Damage Detection Based on the Geometric Interpretation ... - CiteSeerX

Damage Detection Based on the Geometric Interpretation of the Eigenvalue Problem by Frederick A. Just Agosto Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Engineering Science and Mechanics APPROVED

______________________________ Dr. Scott L. Hendricks, Chairman

______________________________ Dr. Harley H. Cudney

______________________________ Dr. Surot Thangjitham

______________________________ Dr. Ricardo A. Burdisso

_______________________________ Dr. Saad A. Ragab

December 1997 Blacksburg, VA

Damage Detection Based on the Geometric Interpretation of the Eigenvalue Problem by Frederick A. Just Agosto Scott L. Hendricks, Chair Engineering Science and Mechanics (ABSTRACT)

A method that can be used to detect damage in structures is developed. This method is based on the convexity of the geometric interpretation of the eigenvalue problem for undamped positive definite systems.

The damage detection scheme

establishes various damage scenarios which are used as failure sets. These scenarios are then compared to the structure’s actual response by measuring the natural frequencies of the structure and using a Euclideian norm. Mathematical models were developed for application of the method on a cantilever beam. Damage occurring at a single location or in multiple locations was established and studied. Experimental verification was performed on several prismatic beams in which the method provided adequate results. The exact location and extent of damage for several cases was predicted. When the method failed the prediction was very close to the actual condition in the structure. This method is easy to use and does not require a rigorous amount of instrumentation for obtaining the experimental data required in the detection scheme.

Acknowledgments I must first give thanks to my Lord and Savior Jesus Christ, without his help nothing is possible. I would also like to express my gratitude to Dr. Scott L. Hendricks for his patience and help in supervising this work. His technical knowledge and philosophy of life is always a source of constant encouragement for me. This gratitude is also extended to Dr. Harley H. Cudney, Dr. Saad A. Ragab, Dr. Robert Heller for their help, time and patience in my career. The education which my professors have given me is second to none. Each is his own way is an encouragement and a mentor for me. I must also thank Dr. Ricardo Burdisso and Dr. Surot Thangjitham for their help in my time of need. I will always be in debt and I am at their order. I can only say thanks to all my friends: Haider, Jay, Chris, Li, Rakesh, Reza, Sergio, David and Hector. This research was funded by the Charles E. Minor Fellowship at Virginia Tech. The support is gratefully acknowledged. Finally, and most importantly, I want to thank my family. This would be my mother who always supported me. My wife, Carmen, who makes my life worth living and my daughter Guinevere who radiates a child’s innocence in my life. Also I can never forget my sister Nomita with her family Willie, Jason and Jr who are always there for me.

para mami God place your hands over mine so I may do the work you desire Place your vision in my eyes so I may see the things you want and Place my feet in the foot steps you want my life to take

iii

Table of Contents I.

Introduction

II

A Literature Review 2.1 Time Domain Methods 2.2 Frequency Domain Methods 2.3 Frequency Measurements to Detect Damage

4 5 6 10

III

Modeling Damage 3.1 A Single Degree of Freedom Model for a Cantilever Beam 3.2 A Two Degree of Freedom Model for a Cantilever Beam 3.3 A Three Degree of Freedom Model for a Cantilever Beam 3.4 A Single Degree of Freedom Model for a Simply Supported Beam 3.5 A Two Degree of Freedom Model for a Simply Supported Beam

14 14

IV.

V.

1

A Damage Detection Scheme 4.1 Mathematical Models used in Structures 4.2 The Eigenvalue Problem 4.3 The Geometric Interpretation of the eigenvalue Problem 4.4 The Convexity of the Geometric Interpretation of Eigenvalue Problem 4.5 An Arbitrary Two Degree of Freedom System Example No I Example No II Example No III Example No IV Example No V 4.6 The Damage Detection Scheme Example No VI A Continuous Cantilever Beam 5.1 The Crack Model 5.2 The Equations of Motion of the Beam 5.3 The Eigenvalue Problem and the Natural Frequencies of the Cracked Cantilever Beam

iv

20 27 37 41

49 49 55 57 62 65 70 71 72 75 78 79 80 86 87 89 94

5.4 Review of the Damage Detection Scheme (Nearest Approximation Method) 5.5 Experimental Verification 5.6 Conclusions for the Cantilever Beam VI.

VII.

97 102 104

A Cantilever Beam with Multiple Damage Locations 6.1 The Cracked Model 6.2 The Equations of Motion of the Beam 6.3 The Eigenvalue Problem and the Natural Frequencies of the Multi-Cracked Cantilever Beam 6.4 Experimental Verification 6.5 Discussion of Results

106 106 108 114

Conclusions 7.1 Future Work and Other Possible Applications

125 127

References

128

Vita

134

v

118 122

List of Figures Figure 3.1: Figure 3.2: Figure 3.3: Figure 3.4: Figure 3.5: Figure 3.6: Figure 3.7: Figure 3.8: Figure 3.9: Figure 3.10: Figure 3.11: Figure 3.12: Figure 3.13: Figure 3.14: Figure 3.15: Figure 3.16: Figure 3.17: Figure 4.1: Figure 4.2: Figure 4.3: Figure 4.4: Figure 4.5: Figure 4.6: Figure 4.7:

The single degree of freedom model 15 The Deflected beam with a small mass element of length .B 16 Graph showing frequency change of a cantilever in which damage 18 is represented as a reduction in stiffness A two beam finite element model used to represent a cantilever beam 20 First Natural Frequency Behavior due to Various Damage Scenarios 25 Second Natural Frequency Behavior due to Various Damage Scenarios 26 A Cantilevered Beam modeled by Three Finite Elements 27 A Volumetric Representation used to describe conditions for a Three 32 Degree of Freedom Model of a Cantilever Beam Frequency Behavior for various Damage conditions holding the state 34 of damage of the third element constant at 0%, 25%, and 50% damage A section of the graph of Figure 3.9 showing the three surfaces 35 corresponding to 0%, 25%, and 50% damage of the third element Second Natural Frequency behavior for the three degree of freedom 36 model Third Natural Frequency behavior for the three degree of freedom 37 model The Simply Supported Beam with a small mass element of length .B 38 Graph showing frequency change of a simply supported beam 41 in which damage is represented as a reduction of stiffness A simply Supported Beam Modeled by Three Beam Elements 43 Behavior of the first Natural Frequency of a simply supported beam to 46 various damage scenarios Behavior of the second Natural Frequency of a simply supported beam 47 to various damage scenarios An Ellipsoid which has been sliced in half in a 3-dimensional Euclidian 59 space Two regions containing line segments in a 2-dimensional Euclidean space 62 A Lumped Mass two Degree of Freedom System 65 Ellipses corresponding to the geometric Interpretation of the Eigenvalue 71 Problem for various scenarios in Example No. I. Geometric Interpretation of the Eigenvalue Problem for Various 73 Damage Scenarios in Example No. II Geometric Interpretation of the Damage Scenarios for Example No. II. 74 Twenty five percent reduction in the first spring for the last case Geometric Interpretation of the Eigenvalue Problem given in Example 76 No. III.

vi

Figure 4.8:

The geometric interpretation of Eigenvalue problem for a Cantilever Beam Figure 4.9: The Geometric Interpretation of the Eigenvalue Problem for a Simple Supported Beam Figure 4.10: Convex hulls of the Eigenvalue Problem Element One has ten, and twenty percent damage respectively Figure 4.11: A graphical Representation of the Damage Detection Scheme for a Cantilever Beam Represented by a Two Degree of Freedom Model Figure 5.1: Schematic of a cantilever beam with a transverse crack Figure 5.2: A Cantilever Beam with a Crack Located at position 6" Figure 5.3: First Natural Frequency of a Cracked Cantilever Beam Figure 5.4: Second Natural Frequency for a Cracked Cantilever Beam Figure 5.5: Third Natural Frequency for a Cracked Cantilever Beam Figure 5.6: Response Space used in Nearest Approximation Method Figure 5.7: The Cantilever Mounting System used for the Experiment Figure 6.1: A Cantilever Beam with Multiple Damage Locations Figure 6.2: The Cantilever Mounting System used in the Experiment Figure 6.3: The First Three Natural Frequencies of the Cantilever Beam Figure 6.4: The Frequency Response of a Damaged Cantilever Beam Around the first Natural Frequency. (20% damage, 4 inches from cantilever) Figure 6.5: The Frequency Response of a damaged Cantilever Beam around the First Natural Frequency (20% damage, 4 inches from cantilever; 10% damage, 16 inches from cantilever) Figure 6.6: Various Values of the Euclidean Norm for Scenarios Corresponding to one Crack on the Beam (actual condition one crack of 10% depth located at 24 in from cantilever)

vii

77 78 82 85 88 89 98 99 100 101 102 108 119 120 120

121

123

List of Tables Table 4.1: Table 5.1: Table 5.2: Table 6.1: Table 6.2:

Natural Frequencies and Corresponding Damage for a Two-Beam 84 Finite Element Model Nearest Approximation Method: Predicted Results and 104 Experimental Measurements Approximation Method: Predicted Results and Experimental Results 104 Location and Magnitude Natural Frequency and Calculated Torsional Spring Value for 107 a Rectangular Aluminum Beam Predicted and Obtained Vibrations Results for two 122 Damaged Cantilever Beams

viii

Chapter One: An Introduction

There are many different procedures available to the engineer for detecting changes in structures. These techniques can range from methods using acoustic emission, ultrasonic and eddy current measurements, x-ray diffraction and even vibration response. Each procedure has advantages and disadvantages, making no particular method superior to all the others. In this dissertation a detection scheme is developed which uses a vibration response technique. The method is based on the behavior of the geometric interpretation of the eigenvalue problem, which is a convex set.

Predicted natural

frequencies of the structure, from a mathematical model, and those obtained from the free vibration response are used in the algorithm The manuscript is divided into seven chapters which are outlined in the paragraphs that follow. Chapter Two is a literature review which is divided into three sections. These sections have categorized the detection schemes found in the scientific literature into three basic categories which are the Time Domain methods, the Frequency Domain Methods, and the use of Frequency Measurements only. Time Domain methods are methods which use the general vibration of the structure and work in the time domain. Frequency Domain methods, on the other hand, take data from the response signal of the

1

structure and transform it to the frequency domain. Literature pertaining to damage detection schemes which use natural frequency measurements only are categorized in the section of Frequency Measurements only. Chapter Three is an introduction to modeling damage in structures. This chapter is intended to acquaint the reader with the effect damage can have on the behavior of the natural frequency of the structure.

The chapter covers several simple mathematical

models which use natural frequency as a means of detecting damage. Two simples cases that of a cantilever beam and a simply supported beam are studied. Chapter Four develops the damage detection scheme used in the dissertation. The beginning of the chapter reviews the assumptions and the procedure used in developing the mathematical models. Upon completion, the eigenvalue problem is discussed along with its geometric interpretation. Convexity is then defined, and the importance of this concept in the geometric interpretation of the eigenvalue problem is shown through several examples. Finally the damage detection scheme is developed and a simple two degree of freedom cantilever beam model using a finite element method formulation is examined. Chapter Five covers the use of the method for a continuous beam with damage occurring only in one location.

Damage is modeled as a torsional spring and a

mathematical model is developed by the use of Hamilton’s method. Several damage scenarios are developed and the damage detection scheme is performed on a steel beam. The results of the use of the method with the experimental verification is discussed. Chapter Six covers the use of the method for a continuous beam. Here damage can be located at two positions simultaneously. The structural damage is produced saw cuts and is modeled as a torsional spring. The mathematical model is developed by

2

Hamilton’s method. The results of the use of the detection scheme with experimental verifications is also discussed. Chapter Seven concludes the research performed. Future work which can be done along with application of the method in other fields is also discussed.

3

Chapter Two: A Literature Review

The need to maintain and repair structures has produced an increased study in the topic of damage detection. Studies from the effect of failure of a member in a truss structure Malla (1993) to the behavior of a bridge structure after damage Salane (1990) have shown the potential effects that damage may produce. There are many ways in which structures may be inspected for damage. A review of the trends in system identification, related to damage Nathe (1990a) classifies detection into two basic groups: the time domain method and the frequency domain method. These groups may be subdivided into different areas depending on the parameters used or the method performed in the damage detection process. This literature review will cover work which has been performed in the two basic groups and then will concentrate on the studies which use frequency as a detection scheme. The change in vibration characteristics due to changes in structural parameters has been studied by various investigators. The determination of the natural frequency of nonuniform beams by the use of a matrix method was proposed by Thomson (1950). This method gave a procedure for the case where a non uniform beam was represented by a series of uniform sections. Measuring the natural period of vibration of buildings to

4

detect changes in the structure before and after an earthquake was suggested by Housner (1963). A method for predicting the effect of a local modification, the removal of a concentrated mass or linear spring, on the vibration characteristics of a linear system was investigated by Weissenburger (1968). Vibration testing as a nondestructive tool for composite materials by measuring damping changes in torsional vibrations was studied by Adams and Cawley (1974).

A general overview of vibration techniques in

nondestructive testing was given by Adams and Cawley (1985).

Global and local

methods which use vibrations as a means of damage detection were discussed. It was noted that damping is often more sensitive to damage than are natural frequencies and that damping is also more insensitive to dimensional variations in the structure. However, damping is much more difficult to measure accurately since care must be taken to minimize damping from the support system used to experimentally measure or hold the structure.

2.1 Time Domain Methods

Time response measurements are used in time domain methods to detect changes in the mass or stiffness of the structure. A general real-time domain technique was developed by Natke (1990b). Response measurements of input-output data were used to determine the time varying system parameters used in simulating the behavior of a structure.

Three time domain methods that estimate multiple-input, multiple-output

models for structural systems were compared by Hollkamp (1992). These methods were the eigen realization algorithm, a time domain method which uses impulse response functions of the system, the backward time series approach, a method which estimates a time series from input-output data using a least squares method, and the state-space

5

realization algorithm. This last method synthesizes a state-vector sequence from the time data of the system using a singular value decomposition method. The study suggested that the eigen realization method performed the best in creating a model for a structural system from input-output data. Another time response method used to predict damage detection was developed by Banks (1996). This method used an interactive least square error minimization to fit various damage model response scenarios to a cantilever beam that was damaged by drilling a hole in it. The model with the best least squares fit was used to predict the damage on the beam.

2.2 Frequency Domain Methods

Frequency domain methods have been used more by investigators than time domain methods. In frequency domain methods either the natural frequencies of the structure or the mode shapes of the structure, or a combination of both are used to detect damage.

A method to modify the stiffness and mass matrices of a finite element

formulation using statistical data, which was estimated by the analysts, was developed by Collins (1974). This procedure did not require the use of all the natural frequencies and mode shapes. The modal function were related to the structural parameters by a Taylor series expansion. Structural connectivity remained in the same form as the original finite element model. The experimental mode shapes which are normally obtained from experimental modal analysis are not usually orthonormal as are the ones expressed by mathematical models describing them. A technique in which the measured mode shapes where forced to satisfy the condition of weighted orthogonality was developed by Barach (1978). This technique is based on the minimization of a Euclidean norm of the errors of the mode

6

shapes subject to the orthogonality requirement.

Once the orthonormal modes are

obtained, a correction to the stiffness matrix is performed. It was suggested that this algorithm could be developed into a damage detection scheme. A method which used natural frequencies and mode shapes to improve the analytical mass and stiffness matrices was obtained by Berman (1983). This procedure did not require the mode shape data to be the same dimension as that of the analytical model. Stiffness updating used the same procedure developed by Barach. A general theory of damage detection in structures was presented by Stubbs (1985). Here changes in the modal properties of the structure are modeled as a series of uncoupled equations used to determine the damage site. This method required a priori knowledge of the natural frequencies of the structure with its modes shapes.

An

analytical model which predicted undamaged modal response was also used. Detection for damage, at a particular point in the structure, is sometimes insensitive to some particular mode shapes and natural frequencies. A damage detection scheme, in which the mass of the system is assumed constant, Chen (1988) is based on measuring the kinetic energy change of the ith mode of vibration before and after damage. A sub matrix approach in correcting the stiffness matrix in an analytical model to modal data was developed by Lim (1990). Here the mass matrix is assumed to remain constant and the stiffness matrix is represented as the summation of sub matrices multiplied by their corresponding scaling factors. The scaling factor is adjusted, using a least squares technique, to match the modal response of the system obtained by experimentation. A similar approach was used by Lin (1991) in structural damage detection. The finite element model and mode shapes must contain the same number of degrees of freedom. The stiffness matrix is modified by the addition of sub matrices representing damage. The fractional mode shape strain energy is used as a means to detect which modes and

7

frequencies are affected by the damage. If all degrees of freedom are not present then a reduction of the analytical model is performed. This procedure yields a system of linear equations which is used for obtaining the reduction factor of each sub matrix representing a particular type of damage. In addition to measuring modal strain energies as a damage detection indicator, an algorithm using residual forces has been developed by Ricles (1992). The residual forces are used to determine the damage location and a separate weighted sensitivity analysis is then used to determine the extent of the change in the mass and stiffness matrix. An energy method for crack size evaluation has been developed by Kam (1994). This method requires the use of an eigencouple, a natural frequency with its corresponding mode shape.

