### Abstract

This paper addresses the stochastic modeling of the stiffness matrix of slender uncertain curved beams that are forced fit into a clamped-clamped fixture designed for straight beams. Because of the misfit with the clamps, the final shape of the clamped-clamped beams is not straight and they are subjected to an axial preload. Both of these features are uncertain given the uncertainty on the initial, undeformed shape of the beams and affect significantly the stiffness matrix associated with small motions around the clamped-clamped configuration. A modal model using linear modes of the straight clamped-clamped beam with a randomized stiffness matrix is employed to characterize the linear dynamic behavior of the uncertain beams. This stiffness matrix is modeled using a mixed nonparametric-parametric stochastic model in which the nonparametric (maximum entropy) component is used to model the uncertainty in final shape while the preload is explicitly, parametrically included in the stiffness matrix representation. Finally, a maximum likelihood framework is proposed for the identification of the parameters associated with the uncertainty level and the mean model, or part thereof, using either natural frequencies only or natural frequencies and mode shape information of the beams around their final clamped-clamped state. To validate these concepts, a simulated, computational experiment was conducted within Nastran to produce a population of natural frequencies and mode shapes of uncertain slender curved beams after clamping. The application of the above concepts to this simulated data led to a very good to excellent matching of the probability density functions of the natural frequencies and the modal components, even though this information was not used in the identification process. These results strongly suggest the applicability of the proposed stochastic model.

Original language | English (US) |
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Title of host publication | Proceedings of the ASME Design Engineering Technical Conference |

Pages | 887-895 |

Number of pages | 9 |

Volume | 1 |

Edition | PARTS A AND B |

DOIs | |

State | Published - 2012 |

Event | ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2012 - Chicago, IL, United States Duration: Aug 12 2012 → Aug 12 2012 |

### Other

Other | ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2012 |
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Country | United States |

City | Chicago, IL |

Period | 8/12/12 → 8/12/12 |

### Fingerprint

### Keywords

- Maximum entropy
- Parametric stochastic model
- Uncertain geometry
- Uncertain preload
- Uncertain stiffness

### ASJC Scopus subject areas

- Mechanical Engineering
- Computer Graphics and Computer-Aided Design
- Computer Science Applications
- Modeling and Simulation

### Cite this

*Proceedings of the ASME Design Engineering Technical Conference*(PARTS A AND B ed., Vol. 1, pp. 887-895) https://doi.org/10.1115/DETC2012-71169

**Stochastic modal models of slender uncertain curved beams preloaded through clamping.** / Avalos, Javier; Richter, Lanae A.; Wang, X. Q.; Murthy, Raghavendra; Mignolet, Marc.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the ASME Design Engineering Technical Conference.*PARTS A AND B edn, vol. 1, pp. 887-895, ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, IDETC/CIE 2012, Chicago, IL, United States, 8/12/12. https://doi.org/10.1115/DETC2012-71169

}

TY - GEN

T1 - Stochastic modal models of slender uncertain curved beams preloaded through clamping

AU - Avalos, Javier

AU - Richter, Lanae A.

AU - Wang, X. Q.

AU - Murthy, Raghavendra

AU - Mignolet, Marc

PY - 2012

Y1 - 2012

N2 - This paper addresses the stochastic modeling of the stiffness matrix of slender uncertain curved beams that are forced fit into a clamped-clamped fixture designed for straight beams. Because of the misfit with the clamps, the final shape of the clamped-clamped beams is not straight and they are subjected to an axial preload. Both of these features are uncertain given the uncertainty on the initial, undeformed shape of the beams and affect significantly the stiffness matrix associated with small motions around the clamped-clamped configuration. A modal model using linear modes of the straight clamped-clamped beam with a randomized stiffness matrix is employed to characterize the linear dynamic behavior of the uncertain beams. This stiffness matrix is modeled using a mixed nonparametric-parametric stochastic model in which the nonparametric (maximum entropy) component is used to model the uncertainty in final shape while the preload is explicitly, parametrically included in the stiffness matrix representation. Finally, a maximum likelihood framework is proposed for the identification of the parameters associated with the uncertainty level and the mean model, or part thereof, using either natural frequencies only or natural frequencies and mode shape information of the beams around their final clamped-clamped state. To validate these concepts, a simulated, computational experiment was conducted within Nastran to produce a population of natural frequencies and mode shapes of uncertain slender curved beams after clamping. The application of the above concepts to this simulated data led to a very good to excellent matching of the probability density functions of the natural frequencies and the modal components, even though this information was not used in the identification process. These results strongly suggest the applicability of the proposed stochastic model.

AB - This paper addresses the stochastic modeling of the stiffness matrix of slender uncertain curved beams that are forced fit into a clamped-clamped fixture designed for straight beams. Because of the misfit with the clamps, the final shape of the clamped-clamped beams is not straight and they are subjected to an axial preload. Both of these features are uncertain given the uncertainty on the initial, undeformed shape of the beams and affect significantly the stiffness matrix associated with small motions around the clamped-clamped configuration. A modal model using linear modes of the straight clamped-clamped beam with a randomized stiffness matrix is employed to characterize the linear dynamic behavior of the uncertain beams. This stiffness matrix is modeled using a mixed nonparametric-parametric stochastic model in which the nonparametric (maximum entropy) component is used to model the uncertainty in final shape while the preload is explicitly, parametrically included in the stiffness matrix representation. Finally, a maximum likelihood framework is proposed for the identification of the parameters associated with the uncertainty level and the mean model, or part thereof, using either natural frequencies only or natural frequencies and mode shape information of the beams around their final clamped-clamped state. To validate these concepts, a simulated, computational experiment was conducted within Nastran to produce a population of natural frequencies and mode shapes of uncertain slender curved beams after clamping. The application of the above concepts to this simulated data led to a very good to excellent matching of the probability density functions of the natural frequencies and the modal components, even though this information was not used in the identification process. These results strongly suggest the applicability of the proposed stochastic model.

KW - Maximum entropy

KW - Parametric stochastic model

KW - Uncertain geometry

KW - Uncertain preload

KW - Uncertain stiffness

UR - http://www.scopus.com/inward/record.url?scp=84884663817&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884663817&partnerID=8YFLogxK

U2 - 10.1115/DETC2012-71169

DO - 10.1115/DETC2012-71169

M3 - Conference contribution

AN - SCOPUS:84884663817

SN - 9780791845004

VL - 1

SP - 887

EP - 895

BT - Proceedings of the ASME Design Engineering Technical Conference

ER -