A novel tension estimation approach for elastic cables by elimination of complex boundary condition effects employing mode shape functions

To establish a rigorous mathematical foundation for a recently developed tension determination method incorporating the mode shape functions, the frequency equations and mode shape functions for the vibrating cables with various types of boundary conditions are first derived in this research. According to these analytical expressions in terms of dimensionless parameters, it is confirmed that the mode shape functions associated with complicated boundary conditions are all dominantly contributed by the sinusoidal components to suggest a more convenient option than the intertwined frequency equations. Based on this discovery, the cable tension can be generally decided with an explicit formula similar to that for the case with hinged boundary constraints at both ends. The effective vibration length for each selected mode in the formula, however, needs to be pre-determined by fitting the corresponding sinusoidal function with the mode shape ratios identified from multiple synchronized measurements. Parametric study is also conducted on the derived mode shapes to systematically investigate the interference effect of the hyperbolic component near the boundaries and provide valuable guidelines for the sensor deployment in engineering applications. Finally, the performance of the developed methodology in the cases with more involved boundary constraints is certified with demonstrative numerical examples and laboratory experiments.