Entries of R L Q R are determined so as to prevent excessively large control moves. If the proposed controller runs for a sufficiently long time, it will yield a stationary Kalman Filter and Regulator for which the values of LQE and LQR gains can be expressed simply by limiting gain values of L and K , respectively.
Examining the stability of the LQE-LQR system, where the combined error and state estimation unforced dynamics of the closed-loop system are expressed by:. Notice here that the observer gain L can be designed independently from the regulator gain K by the separation principle . In Eq.
If we refer to the closed-loop model of the system dynamics presented in Eq. We notice that those frequencies are best estimates of those of the structure. This leads to the next concept in this research which is the prediction of any damage or structural parameter changes that might occur after a certain period of time while the structure is in operation. Keeping the LQG servo control gains the same at all times, any change in the structural parameters will alter the actual values of A , B , C and D.
In that case, a damaged or altered system will have a state-space model of the form similar to the one in Eq. A - , B - , C - and D - are the previously defined matrices of the damaged system. Therefore, the Eigen values of the closed-loop model of the damaged systems will be:. Notice that the damage to the structure will appear in the in closed-loop model estimated response as shifts in some or all of its Eigen values. The shift is expressed as:. The following section shows numerically and experimentally that damage inflicted on the structure can be detected from shifts in the natural frequencies predicted by the response of the closed-loop model.
Table 2 lists the values of Modal frequencies calculated by ANSYS as-well-as the actual ones obtained from experimental impulse force test of the pipe section. Thus, the two aforementioned reasons motivated the use of LQG servo-control and modal reduction to minimize model-plant mismatch. All accelerometers measure the acceleration in the y -direction which is perpendicular to the ground. Measurement obtained from the Right accelerometer is the only feedback signal to the LQG servo controller. Measurements of Middle and Left accelerometers are only used for evaluating the accuracy of model estimates at the corresponding locations.
The implementation of the proposed approach is presented in the schematic of the servo controlled closed-loop system shown in Fig. In the following, the input force is applied to the pipe by the instrumented hummer shown in Fig. Numerical evaluation of the proposed estimation technique is carried out by exciting the structure with multiple hammer hits near the Right accelerometer and collecting the acceleration data at positions a, b and c as indicated in Table 3.
Modelling and Estimation of Damage in Structures is a comprehensiveguide to solving the type of modelling and estimation problems. Modelling and Estimation of Damage in Structures is a comprehensiveguide to solving the type of modelling and estimation problems associated with the.
A sample input-output data is shown in Fig. Experimental setup and data acquisition hardware. Small value of R L Q R is chosen to prevent excessive control input to the model.
On the other hand, large elements of Q L Q R matrix are designed to bring all the states to the unperturbed state values, faster. Data is collected at a sampling frequency of 10 kHz, which is high compared to the highest considered Eigen frequency of the pipe, but was necessary for the stability of the Matlab Solver handling the numerical calculations. The LQE gain matrix is determined such that the error between estimates and actual is minimized.
Entries of Q w are selected to achieve minimal estimation error. Table 3. Accelerometer properties and locations relative to the right end of the pipe. Schematic of the servo controlled system used for acceleration estimation. It is clear from the Figure, that both Modal frequencies and mode shapes poorly match actual ones. Nonetheless, amplitude discrepancies as well as modes that are not in the direction of measurement still appear in the estimates although they might not be detected by actual accelerometer measurement.
Further improvement to the model is carried out using model reduction based on SVD. The latter shows that only eight states have significant contribution to the response in the direction of measurement, at the three prescribed locations where the output is desired.
The closed-loop servo controlled reduced-order model CLROM is implemented and the response spectrums at the Right, Middle and Left accelerometer locations are plotted in Fig. Spectrogram of actual vs. CLFOM acceleration estimates. Hankel singular values of full-order modal model in y - direction. The effect of changes in structural parameters on Modal frequencies is examined experimentally based on the two targeted outcomes listed in Section 4.
To examine the effect of structural damage on Modal frequencies, a mass block MB is retrofitted to the pipe to mimic structural damage such as pipe scale. Further, the effect of damage location relative to sensor location is also tested by varying the MB distance from the feedback sensor along the pipe span.
The Right accelerometer measurement is used as the feedback signal to the CLROM and estimates, rather than actual measurement, are used to predict damage. Estimates at the Middle and Left accelerometer locations are generated by the CLROM and frequencies shift in the estimates spectrum, if any, are evaluated, keeping in mind that actual acceleration measurements at the Middle and Left locations are used only to assess the accuracy of CLROM estimates. The MB has a mass of 5. This choice of a relatively high value for the mass block is aimed at eliminating any possibility of experimental error arising from using a smaller mass.
