A factory floor bay has a lowest structural mode that behaves like a single…

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations · Coupled oscillators, normal modes and Fourier analysis

Modes, Packets and Media, Ch Normal modes of many coupled oscillators: eigenvalues, symmetry and response

Problem

A factory floor bay has a lowest structural mode that behaves like a single oscillator of modal mass

m_1 = 800\,\mathrm{kg}

and natural frequency

6.00\,\mathrm{Hz}

. A compressor bolted to the bay drives it with a force

F\cos\omega t

6.00\,\mathrm{Hz}

. An undamped tuned absorber of mass ratio

\mu = m_2/m_1 = 0.030

is to hold the bay still at that frequency. ((a)) Find the absorber mass

m_2

and its spring stiffness

\kappa_2

. ((b)) Find the two new resonance frequencies of the bay with its absorber attached, in hertz. ((c)) The compressor's speed drifts. Solve the two equations of motion for the bay's steady amplitude and find the band of drive frequencies, in hertz, over which the absorber keeps that amplitude below

F/\kappa_1

, the static deflection the same force would give the bay alone.

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