A non-linear vibration spectroscopy model for structures with closed cracks

• A vibrating structure with a crack is modeled using the Hertz contact theory.
• Damping is modeled using a fractional order-derivative.
• The non-linear equation was solved using a mixed perturbation scheme.
• Multiple Time Scales–Lindsted Poincaré data agrees with the experiment.
• The deformation raised to the third power is found for the Hertz model.

Ensuring an uninterrupted service in critical complex installations requires parameter health monitoring of the vibrating structures. Tools for monitoring structural modifications through changes in the measured dynamic responses are necessary in order to detect the advent and evolution of cracks before the occurrence of catastrophic failures. It is shown, both theoretically and experimentally, that the equation for the modes of vibration of a structure with closed (breathing) cracks and whose surfaces enter into contact during vibration can be modeled using the Hertz contact theory. The damping chosen is a fractional order derivative to investigate the order matching the experimental data. A perturbation solution technique, combining the Multiple Time Scales and Lindsted–Poincaré methods, has been employed to construct analytical approximations to the resulting non-linear equation of vibration. A 3D finite element model of the structure has been employed to compute the eigenvalues of the sound structure, providing a means to validate the measured resonance frequencies and also allowing the visualization of the modal deformations thus giving greater insight into the physics of the problem.