Abstract The dynamic models for different linear or nonlinear beam structures are proposed in this dissertation. The proposed mathematical model for a turbo-disk, which is valid for whatever isotropic or orthotropic turbo-blades with or without shrouds, accounts for the geometric pretwist and taper angles, and considers coupling effect among bending and torsion effect as well. The Kelvin-Voigt internal and external damping effects have been included in the formulation. The effect of fiber orientation on the natural frequencies of a fiber-reinforced orthotropic turbo-blade has also been investigated. The eigenvalue problems of a single pre-twisted taper-blade or a shrouded turbo-blade group are formulated by employing the differential quadrature method (DQM). The DQM is used to convert the partial differential equations of a tapered pre-twisted beam system into a discrete eigenvalue problem. The Chebyshev-Gauss- Lobatto sample point equation is used to select the sample points in these analyses. The effect of the number of sample points on the accuracy of the calculated natural frequencies has also been studied. The integrity and computational efficiency of the DQM in this problem will be demonstrated through a number of case studies. The effects of design parameters, i.e. Kelvin-Voigt internal and linear external damping coefficients, the fiber orientation, and the rotation speed on the dynamic behavior for a pretwisted turbo-blade are investigated. The dynamic response of a nonlinear electrode actuator used in the MEMS has also been formulated and analyzed by employing the proposed DQM algorithm. The transitional responses of the derived nonlinear systems are calculated by using the Wilson¡V method. Results indicated the curve shape of the electrode and the cantilever actuator may affect the pull-in behavior and the residual vibration of the electrostatic actuators significantly. Numerical results demonstrated the validity and the efficiency of the DQM in solving different type beam problems.

This thesis presents the Differential Quadrature Method (DQM) for solving time-dependent or heat conduction problem. DQM discretizes the space derivatives giving a system of ordinary differential equations with respect to time and the fourth order Runge Kutta Method (RKM) is employed for solving this system. Stabilities of the ordinary differential equations system and RKM are considered and step sizes are arranged accordingly. The procedure is applied to several time dependent diffusion problems and the solutions are presented in terms of graphics comparing with the exact solutions. This method exhibits high accuracy and efficiency comparing to the other numerical methods.

It is widely accepted that, for many buckling problems of plates and shells in the plastic range, the flow theory of plasticity either fails to predict buckling or significantly overestimates buckling stresses and strains, while the deformation theory, which fails to capture important aspects of the underlying physics of plastic deformation, provides results that are more in line with experimental findings and is therefore generally recommended for use in practical applications. This thesis aims to contribute further understanding of the reasons behind the seeming differences between the predictions provided by these two theories, and therefore provide some explanation of this so-called ‘plastic buckling paradox’. The study focuses on circular cylindrical shells subjected to either axial compression or non-proportional loading, the latter consisting of combined axial tensile stress and increasing external pressure. In these two cases, geometrically nonlinear finite-element (FE) analyses for perfect and imperfect cylinders are conducted using both the flow and the deformation theories of plasticity, and the numerical results are compared with data from widely cited physical tests and with analytical results. The plastic buckling pressures for cylinders subjected to non-proportional loading, with various combinations of boundary conditions, tensile stresses, material properties and cylinder’s geometries, are also obtained with the help of the differential quadrature method (DQM). These results are compared with those obtained using the code BOSOR5 and with nonlinear FE results obtained using both the flow and deformation theories of plasticity. It is found that, contrary to common belief, by using a geometrically nonlinear FE formulation with carefully determined and validated constitutive laws, very good agreement between numerical and test results can be obtained in the case of the physically more sound flow theory of plasticity. The reason for the ‘plastic buckling paradox’ appears to be the over-constrained kinematics assumed in many analytical and numerical treatments, such as BOSOR5 and NAPAS, whereby a harmonic buckling shape in the circumferential direction is prescribed.

A mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. The nonlinear dynamics of the AFM microcantilever are studied in both of the monostable and bistable phases with the microcantilever tip being, respectively, located in the monostable and bistable regions of the static bifurcation diagram in the reference configuration. Free-vibration responses of the AFM probes, including the microcantilever natural frequencies and mode shapes, are determined. It is found that, for the parameters used in a practical operation of an AFM, the natural frequencies and mode shapes of the AFM microcantilever are almost the same as those of a free-end microcantilever with the same geometry and made of an identical material. A multimode Galerkin approximation is utilized to discretize the nonlinear partial-differential equation of motion and associated boundary conditions governing the cantilever response and obtain a set of nonlinearly coupled ordinary-differential equations (ODE) governing the time evolution of the system dynamics. The corresponding nonlinear ODE set is then solved using numerical integration schemes. A comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors. A point-mass model is also developed based on the single-mode Galerkin procedure to compare with the present distributed-parameter model. In addition, a reduced-order model based on a differential quadrature method (DQM) is employed to explore the dynamics of the AFM probe in the bistable phase where the multimode Galerkin procedure is computationally expensive. We found that the DQM with a few grid points accurately predicts the static bifurcation diagram. Moreover, we found that the DQM is capable of precise prediction of the lowest natural frequencies of the microcantilever with only a few grid points. For the higher natural frequencies, however, a large number of grid points is required. We also found that the natural frequencies and mode shapes of the microcantilever about non-contact equilibrium positions are almost the same as those of the free-end microcantilever. On the other hand, free-vibration responses of the microcantilever about contact equilibrium positions are quite different from those of the free-end microcantilever. Moreover, we used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite-difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region. / Ph. D.

This thesis presents the Differential Quadrature Method (DQM) for solving Helmholtz, modified Helmholtz and Helmholtz eigenvalue-eigenvector equations. The equations are discretized by using Polynomial-based and Fourier-based differential quadrature technique wich use basically polynomial interpolation for the solution of differential equation.

<p>У овој тези проучене су осцилације и стабилност запремински оптерећене правоугаоне<br />нано-плоче уз коришћење Ерингенове теорије еластичности. Запреминско оптерећење<br />је константно са правцем који је у равни плоче. Гранични услови су моделовани као<br />покретна укљештења. Класична теорија плоча и Карманова теорија плоча, које су<br />надограђене Ерингеновом теоријом еластичности, искоришћене су за формирање<br />диференцијалне једначине стабилности и осциловања нано-плоче. Галеркиновом<br />методом одређене су сопствене фреквенције трансверзалних осцилација нано-плоче у<br />зависности од ефеката запреминског оптерећења и нелокалности. Одређене су<br />критичне вредности параметра запреминског оптерећења при којима нано-плоча губи<br />стабилност. Приказан је утицај ефеката запреминског оптерећења и нелокалности на<br />неколико облика осциловања. Верификација резултата извршена је помоћу методе<br />диференцијалних квадратура.</p> / <p>U ovoj tezi proučene su oscilacije i stabilnost zapreminski opterećene pravougaone<br />nano-ploče uz korišćenje Eringenove teorije elastičnosti. Zapreminsko opterećenje<br />je konstantno sa pravcem koji je u ravni ploče. Granični uslovi su modelovani kao<br />pokretna uklještenja. Klasična teorija ploča i Karmanova teorija ploča, koje su<br />nadograđene Eringenovom teorijom elastičnosti, iskorišćene su za formiranje<br />diferencijalne jednačine stabilnosti i oscilovanja nano-ploče. Galerkinovom<br />metodom određene su sopstvene frekvencije transverzalnih oscilacija nano-ploče u<br />zavisnosti od efekata zapreminskog opterećenja i nelokalnosti. Određene su<br />kritične vrednosti parametra zapreminskog opterećenja pri kojima nano-ploča gubi<br />stabilnost. Prikazan je uticaj efekata zapreminskog opterećenja i nelokalnosti na<br />nekoliko oblika oscilovanja. Verifikacija rezultata izvršena je pomoću metode<br />diferencijalnih kvadratura.</p> / <p>In this thesis, the problem of stability and vibration of a rectangular single-layer graphene<br />sheet under body force is studied using Eringen&rsquo;s theory. The body force is constant and<br />parallel with the plate. The boundary conditions correspond to the dynamical model of a<br />nanoplate clamped at all its sides. Classical plate theory and von K&aacute;rm&aacute;n plate theory,<br />upgraded with nonlocal elasticity theory, is used to formulate the differential equation of<br />stability and vibration of the nanoplate. Natural frequencies of transverse vibrations,<br />depending on the effects of body load and nonlocality, are obtained using Galerkin&rsquo;s method.<br />Critical values of the body load parameter, i.e., the values of the body load parameter when<br />the plate loses its stability, are determined for different values of nonlocality parameter. The<br />mode shapes of nanoplate under influences of body load and nonlocality are presented as<br />well. Differential quadrature method is used for verification of obtained results.</p>