A machine of mass 100 kg is subjected to an external harmonic force with a frequency of 40 rad/s. The designer decides to mount the machine on an isolator to reduce the force transmitted to the foundation. The isolator can be considered as a combination of stiffness (K) and damper (damping factor, ζ) in parallel. The designer has the following four isolators: 1) k = 640 kN/m, ζ = 0.70 2) k = 640 kN/m, ζ = 0.07 3) k = 22.5 kN/m, ζ = 0.70 4) k = 22.5 kN/m, ζ = 0.07 Arrange the isolators in the ascending order of the force transmitted to the foundation.
4-3-1-2
Use force transmissibility \(T_R=\sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}}\), where \(r=\omega/\sqrt{k/m}\). The four values are approximately 1.190, 1.331, 0.540 and 0.174. Thus the increasing order is 4–3–1–2.
Important Questions from Transmissibility and Magnification Factor
When ω/ωn = 1, the transmissibility is
where ω = frequency of excitation, rad/s
ωn = Natural frequency, rad/s
The Transmissibility of a vibrating system, for the values of (ω/ωn) >√2, will be
A vibrating system having a mass of 30 kg is supported by 4 springs, each having a stiffness of k. The system runs at 420 rpm. If only 10% of the shaking force is allowed to be transmitted to the supporting structure, then the value of stiffness k will be close to:
For a single degree of freedom viscous damped system, if the frequency ratio is greater than √2, it implies that:
Dynamic Magnification Factor is computed as:
