# Effect of Environmental Parameters on Structural Health Status Assessment Using OMA Techniques
## 8 Correlation of Natural Frequencies with Temperature and Axial Force
### Frequency Correction with Temperature or Axial Force  
Raw frequencies can be corrected using temperature regression parameters as follows:  
\[f_{c,i}=f_{raw,i}-D_{t}(T_{i}-T_{0}) \tag{13}\]  
where:  
* \(f_{c,i}\) is the i-th value of the corrected data set; namely, corrected for temperature effects.
* \(f_{raw,j}\) is the i-th raw frequency value.
* \(D_{t}\) is the temperature drift coefficient.
* \(T_{i}\) is the instantaneous temperature.
* \(T_{0}\) is a temperature value chosen as a reference. The average temperature on the 72-h data set has been chosen as the temperature base point, \(T_{0}=21.29\ ^{\circ}\)C.  
The same parameters \(D_{t}\) and \(T_{0}\), coming from the 72-h dataset (chosen as reference) are then used for the correction of all the other datasets.  
Raw frequencies can also be corrected using axial force regression parameters as follows:  
\[f_{c,i}=f_{raw,i}-D_{f}(F_{i}-F_{0}) \tag{14}\]  
where:  
* \(f_{c,i}\) is the i-th value of the frequency corrected using axial force.
* \(f_{raw,j}\) is the i-th raw frequency value.
* \(D_{f}\) is the force drift coefficient.
* \(F_{i}\) is the instantaneous axial force.
* \(F_{0}\) is an axial force value chosen as a reference. The average force on the 72-h data set has been chosen as the force base values \(F_{0}=8.08\) kN for Beam 1 and \(F_{0}=8.63\) kN for Beam 2.  
The same parameters \(D_{f}\) and \(F_{0}\), coming from the 72-h dataset (chosen as reference), are then used for the correction of all the other datasets.  
A comparison between frequencies corrected using temperature (green curves) and frequencies corrected using axial force (red curves) is shown in Figure 10. These results are related to the 72-h dataset for Mode 1 frequencies extracted with the peak-picking method. Similar results are obtained for the remaining four datasets for the frequencies of modes higher than the first, and for PolyMAX and Cov-SSI results.  
The result of the correction of the raw frequencies using ambient temperature is still strongly correlated to the daily thermal cycles, while the correction using axial force provides a better and more stable result.  
In Beam 1, the 72-h dataset corrected with temperature obtains:  
* average frequency 13.79 Hz
* maximum frequency 13.88 Hz (+0.7% form average)
* minimum frequency 13.68 (\(-0.8\)% from average)
* whereas, if axial force is used, the same values are:
* average frequency 13.78 Hz
* maximum frequency 13.81 Hz (+0.1% form average)
* minimum frequency 13.76 (\(-0.1\)% from average)  
Similar results are obtained for Beam 2.  
A comparison of the corrected frequencies over 12 h of the five analyzed datasets is shown in Figure 11 for the results obtained using peak-picking method for Mode 1. Completely similar results are obtained for higher modes with PolyMAX and Cov-SSI.  
In Beam 1, where the damage is simulated, the frequencies of Mode 1 decrease depending on the magnitude of damage and its location, as will be discussed in the next paragraph.  
In Beam 2, where damage is not present, the force-corrected frequencies lie in a very narrow range of values (red curves), whereas the temperature-corrected frequencies (green curves) show a less stable trend and therefore seem unreliable to detect damage.