This work presents a commentary of the article published by Asl and Ulsoy (2003, ASME J. Dyn. Syst., Meas., Control, 125, pp. 215–223). We show by an example that their method leads to inaccurate results and is therefore erroneous.
The exact solution of Eq. 1 cannot, in general, be obtained. Special cases of Eq. 1 were considered by Chen et al. (1) and exact analytical solutions were obtained. In (2), Asl and Ulsoy offer a closed form solution to the general case in matrix form and compute the stability lobes numerically. Both studies in (1,2) are based on a solution of a transcendental equation expressed in terms of the Lambert function, which was first derived by Briggs in (3). Because the solution involves the Lambert function and it is given as a series, reaching an exact analytical stability bound for Eq. 1 is still a problem.
Method of Asl and Ulsoy
Example and Conclusion
Let us reconsider the case study in (2) with
In Fig. 1, the data points , represented by a diamond sign, are obtained from 8 numerically for the principal branch of the Lambert function. An important observation is that these data points do not match with Fig. 9 of (2). Figure 1 also displays, by the solid curve, the actual stability lobe diagram predicted analytically (5).
Using certain data points from Fig. 1, one can easily verify that 8 is satisfied but 7 is not. It is easy to check that at is such a point. On the other hand, it can also be shown that 7 holds but 8 does not for some data points from Fig. 1. For instance, this can be verified by using at .
I would like to thank the anonymous referee for the valuable suggestions.