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An Efficient Approach for Determining Forced Vibration Response Amplitudes of a MDOF System with Various Attachments
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| Zeitschriftentitel: | Shock and Vibration |
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| Personen und Körperschaften: | , |
| In: | Shock and Vibration, 19, 2012, 1, S. 57-79 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Hindawi Limited
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| Schlagwörter: |
| Zusammenfassung: | <jats:p>The frequency-response curve is an important information for the structural design, but the conventional time-history method for obtaining the frequency-response curve of a multi-degree-of-freedom (MDOF) system is time-consuming. Thus, this paper presents an efficient technique to determine the forced vibration response amplitudes of a multi-span beam carrying arbitrary concentrated elements. To this end, the "steady" response amplitudes<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>of the above-mentioned MDOF system due to harmonic excitations (with the specified frequencies<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math>) are determined by using the numerical assembly method (NAM). Next, the corresponding "total" response amplitudes<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math>of the same vibrating system are calculated by using a relationship between<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msub></mml:mrow></mml:math>obtained from the single-degree-of-freedom (SDOF) vibrating system. It is noted that, near resonance (i.e.,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>e</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mi>w</mml:mi></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>≈</mml:mo></mml:math>1.0), the entire MDOF system (with natural frequency<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>w</mml:mi></mml:math>) will vibrate synchronously in a certain mode and can be modeled by a SDOF system. Finally, the conventional finite element method (FEM) incorporated with the Newmark's direct integration method is also used to determine the "total" response amplitudes<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:mi>Y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math>of the same forced vibrating system from the time histories of dynamic responses at each specified exciting frequency<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:mrow></mml:math>. It has been found that the numerical results of the presented approach are in good agreement with those of FEM, this confirms the reliability of the presented theory. Because the CPU time required by the presented approach is less than 1% of that required by the conventional FEM, the presented approach should be an efficient technique for the title problem.</jats:p> |
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| Umfang: | 57-79 |
| ISSN: |
1070-9622
1875-9203 |
| DOI: | 10.1155/2012/457104 |