The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations. / Ph. D.

We investigate theoretically and experimentally the nonlinear responses of a clamped-clamped buckled beam to a variety of external harmonic excitations and internal resonances. We assume that the beam geometry is uniform and its material is homogeneous. We initially buckle the beam by an axial force beyond the critical load of the first buckling mode, and then we apply a transverse harmonic excitation that is uniform over its span. The beam is modeled according to the Euler-Bernoulli beam theory and small strains and moderate rotation approximations are assumed. We derive the equation of motion governing the nonlinear transverse planar vibrations and associated boundary conditions using the extended Hamilton's principle. The governing equation is a nonlinear integral-partial-differential equation in space and time that possesses quadratic and cubic nonlinearities. A closed-form solution for such equations is not available and hence we seek approximate solutions. We use perturbation methods to investigate the slow dynamics in the neighborhood of an equilibrium configuration. A Galerkin approximation is used to discretize the nonlinear partial-differential equation governing the beam's response and obtain a set of nonlinearly coupled ordinary-differential equations governing the time evolution of the response. We based our theory on a multi-mode Galerkin discretization. To investigate the large-amplitude dynamics, we use a shooting method to numerically integrate the discretized equations and obtain periodic orbits. The stability and bifurcations of these periodic orbits are investigated using Floquet theory. We solve the nonlinear buckling problem to determine the buckled configurations as a function of the applied axial load. We compare the static buckled configurations obtained from the discretized equations with the exact ones. We find out that the number of modes retained in the discretization has a significant effect on these static configurations. We consider three cases: primary resonance, subharmonic resonance of order one-half of the first vibration mode, and one-to-one internal resonance between the first and second modes. We obtain interesting dynamics, such as phase-locked and quasiperiodic motions, resulting from a Hopf bifurcation, snapthrough motions, and a sequence of period-doubling bifurcations leading to chaos. To validate our theoretical results, we ran an experiment, which is a modified version of the experiment designed by Kreider and Nayfeh. We find that the obtained theoretical results are in good qualitative agreement with the experimental results. In the case of one-to-one internal resonance, we report, theoretically and experimentally, energy transfer between the first mode, which is externally excited, and the second mode. / Ph. D.

There is a need for reliable methods to determine approximate solutions of nonlinear continuous systems. Recently, it has been proved that finite-degree-of-freedom Galerkin-type discretization procedures applied to some distributed-parameter systems may fail to predict the correct dynamics. By contrast, direct procedures yield reliable approximate solutions. Starting from these results and extending some of these concepts and procedures, we compare the outcomes of these two approaches (the Galerkin discretization and the direct application of a reduction method to the original governing equations) with experimental results. The nonlinear planar vibrations of a buckled beam around its first buckling mode shape are investigated. Frequency-response curves characterizing single-mode responses of the beam under a primary resonance are generated using both approaches and contrasted with experimentally obtained frequency-response curves. It is shown that discretization leads to erroneous quantitative as well as qualitative results in certain ranges of the buckling level, whereas the direct approach predicts the correct dynamics of the system. / Master of Science

The first objective of this Dissertation is to present a methodology to calculate analytically the mode shapes and corresponding natural frequencies and determine critical buckling loads of mechanically coupled microbeam resonators with a focus on micromechanical filters. The second objective is to adopt a nonlinear approach to build a reduced-order model and obtain closed-form expressions for the response of the filter to a primary resonance. The third objective is to investigate the feasibility of employing subharmonic excitation to build bandpass filters consisting of either two sets of two beams coupled mechanically or two sets of clamped-clamped beams. Throughout this Dissertation, we treat filters as distributed-parameter systems. In the first part of the Dissertation, we demonstrate the methodology by considering a mechanical filter composed of two beams coupled by a weak beam. We solve a boundary-value problem (BVP) composed of five equations and twenty boundary conditions for the natural frequencies and mode shapes. We reduce the problem to a set of three linear homogeneous algebraic equations for three constants and the frequencies in order to obtain a deeper insight into the relation between the design parameters and the performance metrics. In an approach similar to the vibration problem, we solve the buckling problem to study the effect of the residual stress on the static stability of the structure. To achieve the second objective, we develop a reduced-order model for the filter by writing the Lagrangian and applying the Galerkin procedure using its analytically calculated linear global mode shapes as basis functions. The resulting model accounts for the geometric and electric nonlinearities and the coupling between them. Using the method of multiple scales, we obtain closed-form expressions for the deflection and the electric current in the case of one-to-one internal and primary resonances. The closed-form solution shows that there are three possible operating ranges, depending on the DC voltage. For low DC voltages, the effective nonlinearity is positive and the filter behavior is hardening, whereas for large DC voltages, the effective nonlinearity is negative and the filter behavior is softening. We found that, when mismatched DC voltages are applied to the primary resonators, the first mode is localized in the softer resonator and the second mode is localized in the stiffer resonator. We note that the excitation amplitude can be increased without worrying about the appearance of multivaluedness when operating the filter in the near-linear range. The upper bound in this case is the occurrence of the dynamic pull-in instability. In the softening and hardening operating ranges, the adverse effects of the multi-valued response, such as hysteresis and jumps, limit the range of the input signal. To achieve the third objective, we propose a filtration technique based on subharmonic resonance excitation to attain bandpass filters with ideal stopband rejection and sharp rolloff. The filtration mechanism depends on tuning two oscillators such that one operates in the softening range and the other operates in the hardening range. Hardware and logic schemes are necessary to realize the proposed filter. We derive a reduced-order model using a methodology similar to that used in the primary excitation case, but with all necessary changes to account for the subharmonic resonance of order one-half. We observe that some manipulations are essential for a structure of two beams coupled by a weak spring to be suitable for filtration. To avoid these complications, we use a pair of single clamped-clamped beams to achieve our goal. Using a model derived by attacking directly the distributed-parameters problem, we suggest design guidelines to select beams that are potential candidates for building a bandpass filter. We demonstrate the proposed mechanism using an example. / Ph. D.