Explanation Calculation: Since ( c < c c ) which is under damped, the displacements x 1 , x 2 and x 3 may be assumed equal to maximum displacements. The expression for the differential equation of under damping as follows: x = x 0 e − ( c 2 m ) t sin ( ω 0 t + ϕ ) (1) Here, x is the displacement, x 0 is the initial displacement, c is the coefficient of viscous damping, m is the mass of the body, ω 0 is the frequency, t is the time, and ϕ is the phase angle. Apply boundary conditions at the maximum displacements. When t = t n then x = x n . Substitute t n for t , x n for x in the equation (1). x n = x 0 e − ( c 2 m ) t n sin ( ω 0 t n + ϕ ) Substitute 1 for sin ( ω 0 t n + ϕ ) x n = x 0 e − ( c 2 m ) t n ( 1 ) = x 0 e − ( c 2 m ) t n Apply boundary conditions at the next maximum displacements. When t = t n + k then x = x n + k . Substitute t n + k for t , x n + k for x in the equation (1). x n + k = x 0 e − ( c 2 m ) t n + k sin ( ω 0 t n + k + ϕ ) Substitute 1 for sin ( ω 0 t n + 1 + ϕ ) x n + k = x 0 e − ( c 2 m ) t n + k ( 1 ) = x 0 e − ( c 2 m ) t n + k The expression for the time at the maximum displacement and next maximum displacement as follows; ω D t n + k − ω D t n = k 2 π ω D ( t n + k − t n ) = k 2 π t n + k − t n = k 2 π ω D t n − t n + k = − k 2 π ω D Here, ω D is the damped circular frequency. Calculate the ratio of two successive displacements ( x n x n + k ) using the relation: Ratio of two successive displacement = x n x n + k Substitute x 0 e − ( c 2 m ) t n + k for x n + k and x 0 e − ( c 2 m ) t n for x n

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

See similar textbooks

Concept explainers

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…

Question

Chapter 19.5, Problem 19.130P

To determine

Show that the logarithmic decrement can also expressed as (1k)ln(xnxn+k).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 19.5, Problem 19.129P

Chapter 19.5, Problem 19.131P

Students have asked these similar questions

Using the data of Table 19.1, determine the length in inches of a simple pendulum that oscillates with a period of 2 s and an amplitude of 90°. Reference to Table 19.1:

A vibrometer used to measure the amplitude of vibrations consists essentially of a box containing a mass-spring system with a known natural frequency of 120 Hz. The box is rigidly attached to a surface that is moving according to the equation y= δm sin wf t. If the amplitude zm of the motion of the mass relative to the box is used as a measure of the amplitude δm of the vibration of the surface, determine (a) the percent error when the frequency of the vibration is 600 Hz,(b) the frequency at which the error is zero.

Two springs of constants K1 and K2 are connected in series to a block A that vibrates in simple harmonic motion with a period of 5 s. When the same two springs are connected in parallel to the same block, the block vibrates with a period of 2 s. Determine the ratio K1/K2 of the two spring constants.

Chapter 19 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 19.1 - A particle moves in simple harmonic motion....Ch. 19.1 - A particle moves in simple harmonic motion....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 - A simple pendulum consisting of a bob attached to...Ch. 19.1 - Prob. 19.9P Ch. 19.1 - Prob. 19.10P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

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

Solution Summary: The author explains that the logarithmic decrement can also be expressed as underset_(1k)mathrmln

Concept explainers

Show that the logarithmic decrement can also expressed as (1k)ln(xnxn+k).

Want to see the full answer?

Chapter 19 Solutions