Rotating machines have become increasingly powerful and rapid over time, often working now close to, or even above, their critical speeds. When, as a result, resonance or dynamic instability occurs, it can cause high vibration levels in these machines, particularly when rolling bearings are used. Vibration control of rotating systems can be made by viscoelastic dynamic vibration neutralizers (VDVNs), which are relatively cheap passive devices with a wide range of applications. For the control of flexural vibration in dynamic rotors, a translational VDVN is usually employed. It should be attached to the rotor by means of a special support, commonly between bearings, where the amplitude of the vibration mode of concern is high. The particular point where this type of device is attached depends on the mode to be controlled and in machines with reduced internal space, it cannot be placed in a suitable position. Therefore, this paper introduces a new type of VDVN, the angular VDVN, and proposes a methodology for the optimal design of a set of these devices for shaft slope degree of freedom control. This control aims at indirectly reducing flexural vibration. According to this approach, the control device can be installed close to the bearing where the slope degree of freedom presents its highest value at any critical speed. The conceptual design of the angular VDVN is presented to illustrate the proposed methodology, and a numerical example is given to demonstrate the influence of the angular VDVN geometry on the response. The corresponding results are fully discussed.