An energy balance is performed by

equating the damaged maximum strain energy with that of the undamaged maximum strain energy. This difference in measured energy is used with a model of the additional strain energy produced by damage to evaluate the state of damage. A damage detection scheme in which the curvature changes in mode shapes are used was developed by Pandy (1991). The system here is represented by use of the finite element method.

Curvature changes in the modes shape are to be extracted from

experimental results. Using the same idea Pabst (1993), developed an algorithm to detect damage which uses the natural frequencies and undamaged mode shapes of the structure. The crack location is determined by choosing where the mode shape curvature which has changed the greatest. It is assumed that the variation of the mode shape curvature at a damage location is much greater than that of the location without damage. A damage coefficient used to describe the degree of damage in the elements of a finite element model was developed by Linder (1993). The coefficient was determined using a least squares solution for an over determined system. This damage coefficient is calculated for

8

various damage scenarios. The damaged element which maintains a constant value for various mode shapes is chosen to be the damaged location and state in the structure. A method in which anti-optimization is used to detect damage has been developed by several investigators. These algorithms use anti-optimization to generate contrast excitation vectors that are most sensitive to damage. Sensmeier et al. (1995) used contrast maximization vectors in conjunction with neural networks to locate damage in a composite beam. Juneja, Cudney, and Haftka (1996) used a similar approach in which contrast maximization vectors established a data base to locate damage in a 3-D truss. The contrast maximization vector was then compared to the actual damaged structure by use of an average angle. The vector with the lowest average angle would predict the state of the trust structure. Experimental verification of this method was also performed. In addition to the use of modal measurements, natural frequencies and mode shapes, static analysis may also be employed in damage detection. The first method using a static displacement procedure to correct stiffness matrices in finite element models was given by Sheena (1982).

This method used static test results as constraints in the

correction of the stiffness matrix. In order to use this method all degrees of freedom must be given. Spline curves are used for interpolation of the degrees of freedom that are not measured through experimentation.

An approach to damage detecting using static

displacements and modal parameters was developed by Hajela (1990). Structural damage is represented by changes in the stiffness of elements in a finite element model representation. An interactive optimization scheme is used which tries to find a static loading condition that is optimal in the damage location. A unique procedure which uses static loading was developed by Ben-Haim (1992). In this method, the structure is loaded in such a fashion that only certain stiffness elements show sensitivity to damage. The loading is varied along the structure selectively studying the response of certain elements.

9

An extension of this method Ben-Haim (1994) uses the addition of spring elements to the structure to reduce the loading actuators needed in the selective sensitivity technique. Even though many algorithms have been developed to detect damage, underlying modal data issues must be resolved to obtain reasonable results. This area is being studied and was published in a paper by Smith (1992). Issues such as how the level of uncertainty affects damage detection, the measurements that should be used as damage indicators, and which modes and frequencies are best for determining a particular damage location on a structure were discussed.

2.3 Frequency Measurements to Detect Damage

Damage detection schemes which primarily use natural frequencies usually require the least amount of experimental data than other frequency domain methods. One great advantage in using these methods is that practically any point in the structure can be used to obtain the natural frequency information. One method, which is probably the first significant paper in this field, was developed by Adams and Cawley (1978). Drops in the natural frequencies corresponding to the axial vibration modes were used as detection indicators. Damage location was modeled to be an independent continuous function of frequency. Graphs of this function were made for a specific damage extent by varying the location point using computer simulation. At least two graphs were required to predict a location point. This procedure was verified by damaging a beam to a given extent and then changing the damage location by reducing the length of the beam. This method seemed viable from the results obtained. Another procedure developed by Cawley and Adams (1979) used the same principal idea about damage location. Here the function was expanded using a first order Taylor series. A frequency drop ratio was established as

10

the indicator for damage location. Sensitivity Analysis was used to predict the frequency drop as a function of stiffness reduction using the undamaged mode shapes. Chrondros and Dimarogonas (1980) established a procedure using the reduction of the natural frequencies to evaluate damage severity. Damage was assumed to be a crack at the cantilevered end of a beam. experimental analysis.

The crack model was obtained from

A relationship of crack damage to the damaged/undamaged

frequency ratio of a mode was established. Petroski (1981) investigated the behavior of an elastic beam with a crack. The damage location was at the mid span of a simply supported beam. The crack was represented as a pair of concentrated couples. Analytical models suggested that the beam vibration amplitude would increase substantially in the presence of damage. The use of a frequency vibration function for various damage scenarios was used by Ju , Akun et.al. (1982). This function was graphed for various crack depths and positions. The crack position satisfying a given frequency drop of a particular mode shape for a given depth was established. Experimental analysis would yield frequency drops corresponding to undamaged mode shape frequencies. Graphs of crack depth versus position for various mode shape frequencies were made. The point of intersection of the graphs suggested the damage location and state. Bastadzham and Atrakhov (1990) investigated the frequency spectrum change for damage in beams. The damage was simulated as a reduction of the beam section. It was suggested that examination of a wider spectrum of natural frequencies enhances the fault detection diagnosis. This was observed by plotting the frequency variation for various crack positions for a given fault depth. A method to detect damage, which measured vibration amplitude at two different points, was developed by Rizo, Aspragthos and Dimarogonas (1990). The method was only applicable to one dimensional structures and required vibration measurements to be made at natural frequencies.

11

A compliance

function was used to model the crack as a torsional spring. The method developed by Cawley and Adams (1979) was extended by Friswell et.al. (1994).

This variation

included the effect of noise in the model of the structure. The ratio of natural frequencies and a statistical method of identification based on a generalized least squares theory is used to identify the damage extent and location. In addition to damage detection using frequency reduction schemes, Hemez et.al. (1995) use a procedure that minimizes a Euclidean norm of the modal residuals for various damage scenarios. This tries to predict which frequencies produce the greatest variation in a particular mode shape frequency for a specific damage location. The use of rank ordering of eigenfrequency shifts has been suggest by Ben Haim and Braun (to appear). This method is based on the observations that frequency tends to vary in reduction depending upon the location of damage. Thus the order of frequency reduction for various mode shapes may be used to classify damage in areas of the beam. Modeling frequency behavior in beams with damage has been an area of interest in recent years. The objective of research in this field is to develop various models to describe and match vibration patterns of beams with damage such as crack or slots. Sato (1983) used a combination of transfer matrix methods with finite elements to study the effects that cracks have on beams. The crack was modeled as a large and abrupt change in the cross sectional area. A general flexibility matrix was used to model the behavior of a crack in the work of Gudmunson (1983). Christides and Bar (1984) developed an exponential decay function to model the stress intensity around a crack. This was used to formulate the equations of motion for an Euler-Bernoulli beam. Yuen (1985) used a finite element model, consisting of Timoshenko beam elements, to study damage on a beam. Damage was introduced as a reduction of the modulus of elasticity of a particular element. Cawley and Ray (1988) compared the natural frequencies produced by cracks

12

and slots on beams. It was noted that detection schemes which verify their results by producing damage in the form of slots may not work when an actual crack is measured. This is because a slot reduces the stiffness in a structure more than a crack of equal depth. Shen and Pierre (1990) represented damage as symmetric slots in a beam. A Galerkin formulation was used to obtain the natural frequencies. Lee and Ng (1994) used a linear spring to model the continuity condition of a crack in a beam. A torsional spring was also used to model the effect the crack had on the bending stiffness. A Rayleigh-Ritz method was used to obtain the stiffness and mass matrices for the mathematical model. Shen and Pierre (1994) developed the equations of motion of a beam with a crack by variational principals. Several modified stress, strain and displacement fields were used to satisfy compatibility requirements in the vicinity of the crack. Simply supported boundary conditions were used in the development. In addition to beams, frequency methods have been used in predicting damage in other structures. Vandiver (1975) computed changes in natural frequencies to detect damage in structural member of an oil platform. Petroski (1983) studied the effects of cracks on the structural response of piping systems. The use of vibrations to detect the presence of a peripheral crack in a thick wall pipe was investigated by Moshreti et.al.. Belz and Irretier (1993a, 1993b) experimentally investigated the effects of the change in damping and natural frequency for a composite driving shaft.

The more sensitive

indicator for the state of the damage was the change in the natural frequencies, especially the ones related to the bending modes.

13

Chapter Three: Modeling Damage

Mathematical models of structures which describe certain specific characteristics may be obtained by many different means.

Formulations using partial differential

equations may be developed for models which are not too complex. Other methods such as transfer matrix methods, and the finite element method, which is capable of handling more complex structural geometry, are often used. The goal of all these is to produce a mathematical model which will give insight into the behavior of the structure. Structural, geometric, and material properties influence the solution of the mathematical model describing the structure. This chapter will consider several basic models for the behavior of a beam which undergoes damage modeled as a reduction in stiffness to represent a crack. A single degree of freedom model and finite element models consisting of two and three elements will be used to investigate the behavior of an Euler-Bernoulli beam with different boundary conditions

3.1 A Single Degree of Freedom Model for a Cantilever

The simplest model which can be established for a cantilever beam is a system having only one coordinate to describe its behavior. This is a single degree of freedom

14

model. An equivalent spring constant, 5/ Ç and equivalent mass, 7/ Ç will be used to model the beam. This cantilever beam is depicted in Figure 3.1. The beam is of length, 6Ç density 3, and a uniform cross-sectional area (,¹¸2¹Ç in which, ,, is the base of the rectangular area and 2 , is the height.

h b

l

Figure 3.1: The single degree of freedom beam model

The model that will be used to obtain the equation of motion will consist of the following. The beam will be a massless elastic structure with a force of 7/ 1 applied at its end. This force is applied on the beam to simulate the effect of the body force produced by the mass of the original structure under the influence of gravity. The equivalent spring constant, 5/ , for the above model may be found in any standard vibrations book (Humar J. L. 1990) as 3IM¶6$ Æ Here I is the Young’s Modulus of the structure’s material and M is the second moment of area of the beam. In order to establish the equivalent mass, 7/ Ç the deflection curve of the beam during vibration is assumed to be similar in shape to the static defection under the gravitational load of the beam’s mass (Weaver, Timosinko, Young, 1990). Since the deflection curve is a function of the position on the beam, the maximum velocity will occur at the position of the maximum

15

deflection. The beam defection curve with an infinitesimal element, used in obtaining the equivalent mass, 7/ , is depicted in Figure 3.2.

meg l x

x

dx

Figure 3.2: The deflected beam with a small mass element of length .BÆ

The equivalent mass of this system is determined as follows. The velocity of any element at a distance B on the beam may be expressed as a function of the beam’s maximum velocity (@7+B ¹Æ This expression is given as: %6B$  ¸B% € '6# B# ¹ @ é @7+B é †. $6% The maximum kinetic energy of the beam is obtained by integrating the kinetic energy of the infinitesimal element over the domain, 6, of the beam. Let 7 be defined as the mass per unit length of the structure. This integration results in #

6

" %6B$  ¸B% € '6#B# ¹ 7@ ( ö 7+B è ‚ú .B $6% ! #

16

which is evaluated as " "!% # 7@7+B 6. # %!&

¸$Æ"¹

The maximum kinetic energy of a single degree of freedom system with an equivalent mass, 7/ Ç is given by " # 7/ @7+B Æ #

¸$Æ#¹

Equating expressions (3.1) and (3.2) yields the equation for an equivalent mass as 7/ é

"!% 76 %!&

in which 7 and 6 retain their meaning given earlier. The governing equation of a single degree of freedom system under free vibration is 5/ ÆÆ B€ B é !Æ 7/

¸$Æ$¹

The natural frequency, which consist of an expression of the equivalent mass and stiffness, is given by = é °5/ ¶7/ which may be expressed as = é ± "#"&IM "!%76% Æ This single degree of freedom model will yield a specific natural frequency for a given beam. Consider a cantilever beam with the following properties: a length 6 of 2 ft. and a $ rectangular cross-sectional area of ( "' ô 3% ) inch.

The beam is constructed from

aluminum with a modulus I of 10.8 ô 106 psi and a density of &Æ$* =6?1 0>$ Æ The single degree of freedom model developed yields a natural frequency of %"Æ&) Hz.

The

theoretical value of the exact natural frequency for this cantilever beam is 42.78 Hz. The error in the model, for predicting the natural frequency, is 2.805%. A graph showing the .= sensitivity of natural frequency change to stiffness change, ( .5 ¹, may be obtained by /

examining various reductions in stiffness of the equivalent spring. The information given

17

by this graph helps show how damage may be predicted by frequency changes. This graph is shown in Figure 3.3.

Frequency Change (Hz)

0 -2 -4 -6 -8

-10 -12 0

10

20

30

40

50

Damage Represented as a Percent of Stiffness Reduction

Figure 3.3: Graph showing frequency change of a cantilever beam in which damage is represented as a reduction in stiffness. An equation which describes the frequency behavior produced by damage is obtained by taking the derivative of the equation of the natural frequency, = é °5/ /7/ Ç with respect to the equivalent stiffness. This yields the following ¨ .A " " é ƒ Æ ° .5/ # 5/ 7/ „ If the change in stiffness, .5/ , is expressed as a percent change of the original stiffness say, "5/ , then the above equation becomes .A é

" "= è ‚Æ # "!!

18

(3.4)

Equation (3.4) yields an approximation of the behavior of the natural frequency due to stiffness changes. Now that an approximation of frequency change of damage is given, the question to ask is how accurate is the single degree of freedom model representing a cantilever beam? If damage to the beam resulted in a 10 percent reduction in stiffness, the frequency would decrease by 2.08 LD according to our model. This value should be distinguishable by most laboratory frequency analyzers which gives a feasible predication from our model. Care however, should be taken in interpreting the results of the single degree of freedom model. The way in which the damage is modeled must be compatible with the actual structure under consideration and the forms of damage it may receive. The equations and graphs which have been presented up to now, Equation (3.3) and Figure 3.3, consider only the beam’s stiffness as a parameter which may vary to describe damage. If the beam’s physical mass is conserved in this stiffness reduction, the model represents damage as a change in the modulus of elasticity of the structure. A physical interpretation of this is that the damaged beam conserves the same geometric properties of the original structure while the Young’s Modulus has been reduced. This interpretation is considered valid as a means of representing damage. However, no physical insight to the location of the damage is given. The results do not depend on the location of the damage. This implies that the crack or damaged area may occur any where on the structure in consideration. No indication is given if the damage is located in a critical loading area or in an area of less importance. The single degree of freedom model predicts a global reduction in the natural frequency when damage occurs. However it does not in any way consider the damage location along the beam which is of the most importance in any damage detection scheme. One way to overcome this lack information is to use a more elaborate model

19

such as one containing more degrees of freedom. A two degree of freedom model will be discussed next.

3.2 A Two Degree of Freedom Model for a Cantilever Beam

The way in which a two degree of freedom model is developed is of great importance in the representation of the actual beam. One method is to model the beam as a two spring, two mass system. The springs would represent the stiffness of the beam for a given section and the mass would correspond to its equivalent weight. This method sounds reasonable however, the only problem is in determining the values of the springs and masses which would produce reasonable results with actual structures. Another way of representing the beam, which in some way conquers this drawback, is to use the finite element method. A two-beam finite element model will be used to represent the cantilever beam. This model will have four degrees of freedom and is applied to the cantilever beam shown in Figure 3.4.

y

y2

y1 l/2

θ1

l/2

θ2

x l Figure 3.4: A two beam finite element model used to represent a cantilever beam

20

The terms C3 represent translational degrees of freedom and the terms )3 represent rotational degrees of freedom. Since two elements are used to represent a beam of length, 6Ç each element will be of length 6¶2. All Properties of the beam are assumed to be homogeneous and isotropic. The mass of the beam is given by 3,26 and all remaining parameters retain their original meaning given in the discussion of the single degree of freedom system.

