The mass of a single degree damped system is 8 kg and makes 28 free oscillations in 16 secs. TheAmplitude however reduces to 0.3 of its initial value after 5 complete oscillations.

Determine:

i. Stiffness of the spring

ii. Logarithmic decrement

iii. Damping factor.

Given : m = 8 kg

Since 28 oscillations are made in 16 seconds, therefore frequency of free vibrations, fn = 28/16 = 1.75 and $$ \begin{array}{r} \omega_{n}=2 \pi \times f_{n}=2 \pi \times 1.75 \\ =11 \mathrm{rad} / \mathrm{s} \end{array} $$

1.Stiffness of the spring

Let s = stiffness of the spring in N/m.

We know that $$ \begin{aligned} \left(\omega_{n}\right)^{2}=& s / m \text { or } s=\left(\omega_{n}\right)^{2} m \\ &=(11)^{2} 8=968 \mathrm{~N} / \mathrm{m} \end{aligned} $$

2.Logarithmic decrement

Let x1 = Initial amplitude, x6 = Final amplitude after five oscillations = 0.3 x1 (Given)

$$ \begin{gathered} \frac{x_{1}}{x_{6}}=\frac{x_{1}}{x_{2}} \times \frac{x_{2}}{x_{3}} \times \frac{x_{3}}{x_{4}} \times \frac{x_{4}}{x_{5}} \times \frac{x_{5}}{x_{6}}=\left(\frac{x_{1}}{x_{2}}\right)^{5} \\ \ldots\left[\because \frac{x_{1}}{x_{2}}=\frac{x_{2}}{x_{3}}=\frac{x_{3}}{x_{4}}=\frac{x_{4}}{x_{5}}=\frac{x_{5}}{x_{6}}\right] \end{gathered} $$

$$ \begin{gathered} \frac{x_{1}}{x_{2}}=\left(\frac{x_{1}}{x_{6}}\right)^{1 / 5}=\left(\frac{x_{1}}{0.3 x_{1}}\right)^{1 / 5} \\ =1.27 \end{gathered} $$

$$ \delta=\log _{e}\left(\frac{x_{1}}{x_{2}}\right)=\log _{e} 1.27=0.24 $$

3. Damping factor

Let c = damping coefficient for the actual system, and cc = Damping coefficient for the critical damped system. We know that logarithmic decrement (δ)

$$ 0.24=\frac{a \times 2 \pi}{\sqrt{\left(\omega_{n}\right)^{2}-a^{2}}}=\frac{a \times 2 \pi}{\sqrt{(11)^{2}-a^{2}}} $$ $$ 0.0576=\frac{a^{2} \times 39.5}{121-a^{2}} $$ ... (Squaring both sides) $$ 6.9696-0.0576 a^{2}=39.5 a^{2} $$ or $a^{2}=0.17618$ or $a=0.419$ We know that $$ \begin{aligned} &a=c / 2 m \\ &\text { or } \begin{aligned} c=a \times 2 m=0.419 \times 2 \times 8 \\ &=6.7 \mathrm{~N} / \mathrm{m} / \mathrm{s} \\ \text { and } \quad c_{c} &=2 \mathrm{~m} . \omega_{n}=2 \times 8 \times 11\\ &=176 \mathrm{~N} / \mathrm{m} / \mathrm{s} \end{aligned} \\ &\therefore \text { Damping factor }=c / c_{c} \\ &=6.7 / 176=0.038 \end{aligned} $$