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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

Presently, auto manufacturers do not have do not have efficient or accurate methods to determine the amount of refrigerant (R-134a) in an air conditioning system of an automobile. In the research presented, vibration analysis is examined as a possible method to determine this R-134a amount. Initial laboratory tests were completed and experimental modal analysis methods were investigated. This approach is based on the hypothesis that the natural frequency of the accumulator bottle is a function of the mass of refrigerant in the system. Applying this theory to a working automotive air conditioning bench test rig involved using the roving hammer method—forcing the structure with an impact hammer at many different points and measuring the resulting acceleration at one point on the structure. The measurements focused on finding the natural frequency at the accumulator bottle of the air condition system with running and non-running compressor scenarios. The experimental frequency response function (FRF) results indicate distinct trends in the change of measured cylindrical natural frequencies as a function of refrigerant level. Using the proposed modal analysis method, the R-134a measurement accuracy is estimated at ±3 oz of refrigerant in the running laboratory system and an accuracy of ±1 oz in the non-running laboratory system. / Master of Science

Experimental modal analysis (EMA) techniques have become a popular method of studying the dynamic characteristics of structures. A survey of literature available reveals that experimental modal models resulting from EMA may suffer from inaccuracy due to a host of reasons. Every stage of EMA could be a potential source of errors - from suspension of the test structures, transduction to parameter estimation phase. Though time-domain methods are actively being investigated by many researchers and are in use, fast Fourier transform (FFT) methods, due to their speed and ease of implementation, are the most widely used in experimental modal analysis work. This work attempts to quantify errors that result from a typical modal test. Using a simple beam with free-free boundary conditions simulated, three different modal tests are performed. Each test differs from the other chiefly in the excitation method and FRF estimator used. Using finite element models as the reference, correlation between finite element and experimental models are performed. The ability of the EMA process to accurately estimate the modal parameters is established on the basis of level of correlation obtained for natural frequencies and mode shapes. Linear regression models are used to correlate test and analysis natural frequencies. The modal assurance criterion (MAC) is used to establish the accuracy of mode vectors from the modal tests. The errors are further quantified spatially (on a location-by-location basis) for natural frequencies and mode shapes resulting from the EMA process. Finally, conclusions are made regarding the accuracy of modal parameters obtained via FFT-based EMA techniques. / Master of Science

The Laser Doppler Velocimetry technique has been widely used for dynamic measurements and experimental modal analysis. A laser scanning system that provides position accuracy, speed, and flexibility plays a key role in this technique. This thesis gives an overview of various laser scanning techniques and the requirements of a laser scanning system for the LDV and modal testing. The G3B/DE2488, a most-advanced galvanometer-based laser scanning system manufactured by the General Scanning Inc., is one of the most suitable laser scanning systems for the LDV and modal testing. The focus of this work was to test and calibrate such a scanning system to meet the requirements for modal testing. A new method to determine laser scanning angles was introduced. Based on this test method, a laser scanning system test rig was designed and constructed. To determine a laser bealTI scanning angle, the laser and scanner together were translated in a direction perpendicular to the target plane by using a micrometerdriven translation stage. The translation of the scanned laser spot at the target plane due to the translation of the laser-scanner unit was traced by a photodetector and another set of micrometer-driven translation stages that moved in the target plane. The laser beam scanning angle was calculated from the traveled distances of the laser-scanner unit and of the laser spot at the target plane. The test setup was used to determine the overall performance of the G3B/DE2488 which included the scanning time and accuracy. The errors that affected the scanning accuracy were analyzed. Due to the relatively low precision and quality of the cost-constrained equipment used in the test setup, the accuracy of determining a scanning angle was not very high (around 50 µrad). However, if some high-accuracy and high-resolution equipment such as a beam profiler and a set of motor-driven stages are used, this test method has the potential to determine a laser beam scanning angle with an accuracy in the order of microradians. / Master of Science

In this thesis, we apply nonlinear Koopman mode analysis to decompose the swing dynamics of a power system into modes of oscillation, which are identified by analyzing the Koopman operator, a linear infinite-dimensional operator that may be defined for any nonlinear dynamical system. Specifically, power system modes of oscillation are identified through spectral analysis of the Koopman operator associated with a particular observable. This means that they can be determined directly from measurements. These modes, referred to as Koopman modes, are single-frequency oscillations, which may be extracted from nonlinear swing dynamics under small and large disturbances. They have an associated temporal frequency and growth rate. Consequently, they may be viewed as a nonlinear generalization of eigen-modes of a linearized system. Koopman mode analysis has been also applied to identify coherent swings and coherent groups of machines of a power system. This will allow us to carry out a model reduction of a large-scale system and to derive a precursor to monitor the loss of transient stability. / Master of Science