Boundary stiffness and axial tension for a uniform Euler-Bernoulli beam model can be estimated by applying non-linear Gauss regression to measured natural frequencies. However, there are two major drawbacks in this method: it requires many measured natural frequencies to obtain reliable estimates, and for a lengthwise uniform beam cross-section, the boundary stiffness parameters may converge to swapped boundaries without mode shape measurements. This work suggests a simple novel concept to overcome these shortcomings: By attaching an external boundary mass with rotational inertia to one end of the beam, the natural frequencies will change without changing the boundary stiffness and axial tension. Combined with measurements without additional mass, this provides additional information to be used in a regression model, without imposing more unknown parameters: The mass adds a known asymmetry, permitting a distinction between left and right, and for each added mass the number of measured natural frequencies needed to obtain a reliable estimate is roughly halved. Estimation of boundary stiffness and axial tension can be of interest in, e.g., bolted joints. A tightened bolt can be modeled as a beam, with unknown boundary parameters, where the main interest often would be to estimate the axial tension. Previously, an estimation method would be hard to apply in practice, as bolts are often mounted in complex structures with many interacting modes, making it difficult to measure many transverse natural frequencies of the bolt itself. By adding a mass, the method can potentially be applicable for estimating tension in bolted joints. The proposed method is validated using simulated noisy measurement data and tested with experimental measurements obtained for a real bolt tightened in a structure.
- Axial tension
- Beam theory
- Bolt tension
- Boundary conditions
- Structural parameter estimation