A piping system experiences resonance when the pump...

A piping system experiences resonance when the pump supplying power to the system operates at $500 \mathrm{r} / \mathrm{min}$. When a $5-\mathrm{kg}$ absorber tuned to $500 \mathrm{r} / \mathrm{min}$ is added to the pipe, the system's new natural frequencies are measured as 380 and $624 \mathrm{r} / \mathrm{min}$. What is the natural frequency of the piping system and its equivalent mass?

Key Concepts

Natural Frequency

The natural frequency is the inherent oscillation rate of a system without external forcing, determined by the balance of stiffness and mass. This fundamental property is critical because when a system is excited near its natural frequency, even small forces can result in large amplitude oscillations.

Resonance

Resonance occurs when an external force or input frequency matches the natural frequency of a system, leading to significantly amplified vibrations. This phenomenon is important in engineering design because it can cause excessive oscillations and potentially damage the structure if not properly controlled.

Vibration Absorber (Tuned Mass Damper)

A vibration absorber, often referred to as a tuned mass damper, is an additional mass that is deliberately tuned to a specific frequency to counteract vibrational energy. By attaching this absorber to a system, it interacts dynamically to shift the natural frequency and mitigate the effects of resonance, thereby stabilizing the system.

Coupled Oscillators and Frequency Splitting

When a vibration absorber is connected to a system, the two become coupled oscillators. Their dynamic interaction leads to the splitting of the original natural frequency into two separate modes. This phenomenon, known as frequency splitting, reflects the altered stiffness and mass distribution due to the absorber’s presence, resulting in two distinct oscillation frequencies.

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