M.Sc Student | Nachum Sarig |
---|---|

Subject | Vibrations of Beams and Bars with Deterministic and Stochastic Heterogeneity |

Department | Department of Mechanical Engineering |

Supervisor | Professor Emeritus Eli Altus |

Full Thesis text |

Natural frequencies and mode shapes of non-homogeneous (deterministic and
stochastic) rods and beams are studied. The solution is based on the Functional
Perturbation Method (FPM). The frequencies and mode shapes are considered as
functionals of the non-homogeneous properties. The natural frequency and mode
shape of the *k*^{th} order is obtained analytically to any
desired degree of accuracy. Once the functional derivatives (with respect to
the non-uniform property) have been found, the solution for any morphology is
obtained by direct integration without resolving the differential equation. Several
examples with different non-homogeneous properties are solved and compared with
exact and analytical solutions as an accuracy check. The FPM accuracy range for
the frequency *w* and the mode
shape is less than 1% even for high heterogeneities. In the stochastic case the
accuracy of the natural frequencies depends on the stochastic information
used/given, on the correlation distance (roughly the “grain size”), on the
function around which the perturbation is executed, and on whether we are
interested in the properties of *w*
or of *w*^{2}. Moreover,
all frequency modes have the same response to heterogeneity as long as their
wave length is of the order of the heterogeneity’s characteristic distance. In
addition, the heterogeneity effect on the average natural frequencies is minimal
for the fundamental mode.