3.2.2. Numerical Validation

The optimization procedure utilized in the previous section is based on a heuristic approach, which does not prove that *γ*opt and *ζ*2opt provide the maximal possible value of *ψ* ∗ . Therefore, its validity should be verified numerically. The numerical analysis is performed by directly computing eigenvalues of matrix *A* on a grid of the (*ζ*2, *γ*, *ψ*) space for a fixed *ε* value (*ε* = 0.05) and identifying the couple of values *ζ*2, *γ* which provides the maximal *ψ* ∗ . After several trials, the analysis is finally performed on the grid described in Table 1.

**Table 1.** Parameter grid for the optimum search.


This analysis provided the optimal values for *ζ*<sup>2</sup> and *γ*, which are indicated in Table 2 and directly compared with the optimal values obtained analytically.


**Table 2.** Comparison of numerical and analytical results.

Although numerical and analytical optimal parameters do not coincide, their difference is minimal and negligible for most engineering applications. In particular, the optimal *γ* value is practically the same in both cases. The critical velocity, computed utilizing the parameter values indicated in Table 3, has a difference of less than 0.4% in the two cases. We remark that the difference between numerical and analytical computation is not related to the inaccuracy of stability estimation through the LCC, which exactly predicts an equilibrium's stability, but to the heuristic approach utilized for the optimization.

**Table 3.** Numerical input data.


Figure 5, illustrating the curve ∆<sup>3</sup> = 0 for various values of *ζ*2, enables us to understand the reason for the difference between the results obtained with the numerical and analytical approach. The blue line in the figure corresponds to the analytical optimization, while the green line to the numerical one. The yellow and red curves refer to values of *ζ*<sup>2</sup> slightly higher and lower than the optimal ones, respectively. The inaccuracy of the analytical procedure is due to the fact that the peak of the ∆<sup>3</sup> = 0 curve does not exactly lie on Point P (which has a fixed *γ* value and it is not represented in the figure). However, considering the minimal difference found and the practical compactness of Equations (24) and (25), these will be utilized in the continuation of the paper.

**Figure 5.** Vanishing loop of the ∆<sup>3</sup> = 0 curve. The blue line was obtained utilizing the optimal damping as defined by the analytical procedure *ζ*<sup>2</sup> = *ζ*2opt , the green line corresponds to the optimal solution obtained by the numerical procedure and the yellow and red lines are obtained for *ζ*<sup>2</sup> values slightly larger and smaller, respectively, than *ζ*2opt .
