Consider a single degree of freedom spring-mass system of spring stiffness \(k_{1}\) and mass m which has a natural frequency of 10 rad/s. What will be the spring stiffness \(k_{2} \)?
Consider a single degree of freedom spring-mass system of spring stiffness \(k_{1}\) and mass m which has a natural frequency of 10 rad/s. Consider another single degree of freedom spring-mass system of spring stiffness \(k_{2}\) and mass m which has a natural frequency of 20 rad/s. What will be the spring stiffness \(k_{2} \)?
Natural frequency of the spring mass system is given as \[\omega_{n}=\sqrt{\frac{k}{m}}\]
For the first spring
\(\omega_{n_{1}}=\sqrt{\frac{k_{1}}{m}}
\\=10\;rad/s\)
For the second spring
\(\omega_{n_{2}}=\sqrt{\frac{k_{2}}{m}}
\\=20\;rad/s
\\\Rightarrow \frac{\omega _{n1}}{\omega _{n2}}=\sqrt{\frac{k_{1}}{k_{2}}}
\\\Rightarrow \sqrt{\frac{k_{1}}{k_{2}}}=\frac{10}{20}
\\\Rightarrow \frac{k_{1}}{k_{2}}=\frac{1}{4}
\\\Rightarrow k_{2}=4k_{1}\)
