When a 10-kg undamped absorber tuned to 100 rad / s is added to a 1-degree-of-freedom structure

step 1 So if we start off with the equation of motion, J -Nought is equal to J -G plus M -B -squared is

When a 10-kg undamped absorber tuned to 100 rad / s is added...

When a $10-\mathrm{kg}$ undamped absorber tuned to $100 \mathrm{rad} / \mathrm{s}$ is added to a 1-degree-of-freedom structure of stiffness $5 \times 10^6 \mathrm{~N} / \mathrm{m}$, the lowest natural frequency of the structure is $85.44 \mathrm{rad} / \mathrm{s}$. What is the higher natural frequency of the structure?

Key Concepts

Natural Frequency

The natural frequency is an intrinsic property of a system that defines the rate at which it will oscillate when disturbed from its equilibrium position in the absence of any external forcing or damping. It is determined by the system's stiffness and mass distribution, and plays a key role in understanding resonant behavior in mechanical systems.

Degree-of-Freedom Systems

A degree-of-freedom (DOF) system is one in which the motion of the system can be described by a minimum number of independent coordinates. In vibration analysis, a system's DOF determines how many independent motions contribute to its dynamic response, and when additional components such as absorbers are added, the system becomes a multi-DOF system where coupled modes must be analyzed.

Tuned Mass Absorber

A tuned mass absorber is a secondary device attached to a primary structure, designed to mitigate vibrations by transferring energy away from the primary system. It is tuned to a specific frequency to counteract the motion of the structure, often resulting in the splitting of the system's natural frequencies. In its undamped form, it alters the dynamic properties of the coupled system without dissipating energy.

Coupled Vibration Analysis

Coupled vibration analysis involves studying systems where multiple masses and stiffness elements interact, leading to the formation of multiple natural frequencies and mode shapes. The introduction of additional masses, such as a tuned absorber, results in the splitting of the original natural frequency into lower and higher modes due to the coupling between the primary structure and the absorber.

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