The equation of free vibration in a system is \(\ddot x + 36{\pi ^2}x = \;0\). Its natural frequency is
3 Hz
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.
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