PSI - Issue 64

Pavel Ryjáček et al. / Procedia Structural Integrity 64 (2024) 228 – 237 Pavel Ryjacek/ Structural Integrity Procedia 00 (2019) 000 – 000

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7. Dependence of mode shapes and frequencies on temperature Furthermore, the variation of the natural frequencies as a function of temperature was analysed with respect to a reference temperature of 20 °C. The resulting values were converted to a percentage difference, see Tab. 1.

Table 1. The change of the natural frequencies according to the temperature Mode shape Natural frequency for the cooling of 10°C Natural frequency of the reference model (10°C)

Natural frequency for the warming of 10°C

1 st mode shape 2 nd mode shape 3 rd mode shape 4 th mode shape 5 th mode shape 6 th mode shape

+4,17% +3,09% -1,59% +3,88% +3.77% +3.33%

0% 0% 0% 0% 0% 0%

-3.77% -2.94% +2.23% -3.49% -4.32% -3.13%

The 3rd natural frequency is particularly interesting, as it increases with increasing temperature, not like other frequencies. It can be seen that around 32-36 °C, the same frequency occurs for the 3rd and 4th mode shape and as the temperature increases further, the order of the eigenmodes switches. The change of the 4th eigenmode to the 3rd eigenmode was recorded, among others, during the dynamic test in 2013. The dynamic test was carried out at an average air temperature of 35 °C. This dependence of the eigenmodes on temperature can explain some of the anomalies observed in the dynamic load test. see Fig. 6.

a

b

Fig. 6 (a) Calculated 3 rd (a) and 4 th (b) mode shapes at different temperatures compared to the 20 °C

Based on the observed dependence between natural frequencies and temperature, it can be concluded that if the exact temperature of the structure is not known (and given the variability of temperature, this is a difficult variable to measure), an additional uncertainty is introduced into the measurement, which can be estimated at 0.3% for every 1 °C. At the same time, it should be noted that the measurement of mode shapes and frequencies is a process lasting several hours. During the measurement, gradual changes in temperature can occur and cannot be filtered.

Fig. 7 Change in the diagonal members of the modal flexibility matrix [  ] for 15 °C cooling for 86 sections