## 2 Proposed Method
### Proposed Approach for Actuators Positioning  
Starting from the actuation solution in Figure 3, a strategy for actuators placing, in order to maximize effectiveness, while satisfying requirements, was developed.  
As mentioned, one of the main advantages of active damping techniques is their versatility. Using these approaches, it is possible to improve chatter stability in different tool configurations (i.e., overhangs). Since in the proposed approach actuators are integrated on the boring bar, in order to maximize range of available overhangs, the configuration with one actuation section, placed as close as possible to the tooltip, is selected.  
In accordance with the requirements presented in Section 2.1, the integration strategy of the actuators on the boring bar body must guarantee: (i) the actuation in both radial and cutting velocity direction; (ii) maximum effectiveness (i.e., maximize the harm of the forces generated by the actuators; (iii) limited radial size; (iv) dynamic and static equivalence. In the proposed strategy, requirements (i), (ii) and (iii) are achieved by means of a section optimization (presented in this section), while the last requirement is achieved by properly dimensioning protective covers in order to compensate the loss of stiffness due to material removed (Section 2.5).  
In order to find the optimal actuator position, different configurations were evaluated. In Figure 4a, the most effective configuration in case of one actuation direction is presented; however this design provides low actuation forces and a single actuation direction (y). To increase the effectiveness, multiple actuators can be used, either in parallel (Figure 4b), or rather on two sides with opposite actuation forces (Figure 4c), optimal placement to reduce the required space. To achieve a multidirectional actuation, the solution showed in Figure 4d involves the employment of two actuators for each damping direction. In particular, actuators named X1 and X2 act to damp the vibration along the x direction, while Y1 and Y2 damp the vibration along the y direction. Since this solution exploits two actuators at time, effectiveness can be further increase by activating all the four actuators, as presented in the selected configuration (Figure 4e) that includes the four actuators on 45\({}^{\circ}\) position respect to the two directions. Using this configuration, the actuation in x direction is achieved by A and B working with opposite signal of C and D, while actuation in y direction is achieved by A and D working with opposite signal of B and C. As result, even if actuators distance from the tool axis is reduced, using double actuation forces increases the resultant moment.  
Optimal radial position (i.e., radial distance of actuators from tool axis) allows to maximize the arm of the force, while respecting radial imposed size. Moreover, the centroid of modified boring bar section should be aligned with the one of the original sections, in order to avoid different preferential direction of inflection among the different section. This implies that, in case of a boring bar section that is not symmetric within the x and y direction, actuators radial position has to be different (Figure 4e).  
Actuators are then placed at the minimum distance from tool axis that allows to reduce stress concentration due to the shape of the edge, and thickness of each slot was calculated in a way that the inertial centroid of modified section is the same of the one of original section.  
In order to guarantee the same static and dynamic behavior of the original boring bar, the loss of stiffness due to material removed must be compensate. Since the presence of actuators does not provide enough stiffness to compensate material removal, additional components are required. In order to reduce the number of components, protective covers, required to avoid inclusion of chip and debris on the housing slot, can be properly dimensioned to achieve this goal. The methodology for optimal protection cover dimension is presented in Section 2.5.