Hamilton's principle and a third-order shear-deformation theory are used to derive a set of five coupled partial-differential equations governing the nonlinear response of composite plates. The reduction of these equations by using classical plate theory is discussed and the corresponding partial-differential equations governing both rectangular and circular plates are derived. Generalized Levy-type solutions are obtained for the problem of linear free vibrations and linear stability of shear-deformable cross-ply laminated plates. The governing equations are transformed into a set of first-order linear ordinary-differential equations with constant coefficients. The general solution of these equations is obtained by using the state-space concept. Then, the application of the boundary conditions yields equations for the natural frequencies and critical loads. However, a straightforward application of the state-space concept yields numerically ill-conditioned problems as the plate thickness is reduced. Various methods for overcoming this problem are discussed. An initial-value method with orthonormalization is selected. It is shown that this method not only yields results that are in excellent agreement with the results in the literature, but it also converges fast and gives all the frequencies and buckling loads regardless of the plate thickness. Further It is shown that the application of classical plate theory to thick plates yields inaccurate results. The influence of modal interactions on the response of harmonically excited plates is investigated in detail. The case of a two-to-one autoparametric resonance in shear-deformable composite laminated plates is considered. Four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the internally resonant modes are derived using the averaged Lagrangian when the higher mode is excited by a primary resonance. The fixed-point solutions are determined using a homotopy algorithm and their stability is analyzed. It is shown that besides the single-mode solution, two-mode solutions exist for a certain range of parameters. It is further shown that in the multi-mode case the lower mode, which is indirectly excited through the internal resonance may dominate the response. For a certain range of parameters, the fixed points lose stability via a Hopf bifurcation, thereby giving rise to limit cycle solutions. It is shown that these limit-cycles undergo a series of period-doubling bifurcations, culminating in chaos. Finally, the case of a combination resonance involving the first three modes of axisymmetric circular plates is studied. The method of multiple scales is used to determine a set of ordinary-differential equations governing the modulation of phases of the modes involved and that the excited mode is not necessarily the dominant one. Furthermore, it is shown that for a choice of parameters the multi-mode response loses stability through a Hopf bifurcation, resulting in periodically or chaotically modulated motions of the plate. / Ph. D.

On August 27, 2016, Southwest Airlines flight 3472 from New Orleans to Orlando had to perform an emergency landing when a fan blade separated from the engine hub and destroyed the cowling and punctured the fuselage. Initial findings from the metallurgical examination conducted in the National Transpiration Safety Board Materials Laboratory found that the fracture surface of the missing blade showed curving crack arrest lines consistent with fatigue crack growth. Fatigue is often cause by resonate vibrations. Modal analysis is a method that can model the natural frequencies and bending modes of turbomachinery blades. When performing modal analysis with finite element solvers like Mechanical ANSYS, images are often generated to help an engineer identify mode shapes created by nodal displacements. Manually identifying mode shapes from these generated images is an expensive task. This research proposes an automated process to identify mode shapes from gray-scale images of turbomachinery blades within a jet-engine. This work introduces mode shape identification using principal component analysis (PCA), similar to approaches in facial and other recognition tasks in computer vision. This technique calculates the projected values of potentially linearly correlated values onto P-linearly orthogonal axes, where P is the number of principal axes that define a subset space. Classification was performed using support vector machines (SVM). Using the PCA and SVM algorithm, approximately 5300 training images, representative of 16 different modes, were used to create a classifier. A test set was created with approximately 2000 unknown mode images. The classifier achieved on average 98.4% accuracy on the test set when using the bilinear Eigenface method. The accuracy was 98.6% when using the triangle interpolate Eigenface method. In addition, The results suggest that using digital images to perform mode shape identification can be achieved with better accuracy and computation performance compared to previous work. Potential generalization of this method could be applied to other engineering design and analysis applications.