With consideration of the boundary condition, zero slope and

displacement at the cantilevered section, the finite element model yields the following matrix equation ¼ $"# 76 ¾ ! ¾ &% )%! ½  "$

¼ #% )IM ¾ ! ¾  "# 6$ ½ '

! ) "$ $

! ) ' #

&% "$ "&'  ##

 "# ' "# '

 ÆÆC Æ É  "$ ¿Å Å Å 6)"" É É $ Á # Áà ÆÆ Ç €  ## Å C# É Å É Å À 6) É % Ä ## È ÂC Æ Æ É Â ' ¿Å Å Å 6)"" É É Å Å!É É # Á # ! ÁÃ Ç é à ÇÆ ' Å Å C# É É Å Å!É É Å É À ) 6 # % Ä È Ä!È

¸$Æ&¹

#

The solution of the standard eigenvalue problem, (E¾C¿ é =# ¾C¿¹Ç yields four natural frequencies for Equation ¸$Æ&¹. This means that the model is not a two degree of freedom system but a four degree of freedom system. Two of the natural frequencies given by this model are a result of the extra (rotational) degrees of freedom. The model which is of interest right now should only have two natural frequencies corresponding to the translational motion of the beam in the C direction. This can be overcome by using static condensation of the stiffness matrix (Humar J. L. 1990). The degrees of freedom )1 and

21

)2 will be eliminated to yield a representative two degree of freedom model of the cantilever beam. The equations for this static condensed model are given by: 76 $"# ö )%! &%

ÆÆ )IM "$Æ("%$ &% C" ÆÆ † € $ ö úé "&' C#  %Æ#)&( 6

 %Æ#)&( C" ! éé † úé † "Æ("%$ C# !

¸$Æ'¹

Consider the case of a beam with the following dimensions: a length 6 of # ft., a cross $ sectional area of ( "' in. ô $% in.¹,

a density 3 of &Æ$* =6?1 0>$ Ç and a modulus I of

"!Æ) ô "!' psiÆ Solving the eigenvalue problem of Equation ¸3.6 ¹ with values which are given above yields two natural frequencies of 43.29LD and #($Æ" L z. The first natural frequency given by this model is around one percent greater than the theoretical value. Now that a representative model for the two degree of freedom system has been established the modeling of the damage should be considered. The type of damage that will be studied is similar to the one which was discussed in the single degree of freedom model. This type of damage only reduces the stiffness of the beam. The damage may be represented as a given reduction in stiffness of each element. The stiffness of the first element, before the process of static condensation, will be given by the expression [!O/ ». Here O/ is the local stiffness matrix of an individual element and ! is a factor used to represent damage. In other words if ! has a value of one then zero percent damage on the beam corresponding to the element one has occurred If ! is given a value of 0.95 than a damage corresponding to a five percent stiffness reduction is considered. The local stiffness matrix of a beam element used in our mathematical model is given in symbolic form in Equation (3.7). ¼ !O"" ¾ !O#" !O/ é ¾ !O$" ½ !O%"

!O"# !O## !O$# !O%#

22

!O"$ !O#$ !O$$ !O%$

!O"% ¿ !O#% Á Á !O$% !O%% À

¸$Æ(¹

If the corresponding values of the beam that has been studied up to now are substituted into the local stiffness matrix expression given by Equation (3.7), (Meirovitch L, 1975), the stiffness matrix becomes. ¼ "#! $IM ¾ '! !O/ é $ ¾  "#! 6 ½ '!

'! %!  '! #!

 "#!  '! "#!  '!

'! ¿ #! Á Á  '! %! À

¸$Æ)¹

Since the damage under consideration will not affect the mass of the system, the local mass matrix for both beam elements will not change. This methodology can also be applied to the second finite element. The stiffness formulation in the local coordinate system is the same for both elements and the factor, ", will be used to represent the stiffness percent reduction of the second element. In other words, the local stiffness matrix for element number two is represented by "O/ .

Static condensation of the

stiffness matrix for the reduction of the rotational degrees of freedom may be used with this type of formulation. The use of this procedure gives a two degree of freedom system in which the governing equation may be expressed as follows: ¸7"$$ € 7#"" ¹ ö 7#$"

ÆÆ " # # 7#"$ ¸!O%% € "O"" ¹ "O"$ C" € # úé ÆÆ # # ú C# † èö 7$$ "O$" "O$$

" # ¸!O$% € "O"# ¹ ö # "O$#

ö

# " # # "O"% ¸!O%% € "O## ¹ "O#% # úö # # ú "O$% "O%# "O%%

" # # ¸!O%$ € "O#" ¹ "O#$ C" ! # # ú‚é C † é é ! † "O%" "O%$ #

"

¸$Æ*¹

The notation in Equation (3.9), O345 and 7534 is defined as follows. The superscript denotes the element number and the subscript denotes the matrix element position in OÇ

23

or 7, for the local stiffness or mass matrix of that element. By using Equation (3.9) the two beam finite element model may represent different types of damage conditions. This is achieved by substituting the proper values for each of the damaged parameter indicators (! and "¹ for elements one and two receptively. Substitution of (! é "Ç and " é "¹ with the corresponding values of O345 and 7534 for element one and two yields Equation (3.6) as expected. By solving the eigenvalue problem of Equation (3.9), a graph may be established to represent various conditions of damage on the cantilever beam. Changing the parameter ! only, corresponds to damage in the section of the cantilever beam represent by element one. If only the value of " is changed, a case corresponding to damage in the beam section of element two is obtained. Varying both parameters yields a combination of two damage conditions on the beam. Figures 3.5 and 3.6 demonstrate the first and second natural frequencies corresponding to various damage scenarios represented by !, and ". Examination of Figures 3.5 and 3.6 shows that each damage scenario has a particular value for the first and second natural frequencies. This observation may be used as a means of identifying the damage and its corresponding location. The problem of determining the exact position of damage occurring in only one spot has been improved when compared to that of the single degree of freedom model. The exact location is not specifically determined, however it is now isolated to one of two regions The first of these corresponds to the section from the cantilever end of the beam to the center. The other region is found in the section between the center and the free end of the beam. The intention of continuing this type of evaluation with models containing more degrees of freedom seems a reasonable one and a model containing three beams elements should yield more information. A three degree of freedom system developed with three beam finite elements will now be discussed.

24

44 42 40 Frequency (Hz)

38 36 34 32 30 28 5 1015 20 25 Perc ent 30 of S tiffn 35 40 ess 45 Red 50 ucti on E 55 le m 60 en t Two

60

0 105 15 25 20 t One 30 n 35 eme 40 f El o n 45 ctio 50 e du 55 ffness R Sti

Figure 3.5: First Natural Frequency Behavior due to Various Damage Scenarios

25

280

Frequency (Hz)

260

240

220

200

180 5 1015 20 25 Perc ent 30 of S tiffn 35 40 ess 45 Red 50 ucti on E 55 le m 60 en t Two

60

0 105 15 25 20 t One 30 n 35 eme 40 f El o n 45 ctio 50 e du 55 ffness R Sti

Figure 3.6: Second Natural Frequency Behavior due to Various Damage Scenarios

26

3.3 A Three Degree of Freedom Model for a Cantilever

The three degree of freedom model of the cantilever beam will be developed using a similar finite element approach as discussed in the two degree of freedom model. The beam is discretized by three beam elements each of equal length. Figure 3.7 depicts this model applied to the cantilever beam.

y

y1 l/3

y3

y2 l/3

θ1

1

θ2

l/3 θ3 3

2

x

l

Figure 3.7: A Cantilever Beam Modeled by Three Beam Elements When the cantilever beam is modeled in this fashion it has six degrees of freedom. Vertical displacements are represented by C" Ç C# Ç C$ and rotational displacements by )" Ç )# Ç and )$ Æ The finite element description is a mathematical model which is obtained from the local mass and stiffness matrices of the beam elements used to describe the cantilever. These matrices are superimposed according to the degrees of freedom of the model and corresponding boundary conditions to produce the model under study. A mathematical description of this finite element model is given in Equation (3.10).

27

" # ¼ 7$$ € 7"" # ¾ 7"%$ € 721 ¾ # ¾ 7$" ¾ ¾ 7#%" ¾ ¾ ! ½ !

" # ¼ 5$$ € 5"" " # ¾ 5%$ € 521 ¾ ¾ 5# $" ¾ ¾ 5# %" ¾ ¾ ! ½ !

# 7"$% € 7"# 7"%% € 7### # 7$# # 7%# ! !

" # 5$% € 5"# " # 5%% € 5## # 5$# # 5%# ! !

# 7"$ 7##$ # 7$$ € 7$"" # $ 7%$ € 7#" 7$$" 7$%"

# 5"$ # 5#$ # $ 5$$ € 5"" # $ 5%$ € 5#" $ 5$" $ 5%"

7#"% 7##% # 7$% € 7$"# $ 7%#% € 7## $ 7$# $ 7%#

# 5"% # 5#% # $ 5$% € 5"# $ # 5%% € 5## $ 5$# $ 5%#

! ! $ 5"$ $ 5#$ $ 5$$ $ 5%$

! ! 7$"$ $ 7#$ $ 7$$ $ 7%$

ÆÆ Æ C" É ! ¿Â Å Å ÆÆ É Å É 6 Å Á Å $ )" É É ! ÁÅ Å É ÆÆ É C 7$"% Á Á à ÆÆ# Ç € 6 ) É 7$#% Á ÁÅ $ #É Å Å É $ Á Å ÆÆ É 7$% Å Å CÆÆ$ É É Å É 6 $ À 7%% Ä $ ) $ È

Â!É Æ C" Æ ! ¿Â Å É Å É Å Å É Å É 6 Å É Å É Å $ )" É Å!É ! Á É ÁÅ Å É Å Å É É $ Á 5"% Á C # ! $ ÁÃ 6) Ç é Ã!Ç 5#% Å$ #É É Å Å É É ÁÅ É É Å $ ÁÅ É Å É Å ! C É Å É Å 5$% $ É Å É Å Å É 6 $ À ! È Ä ) 5%% Ä $ $ È

(3.10)

5 The notation used 534 Ç 7534 is similar to the one developed in the two degree of freedom

model.

Superscripts refer to the local stiffness matrix or mass matrix of the

corresponding element. Subscripts denote the location in that matrix. Equation (3.10) describes the free vibration of a beam that has no damage. The effect of damage will be introduced into the model by multiplying the local stiffness elements by a reduction factor. The variables, !, ", and > will represent a percentage reduction of the stiffness matrix corresponding to the first, second and third elements respectively. Use of this damage notation will only change the stiffness matrix in Equation (3.10). This new matrix will have the form of " # ¼ !5$$ € "5"" " # ¾ !5%$ € "521 ¾ # ¾ "5$" ¾ # ¾ "5%" ¾ ¾ ! ½ !

" # !5$% € "5"# " # !5%% € "5## # "5$# # "5%# ! !

# "5"$ # "5#$ # $ "5$$ € >5"" # $ "5%$ € >5#" $ >5$" $ >5%"

# "5"% # "5#% # $ "5$% € >5"# # $ "5%% € >5## $ >5$# $ >5%#

! ! $ >5"$ $ >5#$ $ >5$$ $ >5%$

! ¿ ! Á Á $ Á >5"% Á $ Á. >5#% Á $ Á >5$% >5 $ À %%

The mathematical model which has been developed may be used in an eigenvalue formulation to obtain six corresponding natural frequencies. Three of these frequencies

28

are a consequence of the rotational degrees freedom.

Experimentation is usually

performed with accelerometers which measure linear acceleration, which means that frequencies corresponding to rotational motion are usually not obtained. In order to have a model which will just give the natural frequencies that will be measured, static condensation will be used. This produces a three degree of freedom model in which only the translational degrees of freedom are considered. The following equation describes the model with static condensation. "

#

¼ 7$$ € 7"" 7#$" ½ ! "

" # ! ¿Â ÆÆC " Æ ¶¼ !5$$ € "5"" ÆÆ # 7$"$ à C # Ç € "5$" ÆÆ 7$$$ ÀÄ C $ È ·½ !

7#"$ $ 7#$$ € 7"" 7$$" #

¼ !534 € "512 #  "5$# ½ ! "

# "5"% # $ "534 € >5"# $ >5$#

#

¼ !5%$ € "521 # "5%" ½ !

" # ! ¿¼ !5%% € "5## $ # >5"% "5%# $ À½ >5$% !

# "5#$ # $ "5%$ € >5#" $ >5%"

# "5"$ # $ € >5"" "5$$ $ >5$"

# "5#% # $ "5%% € >5## $ >5%#

! ¿¹Â C" Æ Â 0 Æ $ >5#$ à C# Ç é à 0 Ç. $ ÀºÄ C È Ä0È >5%$ $

! ¿ $ >5"$ $ À >5$$

! ¿ $ >5#% $ À >5%%

"

¸$Æ""¹

The eigenvalue problem of Equation (3.11) yields the eigenvalues corresponding to the translational degrees of freedom. These values are used to obtain the natural frequencies of the cantilever structure. The effect that damage may have on the natural frequencies, due to its location and extent, may now be studied. In order to perform this, data corresponding to the cantilever beam of the two degree of freedom model will be used to verify the validity of Equation (3.11). Substitution of the values of !, " , and > corresponding to zero damage in the beam produce the following equation.

29

$"# 76 ¼ &% "#'! ½ ! ")Æ%'"& #(IM ¼  "!Æ'"&% 6$ ½ #Æ(')#

&% $"# &%

 "!Æ'"&% "!Æ!&$)  $Æ'*#$

ÆÆ ! ¿Â C " Æ ÆÆ &% à C # Ç € ÆÆ "&' ÀÄ C $ È #Æ('*# ¿Â C" Æ Â ! Æ  $Æ'*#$ à C# Ç é à ! Ç "Æ'"&% ÀÄ C$ È Ä ! È

¸$Æ"#¹

The beam used in the two degree of freedom model had a length, 6, of 2 ft., a cross 3 sectional area of ( 16 in. ô 34 in.), a density 3 of 5.39 =6?1= 0>$ Ç and a modulus I of

"!Æ) ô "!' :=3. Use of these values in Equation (3.12) produces an equation whose eigenvalue problem yields the three natural frequencies of vibration. When these values are calculated for zero damage in the structure, a first natural frequency of %%Æ%LDÇ a second natural frequency of #)*Æ4LDÇ and a third natural frequency with a value of )!)Æ8LD are obtainedÆ If the first natural frequency obtained through the two beam finite element analysis is compared to the first natural frequency obtained through the three degree of freedom model a difference of 2.59 percent is obtained. Repeating this analysis with the second natural frequency yields a difference of approximately

6 percent.

Because of the relatively low error, (less than ten percent), produced by this model the three element cantilever beam model is assumed to be an adequate representation of the cantilever beam under study. Figures 3.3, 3.5, and 3.6 were used to help visualize how damage can be studied using a single degree of freedom model and a two degree of freedom model respectively. In these cases damage at particular locations along the beam where used as axes to plot the change in the natural frequency. The graphs that were produced where either a line, corresponding to the single degree of freedom model, or a surface, corresponding to the two degree of freedom model. This means of visualization helps suggests a detection scheme that is based upon how frequencies change due to a given damage condition. The

30

scheme developed is dependent upon the way the information given by our modeling is represented. For example if the single degree of freedom model is used, a frequency drop would correspond to a given damage condition on the beam. Thus by examining the graph, which was a line in this case, an overall idea of the extent of damage is determined. However the model used for this representation had the entire length of the beam reduced to a single parameter, 6. This made the detection of a particular damage location on the beam impossible. The two degree of freedom model represented the length of the beam as two distinct regions of equal length. This enabled the prediction of damage in one half the length of the beam or the other half by viewing the graphs corresponding to the first and second natural frequencies predicted by the model. Up to now the visual representation of frequency versus damage extent on a location has produced a means to observe the structure’s frequency behavior due to damage.

A trend of a drop in frequency

corresponding to an increase in damage in all models has been observed. It seems reasonable that a graph using a similar construction technique of the one and two degree of freedom models should be used for the three degree of freedom model. The three degree of freedom model divides the beam into three individual regions of equal length. If these regions are used as damage location sites on the beam a method of describing the actual damaged state of the beam may be established. The individual regions will be used as independent axes. The existing damage conditions on each region are established as variables used to describe the state of the beam. Thus a description which represents any existing damage conditions on the beam is a volume. Figure 3.8 demonstrates this idea by representing various points, corresponding to various damage conditions inside the domain of this volume.

31

Representing data in this fashion may lead to some difficulty interpreting frequency behavior because a point corresponding to a given damage scenario will have values of the first, second, and third natural frequencies corresponding to it. axis of percent damage in third element

Point corresponding to 50% damage of maximum damaged studied in all three elements

Plane corresponding to fifty percent damage in the third element.

Point corresponding to maximum damaged studied in the second element only

axis of percent damage in first element

Plane corresponding to zero percent damage in the third element.

Point corresponding to 0% damage third element maximum damage studied in second and first elements

Figure 3.8 A volumetric representation used to describe damage conditions for a three degree of freedom model of a cantilever beam.

