The prediction of the location and the size of the error agree well, even in the presence of measurement errors and noise. Estimates of the error parameters are quite close to the actual parameters of the test setup error.

106 3.3 Variation of first natural frequency ratios versus defect positions for. different crack depth ratios in the case of a simply supported beam.. with excitation frequencies in the vertical plane. with damping) in a cantilever bracket.

LIST OF TABLES

## NOMENCLATURES

### Introduction

High level of fatigue stresses leads to the development of micro and macro cracks along with inherent structural defects such as mechanical stress raisers - sharp wedge grooves, irregular changes in geometries - notches, slots, etc. With the development of non-contact sensors, the method finds useful application in the case of moving parts of the machine.

### Literature Survey

The free vibration and the influence of the crack on the vibration behavior of the shaft were investigated. Influences of the crack stiffness ratio, the fixed Sommerfeld number (which is the design parameter), the.

### Present Work

Numerically simulated natural frequencies and corresponding mode shapes of the beam with error are used in the algorithm. Since the algorithm is iterative in nature, the crack location is initially obtained for an assumed crack size (i.e., the equivalent crack depth ratio) and is used to estimate the updated crack size by estimating crack ductility coefficients. The excitation for forced vibrations of the beam was provided by a harmonic force of known amplitude and frequency.

The prediction of crack location and size is in good agreement even in the presence of measurement error and noise. The practicality of the developed algorithm is improved by using a new condensation scheme together with conventional condensation schemes.

### Organization of the Thesis

The procedure followed to detect, localize, and quantify the error was similar to that used in earlier phases of the algorithm. The extracted natural frequencies and the amplitude and phase changes of the response were input into the developed algorithm to detect, locate and estimate the error parameters. The accurate measurement of rotational DOFs is also one of the practical problems in implementing the algorithm, and it was proposed to use various condensation schemes to eliminate the inaccessible DOFs.

In Chapter 4, the practical applicability of the developed algorithm is improved by using a new condensation scheme together with conventional condensation schemes. The measured natural frequencies as well as information about the amplitude and phase of the response were used in the developed algorithm.

### Introduction

In Figure 2.2, the hatched part is the part of the beam without defects at the defect site. Let R be the radius of the beam and b be half the width of the beam. The damping matrix, [Dwf]( )e , of the beam element without flaws can be expressed in terms of the element mass and stiffness matrices as.

Kf]( )e is the stiffness matrix of the beam element with fracture, and is given as. The damping matrix of the beam element with fracture, [Df]( )e, can be expressed in terms of the elemental stiffness of the beam and mass matrices with fracture axis.

### Flaw Localization and Sizing Algorithms

The matrix, [ ]S , and the vector, { }B2 , contain all known information (i.e. the beam element model without fault, natural frequencies and corresponding mode shapes) except the fault location. Experimentally measured across natural frequencies and corresponding mode shapes will be used to iteratively obtain the fault location and magnitude. The possible fault location obtained in Step 3 is used to obtain an updated value of the assumed fault depth ratio with the identification algorithm, which gives the equivalent fault size when the fault location is known from measured mode shapes.

With the new value of the fault depth ratio obtained in step 4, steps 2 through 4 are repeated until the convergence of both fault location and size is achieved to the desired level of accuracy. It should be noted that the current identification algorithm gives as a by-product flexibility coefficients of the error (i.e.

### Numerical Examples

Defined in terms of change in the natural frequency due to the defect compared to the intact beam. Each natural frequency of the intact beam (which has two equal natural frequencies for the undamaged configuration in two orthogonal planes) splits into a pair of close frequencies when the fault is present. It should be noted that the reduction in natural frequency depends on the fault location and the vibration mode considered.

The iteration starts uniformly from the first intersection of the fault location for all cases considered in the example. The fault localization was found to be very efficient and accurate for the number of elements.

Conclusions

Introduction

### System Modeling

Details of the mass and stiffness matrices for the Timoshenko beam are given in Appendix D. The system damping has been considered as proportional (i.e. Rayleigh damping) damping; therefore, the damping matrix [Dwf]( )e of the intact beam element can be expressed in terms of the element mass and stiffness matrices as. It should be noted that in the elementary mass and stiffness matrices in equation (3.3) two orthogonal plane motions (x-y and x-z) are taken into account.

The elemental damping matrix for beam with defects, [Df]( )e , can be expressed in terms of beam stiffness and mass element matrices with defects, which. Equations of motion for the complete system can be obtained by summing contributions of elementary equations of motion without and with errors.

