Gyroscopes are used as rotation rate sensors. Conventional gyroscopes are heavy and bulky, which creates important problems regarding their usage in different applications. Micro-gyroscopes have solved these problems due to their small size. The beam micro-gyroscope is one of the popular types of inertial sensors. Their small dimensions and low energy consumption are key reasons for their popularity. In this investigation, the model of an electrostatically actuated beam-based micro-gyroscope is used to study the effect of design parameters on pull-in voltage and fundamental frequency. The micro-gyroscope includes a rotating cantilever beam and a tip mass attached to the free end. DC voltages are applied to both sense and drive electrodes to actuate the system. The tip mass is actuated by an AC voltage in the drive direction to produce oscillations in the sense direction. Equations of motion are solved numerically to study different pull-in and vibrational parameters. Eigenvalues of the uncoupled system are computed to obtain the fundamental frequency of the micro beam for different values of DC voltages and design parameters. The frequencies are computed and validated with those in the literature. The results are beneficial for the design process of micro-gyroscopes.