By what factor does the amplitude change in one period of these damped oscillations? A resistive force is applied to the same mechanical system above, and the amplitude of oscillations decreases 100 times after 1s from the start of oscillations. What is the damping constant? By what factor does the amplitude change in one period of these damped oscillations?**Diagram**:An illustration of a mechanical system is shown, with a mass \( M \) connected to a spring \( K \). This is labeled as "This is mechanical system above."**Equations and Solution**:1. The displacement in a damped oscillation system is given by: \[ x(t) = Ae^{-\beta t} \cos(\omega t - \delta) \quad t = 0 \]2. Examining the change in amplitude over time: \[ \frac{A}{100} = A \cdot e^{-\beta t} \]3. Taking logarithms to solve for \(\beta\): \[ \ln\left(\frac{1}{100}\right) = \ln\left(e^{-\beta t}\right) \]4. Simplifying the equation: \[ + \beta t = \frac{\ln\left(\frac{1}{100}\right)}{-t} \quad t = 1 \]5. Solving for the damping constant \(\beta\): \[ \beta = 4.605 \]The damping constant \(\beta\) is found to be 4.605. This value determines how quickly the amplitude of the oscillations decreases.

Equation of the motion of a damped harmonic oscillator:The general solution of the above equation.…

A resistive force is applied to the same mechanical system above, and the amplitude of oscillations decreases 100 times after 1s from the start of oscillations. What is the dampin constant? By what factor does the amplitude change in one period of these damped oscillations?

A resistive force is applied to the same mechanical system above, and the amplitude of oscillations decreases 100 times after 1s from the start of oscillations. What is the dampin constant? By what factor does the amplitude change in one period of these damped oscillations?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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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 initial displacement, force is removed, and the system vibrates at the natural frequency.

Question

By what factor does the amplitude change in one period of these damped oscillations?

A resistive force is applied to the same mechanical system above, and the amplitude of oscillations decreases 100 times after 1s from the start of oscillations. What is the damping constant? By what factor does the amplitude change in one period of these damped oscillations?

**Diagram**:
An illustration of a mechanical system is shown, with a mass \( M \) connected to a spring \( K \). This is labeled as "This is mechanical system above."

**Equations and Solution**:

1. The displacement in a damped oscillation system is given by:
\[
x(t) = Ae^{-\beta t} \cos(\omega t - \delta) \quad t = 0
\]

2. Examining the change in amplitude over time:
\[
\frac{A}{100} = A \cdot e^{-\beta t}
\]

3. Taking logarithms to solve for \(\beta\):
\[
\ln\left(\frac{1}{100}\right) = \ln\left(e^{-\beta t}\right)
\]

4. Simplifying the equation:
\[
+ \beta t = \frac{\ln\left(\frac{1}{100}\right)}{-t} \quad t = 1
\]

5. Solving for the damping constant \(\beta\):
\[
\beta = 4.605
\]

The damping constant \(\beta\) is found to be 4.605. This value determines how quickly the amplitude of the oscillations decreases.

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