A random vibration analysis was accomplished on the tip-tilt system of the fast steering secondary mirror (FSM) for the Giant Magellan Telescope (GMT). As the FSM was to be mounted on the top end of the secondary truss and disturbed by the winds, dynamic effects of the FSM disturbances on the tip-tilt correction performance was studied. The coupled dynamic responses of the FSM segments were evaluated with a suggested tip-tilt correction modeling. Dynamic equations for the tip-tilt system were derived from the force and moment equilibrium on the segment mirror and the geometric compatibility conditions with four design parameters. Statically stationary responses for the tip-tilt actuations to correct the wind-induced disturbances were studied with two design parameters based on the spectral density function of the star image errors in the frequency domain. Frequency response functions and root mean square values of the dynamic responses and the residual star image errors were numerically calculated for the off-axis and on-axis segments of the FSM. A prototype of on-axis segment of the FSM was developed for tip-tilt actuation tests to confirm the ratio of tip-tilt force to tip-tilt angle calculated from the suggested dynamic equations of the tip-tilt system. Tip-tilt actuation tests were executed at 4, 8 and 12 Hz by measuring displacements of piezoelectric actuators and reaction forces acting on the axial supports. The derived ratios of rms tip-tilt force to rms tip-tilt angle from tests showed a good correlation with the numerical results. The suggested process of random vibration analysis on the tip-tilt system to correct the wind-induced disturbances of the FSM segments would be useful to advance the FSM design and upgrade the capability to achieve the least residual star image errors by understanding the details of dynamics.