These values, which are not viewed, help describe the behavior of the natural frequencies when damage occurs. One way of examining the trends that are occurring is to generate a family of surfaces. This may be produced by holding a given condition on an element constant. The conditions on the remaining elements will be varied to generate the values of the natural frequencies required in the surface plots. The surfaces which are produced, correspond to various damage conditions on the beam with the damage in one of the three elements remaining fixed.

This type of representation will be used to examine the

behavior of the first, second and third natural frequencies in the three degree of freedom model. A damage condition in the third element is held constant and damage in the first

32

and second element is varied.

Figure 3.9 depicts the behavior of the first natural

frequency for three cases. These correspond to zero percent damage in the third element, twenty five percent damage in the third element, and fifty percent damage in the third element. The damage plotted for each case varies according to the extent of damage in the third element. This procedure is performed in order to help visualize the behavior of the first natural frequency to various damage conditions of the third element. When Figure 3.9 is first examined it appears to be one graph with three different shades on it. However this graph is a representation of three different states of damage corresponding to the third element. The damage is held constant in the third element and the range of damage in the first and second element was varied depending upon the given damage level of the third element. The behavior of the surfaces plotted in Figure 3.9 shows a trend in the lack of sensitivity of the first natural frequency to damage occurring in the third element. Figure 3.10 examines a section of the graph of Figure 3.9 to help visualize the three different surfaces corresponding to 0%, 25%, and 50% damage in the third element. This procedure may be repeated to help determine the behavior of the second and third natural frequencies to damage as predicted by the three degree of freedom model of the cantilever beam. Figure 3.11 shows three surfaces which correspond to the behavior of the second natural frequency for three different damage conditions of the third element in our model. The surfaces have been plotted for different ranges corresponding to damage in the first and second elements for three different conditions of the third element.

33

46 44

Zero Percent Damage on Element Three

Twenty Five Percent Damage on Element Three

First Natural Frequency (Hz)

42 40 38 36 34 32 30 10 20 30 40 50 Stiffne ss Redu 60 ction E lement Two

Fifty Percent Damage on Element Three

0 20 10 30 40 60 50 t O ne Elemen n o ti c ss Redu Stiffne

Figure 3.9 Frequency Behavior for Various Damage Conditions Holding the State of Damage of the Third Element Constant at 0%, 25%, and 50% Damage

34

42.0 Twenty Five Percent Damage on Element Three

41.5

Zero Percent Damage on Element Three

First Natural Frequency (Hz)

41.0 40.5 40.0 39.5 39.0 Fifty Percent Damage on Element Three

38.5 38.0 Stiffn es

36 s Red

uction

38 Eleme n

t Two

40

20

18

16

e ss R Stiffn

14

12

10

e nt On e m e l on E educti

Figure 3.10: A Section of the Graph of Figure 3.9 Showing the Three Surfaces Corresponding to 0%, 25%, and 50% Damage of the Third Element.

35

300 Zero Percent Damage on Third Element

Twenty Five Percent Damage on Third Element

Second Natural Frequency (Hz)

280

260

240

220 Fifty Percent Damage on Third Element

200

10 0 20 30 ne 40 ent O m e l 50 E i on 60 educt R s s e Stiffn

10 20 30 Stiffn 40 ess R 50 educt ion E 60 lemen t Two

Figure 3.11 Second Natural Frequency Behavior for the Three Degree of Freedom Model.

Repeating this same procedure with the third natural frequency produces a graph similar in nature to the one shown in Figure 3.11. The graph showing the behavior of the third natural frequency to various damage conditions is shown in Figure 3.12. Again, the surfaces have been plotted for three different damage conditions of the third element. The range of damage in the first and second elements has been varied depending upon the

36

condition of the third element. This was done to help visualize the behavior of the third natural frequency to damage. 850 Zero Percent Damage on Third Element

Twenty Five Percent Damage on Third Element

800

Third Natural Frequency (Hz)

750

700

650

600

Fifty Percent Damage on Third Element

550

10 20 30 S t if f n 40 es s R educt 50 io n E 60 le m e nt Tw o

60

50 S t if f n

40 es s R

e

30

20

em on El i t c u d

10 en t O

0 ne

Figure 3.12 Third Natural Frequency Behavior for the Three Degree of Freedom Model of the Cantilever Beam Up to now a cantilever beam has been modeled by different means to obtain insight in the behavior of the natural frequencies when damage occurs. The principal used in modeling this structural damage has been to reduce the stiffness, by a given

37

factor, and observe how the natural frequency behaves. Figures 3.3, 3.5, 3.6, 3.9, 3.11, and 3.12 show a drop in the frequency versus an increase in the damage. The question of how other beams, with different boundary condition behave may be asked. In what follows an Euler-Bernoulli Beam with simple support conditions will be studied. This beam has some interesting properties which distinguish it from a cantilever beam. At the left support, the beam has geometric and natural boundary conditions present; the cantilever beam only possesses geometric boundary conditions at the left end. The simply supported beam also contains symmetry which is not present in the cantilever beam.

3.4 A Single Degree of Freedom Model of a Simply Supported Beam

The single degree of freedom model developed for the simple supported beam will use the concept of an equivalent spring constant, 5/ Ç and equivalent mass, 7/ Æ This simply supported beam is depicted in Figure 3.13.

x

h b

x

dx

Figure 3.13: The Simply Supported Beam with a Small Mass Element of Length .BÆ

The beam has a length, 6, density, 3, and uniform cross sectional area, ¸,¹¸2¹Ç in whichÇ , and 2, are used to describe the base and height of the beam respectively. The geometric

38

constraints of the beam are zero displacement at the ends which are pinned to the ground supports. An approach similar to the one used on the cantilever beam will be applied to the simply supported beam. In the mathematical model developed the assumption is made that the vibration mode shape is similar in form to the static displacement of the beam.

To produce an equivalent spring for the simply supported beam, the static

displacement described at the center is divided by the force required to produce it. This spring is given by 5/ é

%)IM 6$

in which I is the Young’s modulus of the structure’s material and MÇ is the second moment of area of the beam. The same procedure used in determining the equivalent mass of the cantilever beam will be used for this structure. The deflection curve of the beam during vibration is assumed to be similar in shape to the static deflection produced by the gravitational load of the beam’s mass. This deflection curve of the beam is a function of the position on the beam, thus the maximum velocity occurs at the location of the maximum displacement. Using this concept enables the velocity of an infinitesimal element .B, located at an arbitrary distance B, to be a function of the beam’s maximum velocity @7+B Æ Figure 3.13 depicts the above mentioned element. The equation which describes the velocity of a point on the beam is @é

@7+B "'º#6B$  ¸6$ B € B% ¹» &6%

¸$Æ13¹

In order to obtain the equivalent mass of the beam, 7/ Ç the kinetic energy of an equivalent spring mass system will be set equal to the kinetic energy of the beam. The

39

total kinetic energyÇ O> Ç of the beam is obtained by integrating the kinetic energy of the infinitesimal element over the domain 6. An expression for this is given by #

6 " "'@7+B ¸#6B$  ¸6$ B € B% ¹ O> é 7( é † .B # &6% !

which is evaluated as O> é

" $*') # 7è 6‚@7+B . # ()(&

¸$Æ"%¹

The kinetic energy of an equivalent spring mass system may be expressed as O/; é

" # 7/ @7+BÆ #

¸$Æ"&¹

When Equation (3.14) is set equal to Equation (3.15) an expression for the equivalent mass is obtained. This is given by 7/ é

$*')76 Æ ()(&

The natural frequency of vibration of a single degree of freedom system is obtained by the expression = é °5/ ¶7/ =± #$'#&IM #%)76% Æ Decreasing the value of the equivalent stiffness may be used as a representation of damage. In addition, this leads to a means by which the change in the behavior of the natural frequency due to damage on the structure may be observed. The drawback of this model, as in the case of the cantilever beam, is that the entire length of the beam is used in the development. Thus identifying a specific location of where the damage is occurring is not possible. A graph showing the natural frequency for a range of damage scenarios, for a beam with the same 3 properties: length of 2 ft, cross sectional area of ( 16 ô 34 ) inchÇ density of &Æ$* =6?1= 0>$ Ç and

a Young’s Modulus of 10.8 ô 106 psi, is given in Figure 3.14. It should be noted that the

40

first natural frequency from this model varies only 1.1 percent from the theoretical value of ±97.4

IM 76% Æ

120

Natural Frequency (Hz)

110

100

90

80 0

10

20

30

40

50

Damage Represented as a Percent of Stiffness Reduction

Figure 3.14: Graph W howing Frequency Change of a Simply Supported Beam in Which Damage is Represented as a Reduction in Stiffness.

Observing Figure 3.14 shows a relationship similar to that of the behavior of a single degree of freedom model of a cantilever beam. As damage is increased a reduction in the first natural frequency occurs. This behavior may be explained by the equation used to express the natural frequency of a single degree of freedom model, = é °5/ ¶7/ . A reduction in stiffness will clearly represent a reduction in the natural frequency. This is observed in Figure 3.3 by the negative frequency change for the cantilever beam. The reduction in the natural frequency predicts that damage has occurred in a global sense. No information concerning the location of damage is given other than that it is located in the entire domain of the structure. A mathematical model using more

41

degrees of freedom must be developed in order to obtain more information of where the damage is located. A two degree of freedom model using three beam elements will now be developed.

3.5 A Two Degree Freedom Model for a Simple Supported Beam

The mathematical model used to develop the two degree of freedom model will be performed with the use of finite elements. Since the cantilever beam two degree of freedom model used two beam elements it would seem reasonable to use two elements in developing a two degree of freedom model for the simple supported beam. The model which is developed using two elements will consists of six degrees of freedom. Three of these correspond to translational motion and three correspond to rotational motion. Experimentation that is usually performed only measures the frequencies that correspond to the translational components of our model. A stiffness reduction will be performed so that the model will yield the same frequencies that are measured experimentally. The geometric boundary conditions for the simply supported beam require that the ends don’t translate. If these boundary conditions are imposed on the mathematical model, the model is reduced to one of a single degree of freedom. In order to overcome this problem three beam elements will be used to develop a six degree of freedom model with the imposing boundary conditions. The degrees of freedom corresponding to the rotational degrees of freedom will be eliminated by use of stiffness reduction leaving a two degree of freedom model. This model, which consists of three beam elements, is shown in Figure 3.15. The mathematical model developed by the procedure described above, without the use of stiffness reduction but imposing the boundary conditions, yields a matrix equation of the form

42

ÆÆ ºQ »¾C¿ € ºO»¾C¿ é ¾!¿.

¸$Æ"'¹

y

y2

y3

l/3

l/3 θ1

l/3

θ2

θ3 2

1

θ4

x

3

l

Figure 3.15 A Simply Supported Beam Modeled by Three Beam Elements The matrices ºQ » and ºO» are defined as follows: "

7"#$ 7"$$ € 7#"" # 7"%$ € 7#" 7#$" 7#%" 0

¼ 7## ¾ 7"$# ¾ " ¾7 %# cQ d é ¾ ¾ 0 ¾ ¾ 0 ½ 0

" 7#% " 7$% € 7#"# # 7"%% € 7## 7$#2 7#%2 0

0 7#"$ # 7#$ 7#$$ € 7$"" 74#$ € 7$#" 7$%"

0 # 7"% 7##% 7#$% € 7$"# 7#%% € 7$## 7$%#

0 ¿ 0 Á Á 0 Á Á 73"% Á Á $ Á 7#% 7$%% À

and "

¼ 5## " ¾ 5$# ¾ " ¾5 %# cO d é ¾ ¾ 0 ¾ ¾ 0 ½ 0

" 5#$ " # 5$$ € 5"" " # 5%$ € 5#" # 5$" # 5%" 0

" 5#% " # 5$% € 5"# " # 5%% € 5## # 532 5%#2 0

0 # 5"$ # 5#$ # $ 5$$ € 5"" $ 54#$ € 5#" $ 5%"

0 # 5"% # 5#% # $ 5$% € 5"# # $ 5%% € 5## $ 5%#

0 ¿ 0 Á Á 0 Á Á 3 ÁÆ 5"% Á $ Á 5#% $ À 5%%

ÆÆ The vector ¾C¿ and ¾C¿ represent the displacement and acceleration of the degrees of freedom of the structure obtained by using the three beam elements. These are defined by

43

6)"  Š$ Å Å Å C Å Å Å 6)## Céà $ C$ Å Å Å 6)$ Å Å Å Å 6)$4 Ä $

Æ É É É É É É É Ç É É É É É É É È

ÆÆ Â 6) " Å Å Å ÆÆ$ Å Å C# Å Å Å 6ÆÆ)# ÆÆ C é Ã ÆÆ$ C$ Å Å ÆÆ Å Å )$ 6 Å Å Å Æ Å 6)$Æ Ä

%$È

Æ É É É É É É É É ÇÆ É É É É É É É É È

Static condensation using the Guyan reduction method will be performed for the stiffness reductionÆ This will require matrix manipulations for the stiffness matrix and for the mass matrix. The resultant mass matrix, which will be used in an equation similar to Equation (3.16) having two degrees of freedom, is given by the following expression. " " " " Q é Q>>  Q>) O)) O)> € O>) O)) Q)) O)) O)>  ¸Q>) O)) O) > ¹X

¸$Æ"(¹

were # 7"$$ € 7"" 7#$"

Q>> é ö

7"#$ ¼ " ¾ 7 € 7# #" %$ Q)> é ¾ ¾ 7# ½

O>> é ö

%"

!

" # !5$$ € " 5"" # " 5$"

" !5#$ ¼ ¾ !5 " € " 5 # #" %$ O)> é ¾ ¾ " 5#

½

%"

!

# 7"$# 7"$ Ç Q é > ) ú ö # 7$$ € 7$"" !

" 7$% € 7#"# # 7$#

" ! ¿ ¼ 7## # " Á ¾7 7#$ Á ¾ %# # # ÁÇ Q)) é ¾ ! 7%$ € 7#" $ À ½ ! 7%"

" # !5$# " 5"$ Ç O é > ) ú ö # $ " 5$$ € >5"" !

" ! ¿ ¼ !5## # Á ¾ " " 5#$ ÁÇ O)) é ¾ !5%# # # Á ¾ ! € " 5#" " 5%$ $ À ½ ! >5%"

44

7"#% € 7### # 7%# !

7"%%

# 7"% 7#$% € 7$"#

! 7##% # $ 7%% € 7## 7$%#

" # !5$% € " 5"# # " 5$#

" !5#% " # € " 5## !5%% # " 5%# !

! 7$"% ú ! ¿ ! Á Á 7$#% Á $ À 7%%

# " 5"% # $ " 5$% € >5"#

! # " 5#% # $ € >5## " 5%% $ >5%#

! $ ú >5"% ! ¿ ! Á Á >5#$% Á $ À >5%%

The terms !, ", and > appearing in the matrices O>> Ç O>) Ç O)> Ç and O)) are used to represent damage in the elements one, two, and three respectively. Using the above definitions, the stiffness matrix is given by " O é O>>  O>) O)) O) >

¸$Æ")¹

An equation having the form of Equation (3.16), in which the mass and stiffness matrices are defined by Equations (3.17) and (3.18), may be used to describe the motion of the simply supported beam. The only degrees of freedom that will be used are those which correspond to the translational motion allowed in the elements. This equation may be expressed as follows for a beam with zero damage occurring in the elements. ÆÆ 76 $*$Æ' #%Æ' #(IM C" *Æ'  )Æ% C" é! ö #%Æ' $*$Æ' úé ÆÆC † € $ ö  )Æ% úé *Æ' C# † "#'! 6 #

¸$Æ"*¹

With the use of Equations (3.17) and (3.18), the two degree of freedom model is obtained and may be used to describe the behavior of the structure to various damage scenarios. A beam, with the same characteristics as in the cantilever discussion, will be examined for various damage scenarios. A structure at times may be represented by the behavior of half of its domain. Structures which exhibit this property are said to be symmetric. A simply supported beam is a structure which has this property. The boundary conditions, geometric and natural, are the same at the beginning and end of its domain. The static displacement and natural modes of vibration are symmetric. Because of this symmetry damaging the first element is equivalent to the case of damaging the third element.

