Cable-driven parallel robots (CDPRs) are more convenient to use in large-scale applications than rigid-link robots due to the advantages such as simple structure, high load capacity, and low inertia of cables. However, using cables to drive an end-effector comes with the low equivalent stiffness issue in out-of-plane direction for the CDPRs' planar configuration, which causes long-lasting and high amplitude out-of-plane vibrations under an external disturbance. This article presents a novel method to suppress the out-of-plane vibrations without distorting in-plane positioning control simultaneously for planar CDPRs. The proposed method provides the ease of using only the cable tensioning forces instead of an additional actuator to control the out-of-plane vibrations by decoupling the in-plane and out-of-plane dynamics. The switching stiffness control method is used for out-of-plane vibration control, while a classical control method is used for in-plane dynamics control. These two control inputs are then coupled by a bias torque variation in a torque distribution algorithm. The effectiveness of the proposed method is presented both in simulations and experimental results. The experiments are performed for the specified in-plane positions and an in-plane trajectory tracking study when the planar CDPR is excited to out-of-plane vibration in the normal direction. The out-of-plane vibration suppression time is considerably reduced compared to the free-vibration case, and any distortion on the in-plane positioning control is not observed.