Explanation Given: Concept used: Write the equation of motion a system containing spring, mass and damper. m x ¨ + c x ˙ + k x = 0 ...... (1) Here, m is the mass, k is the spring constant, c is the damping constant, x ˙ is the velocity of damper and x ¨ is the acceleration of mass. For light damping ( c < c c ) the solution for equation (1) is. x = e ( − c 2 m ) t ( c 1 sin ω d t + c 2 cos ω d t ) ...... (2) Here, x is the displacement of system, t is time, ω d is the damped frequency, and c 1 , c 2 are the constants. Write the expression for the damped frequency of system. ω d = ω n 1 − ( c / c c ) 2 ...... (3) Here, ω n is the natural frequency of system and c c is the critical damping coefficient of damper. Write the expression for the vibration diminishing amplitudes. x = x 0 e − ( c 2 m ) t sin ( ω d t + ϕ ) ...... (5) Here, x is the diminished amplitude and ϕ is the phase angle. Write the expression for natural frequency of system. ω n = c c 2 m ...... (6) Calculation: Substitute x n for x and t n for t in equation (5). x n = x 0 e − ( c 2 m ) t n sin ( ω d t + ϕ ) ...... (7) Substitute x n + k for x and t n + k for t in equation (5). x n + k = x 0 e − ( c 2 m ) t n + k sin ( ω d t + ϕ ) ...... (8) Here, k is the number of cycles between readings of maximum displacement. Divide equation (7) by equation (8). x n x n + k = e ( − c 2 m ) ( t n − t n + k ) = e ( c 2 m ) ( t n + k − t n ) Substitute 2 π k ω d for ( t n + k − t n ) in above expression

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 19.130P

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Show that logarithmic decrement can be expressed as 1k(lnxnxn+1).

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Chapter 19.5, Problem 19.129P

Chapter 19.5, Problem 19.131P

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Chapter 19 Solutions

Vector Mechanics for Engineers: Dynamics

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Show that logarithmic decrement can be expressed as 1 k ( ln x n x n + 1 ) .

Solution Summary: The author explains how the logarithmic ratio of two successive maximum displacements can be expressed as 1kleft.

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Show that logarithmic decrement can be expressed as 1k(lnxnxn+1).

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Chapter 19 Solutions