Flexible, lightweight manipulators offer some advantages in contrast to rigid arms, such as compact and lighter drives, energy efficiency, reduced masses and costs. This paper presents a novel approach for vibration damping of a multi-link flexible arm. The strain of the elastic arms is measured with Fiber-Bragg-Grating (FBG) sensors and provides the feedback signal to dampen their flexural dynamics. A dynamic model of a three link arm is derived that accounts for the rigid and flexural dynamics including gravity. The arm vibrations are damped by nonlinear strain feedback. The controller is general and robust and its design does not require a model of the flexural dynamics. In the context of closed loop vibration control FBG sensors offer a better signal to noise ratio compared to strain gauges, which allows a higher static gain in the feedback loop with more efficient dissipation of vibrational energy. The feasibility and effectiveness of the proposed vibration control scheme in conjunction with FBG sensors is verified and analyzed in simulations and confirmed in experiments with a flexible three link robot arm.