Figures (3.16) and

(3.17) show the behavior of the first and second natural frequency to damage occurring in the first two elements. It should be noted that the model developed predicted a first

45

# ± *(Æ764IM natural frequency of =#" é ± *(Æ'#IM % Æ 76% compared to the actual value of =" é

130 125

First Natural Frequency (Hz)

120 115 110 105 100 95 90

10

20

30 40 Stiffnes s Reduc tion Ele m

50

60

ent Two

60

50

40

30

20

E uction d e R s es Stiffn

10 0

t One lemen

Figure 3.16: Behavior of the first natural frequency of a simply supported beam to various damage scenarios

46

540

Second Natural Frequency (Hz)

520

500

480

460

440

420

400

10 Stiffne

20

30 40 50 s s Re d 60 uction Eleme nt Tw o

50

60

40

ss R Stiffne

30

20

10

0

ne ment O le E n eductio

Figure 3.17: Behavior of the second natural frequency of a simply supported beam to various damage scenarios

By examining the graphs (Figures: 3.3, 3.5, 3.6, 3.14, 3.16, and 3.17) produced, both for the cantilever case and the simply supported beam, a trend in the behavior of the natural frequency is observed. Depending upon how the structure is modeled, a one or two degree of freedom system, an estimate of the condition of the structure can easily be

47

determined. As the structure is modeled in a more complex fashion this ability of examining the natural frequency versus all the various damage scenarios graphically becomes impossible. Other means which use information such as the natural frequency or the structures behavior should be applied in determining the condition of the structure. In the following chapter, a norm will be developed which will help determine the condition of the structure and may also be used when more complex modeling is performed.

48

Chapter Four: A Damage Detection Scheme

Mathematical models can be used to predict the vibration response of structures. The solution techniques used in solving these models and the behavior of the system’s response may be used to develop damage detection schemes. In this chapter a method is developed which uses the natural frequencies of the structure to evaluate the damage condition in a structure. The damage location and extent of damage are also predicted by the scheme. 4.1 Mathematical Models Used in Structures Vibration problems are frequently represented by mathematical models using differential equations. Depending upon the model used, discrete or continuous, ordinary or partial differential equations are developed.

In the solution of these equations

synchronous motion is assumed, which in turn is the basis for the development of the eigenvalue problem. The eigenvalue problem is used to obtain the natural frequencies and mode shapes of the structure. The mode shapes describe the spatial domain of the solution. The physical interpretation of the mode shapes is that of the structure’s motion when it is vibrating at a particular frequency. The superposition of these modes are used to describe the general motion of the structure.

49

When structural damage was simulated for the particular models used in chapter three a decrease in the natural frequencies of the models was noticed as damage increased. The way that these frequencies were obtained was by solving the eigenvalue problem of the model under consideration. A basic equation used in modeling a majority of structures is given by ÆÆ Æ cQ de Cf € cG deCf € cO deC f é e0 ¸>¹f.

¸%Æ"¹

In Equation (4.1) the vector {C¿ consists of the generalized coordinates used in describing the motion of the structure. The matrices cQ dÇ cO dÇ and cG d are used to represent the mass, stiffness, and damping in the structure. The dimension of these matrices is 8 ô 8 where 8 corresponds to the number of degrees of freedom of the structure. The elements in these matrices are constants consisting of real coefficients. The vector ¾0 ¸>¹¿ is used to represent generalized forces acting on the structure. The solution of Equation (4.1) depends upon the coefficients in the differential equations of the mathematical model. These coefficients appear as elements in the matrices [Q »Ç ºG»Ç and ºO». The structure of these matrices, i.e. if they are symmetric, positive definite etc., dictates the solution method used in solving these equations. In this dissertation, the equations of motion will be derived from the virtual work associated with nonconservative forces and two scalar functions, the kinetic energy and potential energy. The kinetic energy of a structure may be written in the formÇ X é X# € X" € X0 Æ X2 is a homogeneous quadratic function of the generalized velocities having the form of 8 8 . . X# é "# !! 734 C3 C 4 Æ The other remaining terms, X" and X! Ç describe functions which 3é"4é"

are linear in the generalized velocities or contain no generalized velocities respectively. This discussion, which will be used in the development of Equation (4.1), will be limited

50

to structures having only a X# type kinetic energy description. A detailed discussion of the various forms used in the kinetic energy description of structures motion may be found in Meirovitch (1980). Vibrating structures can dissipate and/or store energy by various means. Energy due to motion is described by kinetic energy. In addition, structures may store energy through elastic forces. This energy can be expressed as a potential energy function, Z Ç which is a function of the generalized coordinates.

The potential energy function,

Z é Z ¸C" Ç C# Ç ÆÆÆÆ, C8 ¹, will be used to describe the elastic energy of the structures considered in this study. Energy dissipation in structures is usually characterized by three methods, these are viscous damping, structural damping, and coulomb friction. Dissipation by viscous damping forces is assumed and Rayleigh’s dissipation function is used to describe it. Rayleigh’s dissipation function is a quadratic function of the generalized velocities and is 8 8 Æ Æ given by Y é "# !! -34 C3 C 4 Æ The coefficients, -34 , are referred to as damping 3é"4é"

coefficients.

These coefficients are generally constant and are symmetric, -34 é -43,

(Meirovitch 1980)Æ The remaining forces, ¾0 ¸>¹¿Ç which do not fall into the above mentioned categories, elastic or damping, will be obtained from the virtual work expression, 8

$ [ é ! 03 $ C3, where $ C3 are the generalized virtual displacements. These generalized 3é"

forces are assumed to be functions of time and not function of generalized displacements or velocities. The Lagrangian of the system is defined by P é X  Z and the equations of motion are derived through Lagrange’s equation which is given by . `P `P `Y € Æ é 03 Ç è Æ ‚ .> `C3 `C 3 `C 3

51

¸3 é "Ç #Ç ÆÆÆÆÆÇ 8¹Æ

¸%Æ#¹

Application of Equation (4.2) will yield a set of 8 second order, nonhomogeneous nonlinear ordinary differential equations. In order to obtain the form given by Equation (4.1), these equations of motion will be linearized about an equilibrium point. Equilibrium points are determined by placing the forces, 0 3 ¸3 é "Ç #Ç ÆÆÆÆ8¹ in Equation Æ (4.2) equal to zero. These points produce a solution of the form C3 é constant, and C3 é ! in the state space plane. This type of solution places a constraint on the potential energy function, Z Ç such that `Z é !Ç 3 é "Ç #Ç ÆÆÆÆÆÇ 8Æ `C3

¸%Æ$¹

By solving Equation (4.3) the equilibrium points are determined. Small motions around the neighborhood of a particular equilibrium point can be established from the linearization of the equations of motion about that point

The

generalized coordinates now describe motions around the vicinity of the point. These equations can be produced from Lagrange’s equation by removing terms in the Lagrangian that are higher than quadratic. The coefficients, 734 , in the kinetic energy term, X# Ç become constant due to this linerization. These coefficients are symmetric and are given by the following equation. 734 é 743 é

` # X# ` # X# é Ç Æ Æ Æ Æ `C3 `C4 £ Cé-98=>Æ `C4 `C3 £ Cé-98=>Æ

3Ç 4 é "Ç #Ç ÆÆÆÆÆÆÇ 8.

(4.4)

The constants given by 734 Ç are the elements of the mass matrix ºQ » given in equation (4.1). In addition to the mass matrix, the damping matrix, ºG»Ç also consists of constant symmetric coefficients.

These coefficients, -34 , can be defined from Rayleigh’s

dissipation function as

52

-34 é -43 é

` #Y ` #Y é Æ Æ £ Ç Æ Æ £ `C 3 `C 4 Cé-98=>Æ `C 4 `C 3 Cé-98=>Æ

3Ç 4 é "Ç #Ç ÆÆÆÆÆÆÇ 8.

(4.5)

The mass and damping coefficients, given by Equations (4.4) and (4.5), have been developed for systems with a specific set of attributes. These vibrating systems have the following properties: 1) 2)

The kinetic energy of the structure is represented by a quadratic function of the generalized velocities only. Damping is produced by viscous forces and is represented with the use of Rayleigh’s dissipation function.

The above mentioned properties establish two terms the X# , kinetic energy term and Rayleigh’s dissipation function. These terms are used to define the elements of the mass and damping matrices respectively.. The remaining matrix in Equation (4.1) is the stiffness matrix, ºO», which is developed from the potential energy function, Z Æ By use of a Taylor series expansion the behavior of the potential energy function in the vicinity of an equilibrium point may be established. This series is expressed by 8

`Z C3 `C 3 3é"

Z ¸C" Ç C# Ç ÆÆÆC8 ¹ é Z ¸C"/ Ç C#/ Ç ÆÆÆÆÆC8/ ¹ € " " 8 8 ` #Z "" C3 C 4 € ø ø ø ø # 3é" 4é" `C3 `C4

¸%Æ'¹

in which C"/ Ç C#/ Ç ÆÆÆÆC8/ Ç are the generalized coordinate= corresponding to the equilibrium point. The partial derivatives in equation (4.') are evaluated at the equilibrium point. The first term in equation (4.') corresponds to the value of the potential energy function evaluated at the equilibrium point and is a constant. This term will have no effect in the equations of motion. The second term is equal to zero because of the constraint given by

53

equation (4.3) at an equilibrium point. Thus the lowest term in the potential energy function required for determining the linear equations of motion is the quadratic term of equation (4.6). Higher order terms in equation (4.6¹ are neglected when linearization is performed. Equation (4.6) may now be modified into a quadratic expression which is given as: " 8 8 ` #Z Z ¸C"Ç C#Ç ÆÆÆÆÆC8¹  é "" C3 C4. # 3é" 4é" `C3 `C4

¸%Æ7¹

The elements which constitute the stiffness matrix, [O», in equation (4.1) may now be obtained by use of equation (4.7). These terms are referred to as stiffness coefficients and are given by 534 é 543 é

` #Z ` #Z é Ç `C3 `C4 £ Cé-98=>Æ `C 4 `C 3 £ Cé-98=>Æ

3Ç 4 é "Ç #Ç ÆÆÆÆÆÆÇ 8.

With the conditions given for the kinetic energy, potential energy and energy dissipation, Equation (4.2), yields a set of 8 linear differential equations 8

"c734 C4 € -34 C4 € 534 C4 d é 0 3Ç

(3 é "Ç #Ç ÆÆÆÆÆ8).

(4.8)

4é"

The elements of the mass, damping, and stiffness matrices in Equation (4.1) are defined by the terms 734 Ç -34 Ç and 534 respectively. These matrices have elements which are made up of real coefficients and are symmetric. The mass matrix is positive definite, by definition, and the damping and stiffness matrices are considered positive semidefinite. The eigenvalues corresponding to the system of equations generated by Equation (4.2) may be characterized by the properties of the mass, damping, and stiffness matrices. In addition to viscous damping forces a proportional damping constraint will be placed on Equation (4.1). Proportional damping occurs when the damping matrix Ç ºG »Ç can be

54

expressed as a linear combination of the mass and stiffness matrices ºQ » and ºO». The importance of this will become apparent in the following section.

4.2 The Eigenvalue Problem

The systems that are considered in this investigation are classified as nongyroscopic systems with proportional damping. When damping is neglected in the mathematical model, the system is classified as a conservative nongyroscopic system. The mathematical model corresponding to this system is a set of 8 linear simultaneous ordinary differential equations with constant coefficients.

These equations may be

represented in the following form ÆÆ ºQ »¾C¿ € ºO»¾C¿ é ¾0 ¸>¹¿Æ

¸%Æ*¹

A common method used to solve these equations is the method of modal analysis which requires the solution of the eigenvalue problem for the system. The eigenvalue problem is developed from the homogeneous representation of Equation (4.9) which has the form of ÆÆ ºQ »¾C¿ € ºO»¾C¿ é !Æ

¸%Æ"!¹

As stated in the introduction of section 4.1, synchronous motion is assumed in establishing the eigenvalue problem. The mathematical significance of this is that a solution of the exponential form eC¸>¹f é /=> e?f

¸%Æ""¹

is considered, here = is a constant scalar and e?f is a constant vector of dimension 8. Substitution of

Equation (4.11) into equation (4.10)Ç performing some algebra, and

establishing the definition, - œ  =#, produces a matrix equation of the form

55

ºO»¾?¿ é -ºQ ¿¾?¿Æ

¸%Æ"#¹

Equation (4.12) represents a set of 8 simultaneous homogeneous algebraic equations in which the unknowns, ?3 ¸3 é "Ç #Ç ÆÆÆÆÆÆ8¹, in the vector ¾?¿ are parameterized by -. The values, -, are determined by admitting a nontrivial solution in Equation (4.12) for the vector ¾?¿Æ This aforementioned procedure establishes the algebraic eigenvalue problem for Equation (4.10)Æ However a form which may yield insight into the behavior of the solution of Equation (4.10) is desired. The eigenvalue problem is generally represented in a standard form given by ºE»¾@¿ é -¾@¿.

¸%Æ"$¹

Several methods are available for the transformation of Equation (4.12) into Equation (4.13). The property of symmetry, given by the matrices ºO» and ºQ », is desired to be retained in the matrix ºE». In order to accomplish this a particular linear transformation will be used. Substitution of the relationship, "

¾?¿ é óQ  # õ¾@¿Ç

(4.14)

into Equation (4.12) produces Equation (4.13) with the desired properties. The matrix, ºE»Ç is now defined as "

"

cAd œ óM # õcO dóQ  # õ.

¸%Æ"&¹

This matrix is a symmetric matrix whose form is defined from the pre and post "

multiplication of a symmetric positive definite matrix, ºQ  # ». Because of the particular similarity transformation given by Equation (4.15), the property of the matrix, ºE» being positive definite or positive semidefinite is determined by the matrix, ºO»Æ The matrix ºO», which is usually considered positive semidefinite, is determined by the potential energy function, Z Æ This function is used to describe the work performed by the

56

conservative forces acting in the system. The assumption will be made that the system, for the time being, will produce a stiffness matrix ºO» that is positive definite. This results in the matrix, ºE»Ç being positive definite. When this occurs, a special geometric interpretation of the eigenvalue problem corresponding to Equation (4.9) may be established.

4.3 The Geometric Interpretation of the I igenvalue Problem

In order to obtain an understanding of the geometric interpretation of the eigenvalue problem corresponding to Equation (4.9) a brief review of quadric surfaces will be presented. This discussion will commence in a 3-dimensional Euclidean space I $ and then proceed to an 8-dimensional Euclidean spaceÇ I 8 . A quadric surface is defined as the subset J in I $ such that J is the set of zeros of the quadratic equation G(?Ç @Ç A¹ é !Æ The variables ?Ç @Ç and A represent a set of arbitrary rectangular coordinates located in I $ Æ The general equation for the quadric surface may be written as +?# € ,@# € -A# € . € #+w @A € #,w ?A € #- w ?@ € #+ww ? € #,ww @ € #- ww A é !

¸%Æ"'¹

in which the coefficients +Ç ,Ç -Ç .Ç +w Ç ,w Ç - w Ç +ww Ç ,ww Ç and - ww are real numbers. When the quadric surface is central, meaning it has a center, and if Equation (4.16) contains no linear terms, the surface can be represented by the following form.

57

¾?

@

¼+ A¿ - w ½ ,w

-w , +w

,w ¿Â ? Æ +w à @ Ç é  . é . w . - ÀÄ A È

(4.17)

All of the properties of the quadric surface are embodied in the real symmetric matrix given in Equation (4.17). If the matrix is positive definite and a suitable rectangular coordinate system, (h ,i ,j ), is used to describe the surface, Equation (4.17) can be expressed as h# i# j# € € é "Æ T# U# V#

¸%Æ")¹

in which T, U and V are constants. Equation (4.18) describes an ellipsoid with a rectangular coordinate system located at its center. For a 2-dimensional Euclidian space, Equation (4.18) reduces to an ellipse. Quadric surfaces may be extended to an 8-dimensional Euclidean space, I 8 , without loss of generality. The subset J is now used to satisfy a quadratic equation which is made up of 8 arbitrary rectangular coordinates. The equations which were used to describe the ellipsoidal surface now contain 8 terms. A quadratic surface in I 8 possessing the requirements for an ellipsoid description (no linear terms and a real symmetric positive definite matrix representation) is referred to as an elliptic quadratic hypersurface. The significance of this is that an 8 ô 8 matrix equation, 8 ˆ %Ç can be used to represent this particular hypersurface. In order to help visualize the preceding discussion consider the ellipsoid shown in Figure 4-1. The origin of two rectangular coordinate systemsÇ (D" Ç D# Ç D$ ) and (?Ç @Ç A¹, are located at the center of the ellipsoid as shown. System (D" Ç D# Ç D$ ) coincides with the → → → principal axes of the ellipsoid and is represented by the directional vectors D" Ç D2 Ç and D3 .

58

→ → → The other coordinate system is defined by the directional vectors ? Ç @ Ç and A . This → system is oriented arbitrarily and is the basis for the position vector < and the gradient → vector f0 . Equation (4.17) is used to represent the ellipsoid in the (?Ç @Ç A) coordinate → system. The tip of the vector < describes any arbitrary point on the ellipsoid’s surface. → The gradient, f0 , is evaluated at this point.

z2 w z3

∇f

r

z1

v

u

Figure 4-1: An Ellipsoid which has been sliced in half in a 3-dimensional Euclidian space

59

An elliptic quadratic hypersurface can be represented by an equation of the form

0 é ¾B"

B#

Ê B8 ¿

ºE»

 ŠB" Æ É Å É B# à Çé" Å Å Ë É É Ä B8 È

¸%Æ"*¹

in which E is a 8 ô 8 real symmetric positive definite matrix and ¾B¿ is a vector containing 8 elements. The vector ¾B¿ is used to represent the components of a vector → → → analogous to < and will be denoted by B . The transpose of the vector B will be T

→ represented by B .

