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

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

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

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

In practice, it is often difficult to determine the logarithmic decrement of a system with light damping defined in Prob. 19.129 by measuring two successive maximum displacements. Show that the logarithmic decrement can also be expressed as (1/k) ln(xn/xn+k), where k is the number of cycles between readings of the maximum displacement.

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 compact object with a mass of 8.80 kg oscillates at the end of a vertical spring with a spring constant of 1.60 ✕ 104 N/m. The motion is damped by air resistance, and the damping coefficient is b = 3.00 N · s/m. (a) What is the frequency (in Hz) of the damped oscillation? Hz (b) By what percentage does the amplitude of the oscillation decrease in each cycle? % (c) Over what time interval (in s) does the energy of the system drop to 5.00% of its initial value? s (d) What If? The atmosphere of Venus is 50 times thicker than that on Earth. If the effect of air resistance on Venus is represented by b = 150 N · s/m, recalculate the answers for parts (a) to (c) for this system if it is set in motion in the atmosphere of Venus. What is the frequency (in Hz) of the damped oscillations? Hz What is the percentage decrease in amplitude in each cycle? % What is the time interval (in s) for the energy to drop to 5.00% of its initial value? s

Chapter 19 Solutions

Vector Mechanics For Engineers

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 - A 20-lb block is initially held so that the...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 - A 5-kg fragile glass vase is surrounded by packing...

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 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.

Concept explainers

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

Want to see the full answer?

Chapter 19 Solutions