PhD_Wallyson

Mr. WALLYSON SILVA

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1 years, 74 days
University of Miskolc
Mechanical Engineering

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I am a Brazilian PhD student in Mechanical Engineering at the University of Miskolc. My research is about: - The vibration of machine tools - Particle Impact Damper in boring bars to internal turning operation - turning hardened materials - passive damping in turning operation - Chatter in Machining process - Mathematical modelling of particle damping systems - Mathematical modelling of boring bar fixation

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These are replies submitted by PhD_Wallyson

@PhD_Wallyson 

Hi 

I followed the recommendation of Professor Rouben Rostamian and studied article number 3. Finally, I did the derivation to got matrices 24, 26 and 27 also 37, 38 and 39 considering the boundary conditions of each system. However, I could plot Figure 2 and Figure 4 of the same article also the eigenfrequencies. 

Question: do you have any tip to give me in order to plot Figure 2 and Figure 4 and find the eigenfrequencies of the mentioned matrices?

here it is my latest attempts:
rascunho_2.mw
rascunho_3.mw


Best regards, 

Wallyson Thomas

@Rouben Rostamian  

how can I share the file with you:
- do you have an email account? mine is wallyson35012726@gmail.com or szmwally@uni-miskolc.hu
Best Regards, 

Wallyson Thomas

@Rouben Rostamian  

Hi Professor, 

[Attachments deleted by moderator, since they clearly stated that redistribution was not permitted.]

Only God can pay you back! 
Thanks for so much help

Hello Everyone, 

I got a lot of knowledge of you and my PhD starts getting well because of your precious help (especially the bright and friendly Professor Rouben Rostamian).

I am here seeking for help. I set up the interface matrix for a Winkler foundation in order to find the eigenfrequencies and plot the mode shapes of Figure 8 (a) and (b) from Article 1. Besides, I tried to expand Professor Rouben Rostamian codes to Figure 2 (a) and (b) - without axial force, Figure 3 from Article 2. Furthermore, I applied that codes in Figure 1(b) and (c), also Figure 3 (b) and (c) from Article 3. However, I could obtain the mode shape of Figure 8 (a) and (b) from Article 1 and I am somewhat mystified by my results. 

Question: how I can obtain the matrix, characteristic equation, eigenfrequencies and the mode shapes of a beam when supported with Winkler foundation in the configurations I have mentioned above?

code: rascunho_1.mw


References:

Book: Vibration of continuous systems-John Wiley & Sons Ltd: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119424284 

Article 1: Cazzani, A., 2013. On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set. International Journal of structural stability and dynamics13(08), p.1350045.(https://doi.org/10.1142/S0219455413500454) In case you need the pdf file, let me know, please!

Article 2: Ghannadiasl, A. and Mofid, M., 2015. An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load. Latin American Journal of Solids and Structures12(13), pp.2417-2438.(https://www.scielo.br/pdf/lajss/v12n13/1679-7825-lajss-12-13-02417.pdf)

Article 3:  Doyle, P.F. and Pavlovic, M.N., 1982. Vibration of beams on partial elastic foundations. Earthquake Engineering & Structural Dynamics10(5), pp.663-674.(https://doi.org/10.1002/eqe.4290100504) In case you need the pdf file, let me know, please!

Best Regards, 
Wallyson Thomas

Hello Everyone, 

I got a lot of knowledge of you and my PhD starts getting well because of your precious help (especially the bright and friendly Professor Rouben Rostamian).

I am here seeking for help. I set up the interface matrix for a Winkler foundation in order to find the eigenfrequencies and plot the mode shapes of Figure 8 (a) and (b) from Article 1. Besides, I tried to expand Professor Rouben Rostamian codes to Figure 2 (a) and (b) - without axial force, Figure 3 from Article 2. Furthermore, I applied that codes in Figure 1(b) and (c), also Figure 3 (b) and (c) from Article 3. However, I could obtain the mode shape of Figure 8 (a) and (b) from Article 1 and I am somewhat mystified by my results. 

Question: how I can obtain the matrix, characteristic equation, eigenfrequencies and the mode shapes of a beam when supported with Winkler foundation in the configurations I have mentioned above?

code: rascunho_1.mw


References:

Book: Vibration of continuous systems-John Wiley & Sons Ltd: https://onlinelibrary.wiley.com/doi/book/10.1002/9781119424284 

Article 1: Cazzani, A., 2013. On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set. International Journal of structural stability and dynamics13(08), p.1350045.(https://doi.org/10.1142/S0219455413500454) In case you need the pdf file, let me know, please!

Article 2: Ghannadiasl, A. and Mofid, M., 2015. An analytical solution for free vibration of elastically restrained Timoshenko beam on an arbitrary variable Winkler foundation and under axial load. Latin American Journal of Solids and Structures12(13), pp.2417-2438.(https://www.scielo.br/pdf/lajss/v12n13/1679-7825-lajss-12-13-02417.pdf)

Article 3:  Doyle, P.F. and Pavlovic, M.N., 1982. Vibration of beams on partial elastic foundations. Earthquake Engineering & Structural Dynamics10(5), pp.663-674.(https://doi.org/10.1002/eqe.4290100504) In case you need the pdf file, let me know, please!

Best Regards, 
Wallyson Thomas

@Rouben Rostamian  

 

Hi Professor 

I did some modifications in your Program so that I can use Krylov-Functions to obtain the same results (eigenfrequencies and mode shapes).

I got the Eigenfrequencies correct but I could get the mode shapes because I do not know how to correct the subroutine error. Do you have some suggestion?

