In this paper, a two-stage method is introduced to design fixed-order data-driven H-infinity controller for flexible mechanical systems. In the first stage of the proposed method, unknown parameters of anti-resonance filter that is added to the forward path of the control loop of the system to minimize resonant peaks, are calculated using frequency domain data obtained from open-loop system identification tests. In the second stage, a fixed-order data-driven H-infinity controller is calculated by solving an optimization problem under convex H-infinity constraints obtained based on the Nyquist diagram. With the proposed method, lower order controllers that meets the performance constraints of classical model-based H-infinity problems can be synthesized without need of a parametric plant model. The method developed in this study is tested experimentally on a military stabilized platform and its performance is compared with a model-based H-infinity controller design method.