12 cycles later.

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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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Find the damping constant ( c ).

Solution Summary: The author calculates the damping constant and the acceleration due to gravity using the time displacement graph from the book.

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Find the damping constant (c).

Find the spring constant (k).

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Chapter 19 Solutions