On August 27, 2016, Southwest Airlines flight 3472 from New Orleans to Orlando had to perform an emergency landing when a fan blade separated from the engine hub and destroyed the cowling and punctured the fuselage. Initial findings from the metallurgical examination conducted in the National Transpiration Safety Board Materials Laboratory found that the fracture surface of the missing blade showed curving crack arrest lines consistent with fatigue crack growth. Fatigue is often cause by resonate vibrations. Modal analysis is a method that can model the natural frequencies and bending modes of turbomachinery blades. When performing modal analysis with finite element solvers like Mechanical ANSYS, images are often generated to help an engineer identify mode shapes created by nodal displacements. Manually identifying mode shapes from these generated images is an expensive task. This research proposes an automated process to identify mode shapes from gray-scale images of turbomachinery blades within a jet-engine. This work introduces mode shape identification using principal component analysis (PCA), similar to approaches in facial and other recognition tasks in computer vision. This technique calculates the projected values of potentially linearly correlated values onto P-linearly orthogonal axes, where P is the number of principal axes that define a subset space. Classification was performed using support vector machines (SVM). Using the PCA and SVM algorithm, approximately 5300 training images, representative of 16 different modes, were used to create a classifier. A test set was created with approximately 2000 unknown mode images. The classifier achieved on average 98.4% accuracy on the test set when using the bilinear Eigenface method. The accuracy was 98.6% when using the triangle interpolate Eigenface method. In addition, The results suggest that using digital images to perform mode shape identification can be achieved with better accuracy and computation performance compared to previous work. Potential generalization of this method could be applied to other engineering design and analysis applications.

No description available.

No description available.

No description available.

In most applications, the structure components have to be tested under different loading conditions before being placed in operation. A reliable and low cost measuring technique is desirable. However, most currently employed measuring approaches can only provide the structural response at several discrete locations. The accuracy of the measurements varies with the location and orientation of the sensors. Practically, it is not possible to place sensors at all the critical locations for different excitations. Therefore, an approach that derives the full field response using a limited set of measured data is desirable. In contrast to experimental full field measurement techniques, the expansion approach involves analytically expanding the limited measurements to all the degrees of freedom of the structure. Among all the analytical methods, the modal expansion method is computationally efficient and thus more suitable for real time expansion of measured data. In this method, the full-field response is approximated by the linear combination of mode shapes. In previous studies, the modal expansion method is limited by errors from mode aliasing, inaccuracy of the calculated mode shapes and the noise in measurements. In order to overcome these limitations, the modal expansion method is enhanced by mode selection and error compensation in this study. First, the key parameters used in modal expansion method were analyzed using a cantilever beam model and a method for optimal placement of sensors was developed. A mode selection method and error compensation method based on operation modal analysis and adaptive compressed sensing techniques were then developed to reduce the effects of mode aliasing, mode shape inaccuracy and measurement noise. The developed approach was further tested virtually using a numerical model of rotor 67. The numerical model was created using a two-way coupled fluid structure interaction technique. By developing these methods, the enhanced modal expansion approach can provide full field response for structures under different load conditions. Compared to the traditional modal expansion method, it can expand the data with high noise and under general dynamic loading. / Doctor of Philosophy / Accurate knowledge of the strain and stress at critical locations of a given structure is crucial when assessing its integrity. However, currently employed measuring approaches can only provide the structural response at several discrete locations. Practically, it is not possible to place sensors at all the critical locations for different excitations. Therefore, an approach that derives the full field response using a limited set of measured data is desirable. Compared to experimental full field measurement techniques, the expansion approach is focused on analytically expanding the limited measurements to all the degrees of freedom of the structure. Among all the analytical methods, the modal expansion method is computationally efficient and thus more suitable for real-time expansion of measured data. The current modal expansion method is limited by errors from mode aliasing, inaccuracy of the mode shapes, and the noise in measurements. Therefore, an enhanced method is proposed to overcome these shortcomings of the modal expansion. The following objectives are accomplished in this study: 1) Develop a method for optimal placement of sensors for modal expansion; 2) Eliminate the mode aliasing effects by determining the significance of participated modes using operational modal analysis techniques; 3) Compensate for the noise in measurements and computational model by implementing the compressed sensing approach. After accomplishing these goals, the developed approach is able to provide full field response for structures under different load conditions. Compared to the traditional modal expansion method, it can expand the data under dynamic loading; it also shows promise in reducing the effects of noise and errors. The developed approach is numerically tested using fluid-structure interaction model of rotor 67 fan blade.

