The free vibrations of a thin elastic orthotropic open cylindrical shell (panel) with one edge free and the other three clamped are considered. Dispersion equations to find the natural frequencies of possible types of vibrations are derived using the classical theory of elasticity. An asymptotic relation between the dispersion equations of the problem under consideration and the similar problem for an orthotropic rectangular plate is established. It is also proved that the dispersion equations of the problem are in asymptotic relationship with the dispersion equations for a semi-infinite open orthotropic cylindrical shell with one free end and two clamped longitudinal edges. Open orthotropic and isotropic shells of different length are considered as an example to obtain approximate values for the dimensionless natural frequency and damping factors for the vibration modes