I hereby declare that the work presented in the thesis entitled “Operational modal analysis of an existing large truss bridge and passive vibration control using an updated model” in fulfillment of the requirement for the award of the degree of Doctor of Philosophy in an authentic report of my own work carried out in the Civil Engineering Department of the Institute. This would not have been possible without the support of the technical staff of the Department of Civil Engineering, IIT Guwahati.
CHAPTER 7 DESIGN OF TMD SYSTEM FOR THE SARAIGHAT BRIDGE
CHAPTER 8 SUMMARY AND CONCLUSIONS 183–188
ABSTRACT
Central trends in mode shapes (representing the aircraft along deck at rail level) identified using three techniques are found to be in good agreement. The FE model of the Saraighat Bridge is updated using the experimental frequencies and mode shapes.
LIST OF TABLES
Expanded form of the experimentally identified mode shapes associated with (a) 1st transverse (b) 2nd transverse (c) 3rd transverse (d) 1st vertical (e) 2nd vertical (f) 3rd vertical (g) 1st torsional and (h) 2nd torsional modes . E ] Direct transit of state space via matrix associated with ith mode [G(ω)] Frequency response function (FRF) matrix.
INTRODUCTION
- PREAMBLE
- STRUCTURAL HEALTH MONITORING
- SYSTEM IDENTIFICATION
- System identification of large Structural systems
- FINITE ELEMENT MODEL UPDATING
- PASSIVE VIBRATION CONTROL
- OBJECTIVE OF THE PRESENT STUDY
- OVERVIEW OF THE THESIS
Some commonly used system identification techniques are mentioned in the next subsection. U(jω) and u(t) are the frequency and time domain counterparts of the input, while Y(jω) and y(t) are such counterparts of the output.
LITERATURE REVIEW
INTRODUCTION
Sensor deployment is an important area of research in the field of structural health monitoring. This study considers in detail sensor placement in the OMA of a large structural system.
LITERATURE REVIEW
- Placement of sensors in operation modal analysis
- Ambient vibration data acquisition
- Identification of modal parameters
- Updating of FE model
- Design of passive control device
Measurement locations considered for (a) test 1 and 2 (b) test 3 during the data acquisition of the Gi-Lu cable-stayed bridge (courtesy of Weng et al. 2008). The model TMD systems were installed in the bridge model to observe the performance of the control devices.
SCOPE OF THE PRESENT STUDY
Therefore, there is great interest in modal identification of the Saraighat Bridge structure. OMA of Saraighat Bridge is proposed using these three techniques for better acceptability in identification of modal parameters. Such statistical representation is intended to be considered in addition to applying multiple techniques for superior acceptance in modal identification of the Saraighat Bridge.
The analytical mass and stiffness matrices for the Saraighat Bridge are planned to be updated using these two techniques mentioned above.
DESCRIPTION OF SAMPLE BRIDGE
- INTRODUCTION
- DESCRIPTION OF THE SAMPLE BRIDGE
- MODELLING OF THE SAMPLE BRIDGE
- DYNAMIC ANALYSIS
- EVALUATION OF MASS AND STIFFNESS MATRICES
- CONCLUDING REMARKS
The present study is primarily associated with system identification based on the dynamic response of the bridge. The following considerations have been made in the selection of the adopted FE model of Saraighat Bridge:. a) The RC substructure/foundation is significantly stiffer than the superstructure, which is made of steel. An important aspect of the structural modeling of the bridge is the modeling of the deck slab at road level.
This approach is followed in modeling the deck slab as shown in Fig.
PLACEMENT OF SENSORS IN OPERATIONAL MODAL ANALYSIS
INTRODUCTION
In addition, planning of suitable sensor locations for a detailed OMA exercise of the Saraighat Bridge is underway.
SYSTEM FORMULATION
A second order system can be decomposed into modal components where each modal component describes the system. It can be shown that modal displacement takes the form as {q} = {q}exp(jωt) for the displacement vector, {x} = {x}exp(jωt). Since matrices [Ξ] and [Ω] are diagonal, Eq. 4.10) can be written as the summation of modal components as.
Therefore, Gi(ω) can only be referred to as modal FRF in an approximate sense for acceleration measurement.
MISO MODELLING OF STRUCTURAL SYSTEM
FRF of MISO model is the appropriate row FRF of MIMO model of a structural system. Similarly, FRF of MISO model related to a mode is the appropriate row of modal FRF matrix (as in Eq. FRF of MISO model, related to all the modes, can be directly evaluated corresponding to kth output as .
The FRF of the MISO model, associated with the ith mode, can be evaluated directly corresponding to the kth output as.
MODAL MEASURES
Contribution of a state to output energy can be estimated if the observability grammian is diagonal. Consequently, contribution of a modal state in output energy can be approximately evaluated with the Eq. Now the contribution of a modal state to an output (DOF) associated output energy can be estimated based on that output (DOF) associated observability grammian.
However, the grammian of the single output based observation shows higher diagonal dominance with lower damping. e) Modal contribution to output energy (MCOE) as a modal measure:.
ISSUES WITH ACCELERATION MEASUREMENT
To analyze the effect of [E], the H∞ rates for the modes are calculated in the following two ways. A good agreement between these two modes of the calculated modal rate H∞ can determine that the feed directly through the matrix [E] is neglected. An approximate relation provided by Gawronski (2004) facilitates estimation of the acceleration-based modal rate using the velocity-based modal rate.
This relationship states that the acceleration-based modal norm for the kth mode, for example, is approximately equal to the product of the velocity-based modal norm for the kth mode and the circular frequency of the kth mode.