A new approach to determining mode vectors is presented which uses predetermined global parameters and an improved residual model to iteratively determine modal residues. The motivation for such a technique is to determine modal parameters rapidly so that, as data acquisition techniques become faster, more structural degrees of freedom can be measured without significantly slowing down the parameter estimation process. The technique requires an accurate determination of the global parameters of natural frequency and damping by means of an FRF curve fit. More than one structural point is recommended to determine the global parameters since they will be used in determining the mode vectors. A structurally damped curve fitter which uses one or two FRFs is described and can be used for determining the global parameters. Examples of curve fitting simulated and measured data are presented and a comparison is made to a commercially available curve-fitter. Once a frequency range-of-interest is selected, frequencies will be chosen at which the mobility is measured using sine excitation. The in-range modal response is represented by a matrix-vector product where the vector contains the residues for the modes of interest. The out-of-range modal content is also represented by a matrix-vector product and forms the improved residual model. The residual content is removed from the measured mobility by an iterative technique which allows for an accurate determination of the residues of interest. An evaluation of the technique is carried out by simulating a dynamic system including the shaker and power supply. The simulated system is closely modeled after a real system used to evaluate the technique on experimental data. Convergence rates are shown for cases of close modes, low amplitude modes and errors in the global parameters. The results of using the technique on experimental data shows that convergence typically occurs in under 15 iterations. Regenerating the FRF from the modal parameters shows close agreement to the original FRF and better agreement than the regeneration from modal parameters derived from a commercially available curve fitter.> / Ph. D.

A criterion using rigid-body modes to verify the conservation of mass inertias is presented. Conservation of rod element mass guarantees convergence to the exact eigensolutions of a rod. Conservation of beam element mass guarantees convergence to the exact eigensolutions of a Bernoulli-Euler beam without rotatory inertia. Conservation of element mass and rotatory inertia guarantees convergence to the exact solutions of a Bernoulli-Euler beam with rotatory inertia. Conservation of mass moment of inertia is not a requirement for convergence, but is important for a beam mass matrix with respect to their accuracy and consistency with various boundary conditions. Based on this criterion, a concept for the formulation of a non-consistent mass matrix is presented. The concept unifies the formulation of various kinds of rod and beam mass matrices, and facilitates the generation of new mass matrices for optimization. To gain more physical insight into the formulation, the shape functions for the non-consistent mass matrices are also introduced. Four examples are considered. The first two examples are used to find the optimized mass matrices for rods and beams and to study their eigensolution errors. The optimized mass matrices minimize the root mean square errors of natural frequencies over a specified range of modes. The results of using a rod optimized mass matrix show that the root mean square error of natural frequencies for the first half of total extractable modes is reduced from 5%, obtained from using the consistent-mass and the lumped-mass matrices, to 1%. The results also show that if equally spaced elements are used for a rod, all the eigenvectors are exact. However, if unequal-length elements are used, both the frequency errors and eigenvector errors increase, and the upper half of total extractable modes are not reliable. The results of using a beam optimized mass matrix show that the root mean square error of natural frequencies is reduced from 0.16%, obtained from using a consistent-mass matrix, to 0.10%. The upper half of the total modes are not reliable. The remaining two examples are used to study the performances of all rod and beam mass matrices (consistent-mass, lumped-mass, and higher-order mass matrices) on a portal arch. According to the results, the higher-order mass matrix generates the most accurate eigensolutions. The use of the higher-order mass matrix in place of the consistent-mass matrix is recommended. The block-diagonal lumped-mass matrix performs better than the diagonal lumped-mass matrices at free ends of a structure. The eigensolution errors for all the mass matrices start to increase significantly after the first one third of the total modes. Finally, a technique for finding the modal reduction mass matrices is proposed. Fully populated modal reduction mass matrices for a rod are successfully extracted. This type of models generate exact natural frequencies and mode shapes for all the extractable modes of a rod problem. Further investigation of this technique is recommended. / Ph. D.

A lumped parameter model is developed for the analysis of dynamic fluid systems and the techniques of modal analysis are applied. An introduction to the lumped parameter modeling approach is accomplished by a thorough review of the dynamic mechanical system. This review of mechanical system analysis introduces terms such as the natural frequency, damping ratio and the frequency response function. For the analysis of more complex mechanical systems the topic of modal analysis is introduced. Proceeding in a manner analogous to that of the review of the mechanical system, the lumped parameter fluid model is introduced. This introduction includes the definition of the dynamic fluid properties and two relatively simple examples of how these properties may be used in the modeling of fluid systems. As an example of this method an analytical model is developed for a compressor system and the techniques of modal analysis are applied in a fluid sense. / Master of Science