Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage

Round 1

Reviewer 1 Report

The submitted paper deals with very interesting issue - the Authors shown the differences in solution (eigenvalue problem) of shell structures with different pattern of damages (perforations or damaged perforation) and material distribution. The results allow to access different homogenization method. 

It should be noted (Authors also not it) that this method of calculation (the FEM model with different damage and material distribution) is time consuming way. It could be good if some recipe or suggestion on the conclusion appear - i.e. in wich cases is enough to use homogenization method and when the proposed method should be used.

However the paper is very good, it need some clarification especially for not very experienced (young, PhD students) scientists, they are as follow:

1) The assumed coordinate system is not clear enougth - it looks like 3D cartensian, but profile function is like for cylindrical. Where the axis x3 is set. Add some explanation and axis of assumed coordinate system in Fig.1. Nevertheless, the rest of explanation according to displacement u,v,w and unit vector e1,e2, and e3 correspond to cylindrical CS are clear. 

2)Why the examples presented in Fig.1 (and in the rest of the text of manuscript) are named parabolic for cylinder and hyberbolic for rotary paraboloid.

3) According to equations 22 nad 23 the presented in Fig.2 pattern should present ellipses holes with a and b. How the h in Fig. 2 is defined or calculated espetially in case of eliptical pattern, maybe the h1 and h2 will be necessary ?

 

Author Response

Thank you for your comments.

We have added a comment in the conclusions on homogenisation. The original version was somewhat negative in tone.

In specific cases where the relevant parameters and their ranges can be reliably identified, homogenisation can be accomplished with less effort.

1) Figure 1 & coordinate systems.

Indeed, the axes were not labeled! We have added the labels and note that x1- and x-coordinates coincide.

2) The classification of the surfaces (parabolic, hyperbolic, elliptic) follows from Gaussian theory of surfaces and is standard use in this class of problems. We had erroneously omitted this from the text and have now added it.

3) We have indicated that there can indeed be h1 and h2. However, we have not modified the text at this point.

Reviewer 2 Report

The paper is well written, have self-contained character, clear language and general readership. It was possible to understand the manuscript without additional reading and a lot of guess work. 

Author Response

Thank you!

Reviewer 3 Report

In this manuscript, the authors present a stochastic study with variational methods and Monte-carlo sampling of perforated shell cylinders with random material and defect distribution. The manuscript sheds light on the high sensitivity of the vibrational behaviour of these shells to material and construction uncertainties and hopes that the methods describe would serve to build a database that enabled the training of systems of damage monitorization and identification. The writing and structure are remarkably good, as well as the mathematical notation, but a few aspects of its content may be still improved. Below there is a list with a few questions, suggestions and doubts raised by this manuscript:

 

Is this a problem with any sort of practical, industrial or engineering relevance, apart from the theoretical interest? In other words, are perforated cylindrical shells of utility anywhere? The introduction does not mention anything of this.

 

Equation 13 probably has a typo. It seems like ro1 is squared when it should not.

 

How are the deformation energies ‘A curled’ related to the stiffness matrix and the potential energies ‘A’? Although this is a known feature of mechanical systems, the mathematical formulation should explain this for clarity.

 

Figure 3: In H(3,2), and H(17,23), how do the numbers between brackets (the Halton grids) do influence the perforation patterns?

 

Section 4.2: Why do you choose to define the damping matrix as a multiple of the identity instead of defining it, for example, as a multiple of the mass and/or the stiffness matrices? Is that realistic?

 

The reviewer awaits the authors’ response to the questions and suggestions proposed by this review report.

Kind regards.

Author Response

Thank you for your comments.

1) We have now mentioned both drying drums ans Trommel screens as industrial examples. It appears we had accidentally removed the mention of drying drums (a very Finnish application).

2) Eq 13 has been fixed. Thanks.

3) We have defined the mechanical energy at the point where it is mentioned the first time.

4) The Halton sequences are pseudo-random. In 2D with some choices of the primes, e.g., H(17,23), the points do spread out without any distinct patterns. With some other choices the sequences degenerate to deterministic ones and lose their applicability here. We have added a comment in the figure caption to emphasize that the sequences have been selected by trial and error.

5) There are many ways to define the damping matrix, often depending on the application. Here we have used the lumping variation just for the convenience. We have added a comment pointing out that we could have used for instance Rayleigh damping where the damping matrix is a linear combination of the mass and stiffness matrices.

Round 2

Reviewer 3 Report

The authors have responded satisfactorily to all my questions, and the manuscript is now apt for publication.

Regards.