On Isospectral Composite Beams
Round 1
Reviewer 1 Report
1.Please specify the innovation of this article.
2.What is the guiding significance of this research for practical engineering?
3.For the structure studied in this paper, can you give its schematic diagram?
No
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors developed a theoretical approach to establish the conditions for which a composite beam is isospectral to another beam, given given the geometrical and elastic characteristics of the latter. The article is well written, and the adopted theoretical considerations are correct.
While the approach seems to be only an application of well known methods to solve wave equations, the applicability off the obtained results justify the publication of this article after some questions are considered by the authors.
1- The definition of a composite beam is loosely given in the text. Only when the reader consider the boundary conditions for the wave equation it becomes clear that the beam is formed by connecting the two identical elastic beams side by side, and not end to end. Please, define this clearly in the text preceding eqs 1 and 2.
2- The derivation of equations 1 and 2 should be given, or at least, referenced.
3- The references are not numbered according to their appearance order in the text, but alphabetically. This must be corrected. Also, the references are given separately when multiple references appear in some point of the text (lines 47, 48, 67, ...). When multiple references are necessary, they should be presented in a single bracket. This is, [7,14,30] instead of [7], [14], [30].
4- The shearing stiffness is considered to be a function of z, the beam coordinate, exclusively. This means that the connection of the beams is thought as a line (rods, for instance), and not as an area. This is a reasonable approximation, however a more realistic model should consider this quantity dependent on the area elements of the connection. Please, comment on this.
5- The presentation of some equations is careless, and should be revised. For instance, in line 58 the equations 5 and 6 are discussed for a pair of boundary conditions for the variable y, while these equations are given for u. Even that the relation between these quantities is direct, the presentation of the ideas should as direct and precise as possible. What is the meaning of the dot product in eq 39?
6- Should it be (?):
- uniquely instead of unique in line 59;
- boundary instead of end in line 65;
- impedance instead of impedence in line 68.
7- The design of the figures 1, 2 and 3 is not very good. The m value and the curve labels should be given as within the plot area. The axis lines and tics are very thin, and somewhat difficult to see. Also, I found strange that in figure 1 k is different to ^k at the maximum of ym (x=1/2). In figure 2, k is equal to ^k for every maximum and minimum of ym, this is, x=1/4, 1/2, 3/4, 1. It seems that k is asymmetrically distributed around the ^k value for m=1, which does not seem to be the case for m=2 and 5. Please, comment on this.
Once these questions are carefully addressed b the authors, the article can be accepted for publication. So, my recommendation is to accept after minor revisions.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
The manuscript presents an analytic investigation on the axial vibrations of composite beams with partial interaction, namely on the possibility of isospectral configurations of the beams, for two and more equal layers.
The objective is fully accomplished, and the carefully written manuscript makes an agreeable reading. I only two minor possible suggestions:
page 2, line 59 : "and half of the eigenvalues of the cantilever spectrum (...) of the composite beam determine unique the shearing stiffness k(x)". I think the authors mean "uniquely"
page 6, Figure 1: why isn't the position of the center node appear to be not exactly the same for the 3 configurations? (I think this must be an issue with the drawing software, not with the analytical solutions)
page 8, Table 1 : why aren't the two first eigenvalues in an ascending order?
page 8, line 232/3 : "We note that c(1) = 0 and c(1) = p1N (1, . . . , 1), namely the strain energy stored inside the (N − 1) connections of the composite system vanish identically" This sentence sounds a little strange to me...
page 9, line 254 : "We have proved that the composite systems fAb,bkgN and fAb, k(i)gN shear the same eigenvalues fl( ni)gn≥1. " I think that the authors mean "share", not "shear".
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
Dear authors
The manuscript describes composite system consisting of two identical straight elastic beams under longitudinal vibration connected by an elastic interface capable of counteracting with its shearing stiffness the relative vibration of the two beams.
The presentation and the mathematical analysis are complete but some points have to be revised:
(1) I see the references in alphabetical order, that it is not correct. The references must be numbered according to the appearance in the text. Additionally in ref [30] I do not see the author Getsezy
(2) The "doi" of the references are missing I suggest to add to each reference, the digital object identifier, to be easier for a reader to find them
(3) Please refer in the conclusions explicitly the novelty of this research
The quality of the language is good and only some minor editing is needed
Author Response
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Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
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