Solution:

Maximum acceleration is $ a_{\text{max}} = A\omega^2 = \alpha $ and maximum velocity is $ v_{\text{max}} = A\omega = \beta $. Dividing them, we get $ \omega = \frac{\alpha}{\beta} $. Thus, the time period is $ T = \frac{2\pi}{\omega} = \frac{2\pi \beta}{\alpha} $.