PhD_Wallyson

Mr. WALLYSON SILVA

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3 years, 229 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

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

Hello everyone, 

 

I am seeking for help!

You can freely access the mentioned file in any of the following links:

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

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

https://www.researchgate.net/publication/334711617_Transverse_Vibration_of_Clamped-Pinned-Free_Beam_with_Mass_at_Free_End

 

I have a/L a parameter. Give it a value and then beta can be calculated as in Table 1. The first five lowest physically possible values are listed only, but the number of beta-s tend to infinity because of the sine/cosine functions. (22) is clearly a nonlinear equation. As such, in general, it can't be solved analytically. (Of course, e.g., sin(x)=0 is also nonlinear but has analytical solution, but (22) is much more complicated.)
At first step, I give a/L  a value, 0.6 , then plot the left side of (22) and try to read the zeros. Here, beta on the horizontal axis is the independent variable. This method is really slow and inaccurate to get all the results of Figure 2. Instead, I should find the roots numerically by using a built-in solver in Maple.

Once I am done with the above, I should add a do/for loop for the parameter a/L. HOWEVER, I DO NOT KNOW HOW TO DO.

In conclusion, the main question is: How could I plot the graph using MAPLE in Figure 2 considering the transcendental Equation (22)?

Best Regards, 

Wallyson Thomas

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