Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section
In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matl...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-02-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/2/153 |
| _version_ | 1827757271747133440 |
|---|---|
| author | Didier Samayoa Helvio Mollinedo José Alfredo Jiménez-Bernal Claudia del Carmen Gutiérrez-Torres |
| author_facet | Didier Samayoa Helvio Mollinedo José Alfredo Jiménez-Bernal Claudia del Carmen Gutiérrez-Torres |
| author_sort | Didier Samayoa |
| collection | DOAJ |
| description | In this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For each case presented, the transversal displacement and natural frequency are analyzed as a Hausdorff dimension function. It is found that the maximum displacement increases as the Hausdorff dimension increases, with the relationship <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msup><mi>k</mi><mrow><mn>0.79</mn><mo form="prefix">ln</mo><msub><mi>d</mi><mi mathvariant="script">H</mi></msub><mo>+</mo><mn>0.37</mn></mrow></msup></mrow></semantics></math></inline-formula>, being <i>k</i> the iteration number of pre-fractal. The natural frequencies increase as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>∼</mo><msup><mi>M</mi><mrow><mn>2.51</mn></mrow></msup></mrow></semantics></math></inline-formula>, whereas the bending stiffness is expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>I</mi><mo>=</mo><mn>1165.4</mn><mo form="prefix">ln</mo><mo>(</mo><msub><mi>d</mi><mi mathvariant="script">H</mi></msub><mo>+</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Numerical examples are given in order to discuss the mechanical implications. |
| first_indexed | 2024-03-11T08:48:37Z |
| format | Article |
| id | doaj.art-2bff406bc9114bebb96ac6dbc1be8943 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T08:48:37Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-2bff406bc9114bebb96ac6dbc1be89432023-11-16T20:36:43ZengMDPI AGFractal and Fractional2504-31102023-02-017215310.3390/fractalfract7020153Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross SectionDidier Samayoa0Helvio Mollinedo1José Alfredo Jiménez-Bernal2Claudia del Carmen Gutiérrez-Torres3SEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, MexicoEngineering Department, Instituto Politécnico Nacional, UPIITA, Av. IPN, No. 2580, Col. La Laguna Ticoman, Gustavo A. Madero, Mexico City 07340, MexicoSEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, MexicoSEPI-ESIME Zacatenco, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos, Mexico City 07738, MexicoIn this manuscript, static and free vibration responses on Euler–Bernoulli beams with a Koch snowflake cross-section are studied. By applying the finite element method, the transversal displacement in static load condition, natural frequencies, and vibration modes are solved and validated using Matlab. For each case presented, the transversal displacement and natural frequency are analyzed as a Hausdorff dimension function. It is found that the maximum displacement increases as the Hausdorff dimension increases, with the relationship <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msup><mi>k</mi><mrow><mn>0.79</mn><mo form="prefix">ln</mo><msub><mi>d</mi><mi mathvariant="script">H</mi></msub><mo>+</mo><mn>0.37</mn></mrow></msup></mrow></semantics></math></inline-formula>, being <i>k</i> the iteration number of pre-fractal. The natural frequencies increase as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ω</mi><mo>∼</mo><msup><mi>M</mi><mrow><mn>2.51</mn></mrow></msup></mrow></semantics></math></inline-formula>, whereas the bending stiffness is expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>I</mi><mo>=</mo><mn>1165.4</mn><mo form="prefix">ln</mo><mo>(</mo><msub><mi>d</mi><mi mathvariant="script">H</mi></msub><mo>+</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Numerical examples are given in order to discuss the mechanical implications.https://www.mdpi.com/2504-3110/7/2/153Euler–Bernoulli beamHausdorff dimensiontransversal displacementvibration modes |
| spellingShingle | Didier Samayoa Helvio Mollinedo José Alfredo Jiménez-Bernal Claudia del Carmen Gutiérrez-Torres Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section Fractal and Fractional Euler–Bernoulli beam Hausdorff dimension transversal displacement vibration modes |
| title | Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section |
| title_full | Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section |
| title_fullStr | Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section |
| title_full_unstemmed | Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section |
| title_short | Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section |
| title_sort | effects of hausdorff dimension on the static and free vibration response of beams with koch snowflake like cross section |
| topic | Euler–Bernoulli beam Hausdorff dimension transversal displacement vibration modes |
| url | https://www.mdpi.com/2504-3110/7/2/153 |
| work_keys_str_mv | AT didiersamayoa effectsofhausdorffdimensiononthestaticandfreevibrationresponseofbeamswithkochsnowflakelikecrosssection AT helviomollinedo effectsofhausdorffdimensiononthestaticandfreevibrationresponseofbeamswithkochsnowflakelikecrosssection AT josealfredojimenezbernal effectsofhausdorffdimensiononthestaticandfreevibrationresponseofbeamswithkochsnowflakelikecrosssection AT claudiadelcarmengutierreztorres effectsofhausdorffdimensiononthestaticandfreevibrationresponseofbeamswithkochsnowflakelikecrosssection |