A 15-kg undamped absorber tuned to 250 rad / s is added to a 150-kg machine mounted on a foundat

step 1 The main system, M1, is 40 kilograms. K1 is 0 .1 mega -newts per meter. So that's 100 ,000 newto

A 15-kg undamped absorber tuned to 250 rad / s is added to a...

A $15-\mathrm{kg}$ undamped absorber tuned to $250 \mathrm{rad} / \mathrm{s}$ is added to a $150-\mathrm{kg}$ machine mounted on a foundation of stiffness $1 \times 10^7 \mathrm{~N} / \mathrm{m}$. At $250 \mathrm{rad} / \mathrm{s}$, the amplitude of the absorber mass is $3.9 \mathrm{~mm}$. What is the amplitude of the machine at $275 \mathrm{rad} / \mathrm{s}$ ?

Key Concepts

Tuned Vibration Absorber

A tuned vibration absorber is a passive device attached to a vibrating structure to mitigate its oscillations by transferring energy from the primary system to the absorber. It is designed to have a natural frequency that matches the excitation frequency of the primary system, thereby reducing the amplitude of vibrations at that targeted frequency through destructive interference.

Mass-Spring Systems and Natural Frequency

Mass-spring systems form the fundamental basis of understanding vibrations in mechanical systems, where the combination of mass and stiffness determines the system's natural frequency. This natural frequency is the rate at which the system tends to oscillate when disturbed from its equilibrium, and it plays a crucial role in the design of vibration mitigation devices such as absorbers.

Frequency Response in Coupled Systems

The frequency response of a system describes how the amplitude of its vibrations changes in reaction to different excitation frequencies. In coupled systems—where a primary system is linked with an absorber—the interaction between the components leads to complex responses that shift with changes in frequency, particularly near resonance, necessitating careful analysis for vibration control.

Detuning Effects in Vibration Absorbers

Detuning refers to the deviation of the excitation or operating frequency from the exact tuned frequency of the vibration absorber. Even slight detuning can significantly affect the efficiency of the absorber, altering the energy transfer between the absorber and the primary system and resulting in variations in vibration amplitudes. Understanding detuning is essential for assessing the performance of vibration control systems under real-world conditions.

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