For a single degree of freedom viscous damped system, if the frequency ratio is greater than √2, it implies that:

Concept:

Machine transmissibility,

\(\epsilon = \frac{F_t}{F_o} \)

\(\Rightarrow \epsilon = \frac{{\sqrt {1 + {{\left( {\frac{{2\zeta \omega }}{{{\omega _n}}}} \right)}^2}} }}{{\sqrt {{{\left( {1 - {{\left( {\frac{\omega }{{{\omega _n}}}} \right)}^2}} \right)}^2} + {{\left( {\frac{{2\zeta \omega }}{{{\omega _n}}}} \right)}^2}} }}\)

Machine transmissibility depends upon 'ω /ωn'→ (within the machine) = q (say)

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ζ = Damping factor

Damping is beneficial up to a ratio of 

\(\frac{\omega }{{{\omega _n}}}\left\langle {\sqrt 2 , \epsilon\;\;will \uparrow ,\;} \epsilon \right\rangle 1,\; \epsilon = machine\;transmissability\)

Whereas no need for damping, less damping, If the ratio of

\(\frac{\omega }{{{\omega _n}}} > \sqrt 2 ,\;\epsilon \;will \downarrow ,\;\epsilon < 1\)

Explanation:

• For all the value of the damping ratio, the transmissibility is less than one when the ratio of natural and forced frequency is greater than root 2 or \(\frac{\omega }{{{\omega _n}}} > \sqrt 2\)

⇒ \(\epsilon = \frac{F_t}{F_o} <1\)

• For all the value of damping ratio, the transmissibility is greater than one when the ratio of natural and forced frequency is less than root 2 and here the chances of getting infinite frequency is also there when the ratio will be 1, so we have to provide damping here only.