Using this notation Equation (4.19) may be expressed as

→ → → 0 é B E B Æ The gradient of the hypersurface, f0 , can now be written as T

T Â → → Æ Å Å É ` B EB É Â Å É `0 Æ Å É Å É `B" Å É Å É `B" É Å Å É T Å É Å É → → `0 → → ` B E B f0 é Ã `B# Ç é Ã `B# Ç é #E B . Å É Ë É Å Å Å `0 É É Å Å Ë É É Å É Å É T Å É Ä `B8 È Å → É Å `→ É Ä B`BE8 B È

¸%Æ#!¹

Analogues to the ellipsoid, the elliptic quadratic hypersurface can be described by a set of 8 principal axes which are normal to its surface. If the gradient vector is evaluated on a surface point coinciding with the intersection of the surface any principal → axis it becomes collinear to the axis and the vector B . This geometric property enables a → scalar multiplication of the vector B to be equivalent to the gradient. By judiciously choosing the scalar as 2- the following equation may be established → → → f0 é 2A B =2- B .

60

¸%Æ#"¹

Equation (4.21) is obviously the eigenvalue problem expressed in standard form. The implication of this result is that the eigenvalue problem for a real symmetric positive definite matrix, E, is equivalent to finding the principal axes of the quadric surface T

→ → described by B E B é "Æ

When the eigenvalue problem is pre multiplied by the T

→ transpose of the eigenvector, B Ç the following is established. T

T

→ → → → B EB é -B B

¸%Æ##¹

T

→ → → The term on the right side of Equation (4.22), B B , is the norm of the vector B squared, → š B š # . Recognizing that the left hand side of this equation is unity and performing some algebra, yields "

-é

→ š B š2

é =#

¸%Æ#$¹

Equation (4.23), establishes a relationship between the vibration of a structure and the geometric interpretation of its eigenvalue problem. This means that the reciprocal of the length of a principal axis in the quadric surface corresponds an eigenvalue which is the square of the natural frequency of vibration of the structure. The geometric interpretation of the eigenvalue problem is an elliptic quadric T

→ → surface which satisfies the equation B E B é ".

In addition to this, the surface

constitutes a mathematical set which has the property of convexity. The properties of convexity, which is discussed in the next section, will be beneficial to the development of a damage detection scheme.

61

4.4 The Convexity of the Geometric Interpretation of the Eigenvalue Problem.

Mathematical entities such as sets, bodies, and functions can exhibit the property of convexity. A region is said to be convex if a line segment joining any two points in the region is located entirely within the region. Consider the two distinct regions which are represented by the closed sets shown in Figure 4-2.

q/

p/

p q

(a)

(b)

Figure 4-2: Two regions containing line segments in a 2-dimensional Euclidean space.

The geometric property of convexity is easily viewed in Figure 4-2 (a). As long as the points : and ; are within the region, the line segment :; will be in the region. However, this is not the case for the nonconvex region shown in Figure 4-2 (b). Even though the points :w ; w are contained in the region, part of the line segment :w ; w is located outside the set. In a 2-dimensional Euclidean space, as seen in Figure 4-2, the convexity of a set can easily be determined. When sets become more complex and are harder to visualize, an algebraic formulation is required.

62

P/> W ,/ + =/> 90 :938>= 38 +8 8  .37/8=398+6 I?-63.+8 =:+-/ I 8 Æ J 9< +8C >A9 :938>= : +8. ; =?-2 >2+> : • W +8. ; • W >2/= + =/= 38 W ./0 38/. =?-2 >2+> ,: € ¸"  ,¹; • W 0 9< +66 @+6?/= 90 , A3>238 >2/ é t1

$(

># >"

6"

P 6" 7¸B¹ ø # 7¸B¹ ø # IM¸B¹ ww # ‡( ö # ¸C 6 ¹ ú.B € ( ö # ¸C < ¹ ú.B  ( ö # ¸C 6 ¹ ú.B ! 6" ! P

( ö 6"

IM¸B¹ ww # " ¸C< ¹ ú.B  O) ¸C# >"

6"

‡  $ ƒ( ö !

P IM¸B¹ ww # IM¸B¹ ww # ¸C6 ¹ ú.B € ( ö ¸C< ¹ ú.B # # 6"

" € O) ¸C# Æ 7ð( èC6 º$ ¸C6 bd¡  (

>"

!

P

># Æ ( èC< º$ ¸C< bd¡  ( >"

6"

># >" >#

>"

ÆÆ C6 $ ¸C6 ¹.>‚.B €

ÆÆ C< $ ¸C< ¹.>‚.Bñ

¸&Æ'¹

The varied pathÇ in a variational method, must be equal to the actual path at >" and ># Æ This means that the terms $ ¸C6 ¹l>" Ç $ ¸C6 ¹l># Ç $ ¸C< ¹l>" and $ ¸C< ¹l># must be equal to zero. Substitution of this restriction into Equation (5.6) yields: (

>#

>"

6

P

" ÆÆ ÆÆ c7C 6 $ aC6 bd.B € ( c7C < $ aC< bd.Bì.> ( ‡

!

6"

¸&Æ(¹

Taking the variation of expression ¸&Æ&¹, integrating by parts twice, and considering the case of a uniform beam (M¸B¹ é M , a constant value along the length of the beam) produces (

># >"

‡ƒ(

!

6"

cIMC6wwww d$ aC6 b.B

P

+(

6"

cIMC#

(

>"

6"

P

ÆÆ ÆÆ wwww wwww ‡ƒ( c7¸C 6 ¹+IMC6 d$ aC6 b.B € ( c7¸C r ¹+IMCr d$ aCr b.B „ !

6"

l1

+

IMC6ww $ aC6w b¢

 0

w

w

l1

IMC6www $ aC6 b¢ 0 w

öaO) aC<  C6 bb$ aC< bú¢

Bé6"

€

P ww w IMC< $ aC< b¢ 6"



P www IMC< $ aC< b¢ 6"

 öaO) aC¹Ç C# é C# ¸BÇ >¹ and C3 é C3 ¸BÇ >¹Ç are used to describe the motion of the beamÆ The function, C" é C" ¸BÇ >¹, describes the motion of the beam on the left side of the first damage location as measured from the cantilevered end. The function C# é C# ¸BÇ >¹, is used to describe the motion of the beam between the damage locations. The remaining segment of the beam, the right hand side after the second damage location, is described by the function C3 é C3 ¸BÇ >¹Æ All beam properties are assumed to isotropic and homogenous. Throughout the derivation the following notation will be used: ( ø ¹ will represent

` `>

and ( )w will represent

` `B Æ

Figure 6.1 shows a schematic of the described

beam.

l1

l2 h b

L Figure 6.1: A Cantilever Beam with Multiple Damage Locations

108

Hamilton’s principle will be used to derive the equations of motionÆ The kinetic energy of the beam is given by X ¸B¹ é (

6"

!

6# P 7¸B¹ Æ # 7¸B¹ Æ # 7¸B¹ Æ # ¸C ¹ .B € ¸C ¹ .B € ¸C " ¹ ú.B ( ( ö ö ö " ú " ú # # # 6" 6#

in which 7¸B¹ is used to describe the mass per unit length of the beam. The potential energy for this beam can be expressed as Z ¸B¹ é (

6"

!

6# P IM¸B¹ ww # IM¸B¹ ww # IM¸B¹ ww # ¸C" ¹ ú.B € ( ö ¸C# ¹ ú.B € ( ö ¸C$ ¹ ú.B ö # # # 6" 6#

" " € OM ¸C#w  C"w ¹# ¢ € OMM ¸C$w  C#w ¹# ¢ Æ # # Bé6" Bé6# The first three terms in the potential energy expression represent the potential energy due to the bending of the beam. The remaining terms in the expression are used to describe the potential energy of the torsional springs used to represent the damage produced by the two saw cuts respectively. Hamilton’s principle is $(

>#

>"

$(

>#

‡(

>"

(

!

6"

6" !

¸X Z ¹.> é

6# P 7¸B¹ Æ # 7¸B¹ Æ # 7¸B¹ Æ # ¸C ¹ .B € ¸C ¹ .B € ¸C " ¹ ú.B ( ö ( ö ö " ú " ú # # # 6" 6#

6# P IM¸B¹ ww # IM¸B¹ ww # IM¸B¹ ww # ¸C ¹ .B  ¸C ¹ .B  ¸C$ ¹ ú.B ( ö ( ö ö " ú # ú # # # 6" 6#

" "  OM ¸C#w  C"w ¹# ¢  OMM ¸C$w  C#w ¹# ¢ ì.> é ! # # Bé6" Bé6#

109

¸'.2)

Since the operators $ , ( ø ¹Ç and ( ) w commute and the integration with respect to > and B are interchangeable, different parts of

Equation (6-2) may be broken up into the

following terms. (

># >"

‡$ ƒ(

6" !

6# 7¸B¹ Æ # 7¸B¹ Æ # ¸C" ¹ ú.B € ( ö ¸C " ¹ ú.B ö # # 6"

€(

P

6#

ö

7¸B¹ Æ # ¸C" ¹ ú.B‚†.> #

¸'.3)

and (

>#

>"

‡  $ ƒ(

6"

!

ö

6# P IM¸B¹ ww # IM¸B¹ ww # IM¸B¹ ww # ¸C" ¹ ú.B € ( ö ¸C# ¹ ú.B € ( ö ¸C$ ¹ ú.B # # # 6" 6#

" " € OM ¸C#w  C"w ¹# ¢ € OMM ¸C$w  C#w ¹# ¢ „ì.> é ! # # Bé6" Bé6#

¸6.4)

Performing the variation on expression (6.3), integrating by parts, and assuming that 7¸B¹ é 7Ç (a constant), throughout the length of the beam, results in 7(

6" !

>#

>

>"

>"

# ø $ ¸C" ¹¢  ( ÆÆ C C " $ ¸C" ¹.>ñ.B € 7( ð "

€ 7(

P 6#

6# 6"

>#

>

>"

>"

>#

>

>"

>"

# ø $ ¸C# ¹¢  ( ÆÆ C C # $ ¸C# ¹.>ñ.B ð #

# ø $ ¸C$ ¹¢  ( ÆÆ C C $ $ ¸C$ ¹.>ñ.B. $ ð

¸'Æ&¹

Since the varied path must be equal to the actual path at >" and ># , the terms $ ¸C" ¹¡ Ç >"

$ ¸C2 ¹¡ Ç $ ¸C$ ¹¡ Ç $ ¸C" ¹¡ Ç $ ¸C# ¹¡ and $ ¸C$ ¹¡ must be equal to zero. Substitution of this >"

>"

>#

>#

>#

constraint into expression (6.5) yields: (

>#

>"

6

6

P

" # ÆÆ ÆÆ ÆÆ $ c 7C ¸C ¹ d .B € c7C # $ ¸C# ¹d.B € ( c7C $ $ ¸C$ ¹d.B„.> ( ( " " ƒ

!

6"

6#

110

¸'Æ'¹

Taking the variation of expression (6.4), integrating by parts twice, and considering the case of a uniform beam (M (B¹ é M¹Ç produces (

># >"

‡ƒ(

6"

!

IMC"wwww $ ¸C" ¹.B

€(

6# 6"

IMC#wwww $ ¸C# ¹.B

P

€ ( IMC#wwww $ ¸C# ¹.B„ 6#

6"

6"

6#

6#

!

!

6"

6"

€ IMC"ww $ ¸C"w ¹¢  IMC"www $ ¸C" ¹¢ € IMC#ww $ ¸C#w ¹¢  IMC#www $ ¸C# ¹¢ P

P

6#

6#

IMC$ww $ ¸C$w ¹¢  IMC$www $ ¸C$ ¹¢ € OM ¸C#w  C"w ¹º$ ¸C#w ¹  $ ¸C"w ¹»¢

€ OMM ¸C$w  C#w ¹º$ ¸C$w ¹  $ ¸C w# ¹»¢

Bé6#

Bé6"

ì.>.

¸'Æ(¹

The mathematical statement of Hamilton’s principle, for the beam under consideration, is obtained by the substitution of expressions (6.6) and (6.7) into Equation (6.3). This results in

(

># >"

6"

6

# ÆÆ ÆÆ wwww a 7C + IMC b $ ¸C ¹.B € a7C # +IMC#wwww b$ ¸C# ¹.B ( ( " " " ‡

!

6"

6

6"

6"

6"

!

!

6#

# ÆÆ € ( a7C $ +IMC$wwww b$ ¸C$ ¹.B € IMC"ww $ ¸C"w ¹¢  IMC"www $ ¸C" ¹¢ € IMC#ww $ ¸C#w ¹¢

6#

P

P

6"

6#

6#

 IMC#www $ ¸C# ¹¢ € IMC$ww $ ¸C$w ¹¢  IMC$www $ ¸C$ ¹¢

€ OM ¸C#w  C"w ¹º$ ¸C#w ¹  $ ¸C"w ¹»¢

Bé6"

€ OMM ¸C$w  C#w ¹º$ ¸C$w ¹  $ ¸C w# ¹»¢

111

Bé6#

ì.>. ¸'Æ)¹

Because the virtual displacements $ (y1 ), $ (y2 ) and $ (y3 ) are arbitrary and independent through out their domains (0 ì B ì 6" È 6" ì B ì 6# È 6# ì B ì P¹ respectively, the following conditions must be satisfied ÆÆ 7C " € IMC"wwww é !

¸'Æ*¹

ÆÆ 7C # € IMC#wwww é !

¸'.10)

ÆÆ 7C $ € IMC$wwww é !.

¸'Æ""¹

and

Equations (6.9), (6.10), and (6.11) are partial differential equations that describe the motion of the beam on the leftÇ middle and right hand side of the damage locations respectively. The remaining terms in Equation (6.8) describe the boundary conditions that must be satisfied. These are given by the following equation.  IMC"ww $ ¸C"w ¹¢

Bé!

 IMC"www $ ¸C" ¹¢

Bé!

 IMC#www $ ¸C# ¹¢

Bé6#

aIMC"ww  OM ¸C#w  C"w ¹b$ ¸C"w ¹¢

Bé6"

aIMC#ww  OMM ¸C$w  C#w ¹b$ ¸C#w ¹¢

Bé6#

€ IMC$ww $ ¸C$w ¹¢

BéP

 IMC"www $ ¸C" ¹¢

Bé6"

€ IMC$www $ ¸C$ ¹¢

€ IMC#www $ ¸C# ¹¢

Bé6#

€ a  IMC#ww € OM ¸C#w  C"w ¹b$ ¸C#w ¹¢

Bé6"

€ a  IMC$ww € OMM ¸C$w  C#w ¹b$ ¸C$w ¹¢

 IMC$www $ ¸C$ ¹¢

Bé6"

BéP

é!

Bé6#

¸'Æ"#¹

Equation (6.12) can be used to determine the boundary conditions. The functions C" Ç C# and C2 Ç C$ are spliced at the damage locations 6" and 6# respectively. This constraint requires that the variation= $ (C" ¹ and $ (C# ¹ be equivalent at B é 6" and the variations

112

$ (C2 ¹ and $ (C$ ¹ be equivalent at B é 6# Æ The boundary conditions for this structure can now be determined by the following manner. The last two terms in Equation (6.12) imply that C$ww é C$www é !¢

¸'Æ"$¹

BéP

were as the seventh, eighth, ninth, and tenth terms imply IM ww C € C"w é C#w ‚ ¢ OM " Bé6"

¸'Æ"%¹

IM ww C € C#w é C$w ‚ ¢ . OMM # Bé6#

¸'Æ"&¹

C"ww é C#ww ¢

Èè

C#ww é C$ww ¢

Èè

Bé6"

and

Bé6#

The third, fourth, fifth, and sixth terms in Equation (6.12) specify the constraint on the shear forces at the damage locations. These restrictions may be expressed as C2www é C$www ¢

Bé6#

È C"www é C#www ¢

Bé6"

Æ

¸'Æ"'¹

The remaining first two terms can be used to describe the geometric boundary conditions of the beam at the cantilevered end. These restrictions may be expressed as C" é C"w é !¢

¸'Æ"(¹

.

Bé!

In summary, three differential equations are used to describe the spacial motion of the beam. These are given as: ÆÆ ÆÆ ÆÆ 7C " € IMC"wwww é !; 7C # € IMC#wwww é !; and 7C $ € IMC$wwww é !.

