Modification of natural frequencies of an automotive belt drive system based on eigen-sensitivity analysis of its configuration parameters

- A method is presented to modify natural frequencies of an automotive belt drive system via configuration modification.
- The configuration modification is regarded as an inverse eigenvalue problem.
- Necessary changes of a set of configuration parameters are determined for achieving desired natural frequencies.
- Resonance conditions of the belt drive system are successfully avoided.

A methodology is presented to simultaneously modify natural frequencies of an automotive belt drive system via configuration modification to match natural frequencies of the system with its working conditions. In this methodology, the configuration modification is regarded as an inverse eigenvalue problem in which necessary changes of a set of configuration parameters are determined for achieving a desired natural frequency variation relative to the original design value. Natural frequencies and mode shapes of the system are obtained by solving an eigenvalue problem that is directly derived from established governing equations of motion of a typical automotive belt drive system. Explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters are derived by differentiating the eigenvalue equation. Several configuration parameters are selected as design variables and their corresponding modification regions and directions are given based on frequency sensitivity analysis. In the inverse eigenvalue problem, variations of natural frequencies caused by changes of multiple design variables are approximated using the first and second order Taylor series expansions, which are based on explicit expressions of the first and second order derivatives of natural frequencies and eigenvectors with respect to the configuration parameters. By solving the inverse eigenvalue problem, necessary changes of design variables are determined and the desired natural frequencies are achieved so that resonance conditions are successfully avoided.