A 50-kg machine is mounted on a table of stiffness 1 ×10^5 N / m. During operation it is subjec

step 1 Determine the amplitude of vibration. The equation for amplitude of vibration, A, is E m omega s

A 50-kg machine is mounted on a table of stiffness 1 ×10^5...

A $50-\mathrm{kg}$ machine is mounted on a table of stiffness $1 \times 10^5 \mathrm{~N} / \mathrm{m}$. During operation it is subjected to a harmonic excitation of magnitude $1200 \mathrm{~N}$ at $45 \mathrm{rad} / \mathrm{s}$. What is the required stiffness of a $4-\mathrm{kg}$ absorber to eliminate steady-state vibrations of the machine during operation?

Key Concepts

Tuning of Absorbers

Tuning a vibration absorber involves selecting parameters such as mass and stiffness such that the absorber's natural frequency aligns with the excitation frequency. The relationship k = m?², where k is stiffness, m is mass, and ? is the angular frequency, is central to this process. Proper tuning ensures that the absorber effectively cancels out the steady-state vibrations, leading to improved stability and performance of the primary system.

Dynamic Vibration Absorber

A dynamic vibration absorber is a device used to reduce or eliminate unwanted vibrations in a primary system by attaching an auxiliary mass–spring system. When properly tuned to the excitation frequency, the absorber creates a counteracting force that minimizes the amplitude of the vibration in the main structure. The effectiveness of a vibration absorber depends on accurate matching of its natural frequency to the frequency of excitation.

Harmonic Excitation and Forced Response

This concept involves systems subjected to periodic external forces. In such systems, the steady-state response is governed by the characteristics of the forcing function and the dynamic properties of the system, such as mass and stiffness. Understanding how the amplitude and frequency of an applied harmonic force influence the system's vibrations is essential in designing methods to mitigate unwanted oscillations.

Natural Frequency

The natural frequency of a system characterizes the rate at which it tends to oscillate in the absence of damping or external forcing. It is determined by the system’s mass and stiffness, typically through the relationship ? = ?(k/m). Knowing the natural frequency is crucial in applications like vibration absorbers, where resonance conditions need to be avoided or exploited for control purposes.

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