¸'Æ")¹

In addition, twelve boundary conditions must be satisfied by these equations. These correspond to two geometric boundary conditions, C" é C"w é !¢

Bé!

113

, at the cantilevered

end of the beam, four continuity relationships at the first damage location, C" é C# ¢

Bé6"

ww w w ê IM OM C" € C" é C# ý ¢

Bé6"

; C"ww é C#ww ¢

Bé6"

; C"www é C#www ¢

C2 é C$ ¢

the second damage location,

Bé62

C#www é C$www ¢

Bé6#

È

, four continuity relationships at Bé6"

ww w w È ê IM OM C# € C$ é C$ ý ¢

and two natural boundary conditions, C$ww é C$www é !¢

Bé6#

; C#ww é C$ww ¢

; Bé6#

at the free end of BéP

the beam.

6.3 The Eigenvalue Problem and Natural Frequency of the Multi-Cracked Cantilever Beam

The partial differential equations, given by Equations (6.9), (6.10) and (6.11), describe the vibrations of the multi-cracked cantilever beam with two damage locations at positions 6" and 6# respectively. In order to obtain the natural frequencies of this beam, synchronous motion will be assumed. This implies that the solution describing the motion of the beam is made up of functions that are separable in space and time, thus C" é C" ¸BÇ >¹ é ]" ¸B¹J ¸>¹

¸'Æ"*¹

C# é C# ¸BÇ >¹ é ]# ¸B¹J ¸>¹

¸'Æ#!¹

C$ é C$ ¸BÇ >¹ é ]$ ¸B¹J ¸>¹.

¸'Æ#"¹

and

Substitution of Equations (6.19), (6.20) and (6.21) into Equations (6.9), (6.10) and (6.11) respectively and performing separation of variables yields the following four equations: ÆÆ J ¸>¹ € =# J ¸>¹ é !

114

¸'Æ##¹

wwww

] " ¸>¹  wwww

] # ¸>¹ 

=# 7 ]" ¸>¹ é ! IM

¸'Æ#$¹

=# 7 ]# ¸>¹ é ! IM

¸'Æ#%¹

=# 7 ]$ ¸>¹ é !. IM

¸'Æ#&¹

and wwww

] $ ¸>¹ 

Solution to Equations (6.23), (6.24) and (6.25) are given by B B B B ]" ¸B¹ é E" -9=2¸- ¹ € E# =382¸- ¹ € E$ -9=¸- ¹ € E% =38¸- ¹ P P P P

¸'Æ#'¹

B B B B ]# ¸B¹ é F" -9=2¸- ¹ € F# =382¸- ¹ € F$ -9=¸- ¹ € F% =38¸- ¹ P P P P

¸'Æ#(¹

B B B B ]$ ¸B¹ é H" -9=2¸- ¹ € H# =382¸- ¹ € H$ -9=¸- ¹ € H% =38¸- ¹ P P P P

¸'Æ#)¹

and

These equations describe the displacement contribution in the solution of the beam’s equation of motion under free vibration. In Equations (6.26), (6.27) and (6.28), the term % = 3EP - is defined by -=± IM Æ The mass per unit length, 7, has been replaced by the #

%

relationship 7 é 3E in which the terms 3 and E are the density of the beam’s material and the cross-sectional area of the beam respectivelyÆ Substitution of Equations (6.26), (6.27), and (6.28) into the continuity relationships and the boundary conditions

115

establishes ¼ ¾ ¾ ¾ ¾É ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾ ¾É ¾ ¾ ½

The

G ¸"Ç "¹ É

É Ë Ë G ¸#Ç "+¹ Ë Ë É É G ¸$Ç "¹

sub

matrices

Ë Ë Ë G ¸"Ç #¹ Ë Ë Ë É Ë É É É É Ë Ë Ë Ë G ¸#Ç ",¹ Ë G ¸#Ç #¹ Ë Ë Ë É Ë É É É Ë Ë Ë Ë Ë G ¸$Ç #+¹ Ë G ¸$Ç #,¹ Ë Ë Ë Ë

Â Æ Å Å E" É É ¿Å É Å E #É É ÁÅ Å Å E$ É É ÁÅ É É ÁÅ É E É É ÁÅ Å É % ÁÅ ÅF É É Å ÁÅ " É É Á F# Áà Ç= !Æ ÁÅ F$ É G ¸#Æ$¹ ÁÅ É ÁÅ F% É É Å ÁÅ ÅH É É É É ÁÅ "É Å É É ÁÅ Å H Å #É É ÁÅ É Å G ¸$Ç $¹ ÅH É É ÀÅ H$ É Ä %È

G ¸"Ç $¹ É

É

G¸"Ç "¹Ç G¸"Ç #¹Ç G¸"Ç $¹Ç G¸#Ç "+¹Ç G¸#Ç ",¹Ç G¸#Ç #¹Ç G¸#Ç $¹Ç

G ¸$Ç "¹Ç G¸$Ç #+¹Ç G¸$Ç #,¹ and G¸$Ç $¹ are defined by the following: " ¼ ¾ ! G¸"Ç "¹ é ¾ ¾ -9=2¸ -6" ¹ P ½ =382¸ -P6" ¹ ! ¼ ¾ ! G¸"Ç #¹ é ¾ ¾  -9=2¸ -6" ¹ P ½  =382¸ -P6" ¹

! " " ! -6" =382¸ P ¹ -9=¸ -P6" ¹ -9=2¸ -P6" ¹ =38¸ -P6" ¹ ! !  =382¸ -P6" ¹  -9=2¸ -P6" ¹

¼! ¾! G¸"Ç $¹ é G¸$Ç "¹ é ¾ ! ½! -9=2¸ -P6" ¹ ¼ ¾ ¾ -9=2¸ -P6" ¹ ¾ƒ G¸#Ç "+¹ é ¾ € OM P =382¸ -6" ¹ „ IM P ¾ ¾ ! ½ !

116

! ¿ Á " Á -6" =38¸ P ¹ Á  -9=¸ -P6" ¹ À

! !  -9=¸ -P6" ¹  =38¸ -P6" ¹ ! ! ! !

! ! ! !

! ¿ Á ! -6" Á  =38¸ P ¹ Á -9=¸ -P6" ¹ À

!¿ !Á Á ! !À

=382¸ -P6" ¹

¿ Á Á =382¸ -P6" ¹ ƒ € OM P -9=2¸ -6" ¹ „ Á Á IM P Á Á ! À !

 -9=¸ -P6" ¹ ¼ ¾ ¾  -9=¸ -P6" ¹ ¾ƒ G¸#Ç ",¹ é ¾  OM P =38¸ -6" ¹ „ IM P ¾ ¾ ! ½ ! ¼  -9=2¸ -P6" ¹ ¾  OM P =382¸ -6" ¹ ¾ IM P G¸#Ç #¹ é ¾ -6# ¾ -9=2¸ P ¹ ½ =382¸ -P6# ¹ ! ¼ ¾ ! G¸#Ç $¹ é ¾ ¾  -9=2¸ -6# ¹ P ½  =382¸ -P6# ¹

 =382¸ -P6" ¹ -6" MP O IM - -9=2¸ P ¹ =382¸ -P6# ¹ -9=2¸ -P6# ¹ ! !  =382¸ -P6# ¹  -9=2¸ -P6# ¹

-9=2¸ -P6# ¹ ¼ ¾ ¾ -9=2¸ -P6# ¹ ¾ƒ G¸$Ç #+¹ é ¾ € OMM P =382¸ -6# ¹ „ IM P ¾ ¾ ! ½ !  -9=¸ -P6# ¹ ¼ ¾ ¾  -9=¸ -P6# ¹ ¾ƒ G¸3Ç 2,¹ é ¾  OMM P =38¸ -6# ¹ „ IM P ¾ ¾ ! ½ ! and

117

 =38¸ -P6" ¹

¿ Á Á  =38¸ -P6" ¹ ƒ € OM P -9=¸ -6" ¹ „ Á Á IM P Á Á ! À ! -9=¸ -P6" ¹ OM P -6" IM - =38¸ P ¹ -9=¸ -P6# ¹ =38¸ -P6# ¹ ! !  -9=¸ -P6# ¹  =38¸ -P6# ¹

=38¸ -P6" ¹ ¿ OM P -6" Á  IM - -9=¸ P ¹ Á Á Á =38¸ -P6# ¹  -9=¸ -P6# ¹ À ! ¿ Á ! -6# Á  =38¸ P ¹ Á -9=¸ -P6# ¹ À

=382¸ -P6# ¹

¿ Á Á =382¸ -P6" ¹ ƒ € OMM P -9=2¸ -6# ¹ „ Á Á IM P Á Á ! À !  =38¸ -P6# ¹

¿ Á Á  =38¸ P ¹ ƒ € OMM P -9=¸ -6# ¹ „ Á Á IM P Á Á ! À ! -6#

- 6#

¼  -9=2¸ ¹ -# MM ¾ G¸#Ç #¹ é ¾  - =382¸ ¹ -9=2¸-¹ ½ =382¸-¹ P

O

P

IM

6

P

 =382¸ -P6# ¹  OIMMM -P -9=2¸ -P6# ¹ =382¸-¹ -9=2¸-¹

-9=¸ -P6# ¹ OMM P =38¸ -P6# ¹ IM  -9=¸-¹ =38¸-¹

=38¸ -P6# ¹ ¿  OIMMM -P -9=¸ -P6# ¹ Á ÁÆ  =38¸-¹  -9=¸-¹ À

The natural frequencies are determined by finding the values of = that make the determinant of the matrix C go to zero. This determinant is a transcendental equation whose solution is a function of the damage locations, damage depth, and other beam properties.

Since the Nearest Approximation Method will be used, several damage

scenarios will be established. The scenarios will correspond to a $ft cantilever aluminum beam with dimensions of

$ "' in

ô "in. The aluminum, 6061-T6, has a modulus of

elasticity of 10(106 )psiÆ and a density of &Æ#' =6?1= 0>$ Æ

6.4 Experimental Verification

An experimental set up consisting of a 6 ft. aluminum beam mounted on top of a VTS VG100-B shaker powered by an Techron 5507 power supply amplifier was used to model a cantilever beam. The beam was mounted with an adhesive to a Kristler 8638B5 accelerometer connected to the shaker. Two Bruel & Kjaer 8309 accelerometers were placed six inches away from the ends of the beam.

The signal conditioning was

performed with a Kristler 5118A2 power supply coupler and an B&K 2635 Charge type amplifier. These signals were read with an HP 35660 A dynamic signal analyzer to produce the frequency response function of the structure. The dynamic signal analyzer produced a random signal which was sent to the power supply amplifier to excite the structure. This mounting set up is shown in Figure 6.2.

118

accelerometer

signaltoB&K Charge Amplifier

beam

hp dynamic signal analyzer

accelerometer

shaker and amplifier

Figure 6.2: The Cantilever Mounting System Used in the Experiment Damage scenarios corresponding to three different slot depths, (10%, 20%, and 30%) were simulated for the nearest approximation method. A maximum of two damage conditions could exit on the beam at one time. Damage was simulated at all possible four inch intervals. These results where placed on a spread sheet that was programmed to perform the nearest approximation method from data obtained from an actual beam vibration. Two beams where damaged corresponding to five experimental runs. The first beam was damaged at a location of four inches from the cantilever with a slot depth of twenty percent of the beam’s height. The structure was then vibrated and the first four natural frequencies were recorded. The beam was removed and another state damaged was situated at a new location. This new slot was located sixteen inches from the cantilever end and had a corresponding depth of ten percent. Vibration was repeated and the first four natural frequencies were taken. Damage was augmented at the sixteen inch location to a new slot depth of twenty percent.

The vibration experiment was then

performed one more time. Figures 6.3, 6.4, and 6.5 show the frequency response function for the above mentioned beam. Figure 6.3 is a scanned output of the dynamic signal analyzer showing the first three natural frequencies before any damage is applied to the structure. Figures 6.4 and 6.5 are scanned images of the analyzer output for the first two damage states of the beam (20% depth 4 inches from the cantilever and 20% depth 4 inches form the cantilever with 10% depth 16 inches from the cantilever). Here the

119

analyzer was used in a zoom mode to obtain a better resolution of the first four measured natural frequencies.

Figure 6.3: The First Three Natural Frequencies of the Cantilever Beam.

Figure 6.4: The Frequency Response of a Damaged Cantilever Beam Around the First Natural frequency. ( 20% Damage, 4 Inches from Cantilever)

120

Figure 6.5: The Frequency Response of a Damaged Cantilever Beam around the First Natural frequency. ( 20% Damage, 4 Inches from Cantilever; 10% Damage, 16 Inches from Cantilever) In addition to this experiment, a beam with a damage condition that did not appear as one of the nearest approximation method scenarios was investigated. The beam was first damaged to an extent of 10% at a location of 24 inches from the cantilever end. As before, the beam was vibrated with the shaker using random noise and the first four natural frequencies were taken. Later the beam damage was augmented to 20% and a new slot was placed at a location 2 inches from the cantilever with damage depth of 30%. The beam was again vibrated and the first four natural frequencies were taken. Table 6.2 shows the experimental results of both beams along with the predicted values of the nearest approximation method model. Figure 6.6 show values of the Euclidean norm corresponding to scenarios for the case in which the cantilever beam has one location of damage. The actual condition of the beam corresponds to a slot with a depth of 10%

121

located at 24 inches from the cantilevered end. The frequencies which were measured experimentally can be read from Table 6.2.

Table 6.2 Predicted and Obtained Vibrations Results for Two Damaged Cantilever Beams Damage Location from Cantilevered End (in)

Slot Depth (%)

4

20

4 16

20 10

4 16

20 20

24

10

2 24

30 20

Natural Frequency Hz predicted experimental percent error predicted experimental percent error predicted experimental percent error predicted experimental percent error predicted experimental percent error

1st

2nd

3rd

4th

23.801 23.75 0.22 23.75 23.625 0.52 23.71 23.5 0.89 24.21 24 0.88 n/a 21.125 n/a

151.27 151 0.18 150.29 150.75 0.31 149.6 149.5 0.07 151.32 150.125 0.8 n/a 146 n/a

419.45 420 0.13 419.04 419.5 0.11 418.76 419.5 0.18 416.58 416.5 0.019 n/a 404.5 n/a

818.09 818.5 0.05 814.2 816 0.22 811.41 811.75 0.04 818.04 818.125 0.01 n/a 801.75 n/a

6.5 Discussion of the Results

The nearest approximation method is based upon the closest response of a mathematical model to the actual response of the structure. Even though noise and experimental error were not considered in the analysis, the nearest approximation method performed adequately. The method predicted the condition of the first beam correctly for the first and third damage states which correspond to a 20% deep slot at a location of 4 inches from the cantilever and the simultaneous damage conditions of two 20% deep slots located at positions of 4 and 16 inches from the cantilever end respectively. In the second damage state, the method predicted a condition of two 10% deep slots. One was located

122

18 10% slot depth

16

20% slot depth 30% slot depth

Euclidean Norm Value

14 12 10 8 6 4 2 0 0

4

8

12

16

20

24

28

32

Damage Location from Cantilever (in.)

Figure 6.6: Various Values of the Euclidean Norm for Scenarios Corresponding to One Crack on the Beam (Actual Condition One Crack of 10% Depth Located at 24 in. from Cantilever)

at a position of 4 inches from the cantilever end and the other at a location of 20 inches from the cantilever end. It should be noted that three damage scenarios condition were very close in being selected as the condition of the beam. One of these conditions was the actual state of the beam which was a 20% deep slot 4 inches from the cantilever end and a 10% deep slot 16 inches from the cantilever end. On the second beam the method predicted the first condition of damage which was a 10% deep crack located 24 inches from the cantilevered end. Damage was augmented to 20% and a new slot, with a depth of 30%, was placed 2 inch from the cantilevered end. The method predicted a condition that was very close to the actual state which was the condition of two 20% deep cracks.

123

One crack is located at the cantilever end and the other is located 24 inches from the cantilever end. This prediction is very close to the actual state considering that the condition placed on the beam was not modeled in a damage scenario.

124

Chapter Seven: Conclusions

A damage detection scheme that is based on the geometric interpretation of the eigenvalue problem was presented. This scheme takes advantage of the fact that the geometric interpretation of undamped mathematical models with symmetric stiffness matrices is a convex set.