### Flaw Localization and Sizing Algorithms

For given system properties (i.e. [M], []D and []K, which include the beam element model without and with fault, the fault location and its magnitude), the response { }Q in frequency domain can be simulated from equation ( 3.12 ) which corresponding to a given force{ }F. The fault parameters such as fault flexibility coefficients and the equivalent fault depth ratio were estimated together with the fault location. Since the matrix [Kf] is of the same size as the reduced composite stiffness matrix [ ]K in equation (3.12) and it contains non-zero terms corresponding only to the error nodal DOFs { }Qf ( )e.

Equation (3.15) can be used to obtain the unknown flexibility coefficients contained in the matrix [Kf]( )e , which in turn will be used to obtain the equivalent depth of the defect. The equivalent defect depth ratio is obtained from equation (2.30), while the location of the defect is known.

### Numerical Examples

This frequency is taken as the experimentally measured natural frequency used in the identification algorithm to find the fault location. The procedure shown in Figure 2.8 for finding the fault location remains the same, but will apply to the Timoshenko beam. For the assumed parameters, the natural frequency of the intact cantilever beam is found to be 7.19 Hz.

For the defective beam, the natural frequency is found to be 7.1 Hz; for defect depth ratio (a) and defect location ratio (x) of 0.8 and 0.3, respectively. Since the presence of the defect in a structure results in a decrease in the natural frequency, for the intact beam, there is no change (decrease) in the natural frequency.

Conclusions

Introduction

### System Modeling

The standard dynamic condensation could be used during development of the identification algorithm for fault detection, localization and dimensioning.

### Flaw localization and sizing algorithms

To apply dynamic condensation while developing the defect size identification algorithm, equation (4.1) can be rearranged as Since the transverse rotational DOFs of the defective beam element are not eliminated due to difficulties in maintaining the defect flexibility coefficients during condensation, the resulting equation (i.e., equation (4.4)) contains both linear transverse DOFs and rotators related to the beam element assumption. The stiffness matrix [ ]K in equation (4.4) can be split into the [ ]K f matrix, which contributes to the nodal DOFs of the flawed beam, and the []K wf matrix, which contributes to all the elements other nodal DOF.

Equation (4.8) now contains only the translation DOFs, of the beam element with fault at the assumed location, in the state vector and will be used for the estimation of fault flexibility coefficients. The equivalent fault depth ratio was obtained from equation (2.30) from forced responses for the known location of the fault.

### Numerical Examples

The measurement error is introduced into the natural frequency and the measurement noise is introduced into forced responses to check the robustness of the current algorithm. The phase change will be a better indication of the natural frequency location apart from the peak in the amplitude. To check the robustness of the present algorithm, the measurement error in the natural frequency and the measurement noise in forced responses are taken into account.

Figure 3.5 shows the variation of fundamental natural frequency for different fault depths and for various locations of the fault for the cantilever beam. It is found that the fault localization is almost the same in terms of the element number.

Conclusions

Introduction

Model Descriptions and Support Conditions

### Experimental Setup and Instrumentations

The rotation of the beam was prevented by a stinger (pushrod), which served to connect the exciter to the beam. The following are the characteristics of the vibration exciter (B&K, type 4808) used in this experiment (see Figure 5.6). In excitation devices, the shaker drive platform must be connected to a structure that usually includes a force transducer.

In this experiment, a single sine wave waveform was used in combination with the swept mode. The following are the characteristics of the force transducer (B&K, model 2311-1) used in this experiment (see Figure 5.7).

### Test Procedure

One end of the stinger was firmly attached to the exciter and the other end was connected to the beam through a force transducer as shown in Figure 5.7. The force transducer was attached to the beam via a screw connection which was glued to the beam and the signal output of the transducer was connected to the signal processing amplifier. The force signal output (one) from the signal processing amplifier and the displacement signal outputs (four) from the proximinator were connected to signal input channels (five in total) of the data acquisition (DAQ) system of the Pulse system.

Such samples were selected over the entire frequency range of the data and amplitude and phase data (ie, the frequency response function) were extracted. The extracted information (from all captured data) was used to obtain the force-response relationship with respect to frequencies.

### Results and Discussions

In these data, resonance conditions were observed from the amplitude peaks and the associated sudden change in phase on the order of 1800. The above procedure was repeated for two beams with defects, as well as for the intact beam. The measured and processed data will be used for the estimation of damping coefficients, updating the FE model and the estimation of the error parameters.

The approach to numerical predictions of the behavior of a physical system is limited by the assumptions used in developing the mathematical model. The extent to which a numerical model can be improved by updating depends on the wealth of information about the test structure contained in the measurements.