In an elevator system, so as to avoid unwanted safety gear tripping, the kinetic energy, which is caused by inertia of the overspeed governor rope to the lever arm, is dissipated by implementing fluid viscous damping to dampen the rotary movement of the spindle shaft to prevent unwanted safety gear tripping in the event when the upwards movement of the moving mass is decelerated by a machinery brake to perform a quick stop of the moving mass. The fluid viscous damping is effected by a viscous fluid damper which is arranged in the synchronization linkage mounted to the moving mass.