Owing to the flexible structures, the mechanical vibration excited during the operation of the flexible arm impacts the control precision and stability of the flexible arm. To eliminate vibrations, the viscous damper can be installed in a flexible arm. Damping and mass are introduced into the system. However, without a stiffness element to support the mass, the damper cannot effectively function against gravitational force. Given this, a design of a viscous damper based on the principle of viscous damping and quasi-zero stiffness is proposed. The rhombic mechanism, which possesses quasi-zero stiffness, can counteract the gravitational impact on the mass. The structural parameters of the damper are optimized based on the optimal damping coefficient obtained from the frequency response function. Modal tests demonstrate a significant reduction in the amplitude of the host structure due to the viscous damper.