Vibrating System Natural Frequency Calculation

Understanding the natural frequency of a vibrating system is crucial in mechanical engineering. This problem asks us to calculate the natural frequency of vibration for a given system, which consists of a specific mass and a spring with a defined stiffness.

System Parameters for Vibration Analysis

We are provided with the following key parameters for the vibrating system:

To accurately calculate the natural frequency of vibration, it is essential to ensure all units are consistent, preferably in the International System of Units (SI). The mass is already in kilograms (kg), but the spring stiffness is given in Newtons per millimeter (N/mm), which needs to be converted to Newtons per meter (N/m).

Converting Spring Stiffness Units for Vibration

The spring stiffness \(k\) needs to be converted from N/mm to N/m. We know that 1 meter (m) is equal to 1000 millimeters (mm). Therefore, to convert from N/mm to N/m, we multiply the value by 1000:

\[k = 80 \text{ N/mm} \times \frac{1000 \text{ mm}}{1 \text{ m}} = 80000 \text{ N/m}\]

Now, we have the mass in kilograms (kg) and the spring stiffness in Newtons per meter (N/m), which are the standard SI units required for the natural frequency formula.

Natural Frequency of Vibration Formula

For a simple undamped single degree of freedom vibrating system, the natural frequency of vibration (\(\omega_n\)) is determined by the square root of the ratio of the spring stiffness (\(k\)) to the mass (\(m\)). The fundamental formula for natural frequency is:

\[\omega_n = \sqrt{\frac{k}{m}}\]

where:

• \(\omega_n\) is the natural frequency in radians per second (rad/s)
• \(k\) is the spring stiffness in Newtons per meter (N/m)
• \(m\) is the mass in kilograms (kg)

Calculating Natural Frequency of the System

Now, we will substitute the converted spring stiffness and the given mass into the natural frequency formula to find the natural frequency of vibration.

• Step 1: Identify given values with consistent units
  • Mass, \(m = 200 \text{ kg}\)
  • Spring stiffness, \(k = 80000 \text{ N/m}\) (after conversion)
• Step 2: Apply the natural frequency formula

  \[\omega_n = \sqrt{\frac{k}{m}}\]

• Step 3: Substitute the values and calculate

  \[\omega_n = \sqrt{\frac{80000 \text{ N/m}}{200 \text{ kg}}}\]

  First, perform the division inside the square root:

  \[\omega_n = \sqrt{400 \text{ rad}^2/\text{s}^2}\]

  Then, take the square root:

  \[\omega_n = 20 \text{ rad/s}\]

Resulting Natural Frequency

The natural frequency of vibration for the given vibrating system is 20 rad/s.

This value represents the frequency at which the system would oscillate if it were disturbed and allowed to vibrate freely without any external forces or damping effects. This natural frequency is a critical parameter in designing systems to avoid resonance and ensure stable operation.