Nondestructive methods of detecting cracks in structural components and machinery are important, both in preventing failures and in establishing maintenance procedures. This thesis considers how the vibration behavior of cracked members can be modelled mathematically and how these mathematical models may lead to advancements in crack detection procedures. Two separate cases are considered: the longitudinal vibration of a cracked bar and the coupled vibrations of a cracked rotating shaft. In the longitudinal vibration study, the equation of motion is developed for a cantilevered bar with a symmetric surface crack. Next, Galerkin's Method is used to obtain one- and two-term approximate solutions. Both forced and free vibrations of the bar are analyzed. Graphical results showing the relationships between displacement and crack size, crack position, and forcing frequency are presented and discussed. Spectral analysis is used to compare uncracked and cracked bar behavior. Finally, a sensitivity analysis of the forced vibration case is conducted to observe how the forcing frequency affects the rate of change of steady-state response at the onset of cracking. In the second part of the thesis, a similar analysis is conducted for a cracked, simply-supported Timoshenko shaft rotating at a constant angular speed. The equations of motion derived by Wauer (b) are used as the basis of the study. Again, Galerkin's Method is applied to obtain approximate solutions. Time histories and spectra are used to observe how changes in various parameters influence the vibration behavior. The effects of mass eccentricity and gravity are studied. Finally, the effect of a periodic axial impact load is considered. / Master of Science

A theoretical technique to extract drilling-angular velocities from a nodal-spatial array of in-plane translational velocities is developed. The technique utilizes numerical methods for simulation of solutions. A finite element method using I-DEAS 4.1 is used for calculation of in-plane translational velocities and a MATLAB code is written for extraction of the drilling-angular velocities. The case of data with noise content is also considered. All numerical results are compared to a closed-form theoretical solution which is used as a reference for accuracy. Recommendations are made for future testing and experimental applications of this technique. / Master of Science

Rotating machines such as motors, generators, turbines, etc. are crucial mechanical components of modern industrial and power generation facilities. For proper functioning of these facilities during and after an earthquake, it is essential that the rotating machines in these facilities also function as desired. The dynamics of a rotating machine is quite complex. It is further complicated by the presence of earthquake induced base motions. The response spectrum methods, which are now commonly used for calculating seismic design response of civil structures, cannot be used as such for calculating the design response of rotating machines. In this thesis, a response spectrum method which can be applied to the rotating machines is developed. To develop the response spectrum approach, a generalized modal superposition method is utilized. The random vibration analysis is applied to incorporate the stochastic characteristics of the seismic inputs. The applicability of the proposed response spectrum approach is verified by a simulation study where fifty sets of acceleration time histories are used. The proposed method considers the fact that earthquake induced base motions have several components, including rotational inputs. To define the correlation between the rotational and translational input components of the excitation, the correlation matrix and a travelling seismic wave approaches are used. The numerical results are obtained to evaluate the effect of rotational input components on the response of a rotating machine. It is observed that the rotational components are important only when they are very strong. In actual practice, such strong rotational inputs are not expected to excite rotors which are either directly placed on ground or are placed in common buildings. In the proposed spectrum approach, nevertheless, the effect of rotational input components can be easily incorporated if the correlation between various excitation components is specified. / Master of Science

Halo currents (HCs) in post-disruption plasmas can be large enough to exert significant electromagnetic loads on structures surrounding the plasma. These currents have axisymmetric and non-axisymmetric components, both of which pose threats to the vacuum vessel and other components. However, the non-axisymmetric forces can rotate, amplifying the displacements they cause when the rotation is close to the structures’ resonant frequencies. A new physically motivated scaling law has been developed that describes the rotation frequencies of these HCs and has been validated against measurements on HBT-EP, Alcator C-Mod, and other tokamaks. This scaling law can describe the time-evolution of the asymmetric HC rotation throughout disruptions on HBT-EP as well as the time-averaged rotation on C-Mod. The scaling law can also be modified to include the edge safety factor at the onset of rotation (𝒒_𝑜𝑛𝑠𝑒𝑡), which significantly improves its validity when applied to machines like C-Mod, where 𝒒_𝑜𝑛𝑠𝑒𝑡 changes frequently. The 𝒒_𝑜𝑛𝑠𝑒𝑡 dependence is explained by the relationship between the poloidal structure of the HC asymmetries and the MHD instabilities that drive them, which has been observed experimentally for the first time using a novel set of current sensing limiter tiles installed on HBT-EP. The 1/𝑎² and 𝒒_𝑜𝑛𝑠𝑒𝑡-dependence of the rotation suggest that the HCs predominantly rotate poloidally. This remains consistent with the toroidal rotation observed on HBT-EP and other tokamaks through the “Barber Pole Illusion” and the direction of rotation’s dependence on the direction of 𝐼_𝑝. This scaling law is used to make projections for next generation tokamaks like ITER and SPARC, which predicts that rotation will be resonant on ITER. However, resonant effects can still be avoided if the duration of the disruption is kept short enough to prevent two rotations from being completed.