VALIDATION WITH BEAM PROBLEM
2 sensors)
3 sensors)
4 sensors)
- SENSOR PLACEMENT FOR SARAIGHAT BRIDGE
- CONCLUDING REMARKS
Hence the modal participation profile based on the H∞ modal. rate can be considered as similar to the MHSV-based modal share profile for measuring acceleration as well. The observability Gramian matrix profile based on the MIMO model and corresponding to the modal states according to Eq. Sensor locations are identified based on the modal participation profiles for each of the target modes, and the participation profiles are evaluated using MCOE.
MCOE therefore establishes itself as a suitable modal measure for sensor placement in OMA based on the modal approach.
SYSTEM IDENTIFICATION: A STATISTICAL APPROACH
INTRODUCTION
SENSOR AND DATA ACQUISITION SYSTEM
On the other hand, a 48-channel dynamic data acquisition system (model: MGCplus; . make: HBM GmbH, Germany) is used in data acquisition. Maximum sampling rate for data acquisition by MGCplus is and 9600 Hz associated with different sets of channels. In addition, a notebook computer is used to control the data acquisition system and store the measured data.
Two photographs of sensors and the data acquisition system during data recording are shown in Fig.
APPLIED IDENTIFICATION TECHNIQUES
The displacement vector associated with the equation of motion as {x(t)} at time t is related to the modal coordinate as {q(t)} in the form of the following equation. where, n, and r represent the number of DOF, modal matrix and rth mode shape vector respectively. The cross-correlation function Rijk(T) is defined as the expected value of the product of two responses evaluated at a time separation of T as follows. Consequently, the cross-correlation function is a sum of decaying sinusoids of the same form as the impulse response function of the original system as expressed in the Eq.
The key step in FSDD is the singular value decomposition of the estimated output PSD as shown below.
IDENTIFICATION OF MODAL PARAMETERS
IDENTIFIED MODAL PARAMETERS: CENTRAL TENDENCY AND DISPERSION
Central tendencies, distribution and confidence limits for the identified damping ratios using NExT-ERA, SSI and FSDD are presented in Table 5.2. It can be noted that FSDD estimated the damping ratios as comparatively lower than those estimated with NExT-ERA or SSI. However, the trend of damping ratios from lower to higher modes, identified with each of the techniques, shares some similarities.
On the other hand, the identified damping ratios showed a higher dispersion compared to the natural frequency case.
CONCLUDING REMARKS
Environmental vibration data in the form of acceleration response are used for modal identification. Furthermore, it appears that the 95% confidence limits for the natural frequencies lie within a narrow range. e) However, larger variations are observed for the central trends of the damping ratios identified using three techniques. Thus, these parameters identified by each of these three techniques can be taken into account for the update of the FE model.
However, SSI-based parameters are considered for the update in light of key features of SSI such as: sophisticated mathematical formulation, less likely influence of spectral leakage.
DIRECT UPDTING OF THE FE MODEL
INTRODUCTION
UPDATING TECHNIQUES
The measured mode shape vectors must be equal to the size of the analytical system matrices. It is known that the mass and stiffness matrices can be accurately reconstructed using all modal frequencies and all mode shapes when (a) the mode shapes are mass-normalized and (b) all DOFs are preserved in the mode shapes. The matrix mixing technique demonstrates that computational difficulties can be reduced if the estimated mode shapes are extended to the size of the matrices of the analytical system.
SEREP uses the forms of the analytical mode to produce the transformation matrix that enables the expansion process.
DIRECT UPDATING OF THE FE MODEL
Selected Analytical Mode Shapes with Modal Shift Model as: (a) Transverse 1 (b) Transverse 2 (c) Transverse 3 (d) Vertical 1 (e) Vertical 2 (f) Vertical 3 (g) Torsional 1 and (h) 2 twisters. At first, eight experimental mode shapes are expanded using SEREP to the size of the analytical mode shape. A further observation was made on the correlation between analytical and experimental mode shapes.
The MAC-based correlation between experimental and analytical mode shapes is shown in Figs.
RESULTS AND DISCUSSION
Furthermore, mode shapes are also found to be in complete agreement with the measured shapes. Therefore, mode shapes obtained based on the updated system matrices are the same as corresponding expanded measured shapes (as in Fig. 6.3). The MAC between these experimental and the updated mode shapes is evaluated and presented in Table 6.2.
Updated mode shapes corresponding to (a) 1. transverse (b) 2. transverse (c) 3. transverse (d) 1. vertical (e) 2. vertical (f) 3. vertical (g) 1. torsional and (h ) 2. torsional modes.
CONCLUDING REMARKS
DESIGN OF TMD SYSTEM FOR THE SARAIGHAT BRIDGE
- INTRODUCTION
- DESIGN OF MTMD SYSTEM: A MODAL FRF BASED APPROACH
- SELECTION OF TARGET MODES FOR THE SARAIGHAT BRIDGE
- DESIGN OF MTMD SYSTEMS FOR THE SARAIGHAT BRIDGE
The mass ratio of the total mass of the MTMD system with respect to the ith modal mass (Mi) is denoted as i. For each of the possible values of i associated with the ith mode, the optimal values of fi,. i and Ti are evaluated together with the corresponding minimum norm values. In this study, the first two modes are considered in the case of Saraighat Bridge, each in the horizontal (transverse) and vertical directions.
The locations of each of the MTMD systems are considered along the DOF corresponding to the maximum modal deformation ensuring maximum observability.
