Natural Frequency Calculation

This section explains how to determine the natural frequency of a system given its free vibration equation. We will analyze the provided equation, \(\ddot x + 36{\pi ^2}x = \;0\), and calculate its natural frequency.

Understanding the Vibration Equation

The general form of a second-order linear differential equation representing free, undamped vibrations is:

$$ \ddot x + \omega_n^2 x = 0 $$

In this equation:

• \(\ddot x\) represents the acceleration (second derivative of displacement with respect to time).
• \(x\) represents the displacement from the equilibrium position.
• \(\omega_n\) is the natural angular frequency of the system, measured in radians per second (rad/s).

Deriving Natural Angular Frequency

We are given the specific free vibration equation:

$$ \ddot x + 36{\pi ^2}x = \;0 $$

By comparing this equation to the standard form \(\ddot x + \omega_n^2 x = 0\), we can identify \(\omega_n^2\):

$$ \omega_n^2 = 36{\pi ^2} $$

To find the natural angular frequency \(\omega_n\), we take the square root of both sides:

$$ \omega_n = \sqrt{36{\pi ^2}} $$

$$ \omega_n = 6\pi \text{ rad/s} $$

Converting to Natural Frequency (Hz)

The question asks for the natural frequency, typically measured in Hertz (Hz). The relationship between angular frequency (\(\omega_n\)) and frequency (\(f_n\)) is:

$$ \omega_n = 2\pi f_n $$

To find \(f_n\), we rearrange the formula:

$$ f_n = \frac{\omega_n}{2\pi} $$

Now, substitute the value of \(\omega_n\) we found:

$$ f_n = \frac{6\pi}{2\pi} $$

$$ f_n = 3 \text{ Hz} $$

Conclusion on Frequency

The natural frequency of the system described by the equation \(\ddot x + 36{\pi ^2}x = \;0\) is calculated to be 3 Hz.