Krylov_function_free_pinned_pinned_free_beam_v2.mw

Hi 

Professor I am still trying to plot the first three mode shapes. Any suggestions?

Krylov_function_free_pinned_pinned_free_beam_v2.mw


Best Regards, 

Wallyson Thomas

@acer 

HI Professor

I know that I am poor at programming, I am sorry! I should pay more attention to the missing parameters like params! And you are right, I have done the corrections already, but another error comes up! Do you have any suggestion about what could it be?

Krylov_function_free_pinned_pinned_free_beam_v2.mw

Hi 

Considering the previous 3 questions, I got 2 results (matrix of the system and the eigenfrequencies correctly) because of the precious collaboration of Professor  Rouben Rostamian!

The remaining question is how can I got the mode shapes! I tried but the following error appears. What can you recommend me to do?  

Krylov_function_free_pinned_pinned_free_beam_v2.mw

@Rouben Rostamian  

Hi Professor 

I really want to solve that problem properly, because of this so many questions!

I have included the variables which were missing (EI, kt and kr) and I could get the correct Eigenfrequencies now (mu_1:=0.0816176, mu_2:=0.208548, mu_3:=0.34993).

However, I could not plot the MODE SHAPES because of the negative values as you can see in the program. Do you have any suggestions?

9_07_2020_Mode_shapes_matrix_12x12_Figure_3.23_artigo_2.mw

Best Regards, 

Wallyson Thomas 

@Rouben Rostamian  

Hi Professor

As I told you before and demonstrate in the big sketch at the beginning of this conversation that the L/D ratio varies and it is quite important to my PhD findings. Because of this, I would like to know how can I insert the most important lengths for me (rho) and extract the eigenfrequencies values (lambda) from the graph in an automatic way? It is possible or should I do manually one by one? 

29_07_2020_Eigenvalues_3_span_beam.mw

@Rouben Rostamian  

Hi Professor,

I think I understand what you mean by "write them symbolically in terms of the parameters". And after my modifications, I am still trying to plot the mode shapes. how can I plot it and also see the eigenfrequencies that come from the characteristic equation as I highlighted in the following Figures and also showing in the programme code?

9_07_2020_Mode_shapes_matrix_12x12_Figure_3.23_artigo_2.mw

 

@Rouben Rostamian  

 

I did the correction of the boundary conditions as you can see below and in the program but I do not know why the problem persists! My supervisor (Professor Dr Attila https://gepesz.uni-miskolc.hu/staff.php?id=224 ) and I know that you are helping it a lot Professor Rouben Rostamian. I am sorry for bothering you again! 

bcs := 
(D@@2)(X[1])(0) = 0,   # zero moment at x=0
(D@@3)(X[1])(0) = 0,   # zero shear  at x=0

X[1](L[1]) = X[2](L[1]),        # continuous displacement at x=L[1]
D(X[1])(L[1]) = D(X[2])(L[1]),  # continuous slope at x=L[1]
(D@@2)(X[1])(L[1]) = -(1.422*10^4)*(D(X[1])(L[1]))+(D@@2)(X[2])(L[1]),     # equilibrium condition bending moment at x=L[1]
(D@@3)(X[1])(L[1]) = +(4.881*10^9)*((X[1])(L[1]))+(D@@3)(X[2])(L[1]),  # equilibrium condition shear force at x=L[1]

X[2](L[1]+L[2]) = X[3](L[1]+L[2]),        # continuous displacement at x=L[1]+L[2]
D(X[2])(L[1]+L[2]) = D(X[3])(L[1]+L[2]),  # continuous slope at x=L[1]+L[2]
(D@@2)(X[2])(L[1]+L[2]) = +(1.422*10^4)*(D(X[2])(L[1]+L[2]))+(D@@2)(X[3])(L[1]+L[2]),     # equilibrium condition bending moment at x=(L[1]+L[2])
(D@@3)(X[2])(L[1]+L[2]) = -(4.881*10^9)*((X[2])(L[1]+L[2]))+(D@@3)(X[3])(L[1]+L[2]),  # equilibrium condition shear force at x=(L[1]+L[2])

(D@@2)(X[3])(L[1]+L[2]+L[3]) = 0,   # zero moment at x=L[1]+L[2]+L[3]
(D@@3)(X[3])(L[1]+L[2]+L[3]) = 0:   # zero shear  at x=L[1]+L[2]+L[3]
 

28_07_2020_Mode_shapes_matrix_12x12_Figure_3.23_artigo_2.mw

@Rouben Rostamian 

How can I insert the constants later on? Unfortunately, I failed in this task. Could you help me showing me how can I get the first 3 eigenfrequencies and 3 natural frequencies of this problem?

Given data:

kt:= 4.881*10^9

kr:=1.422*10^4

A:=1.1933*10^-3

I:=1.1385*10^-7

pho:=7850

E:=2.05*10^11


My supervisor used FEM (Finite Element Method) just to compare with the numerical results from MAPLE. 

FEM results:

mu_1:=8.007

mu_2:=20.114

mu_3:=33.252

f_1:=508.99925 Hz

f_2:=3211.9914 Hz

f_3:=8778.3528 Hz

 

@Rouben Rostamian  

Hi Professor

I would like to compare the mu (characteristic equation) and Lambda/L2 (Eigenvalues) like in Picture_A. For that reason, do you know any command to take from the graph the first 5 modes (lambda) with precision when I define a value for rho?

When rho = 4.3 (for example)

lambda_1= 
lambda_2=

lambda_3= 
lambda_4=

lambda_5= 
 

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