Natural Frequency Perturbations Using a Scalar Expression with Reference Plots to Predict Associated Errors

Abstract

Purpose Eigenvalues are the natural frequencies of system squared. When designing a system it is important to know the natural frequencies, because if the system is forced near one of these natural frequencies the magnitude of vibration becomes very large. The eigenvalues are typically determined by solving an eigenvalue problem, which is an iterative produce that is expensive for larger systems. If multiple perturbations to the system are made or tested re-solving an eigenvalue problem every time becomes prohibitive. Perturbation methods exist to predict perturbed eigenvalues more quickly. However, these methods typically require matrix–vector products and do not quantify what is considered a small enough perturbation to use these methods. Methods This paper looks to address these issues using a scalar perturbed eigenvalue expression that avoids calculating matrix–vector products for every perturbation and developing reference plots that can be used to predict the associated error. The reference plots may be used to predict errors in the approximated natural frequencies from nominal modal parameters. The scalar perturbed eigenvalue expression and reference plots for errors were tested using numerical examples. Results In every case tested the plots were able to accurately predict the expected errors, to be within a predicted range. Conclusion The proposed method allows one to use the developed scalar expression to predict perturbed eigenvalues, and the developed reference plots may be used to predict the errors associated with using the proposed expression.