### Abstract

A formula to approximate the fundamental period of a fixed-free mass-spring system with varying mass and varying stiffness is formulated. The formula is derived mainly by taking the dominant parts from the general form of the characteristic polynomial, and adjusting the initial approximation by a coefficient derived from the exact solution of a uniform case. The formula is tested for a large number of randomly generated structures, and the results show that the approximated fundamental periods are within the error range of 4% with 90% of confidence. Also, the error is shown to be normally distributed with zero mean, and the width of the distribution (as measured by the standard deviation) tends to decrease as the total number of discretized elements in the system increases. Other possible extensions of the formula are discussed, including an extension to a continuous cantilever structure with distributed mass and stiffness. The suggested formula provides an efficient way to estimate the fundamental period of building structures and other systems that can be modeled as mass-spring systems.

Original language | English |
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Pages (from-to) | 717-732 |

Number of pages | 16 |

Journal | Structural Engineering and Mechanics |

Volume | 25 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2007 Apr 20 |

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### All Science Journal Classification (ASJC) codes

- Civil and Structural Engineering
- Building and Construction
- Mechanics of Materials
- Mechanical Engineering

### Cite this

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*Structural Engineering and Mechanics*, vol. 25, no. 6, pp. 717-732. https://doi.org/10.12989/sem.2007.25.6.717

**An approximate formula to calculate the fundamental period of a fixed-free mass-spring system with varying mass and stiffness.** / Kim, Juwhan; Collins, Kevin R.; Lim, Yun Mook.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An approximate formula to calculate the fundamental period of a fixed-free mass-spring system with varying mass and stiffness

AU - Kim, Juwhan

AU - Collins, Kevin R.

AU - Lim, Yun Mook

PY - 2007/4/20

Y1 - 2007/4/20

N2 - A formula to approximate the fundamental period of a fixed-free mass-spring system with varying mass and varying stiffness is formulated. The formula is derived mainly by taking the dominant parts from the general form of the characteristic polynomial, and adjusting the initial approximation by a coefficient derived from the exact solution of a uniform case. The formula is tested for a large number of randomly generated structures, and the results show that the approximated fundamental periods are within the error range of 4% with 90% of confidence. Also, the error is shown to be normally distributed with zero mean, and the width of the distribution (as measured by the standard deviation) tends to decrease as the total number of discretized elements in the system increases. Other possible extensions of the formula are discussed, including an extension to a continuous cantilever structure with distributed mass and stiffness. The suggested formula provides an efficient way to estimate the fundamental period of building structures and other systems that can be modeled as mass-spring systems.

AB - A formula to approximate the fundamental period of a fixed-free mass-spring system with varying mass and varying stiffness is formulated. The formula is derived mainly by taking the dominant parts from the general form of the characteristic polynomial, and adjusting the initial approximation by a coefficient derived from the exact solution of a uniform case. The formula is tested for a large number of randomly generated structures, and the results show that the approximated fundamental periods are within the error range of 4% with 90% of confidence. Also, the error is shown to be normally distributed with zero mean, and the width of the distribution (as measured by the standard deviation) tends to decrease as the total number of discretized elements in the system increases. Other possible extensions of the formula are discussed, including an extension to a continuous cantilever structure with distributed mass and stiffness. The suggested formula provides an efficient way to estimate the fundamental period of building structures and other systems that can be modeled as mass-spring systems.

UR - http://www.scopus.com/inward/record.url?scp=33947703002&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947703002&partnerID=8YFLogxK

U2 - 10.12989/sem.2007.25.6.717

DO - 10.12989/sem.2007.25.6.717

M3 - Article

AN - SCOPUS:33947703002

VL - 25

SP - 717

EP - 732

JO - Structural Engineering and Mechanics

JF - Structural Engineering and Mechanics

SN - 1225-4568

IS - 6

ER -