To examine the effect of the location of the MB relative to the feedback accelerometer of the Modal frequencies shift, two cases are examined where in Case I the MB is attached at the Left location while in Case II the MB is attached at the middle as shown in Fig. The following presents experimental validation of the proposed method. Table 4. Actual modal frequencies of the healthy pipe compared to estimates generated using full-order and reduced-order modal models.
In this case, MB was attached to the pipe in the vicinity of the Left accelerometer and acceleration is estimated by CLROM at both left and middle locations.
The feedback to the servo controller is the actual acceleration measured by the Right accelerometer. Examining the mode shapes of Fig.
This also validates the effectiveness of the model reduction technique which eliminated all the modes that have minimal contribution to the response in the y -direction. Notice that the remaining modes are predominantly in the lateral and horizontal directions, thus sift in their Modal frequencies is minimal in the vertical direction.
It should be noticed that in Fig. However, Fig. Damaged pipe acceleration estimates vs. Spectrum of acceleration estimates for healthy and damaged pipes in y -direction. MB at the left location. Results in Tables 5 and 6 indicate that Modal frequencies found from actual acceleration measurements at the Right are in good agreement with those found from estimates at the Middle and Left.
It also shows that the shifts found from estimates are almost identical to those found from actual measurements which is an indication that acceleration estimates are just as reliable as actual ones as per the approach proposed in this work. The experiment was replicated three times and the outcome was similar to the results presented, each time. It is worth mentioning that all frequency shifts where towards the left of the graph indicating a drop in the Modal frequencies. This is in-line with expectations since mass was added to the structure. It can also be concluded that a change in stiffness due to cracks or loss of structural continuity would cause a shift in Modal frequencies in the direction of reduced stiffness.
Table 5. Modal frequencies shift in the y -direction found from right acceleration spectrum. MB at left. Table 6. Eigen frequency shifts in the y -direction. CLROM estimates at middle and left acceleration spectrums. In this case, MB was attached to the pipe in the vicinity of the Middle accelerometer and estimates where obtained both at the Middle and Left locations using the CLROM while using the Right measurement as the only feedback signal to it.
However, the nature of the shift and the magnitudes are different from those given in Case I. Tables 7 lists the actual Modal frequencies shift as detected by the Right measurement and its spectrum. The latter two tables show good agreement between actual and estimated shifts. Experimental results have shown that adding MB to the structure has caused modal frequencies shifts in some modes while suppressing other modes.
This is so because the structural parameters have changed, although the change was a localized one. Modes affected by the addition of MB are those having a mode shape involving bending the y - z plane such as Modes 3, 6, 7 and Other modes affected are the ones involving rotation about z - axis such as mode 8 and 9. It is noticed also that higher order modes are affected more than lower order modes, for example the shift in Mode 10 is — The effect of MB on the frequency shift of Mode 10 was half of that when MB was placed at the Left because it was place very close to the modal node.
MB at the middle location. Table 7. MB at middle. Table 8. It is also clear from the Tables that the distance between the localized damage and the feedback sensor has affected the spectrum in different ways and thus, in addition to detecting the damage, this method can be used to detect location of the damage relative to the feedback sensor s , which is important since limited number of sensors are needed for the CLROM.
The manner by which the type of damage and its distance from the sensor is determined is not discussed in this research work due to random nature of failure, which will require extensive statistical analysis of the results presented in this work. The distribution of power into frequency components of the signal Amplitudes in Fig.
The reason for not taking the amplitudes into consideration is that the excitation force applied to the structure was applied manually which made it difficult to replicated test conditions. Modal damping is not considered since damping has not been explicitly altered in the experiment. Anti-resonance or zeros of the system has shown shifts as well since the feedback signal location and locations of estimates are at different distances from modal nodes.
This work only considered the first ten modes of vibration to construct the state-space model from FEA data. Jonathan M. Nichols received the B. His research interests include damage identification in structures, modelling and analysis of infrared imaging devices, signal and image processing, and parameter estimation. Kevin D. Murphy received the B.
Mechanical Engineering and M. Applied Mechanics degrees from the University of Michigan in and respectively.? He received his Ph. His research focuses on the nonlinear mechanics, vibrations, and stability of structures for a broad variety of applications. Modelling and Estimation of Damage in Structures is a comprehensiveguide to solving the type of modelling and estimation problems associated with the physics of structural damage.
Provides a model-based approach to damage identification Presents an in-depth treatment of probability theory and random processes Covers both theory and algorithms for implementing maximum likelihood and Bayesian estimation approaches Includes experimental examples of all detection and identification approaches Provides a clear means by which acquired data can be used to make decisions regarding maintenance and usage of a structure. Read more Read less.