This paper presents a design of oscillation damping control for the cable-Suspended Aerial Manipulator (SAM). The SAM is modeled as a double pendulum, and it can generate a body wrench as a control action. The main challenge is the fact that there is only one onboard IMU sensor which does not provide full information on the system state. To overcome this difficulty, we design a controller motivated by a simplified SAM model. The proposed controller is very simple yet robust to model uncertainties. Moreover, we propose a gain tuning rule by formulating the proposed controller in the form of output feedback linear quadratic regulation problem. Consequently, it is possible to quickly dampen oscillations with minimal energy consumption. The proposed approach is validated through simulations and experiments.