### Abstract

A new technique which employs both Picard iteration and expansion in shifted Chebyshev polynomials is used to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. As in previous studies, the integration and product operational matrices associated with the Chebyshev polynomials are employed. However, the need to algebraically solve for the Chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized. Since these coefficients are expressed as homogeneous polynomials of the system parameters, closed form approximations to the true solutions may be obtained. Also, because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. Two formulations are proposed. The first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix. Three different example problems, including a double inverted pendulum subjected to a periodic follower force, are included and CPU times and convergence results are discussed.

Original language | English (US) |
---|---|

Pages (from-to) | 61-85 |

Number of pages | 25 |

Journal | Journal of Sound and Vibration |

Volume | 206 |

Issue number | 1 |

State | Published - Sep 11 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Engineering(all)
- Mechanical Engineering

### Cite this

*Journal of Sound and Vibration*,

*206*(1), 61-85.

**Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems.** / Sinha, S. C.; Butcher, Eric.

Research output: Contribution to journal › Article

*Journal of Sound and Vibration*, vol. 206, no. 1, pp. 61-85.

}

TY - JOUR

T1 - Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems

AU - Sinha, S. C.

AU - Butcher, Eric

PY - 1997/9/11

Y1 - 1997/9/11

N2 - A new technique which employs both Picard iteration and expansion in shifted Chebyshev polynomials is used to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. As in previous studies, the integration and product operational matrices associated with the Chebyshev polynomials are employed. However, the need to algebraically solve for the Chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized. Since these coefficients are expressed as homogeneous polynomials of the system parameters, closed form approximations to the true solutions may be obtained. Also, because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. Two formulations are proposed. The first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix. Three different example problems, including a double inverted pendulum subjected to a periodic follower force, are included and CPU times and convergence results are discussed.

AB - A new technique which employs both Picard iteration and expansion in shifted Chebyshev polynomials is used to symbolically approximate the fundamental solution matrix for linear time-periodic dynamical systems of arbitrary dimension explicitly as a function of the system parameters and time. As in previous studies, the integration and product operational matrices associated with the Chebyshev polynomials are employed. However, the need to algebraically solve for the Chebyshev coefficients of the fundamental solution matrix is completely avoided as only matrix multiplications and additions are utilized. Since these coefficients are expressed as homogeneous polynomials of the system parameters, closed form approximations to the true solutions may be obtained. Also, because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. Two formulations are proposed. The first is applicable to general time periodic systems while the second approach is useful when the system equations contain a constant matrix. Three different example problems, including a double inverted pendulum subjected to a periodic follower force, are included and CPU times and convergence results are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0031237138&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031237138&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031237138

VL - 206

SP - 61

EP - 85

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

SN - 0022-460X

IS - 1

ER -