3 Temperature Variation Modelling of an Assembled Three-Storey Structure 27 second updating kept the aluminium at its nominal value while varying the steel’s Young’s modulus. This updating resulted in Young’s modulus of the steel to be 210.24 GPa, nearly identical to the nominal value. Despite only varying the steel properties, this updated value produced errors less than 1.25%. For the high-fidelity model, there appeared to be more dependency on the aluminium properties. Because of this, both materials were calibrated based on the first three bending frequencies. This resulted in the steel’s Young’s modulus to become 177.27 GPa and the aluminium to be 47.58 GPa. Both of the stiffness decreased compared to the nominal value. This is partially believed to be an inherent feature of the stiffness for the continuum elements compared to the beam elements due to reduced integration and hourglass adjustment. 3.5 Beam FEA Modelling Once the nominal material properties are defined, then two approaches of temperature modelling were applied to the FEA model. The first approach is the industrialist’s methodology; this applies a tie connection between the parts and sets the thermal expansion coefficient to account for the temperature dependency. To accurately account for the variations, the coefficient of thermal expansion was then calibrated to the change in natural frequencies with respect to temperature. To quantify this change, a linear regression is made on the mean values found experimentally. Figure 3.4 shows the experimentally found mean and 95% in blue and the linear regression shown in red. The main quantity used in calibrating the thermal expansion coefficient is the slope found from the regression (−1.967e−3 Hz/C for the fundamental frequency). To calibrate the thermal expansion coefficient, the slope found from the linear regression for the first three frequencies was taken in an ordinary least squares (similar to Young’s modulus calibration). This calibration results in the steel coefficient being 6.859e−5 C−1 and the aluminium coefficient of 7.338e−5 C−1. This resulted in six times the nominal value for the steel and three times for the aluminium. While this multitude of difference is large, the dynamic modelling of thermal strain is difficult to express via a tied interface. This tied connection enforces the displacement but not the strain. Since the material mismatch is expressed via a strain, the connection is not fully adequate to express thermal dynamic characteristic. Because of this inadequacy for the connection between the materials, the second approach replaces with tied connection with a temperature-dependent elastic connection. For the beam FEA, this reduces down to a temperature-dependent spring between the steel and aluminium. These springs were calibrated to the experimental data collected. This is believed to be reasonable since the system is designed to be a twin, so a large amount of data is collected. In a full deployment, there is a large amount of training data to ensure that the springs are able to account for any of the environmental conditions that the system experiences. In order to calibrate the temperature-dependent spring, the natural frequencies given from the sensors were used in a least-squares sense to determine the stiffness for that given result. The calibration used a gradient decent optimisation within Python, and one example of this for the 35Ccase is shown in Fig. 3.5. Fig. 3.4 Fundamental frequency temperature variations with linear regression
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