Period of beating 

Using the notation:

\(r=\) frequency ratio \(=\frac{\omega}{\omega_{n}}\)
\(\omega=\) forcing frequency \(\omega_{n}=\) natural frequency \(\zeta=\) damping ratio

a. \(\frac{2 \pi}{\omega_{n}-\omega}\)

b. \(\left[\frac{1+(2 \zeta r)^{2}}{\left(1-r^{2}\right)^{2}+(2 \zeta r)^{2}}\right]^{1 / 2}\)

c. \(\frac{\omega_{n}}{\omega_{2}-\omega_{1}}\)

d. \(\frac{1}{1-r^{2}}\)

e. \(\omega_{n} \sqrt{1-\zeta^{2}}\)

f. \(\left[\frac{1}{\left(1-r^{2}\right)^{2}+(2 \zeta r)^{2}}\right]^{1 / 2}\)