Incorporating Boundary Nonlinearity into Structural Vibration Problems
This paper presents a methodology for accurately incorporating the nonlinearity of boundary conditions (BCs) into the mode shapes, natural frequencies, and dynamic behaviour of analytical beam models. Such models have received renewed interest in recent years as a result of their successful implemen...
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MDPI AG
2024-10-01
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| Series: | Vibration |
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| Online Access: | https://www.mdpi.com/2571-631X/7/4/50 |
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| author | Alex J. Elliott Andrea Cammarano |
| author_facet | Alex J. Elliott Andrea Cammarano |
| author_sort | Alex J. Elliott |
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| description | This paper presents a methodology for accurately incorporating the nonlinearity of boundary conditions (BCs) into the mode shapes, natural frequencies, and dynamic behaviour of analytical beam models. Such models have received renewed interest in recent years as a result of their successful implementation in state-of-the-art multiphysics problems. To address the need for this boundary nonlinearity to be more completely captured in the equations of motion, a nonlinear algebra expansion of the classical linear approach for developing solvability conditions for natural frequencies and mode shapes is presented. The method is applicable to any BC that can be accurately represented in polynomial form, either explicitly or through the application of a Taylor expansion; this is the only assumption made in removing the need for the use of analytical approximations of the dynamics themselves. By reducing the BCs of the beam to a system of polynomials, it is possible to utilise the tensor resultant to develop these solvability conditions analogous to the conditions placed on the matrix determinant in linear, classical cases. The approach is first derived for a general set of nonlinear BCs before being applied to two example systems to investigate the importance of including nonlinear tip behaviour in the BCs to accurately predict the system response. In the first, a theoretical, symmetric system, in which a beam is supported by nonlinear springs, is used to explore both the applicability of the methodology and the improvements it can make to the accuracy of the model. Then, the more practical example of a cantilever beam with repulsive magnetic interaction at the tip is used to more explicitly assess the importance of properly incorporating boundary nonlinearity into multiphysics problems. |
| format | Article |
| id | doaj-art-c9d474e846e84566866a3b6e7b6c78c9 |
| institution | Kabale University |
| issn | 2571-631X |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Vibration |
| spelling | doaj-art-c9d474e846e84566866a3b6e7b6c78c92024-12-27T14:58:47ZengMDPI AGVibration2571-631X2024-10-017494996910.3390/vibration7040050Incorporating Boundary Nonlinearity into Structural Vibration ProblemsAlex J. Elliott0Andrea Cammarano1Faculty of Engineering and Applied Sciences, Cranfield University, Cranfield MK43 0AL, UKSchool of Engineering, University of Southampton, Southampton SO16 7QF, UKThis paper presents a methodology for accurately incorporating the nonlinearity of boundary conditions (BCs) into the mode shapes, natural frequencies, and dynamic behaviour of analytical beam models. Such models have received renewed interest in recent years as a result of their successful implementation in state-of-the-art multiphysics problems. To address the need for this boundary nonlinearity to be more completely captured in the equations of motion, a nonlinear algebra expansion of the classical linear approach for developing solvability conditions for natural frequencies and mode shapes is presented. The method is applicable to any BC that can be accurately represented in polynomial form, either explicitly or through the application of a Taylor expansion; this is the only assumption made in removing the need for the use of analytical approximations of the dynamics themselves. By reducing the BCs of the beam to a system of polynomials, it is possible to utilise the tensor resultant to develop these solvability conditions analogous to the conditions placed on the matrix determinant in linear, classical cases. The approach is first derived for a general set of nonlinear BCs before being applied to two example systems to investigate the importance of including nonlinear tip behaviour in the BCs to accurately predict the system response. In the first, a theoretical, symmetric system, in which a beam is supported by nonlinear springs, is used to explore both the applicability of the methodology and the improvements it can make to the accuracy of the model. Then, the more practical example of a cantilever beam with repulsive magnetic interaction at the tip is used to more explicitly assess the importance of properly incorporating boundary nonlinearity into multiphysics problems.https://www.mdpi.com/2571-631X/7/4/50nonlinear vibrationsGalerkin methodfinite element analysisnonlinear boundary conditions |
| spellingShingle | Alex J. Elliott Andrea Cammarano Incorporating Boundary Nonlinearity into Structural Vibration Problems Vibration nonlinear vibrations Galerkin method finite element analysis nonlinear boundary conditions |
| title | Incorporating Boundary Nonlinearity into Structural Vibration Problems |
| title_full | Incorporating Boundary Nonlinearity into Structural Vibration Problems |
| title_fullStr | Incorporating Boundary Nonlinearity into Structural Vibration Problems |
| title_full_unstemmed | Incorporating Boundary Nonlinearity into Structural Vibration Problems |
| title_short | Incorporating Boundary Nonlinearity into Structural Vibration Problems |
| title_sort | incorporating boundary nonlinearity into structural vibration problems |
| topic | nonlinear vibrations Galerkin method finite element analysis nonlinear boundary conditions |
| url | https://www.mdpi.com/2571-631X/7/4/50 |
| work_keys_str_mv | AT alexjelliott incorporatingboundarynonlinearityintostructuralvibrationproblems AT andreacammarano incorporatingboundarynonlinearityintostructuralvibrationproblems |