For a vibrating system, the successive amplitudes of vibration obtained under free effects are 0.70, 0.28, 0.25, 0.23, and 0.067 respectively. The value of the damping ratio of the system is given by:

Concept:

Ratio of successive amplitude is given by

\(\frac{{{{\bf{X}}_0}}}{{{{\bf{X}}_1}}} = \frac{{{{\bf{X}}_1}}}{{{{\bf{X}}_2}}} = \frac{{{{\bf{X}}_2}}}{{{{\bf{X}}_3}}} = \ldots \ldots \ldots \ldots \ldots \ldots = \frac{{{{\bf{X}}_{\bf{n}}}}}{{{{\bf{X}}_{{\bf{n}} + 1}}}} = {{\bf{e}}^{\bf{δ }}}\)

where δ = Logarithmic amplitude

Logarithmic decrement is given by

\({\bf{δ }} = \frac{1}{{\bf{n-1}}}{\bf{ln}}\left( {\frac{{{{\bf{X}}_{\bf{o}}}}}{{{{\bf{X}}_{\bf{n}}}}}} \right)=\frac{1}{{\bf{n}}}{\bf{ln}}\left( {\frac{{{{\bf{X}}_{\bf{o}}}}}{{{{\bf{X}}_{\bf{n+1}}}}}} \right)\)

Damping ratio(ζ) is given by

\(ζ = \frac{δ }{{\sqrt {4{\pi ^2}\; + \;{δ ^2}} }}\)

Calculation:

Given:

X₀ = 0.7, X₁ = 0.28, X₂ = 0.25, X₃ = 0.23, X₄ = 0.067, ζ = ?

Now, we know that

\({\bf{δ }} = \frac{1}{{\bf{n}}}{\bf{ln}}\left( {\frac{{{{\bf{X}}_{\bf{o}}}}}{{{{\bf{X}}_{\bf{n}}}}}} \right) = \frac{1}{4}ln\left( {\frac{{0.7}}{{0.067}}} \right)\)

∴ δ = 0.587

The damping ratio is given by

\(ζ = \frac{\delta }{{\sqrt {4{\pi ^2}\; + \;{\delta ^2}} }} = \;\frac{{0.587}}{{\sqrt {4{\pi ^2}\; + \;{{0.587}^2}} }}\)

∴ ζ = 0.0928