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Application of the Homotopy Analysis Method for Determining the free Vibrations of Beam

Krzysztof GROMYSZ
Krzysztof GROMYSZ
 and 
Damian SŁOTA
Damian SŁOTA
   | Apr 01, 2019
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Published Online: Apr 01, 2019
Page range: 61 - 71
Received: Oct 24, 2017
Accepted: Feb 19, 2018
DOI: https://doi.org/10.21307/acee-2018-006
Keywords
© 2018 Krzysztof Gromysz et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1.

Reduction of the beam with the continuous distribution of mass into the system with one degree of freedom a) system with the continuous distribution of mass m′, b) substitute mass m lumped in the point in which the arrow of vibrations of the system with continuous distribution of mass occurs
Reduction of the beam with the continuous distribution of mass into the system with one degree of freedom a) system with the continuous distribution of mass m′, b) substitute mass m lumped in the point in which the arrow of vibrations of the system with continuous distribution of mass occurs

Figure 2.

Oscillator of parameters k and m as the substitute system for the reinforced concrete beam
Oscillator of parameters k and m as the substitute system for the reinforced concrete beam

Figure 3.

The squared residual E 9
The squared residual E 9

Figure 4.

Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n = 6 (a) and n = 10 (b)
Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n = 6 (a) and n = 10 (b)

Figure 5.

The squared residual E 9
The squared residual E 9

Figure 6.

Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n=10 (a) and n=15 (b)
Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n=10 (a) and n=15 (b)

Figure 7.

Impact of plastic character implying the initial conditions y(0)=0 and ẏ (0)=va (a, b – described in text)
Impact of plastic character implying the initial conditions y(0)=0 and ẏ (0)=va (a, b – described in text)

Figure 8.

Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n=10 (a) and n=15 (b)
Distribution of error (|ye (x) – ŷn (x)|) of the exact solution approximation for n=10 (a) and n=15 (b)

Figure 9.

The squared residual E 20
The squared residual E 20

Figure 10.

Plot of the residual Rsn for n=10 (a) and n=20 (b)
Plot of the residual Rsn for n=10 (a) and n=20 (b)

Figure 11.

Comparison of the approximate solutions (solid line – solution ŷ20 obtained by the homotopy analysis method, dots – approximate solution obtained numerically in Mathematica software)
Comparison of the approximate solutions (solid line – solution ŷ20 obtained by the homotopy analysis method, dots – approximate solution obtained numerically in Mathematica software)

Figure 12.

The squared residual E 20
The squared residual E 20

Figure 13.

Plot of the residual R ^ s n for n=10 (a) and n=20 (b)
Plot of the residual R ^ s n for n=10 (a) and n=20 (b)

Figure 14.

Comparison of the approximate solutions (solid line – solution ŷ20 obtained by the homotopy analysis method, dots – approximate solution obtained by the numerical method available in Mathematica software)
Comparison of the approximate solutions (solid line – solution ŷ20 obtained by the homotopy analysis method, dots – approximate solution obtained by the numerical method available in Mathematica software)

Maximal absolute differences (Δ n ) between the approximate solutions obtained by the homotopy analysis method and the numerical methods available in Mathematica software

n 1 3 5
Δ n 3.35·10-2 4.98·10-3 7.12·10-5
n 10 15 20
Δ n 1.40·10-6 2.99·10-8 3.71·10-9

Values of errors in the reconstruction of the exact solution (Δ n = ||ye – ŷn ||)

n 1 2 3 4 5
Δ n 3.367·10-2 2.084·10-2 1.741·10-3 1.374·10-3 4.188·10-4
n 6 7 8 9 10
Δ n 5.135·10-5 1.785·10-5 9.227·10-5 2.483·10-6 4.230·10-7
n 11 12 13 14 15
Δ n 6.678·10-8 2.612·10-8 1.181·10-8 3.571·10-9 8.398·10-10
eISSN:
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Language:
English
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Journal Subjects:
Architecture and Design, Architecture, Architects, Buildings
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