An Analytic Solution for the Dynamic Behavior of a Cantilever Beam with a Time-Dependent Spring-like Actuator

Round 1

Reviewer 1 Report

See comments in the attached file

Comments for author File: Comments.pdf

Author Response

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Reviewer 2 Report

The paper investigates the forced vibration of a cantilever beam supported by a spring with time-dependent spring stiffness. By applying the shifting-function method, the boundary value problem is reduced to the problem with the homogeneous boundary condition. The expansion theorem method can then be applied to obtain the analytical solution in form of a series. Although the paper offers no great innovations in methods, it is clearly written and deserves publication in Axioms.

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Reviewer 3 Report

The authors present an interesting of a beam with a spring with time dependent stiffness attached to one end. 

The work follows along standard lines, and the archival strength of the work is rather weak.  

There are vibration textbooks, in which beams with different boundary conditions have been studied.  It would be worth looking into them; for example, the Cambridge University Press book entitled ``Vibrations".

There are many examples in the literature where the contact interactions with a beam like structure gives rise to varying boundary conditions at the end, which are described by implicit time-dependent functions; see for example, the AFM literature. 

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Reviewer 4 Report

In general, the manuscript is written well, and the topic has the scientific standards to be published. As an analytical paper, it clearly explains the step-by-step procedure of the exact solution. The validity of the developed closed-form solution is confirmed for two special classic cases of free-end with and without spring. Two comments are:

1- Although the analytical solution for the case of a time-dependent spring has not been reported in the literature yet, however, it would be better to validate the solution if any numerical solution exists in the literature or if not, to compare the developed formulation to a finite-element commercial simulation for one case at least.

2- Please double check whole the manuscript to be sure all the parameters are written: "italics". For example "n" on page 14 and the frequencies "omega",  in the figures.

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Reviewer 5 Report

The paper  titled An Analytic Solution for the Dynamic Behavio using the keywords: spring-like actuator, time-dependent spring, beam vibration, shifting function method, analytical solution was submitted to Axioms. The  goal of this  submitted paper was  to analytically solve the dynamic behavior of a cantilever beam with a time-dependent spring support at the free end using the shifting function method combined with the expansion theorem method

1.               What is the main question addressed by the research?

the authors gave the basic concepts in the introduction.After it the authors dealt with the adopted Mathematical Modeling and then  to the Solution Methodology: The Shifting Function Method, The Expansion Theorem Method, The Complete Solution and the Extreme Case Study. Then, the authors presented the topic Harmonic Excitation and Harmonic Type of a Time-Dependent Spring Support and Numerical Results and Discussions.  Finally  Conclusions and  the of refedrenes  were listed. The  main question addressed  ewas by the  proposed research

The main question addressed by the research, according to the asuthors: (1) The proposed approach combining the shifting function method and the expansion theorem method can obtain the analytical solution for the dynamic behavior of a cantilever beam with a time-dependent spring-like actuator. (2) The deflection of a cantilever beam with a time-dependent spring support is between that of two extreme cases of a pure cantilever beam and a cantilever beam with a constant spring coeffi[1]cient. (3) When the magnitude or the frequency of the spring stiffness is greater than a critical value, the divergence instability occurs in the first mode at even multiples of 0 t = p /w . 17 (4) The important new result is that the resonance frequency significantly depends on the magni[1]tude and the frequency of the spring-like actuator in the first two modes. The magnitude and/or the frequency of the time-dependent spring stiffness can be adjusted, thereby distributing the first two modes to avoid causing unsafe vibrations or even resonances of the beam system

2. Do you consider the topic original or relevant in the field? Does it

address a specific gap in the field?

The topic is relevant  because  the asuthors proposed  new strategies ( see the Topic 1)

3. What does it add to the subject area compared with other published

material?

From both  presented approaches, it was  evident that  the newly developed schemes are more stable, robust and competent in their respective domains. ( see the Topic 1)

4. What specific improvements should the authors consider regarding the

methodology? What further  should be considered?

The Numerical simulations  must show the efficiency of the control methods,  as well as the sensitivity of each strategy to parametric errors.

 Are the conclusions consistent with the evidence and arguments presented

and do they address the main question posed?

See question 3.

6. Are the references appropriate? Yes

7. Please include any additional comments on the tables and figures.

Numerical Simulations results  were well written  and I did not see any mistakes ,  the text is concise and clear . and the concluding remarks were supported  by the data.

 The paper is relevant and interesting.It ssems to be original , according to other results from current literature

It seems to be original , according to other results from current literature mention on the proposed manuscripit .

 I t deserves  publication after  revisions:

  The authors must include  the influence of parametric errors on the proposed approach  could be also include some minor comments

The state of the art needs include international authors.

What is  new?

Why the authors did not use  mdethod as Mulltiple  Scales, Averaging, Harmonic  Balance? and then compare them with the proposed methodology

Author Response

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Round 2

Reviewer 3 Report

The revised manuscript is in reasonable form.  However, the archival content of the manuscript remains weak.

Reviewer 5 Report

The authors satisfactorily answered the questions proposed by the reviewers.

The proposed paper is well written, free of mistakes and the concluding remarks are consistent with the  presented theory 

The work is in its current form in conditions of approval