# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 3 Three DOF Mass-Spring-Damper System  
Three Degree of Freedom (DOF) spring-mass system, as shown in Figure 1, is considered to illustrate the proposed stiffness tracking algorithm. Stiffness change \(\Delta K\) is replaced by internal input forces \(\Delta u(t)\) to the DOFs, where the structural member is connected. Thus, stiffness change in spring K2 will result in additional input forces at DOF 1 and 2 and stiffness change in spring K3 will result in additional input forces at DOF 2 and 3. Since the other end of spring K1 is fixed, the stiffness variation in spring K1 will result in additional input force at DOF 1 only. Initial stiffness of each spring is assumed to be 100 N/m, and mass at each node is assumed to be 1 Kg. Proportional damping is considered, though non-proportional damping can be considered without loss of generality. To monitor the stiffness change in all three springs, the system has three inputs (\(ui\)) and three output sensors (Y\(i\)) at nodes 1, 2 and 3. Figure 2 plots the stiffness variation in individual springs used for simulating input-output data. As it can be observed, initially up to 10 s, a spring-mass damper system is considered to be healthy. Stiffness degradation in an individual spring is considered after 10 s. The stiffness of spring K1 is assumed to be 90 N/m between 10 and 20 s and 80 N/m after 20 s, whereas the stiffness of spring K2 is assumed to be 80 N/m between 10 and 20 s and 60 N/m after 20 s. Spring K3 is assumed to be undamaged till 20 s, and stiffness of 70 N/m is assumed after 20 s. A series of input error function observers is designed for each spring. The number of observers will depend upon the accuracy of the stiffness estimation. Figure 3 plots the data from the input error function observers. Non-zero error is an indication of mismatch between assumed stiffness and actual stiffness of spring, whereas zero error indicates the correct observer and corresponding stiffness of the spring is the estimated correct stiffness of the spring. For brevity, observers for all three springs are included in Figure 3 in increments of 10 N/m beyond the stiffness value of 60 N/m only. Zero input error for a stiffness value of 100 N/m till 10 s is observed for spring K1. Zero input error for a stiffness value of 90 N/m for spring K1 is observed between 10 and 20 s, and finally, zero input error for a stiffness value of 80 N/m for spring K1 is observed beyond 20 s. Thus, it can be concluded that spring K1 is undamaged till 10 s and has a stiffness of 90 N/m between 10 and 20 s and stiffness of 80 N/m beyond 20 s. Zero input error correctly follows the stiffness change shown in Figure 2 for spring K1. Similarly, it can be concluded that spring 2 is undamaged till 10 s and has a stiffness of 80 N/m between  
Figure 1: Schematics of the three DOF spring-mass-damper system. 10 and 20 s and stiffness of 60 N/m beyond 20 s. Spring K3 is undamaged till 20 s and has a stiffness of 70 N/m beyond 20 s. Input error function observers have decoupled the influence of each individual spring on the output measurements and are able to track the stiffness variation in all three springs in real-time. Stiffness variation in individual springs, as shown in Figure 2, is arbitrarily chosen to underline the robustness of the proposed methodology using a set of observers. For brevity, only a limited number of observers are included in Figure 3. Since the stiffness variation considered is of the order of 10 N/m, observers presented in Figure 3 are in increments of 10 n/m. Without loss of generality, any stiffness variations in individual springs can be considered. The number of observers per structural member should be determined on the level of accuracy with which stiffness variation needs to be tracked and the level of accuracy that can be achieved for practical applications. The presence of unwanted noise in input-output measurements will affect the accuracy of the tracked stiffness estimations.