Recently, in this journal an interesting study was published [1] which developed a procedure that allows the determination of the location of the nodes and anti-nodes of a complicated structure. The basis of the method is to attach a virtual element (lumped mass or a grounded spring) to the system and to plot the frequency curves of the combined system against the location of the virtual element. It was shown that for virtual lumped mass, the nodes and anti-nodes correspond to the local maxima and minima of the frequency curves, respectively, while for the virtual spring, they correspond instead to the minima and maxima of the frequency curves.
The method is illustrated by examining a uniform cantilevered Bernoulli-Euler beam with various lumped attachments. It is stated, however, that the method is sufficiently general such that it can be easily extended to locate the nodes and anti-nodes of other combined dynamical systems. The aim of this letter is to make a positive comment on this interesting study and to strenghten the authors' statement on the general applicability of their method by giving an example from the field of longitudinally vibrating rods restrained by a linear spring in-span. In an earlier study, the authors also considered a longitudinally vibrating rod, carrying a lumped mass in-span [2]. They applied the virtual mass approach. Here, in contrast, the virtual grounded spring approach will be employed.
The ''original'' mechanical system to be investigated is shown in Figure 1(a). It consists of a fixed-free longitudinally vibrating elastic rod of length L and axial rigidity EA which is restrained by a linear spring of spring coefficient k. The mass per unit length and location of the spring attachment point are m and hL, respectively. The aim is to locate the nodes and the anti-nodes of this ''original'' system. To this end, according to the virtual spring approach, a virtual grounded spring of spring coefficient k 1 is attached to the system as in Figure 1(b) and then, the frequency curves of the resulting ''combined'' system are plotted against the constraint location parameter h 1 . The nodes and anti-nodes of the original system correspond to the local minima and maxima of these curves. Unlike those in reference [1], the calculations here will be based on expressions of the ''exact'' frequency equations which are derived and given in the appendix.