This property, convexity, enables the development of a

Euclidean norm which is used to detect the failure set most representative of the structures response, hence the Nearest Neighbor Approximation. Damage modeling was discussed for a cantilever beam and a simply supported beam. The mathematical models developed consisted of a single degree and multidegree of freedom systems. These models where obtained using either a single degree of freedom model or the finite element method. The effect of damage on the structure’s vibration response was studied for each model. It was observed that as damage was introduced into the structure, a drop in the natural frequencies occurred. This behavior was examined using line and surface plots. These plots could be used to locate damage for very simple models, however they did not give adequate insight for models with more that two damage locations. In order to overcome this problem, the eigenvalue problem was studied for undamped positive definite systems. These systems have a geometric interpretation for the eigenvalue problem which is a convex set. It was shown that as damage occured in an arbitrary two degree of freedom system the representation of the eigenvalue problem

125

corresponding to the damaged state will become the convex hull of the preexisting condition of the system. This property was demonstrated by several examples and the geometric interpretations were plotted for the undamaged and damaged states. From the graphs, it was shown that each geometric representation was unique and properties which describe them, (the principal axes), may be used to develop a damage detection scheme using a Euclidean norm. This norm is used to measure the response of the system to various damage scenarios which are considered as failure sets. The smallest norm is used to represent the condition of the structure. The theory was developed for discrete systems, however as the modeling uses more and more degrees of freedom, it eventually becomes equivalent to a continuous system. Using this concept damage on a continuous cantilever beam was examined with a continuous model. The equations of motion were developed using Hamilton’s principal and damage was modeled as a torsional spring. Damage scenarios were developed and experimental verification was performed. Here a cantilever beam was damaged at the center and the vibration response was measured for three damage conditions.

The

algorithm detected damage at two locations with no error. The predicted location in the third damage condition was only off by two percent. Another model was developed, this model represented a cantilever beam with multiple damage locations.

As before, Hamilton’s principal was used to obtain a

continuous model. Damage was produced by a saw cut and modeled as a torsional spring. The model permitted a maximum of two damage locations on the beam. This model was experimentally verified with five different experiments. At first a single damage location was verified, then multiple damage locations were performed. The damage detection method predicted damage locations that were either at the exact position or very close to the exact position. The method works well when a good

126

mathematical model of the structure is used. An advantage of this technique is that it is easy to use and does not require a lot of instrumentation to obtain the required data.

7.1 Future Work and Applications

The method was developed using the geometric interpretation of the eigenvalue problem for systems that have a standard form representation. The matrix used for this representation was a symmetric matrix. Investigation for the geometric representation of matrices that are not symmetric should be performed. These representations may have geometric properties that can be used to develop other schemes for damage detection. In addition the method was established for a mathematical representation therefore it should be able to detect changes in others systems that are not structures. As long as the new system has the same representation, the method can be applied. An example of this may be in heat conduction. Heat conduction problems can use the eigenvalue problem in their solution. If it is possible to physically interpret and measure properties of the eigenvalue problem with the physical system, then the method may be used as a means to detect changes in a system.

127

REFERENCES Adams R. D., Cawley P., Pye C. J., Stone B. J., (1978), "A Vibration Technique for Non-Destructively Assessing the Integrity of Structures", Journal Mechanical Engineering Science, Vol. 20, No. 2, pp. 93-100 Adams R. D. and Cawley P., (1985), "Vibration Techniques in Non Destructive Testing", Research Techniques in Non Destructive Testing, Vol. VIII., Academic Press, pp. 304360 Banks H. T., Inman D. J. and Leo D. J., Yun Wang, (1996) "An Experimentally Validated Damage Detection Theory in Smart Structures", Journal of Sound And Vibration, (to appear in the year 1996) Baruch M. and Bar Itzhack Y. I., (1978), "Optimal Weighted Orthogonalization of Measured Modes", AIAA Journal, Vol. 16, No. 4, pp. 346-351 Bastadzhan A. Z., Strakhov G. I. Chunareva N. N., (1990), "Diagnostics of Beam Elements Using Calculated Frequencies and Modes", Vibration Engineering, Translated from Vibrotekhnika, Vol. 4, No. 1, pp. 31-36 Ben-Haim Y., (1992), "Adaptive Diagonosis of Faults in Elastic Structures by Static Displacement Measurement: The Method of Selective Sensitivity", Mechanical Systems and Signal Processing, Vol. 6, No. 1, 85-96 Ben-Haim Y., (1994), "Model Updating of Linear Systems by Selective Sensitivity with Deliberate Structural Modification", The International Journal of Analytical and Experimental Modal Analysis, Vol. 9, No. 4, pp. 287-301 Ben-Haim Y., Armon D. and Braun S., (to appear), "Crack Detection in Beams by RankOrdering of Eigen Frequency Shifts", To appear in Mechanical Systems and Signal Processing

128

Berman A. and Flannelly W. G., (1971), "Theory of Incomplete Models of Dynamic Structures", AIAA Journal, Vol. 9, No. 8, pp. 1481-1487 Berman A. and Nagy E. J., (1983), "Improvement of a Large Analytical Model Using Test Data", AIAA Journal, Vol. 21, No. 8, pp.1168-1173 Cawely P. and Adams R. D., (1979), "The Location of Defects in Structures from Measurements of Natural Frequencies", Journal of Strain Analysis, Vol. 14, No. 2, pp. 49-57 Cawley P. and Ray R., (1988), "Comparison of the Natural Frequency Changes Produced by Cracks and Slots", Journal of Vibration, Acoustics, Stress and Reliability in Design, July, Vol. 110, pp. 266-370 Ceravolo R. and De Stefano A., (1995), "Damage Location in Structures Through a Connectivistic Use of FEM Modal Analyses", The International Journal of Analytical and Experimental Modal Analysis, Vol. 10, No. 3, pp. 178-186 Chen C. J. and Garba J. A., (1988), "On-Orbit Damage Assessment for Large Space Structures", AIAA Journal, Vol. 26, No. 9, pp. 1119-1126 Christides S. and Barr A. D. S., (1983), "One-Dimensional Theory of Cracked BernoulliEuler Beams", International Journal of Mechanical Sciences, Vol. 26, No 11/12, pp. 639648 Chondros T. G. and Dimarogonas A. D., (1980), "Identification of Cracks in Welded Joints of Complex Structures", Journal of Sound and Vibration, Vol. 69, No. 4, pp. 531538 Collins J. D., Hart G. C., Hasselman T. K. and Kennedy B., (1974), "Statistical Identification of Structures", AIAA Journal, Vol. 12, No. 2, pp. 185-190 Diamarogonas A. S., Paipetis Stephen A., (1983), "Analytical Methods in Rotor Dynamics", Applied Science Publishers Ltd., New York USA, pp. 171-178 Friswell M. I., (1994), "Using Vibration Data And Statistical Measures to Locate Damage in Structures", The International Journal of Analytical and Experimental Modal Analysis, Vol. 9, No. 4, pp. 239-254 Gudmundson P., (1983) "The Dynamic Behavior of Slender Structures with CrossSectional Cracks", Journal of the Mechanics and Physics of Solids, Vol. 31, No. 4, pp. 239-345

129

Guyan R. J., (1964), "Reduction of Stiffness and Mass Matrices", AIAA Journal, Vol. 3, No. 2, pg. 380 Hajela P. and Soeiro F. J, (1990a), "Structural Damage Detection Based On Static and Modal Analysis", AIAA Journal, Vol. 28, No. 6, pp. 1110-1115 Hajela P. and Soeiro F. J., (1990b), "Recent Developments in Damage Detection Based on System Identification Methods", Structural Optimization, Vol. 2, pp. 1-10 Hemez F. M., Paris E. C., Farhat C., (1995), "Structural Damage Detection via a Finite Element Model Updating Methodology", The International Journal of Analytical and Experimental Modal Analysis, Vol. 10, No. 3, pp. 152-166 Hollkamp J. J., (1992), "Identification of Structural Systems From A Short Set of InputOutput Data: A Comparison Study", A Collection of Technical Papers, AIAA Dynamics Specialist Conference, Dallas, TX, AIAA-92-2134-CP, pp. 542-549 Idichandy V. G. and Ganapathy C., (1990), "Modal Parameter for Structural Integrity Monitoring of Fixed Offshore Platforms", Experimental Mechanics, Vol. 30, No. 4, pp. 382-391 Irretier H., (1993a), "Crack Detection in Composite Driving Shafts by Experimental Modal Analysis", Vibration, Shock, Damage and Identification of Mechanical Systems, ASME 1993, DE-VOl 64, 14th Biennial Conference on Mechanical Vibration and Noise, Albuquerque, New Mexico, pp. 47-54 Irretier H., Belz J., Reutter F., (1993b), "On the Crack Detection in Composite Driving Shafts by Frequency and Damping Tests", The Institute of Physics Stress Analysis Group conference on Modern Practice in Stress and Vibration Analysis, pp. 109-122 Ju F. D. and Akgun M., Wong E. T., Lopez T. L., (1982), "Model Method in Diagnosis of Fracture Damage in Simple Structures", Productive Applications of Mechanical Vibrations, The Winter Annual Meeting of the American Society of Mechanical Engineers, AMD-Vol. 52, pp. 113-125 Juneja Vikas, (1996), "Locating Structural damage in Real Time Using Contrast Maximization", Ph D Dissertation, Virginia Tech Kam T. Y. and Lee T. Y., (1994), "Identification of Crack Size Via an Energy Approach", Journal of Nondestructive Evaluation, Vol. 13, No. 1, pp. 1-11 Kashangaki T. A., Smith S. W., Lim T. W., (1992) "Underlying Modal Data Issues for Detecting Damage in Truss Structures", AIAA Paper, AIAA-92-2264-CP, pp. 1437-1446

130

Kim H. M. and Bartkowicz, (1993), "Damage Detection and Health Monitoring of Large Space Structures", Sound and Vibration, June Lee H. P. and Ng T. Y., (1994), "Natural Frequencies and Modes for the Flexural Vibrations of a Cracked Beam", Applied Acoustics, vol. 42, pp. 151-163 Lim T. W., (1990), "Submatrix Approach to Stiffness Matrix Correction Using Modal Test Data", AIAA Journal, Vol. 28, No. 6, pp. 1123-1130 Lim T. W., (1991a), "Analytical Model Improvement Using Measured Modes and Submatrices", AIAA Journal, Technical Notes, Vol. 29, No. 6, pp. 1015-1020 Lim T. W., (1991b), "Structural Damage Detection Using Modal Test Data", AIAA Journal, Technical Notes, Vol. 29, No. 12, pp. 2271-2274 Lin C. C., Soong T. T. and Natke H. G. (1990), "Real-Time System Identification of Degrading Structures", Journal of Engineering Mechanics, Vol. 116, No. 10 pp. 22582274 Linder D. K., Twitty G. B. and Osterman S., (1993), "Damage Detection for Composite Materials Using Dynamic Response Data", Adaptive Structures and Materials Symposium, 1993 ASME Winter Annual Meeting, New Orleans, LA Liu P. L., (1995), "Identification and Damage Detection of Trusses Using Modal Data", Journal of Structural Engineering, Vol. 121, No. 4, pp. 599-608 Malla R. B. and Nalluri B. B., (1993), "Dynamic Effects of Member Failure on the Response of Truss Type Space Structures", 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA/ASME Adaptive Structures for Forum, AIAA 93-1434 La Jolla CA, pp. 1-10 Meirovitch Leonard, (1967a), "Analytical Methods in Rotor Dynamics", Macmillan Publishing Co. Inc., New York, pp. 42-47 Meirovitch Leonard, (1967b), "Analytical Methods in Vibrations", Macmillan Publishing Co. Inc., New York, pp. 126-141, 161-166 Meirovitch Leonard, (1980), "Computational Methods in Structural Dynamics", Sijthoff & Noordhoff, Rockville, Maryland, U.S.A. pp.50-102 Meirovitch Leonard, (1986), "Elements of Vibration Analysis", McGraw-Hill, Inc., New York, U.S.A., pp. 300-344

131

Meirovitch Leonard, (1997), "Principles and Techniques of Vibrations", Prentice Hall, New Jersey, U.S.A., pp. 268-360 Moshrefi N., Sreshta H. A. and Springer N. J., (1985), "The Transverse Vibration Characteristics of an Externally Damaged Pipe", Vibration Analysis to Improve Reliability and Reduce Failure, ASME, pp. 23-29 Mottershead J. E., (1990), "Theory for the Estimation of Structural Vibration Parametes from Incomplete Data", AIAA Journal, Vol. 28, No.7, pp. 1326-1328 Pabst U. and Hagedorn P., (1993), "On the Identification of Localized Losses of Stiffness in Structures", Structural Dynamics of Large Scale and Complex Systems , DE-Vol. 59, ASME Panday A. K., Biswas M., Samman M. M., (1991), "Damage Detection From Changes in Curvature Mode Shapes", Journal of Sound and Vibration, Vol. 145, No. 2, pp. 321-332 Petroski H. J., (1981), "Simple Static and Dynamic Models for the Cracked Elastic Beam", International Journal of Fracture, Reports of Current Research Announcements, Vol. 17, No. 4, R71-R76 Petroski H. J., (1983), "Structural Dynamics of Piping with Stable Cracks: Some Simple Models", The International Journal of Pressure Vessels and Piping, Vol. 13, pp. 1-18 Ricles J. M. and Kosmatka J. B., (1992), "Damage Detection in Elastic Structures Using Vibratory Residual Forces and Weighted Sensitivity", AIAA Journal, Vol. 30, No. 9, pp. 2310-2316 Rizos P. F. and Aspragathos N. and Dimarogonas A. D., (1990), " Identification of Crack Location and Magnitude in a Cantilever Beam from The Vibration Modes", Journal of Sound and Vibration, Vol. 138, No. 3, pp. 381-388 Salane H. J. and Baldwin J. W., (1990), "Changes in Modal Paramters of a Bridge During Fatigue Testing", Experimental Mechanics, Vol. 30, No. 2, pp. 109-113 Sanayei M. and Scampoli S. F., (1991), "Structural Element Stiffness Identification from Static Test Data", Journal of Engineering Mechanics, Vol. 117, No. 5, pp. 1021-1036 Sato H., (1983), "Free Vibration of Beams with Abrupt Changes of Cross-Section", Journal of Sound and Vibration, Vol. 89, No. 1, pp. 59-64

132

Sensmeier, M. D., Sensharma, P. K., Haftka, R. T., Griffin, O. H., and Watson, L. T., "Experimental Validation of Anti-optimization Approach for Detecting Delamination Damage", Proceedings of 36th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, New Orleans, April 1995, pp. 3171-3184. Sheena Z., Unger A., Zalmanovich A., (1982), "Theoretical Stiffness Matrix Correction by Using Static Tests Results", Israel Journal of Technology, Vol. 20 pp. 245-253 Shen M. H. H. and Pierre C., (1990), "Natural Modes of Bernoulli-Euler Beams with Symmetric Cracks", Journal of Sound and Vibration, Vol. 138, No. 1, pp. 115-134 Shen M. H. H. and Pierre C., (1994), "Free Vibrations of Beams with a Single-Edge Crack", Journal of Sound and Vibration, Vol. 170, No. 2, pp. 237-259 Smith S. W., Kim H. M., Bartkowicz T. J., and Zimmerman D.C., (1995), "Health Monitoring of Large Structures", Sound and Vibration, Vol. 29, No.4, pp.18-21 Stubbs N., (1985), "A General Theory of Non-Destructive Damage Detection in Structures", Structural Control, Martinus Nijhoff Publishers, Proceedings of the Second International Symposium on Structural Control, University of Waterloo, Ontario. pp. 694713 Wei F. S., (1990), "Mass and Stiffness Interaction Effects in Analytical Model Modification", AIAA Journal, Technical Notes, Vol. 28, No. 9, pp. 1686-1688 Yuen M. M. F., (1985), "A Numerical Study of the Eigen Parameters of a Damaged Cantilever", Journal of Sound and Vibration, Vol. 103, No. 3, pp. 301-310 Zimmerman D. C. and Kaouk M., (1992), "Eigenstructure Assignment Approach for Structural Damage Detection", AIAA Journal, Vol. 30, No. 7, pp. 1848-1855 Zimmerman D. C. and Kaouk M., (1994), "Structural Damage Detection Using Minimum Rank Update Theory", Journal of Vibrations and Acoustics, Vol. 116, No. 2 pp. 222-231

133

Vita

Frederick A. Just Agosto was born on June 8 1959 at Patrick Air Force Base, Florida. He Graduated on June 1985 with a Bachelor of Science in Mechanical Engineering from the University of Puerto Rico at Mayaguez, Puerto Rico. Upon graduation he continued studies where he completed a Master of Science in Mechanical Engineering at Mayaguez Puerto Rico in May 1988. Frederick worked as an instructor, at the University of Puerto Rico in the Department of Mechanical Engineering upon completion of his degree. In August 1990 he enrolled in the Ph.D. program in the Department of Engineering Science and Mechanics at Virginia Tech. He is now currently working at the Univeristy of Puerto Rico in the Department of Mechanical Engineering.

134