Explanation Given: Refer to Fig.19.11; the displacements x 1 , x 2 , x shown are equal to the maximum displacements. 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 + 1 for x and t n + 1 for t in equation (5). x n + 1 = x 0 e − ( c 2 m ) t n + 1 sin ( ω d t + ϕ ) ...... (8) Divide equation (7) by equation (8). x n x n + 1 = e ( − c 2 m ) ( t n − t n + 1 ) = e ( c 2 m ) ( t n + 1 − t n ) Substitute 2 π ω d for ( t n + 1 − t n ) in above expression

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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 19.129P

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Show that the logarithmic ratio of two successive maximum displacement is equal to 2π(c/cc)1−(c/cc)2.

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Chapter 19.5, Problem 19.128P

Chapter 19.5, Problem 19.130P

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Show that the logarithmic ratio of two successive maximum displacement is equal to 2 π ( c / c c ) 1 − ( c / c c ) 2 .

Solution Summary: The author explains that the logarithmic ratio of two successive maximum displacements is equal to 2pi.

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Show that the logarithmic ratio of two successive maximum displacement is equal to 2π(c/cc)1−(c/cc)2.

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