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 questions asked by PhD_Wallyson

Hello friends

I am a PhD student at the University of Miskolc (Hungary). I am writing to asking for help.

considering that my matrix is correct (based on Article 1) and I have all the values needed:

-  From Article 2, How can I plot the graph of Figure 2 and Figure 3 in a logarithmic scale (X-axis) and linear scale (Y-axis). Also, how can I plot the mode shapes of Figure 7 and Figure 8?
 

I tried to do it (below) but I am not sure it is correct.

Graphs_Article_2.mw

Reference:
Article 1 - Doyle, Paul F., and Milija N. Pavlovic. "Vibration of beams on partial elastic foundations." Earthquake Engineering & Structural Dynamics 10, no. 5 (1982): 663-674.

Article 2 - Cazzani, Antonio. "On the dynamics of a beam partially supported by an elastic foundation: an exact solution-set." International Journal of structural stability and dynamics 13, no. 08 (2013): 1350045.

Hi Everyone!



Considering the Figure (3_span_elastic_support) given by the paperwork (https://www.diva-portal.org/smash/get/diva2:837061/FULLTEXT01.pdf) I try to build the matrix based on the following references:
- https://www.sciencedirect.com/science/article/pii/S0045794913000138
- https://www.sciencedirect.com/science/article/pii/S0022460X07009017
- BOOK: Moving Loads - Dynamic Analysis and Identification Techniques_ Structures and Infrastructures Book Series, Vol. 8-CRC Press (2011)

My questions are:

- when I consider the coefficients kt = 4.881*10^9 and kr= 1.422*10^4 the following message appears. What Is the limit of MAPLE? 10^6?


- Because of this "fsolve" take to a long time to compute the values. If the message above appears I can trust in the "fsolve" values?

 

9_07_2020_Mode_shapes_matrix_12x12_Figure_3.23_artigo_2.mw

Hi Everyone

Just to put you in the context: during an internal turning operation, the overhang (ratio Length/Diameter of the tool [L/D]) is really important to guarantee the stability of the process (minimal vibration as possible). Having said that, it is desirable to increases the overhang to do deep holes, because of this the ratio L/D varies depending on the necessity and consequently the natural frequency of the tool will change.

As you can see in the attached Picture_A and B, I am trying to find the Eigenvalues when the overhang (ratio L/D) changes. Is it possible like in Figure 2 in the attached paper (link below)?

https://www.mdpi.com/2076-3417/9/15/2996/pdf

Consider:

L1 = ratio L/D (changeable);
L2 = Fixation of the tool (content)
L3 = the remaining part of the tool out of the fixation (changeable)

Tool length is content = L1 + L2 + L3

 

13_07_2020_Eigenvalues_3_span_beam.mw

Hello Everyone!

I have one more challenge for you.

How can I find for a Free-Pinned-Pinned-Free (3-span) beam (Picture A below) using the Krylov–Duncan Method (Literature links and references below):

- the matrix of the system?

- the transcendental equation in order to determine the natural frequencies?

- the first three mode shapes?

I tried to do it as you can see from my MAPLE (file below), but I got stuck when I use the command "determinant" and it did not find the transcendental equation.

Krylov_function_free_pinned_pinned_free_beam.mw

 

Picture_A

References:

Krylov–Duncan Method
https://link.springer.com/content/pdf/10.1007%2F978-1-4419-1047-9_14.pdf
 

Krylov–Duncan Functions - page 96

You can find that book using the https://libgen.is/
https://www.amazon.com/Formulas-Structural-Dynamics-Tables-Solutions/dp/0071367128

Hi Everyone!

I would like your help again.

Considering a Free-Pinned-Pinned-Free beam (page 88 in the pdf file). In case of a matrix 12x12 how could I find the coefficients (a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4) using MAPLE in order to plot the mode shapes of the Figure 3.22 (a) (page 70 in the pdf file)? in case of a matrix 16x16 and  20x20, the procedure is the same?

I tried to plot the mode shapes but I failed because I believe they should be similar to Figure 3.22 (a) (page 70 in the pdf file)

 

sys_GE := {-a1 + a3 = 0, -a2 + a4 = 0, 0.9341161484*a1 + 0.3569692163*a2 + 1.519943120*a3 + 1.819402948*a4 = 0, 0.6669014188*b1 - 0.7451459573*b2 + 5.530777989*b3 + 5.620454178*b4 = 0, 0.9938777922*c1 - 0.1104849953*c2 + 62.17096851*c3 + 62.17901032*c4 = 0, -0.9341161484*a1 - 0.3569692163*a2 + 1.519943120*a3 + 1.819402948*a4 + 0.9341161484*b1 + 0.3569692163*b2 - 1.519943120*b3 - 1.819402948*b4 = 0, -0.3569692163*a1 + 0.9341161484*a2 + 1.819402948*a3 + 1.519943120*a4 + 0.3569692163*b1 - 0.9341161484*b2 - 1.819402948*b3 - 1.519943120*b4 = 0, 0.3569692163*a1 - 0.9341161484*a2 + 1.819402948*a3 + 1.519943120*a4 - 0.3569692163*b1 + 0.9341161484*b2 - 1.819402948*b3 - 1.519943120*b4 = 0, -0.7451459573*b1 - 0.6669014188*b2 + 5.620454178*b3 + 5.530777989*b4 + 0.7451459573*c1 + 0.6669014188*c2 - 5.620454178*c3 - 5.530777989*c4 = 0, -0.6669014188*b1 + 0.7451459573*b2 + 5.530777989*b3 + 5.620454178*b4 + 0.6669014188*c1 - 0.7451459573*c2 - 5.530777989*c3 - 5.620454178*c4 = 0, 0.7451459573*b1 + 0.6669014188*b2 + 5.620454178*b3 + 5.530777989*b4 - 0.7451459573*c1 - 0.6669014188*c2 - 5.620454178*c3 - 5.530777989*c4 = 0}

 

solve(sys_GE, {a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4});
        {a1 = 0.1014436637 c4, a2 = -0.1143870369 c4, 

          a3 = 0.1014436637 c4, a4 = -0.1143870369 c4, 

          b1 = 0.07095134140 c4, b2 = -0.1260395712 c4, 

          b3 = 0.1510591164 c4, b4 = -0.1737777468 c4, 

          c1 = 0.2102272829 c4, c2 = -0.2816561313 c4, 

          c3 = -1.003990622 c4, c4 = c4}

26_06_2020_Transcedental_equation_matrix_12x12_artigo_2.mw

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