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Study of nonlinear dynamic systems is extremely important, both experimentally and analytically to accurately characterize their behavior. Traditional experimental methods and techniques used for experimental modal analysis have many limitations when the system exhibits nonlinear behavior. Similarly, analytical techniques such is finite element methods (FEM) cannot be directly employed for analyzing nonlinear vibration of complex systems. In this dissertation, a nonlinear modal testing technique using a software phase-locked loop (sPLL) system is fully explored for a system exhibiting geometric nonlinearity. The work also discusses applicability and limitations of traditional modal testing hardware to nonlinear dynamic systems and also highlights the need for a novel excitation technique that can be used for nonlinear modal testing of extremely flexible and thin structures. A framework for updating corresponding FE models for the dynamic system is explored using linear and nonlinear model updating techniques. Several common assumptions and simplifications used in traditional modeling of dynamic systems are explored and analyzed from a nonlinear perspective, including the accuracy of the various nonlinear modeling types, and the merits and demerits of using these different modeling types. A new metric for correlating nonlinear systems responses obtained experimentally and from corresponding FE models is also formulated, and the performance of this metric is thoroughly explored using nonlinear modal testing data and results from their corresponding FE models.

This dissertation aims to 1) design microsystems with strong intentional geometrical nonlinearity, 2) analyze various rich nonlinear characteristics/resonances using experimental characterization, theoretical modeling, and numerical simulation thoroughly, and 3) tailor the design to manipulate the level of nonlinearity. The knowledge obtained through this research provides strategies to manipulate and optimize the nonlinear properties in the design of nonlinear resonant MEMS/NEMS. A novel non-prismatic heterogeneous micro resonator design is proposed which consists of single/double silicon micro cantilever(s)-polymer component by developing the batch fabrication process that can integrate two dissimilar materials in a free-standing fashion. The variation in geometry as well as the material properties enables to have more freedom to control the desired level of nonlinearity in the system. A comprehensive experimental, analytical, and numerical methodology is developed to investigate nonlinear resonances. A novel finite element simulation approach is proposed to provide insights about the detailed nonlinear mechanisms involved, and to obtain the nonlinear coefficients. The results show that both single and double micro cantilever designs are capable of introducing the strong nonlinear hardening resonances into the dynamics owing to the stretching effect within the midplane of the polymer coupling. It is found that the results from FE simulation, experiment, and theoretical modeling are in good agreement. Moreover, the effects of the variation in Young’s modulus and length ratio are studied on the level of nonlinearity in the systems. Besides single mode nonlinear resonances, the nonlinear designs were also able to exhibit intermodal coupling in the dynamic response. This remarkable nonlinear feature can be triggered by deliberately imposing the commensurate (1:n ratio) relation between the modal frequencies with one another (i.e. internal resonance) and/or with driving frequency (combination resonance). These two nonlinear mechanisms can couple modes internally which can, in turn, flow the energy from the driven to the internally coupled modes. 1:2 and 2:1 internal resonances were successfully realized between the second and third modes in experimental results when any of the modes were driven. An analytical model is developed for internal resonance in a nonlinear quadratic system and a comprehensive parametric study is conducted on internal resonance line-shapes and intermodal energy transfer. The experimental results and analytical predictions match well qualitatively. The model also provides the equation for the nonlinear coupling coefficient to design optimal internal resonance systems. Additionally, combination resonances were successfully realized between the first three flexural modes of the nonlinear system by carefully designing the second mode to be in the vicinity of the average of the first and third modes. When driving around the second mode, following the typical Duffing-type hardening resonance, resonance bandwidth expansion occurred in all three internally coupled modes as a result of combination resonance-induced modal coupling. A theoretical model with cubic nonlinear terms is also developed to obtain a deeper insight about this phenomena, and the results from the analytical prediction agree well with the experimental results.

In this thesis the dynamic response of structures that are allowed to uplift, roll, and rock when subjected to ground excitation is investigated. High potential for successful mitigation of the harmful effects of earthquake ground excitation is associated with rocking structures and has been an important topic of researcher for decades. However, the lack of relatively simple rocking structure design models, as opposed to complex response simulation models, deters from widespread use of rocking structures in earthquake engineering practice. The aim of this thesis is to try to close the gap between practicing engineers and researchers and to contribute to the understanding of the fundamental dynamics of rocking systems. A new rolling-and-rocking solution for a rocking oscillator is developed, achieved by extending its base.

This thesis proposes an extended elastoplastic analysis of nonlinear structures that consider the influences of uncertain data (i.e. material properties and forces). The uncertainties are modeled as intervals (or convex models), which are deterministic but lie within known upper and lower bound ranges. The analysis determines the most maximum and the most minimum values of some specified (e.g. displacement at some location) variable. The proposed scheme uses a step-by-step interval holonomic analysis of structures under load control. Each incremental step involves formulation and solution of a pair of nonstandard optimization problems (with one problem capturing the most maximum response and the other the most minimum). These problems are called interval mathematical programs with equilibrium constraints (or interval MPECs). In addition to the difficulties underlying standard MPECs, the presence of interval data makes the problems very challenging to solve. Common searching techniques (such as Monte Carlo simulations) by simply predefining the intervals as some feasible deterministic parameters and exhaustively processing the associated (noninterval) mixed complementarity problem (MCP) do not guarantee accurate interval response bound solutions. A novel and simple technique is proposed to determine in a single step the most maximum response in one case and the most minimum in the other case for a given load step. The algorithm first transforms the governing interval MPECs into noninterval MPECs simply by replacing the intervals with unknown variables and additional constraints describing the bounds of these variables. Such unknown variables describe the critical interval parameters associated with each of the maximum and minimum response limits, and can be obtained directly from the optimization process. The reformulated (noninterval) MPECs are processed using some regularized nonlinear programming (NLP) approaches. The efficiency and robustness of the proposed scheme are illustrated through various engineering applications. The main ones are the analysis of interval structures considering elastoplastic materials, semi-rigid beam-to-column steel connections and/or geometric nonlinearity. This direct numerical tool fruitfully assists engineers to assess the effects of interval parameters and hence the safety of various realistic structures with uncertainties.

Reinforced concrete (RC) shear wall buildings constitute a significant portion of the building inventory in many earthquake-prone regions. A similar type of structural system is fully-grouted reinforced masonry (RM) shear wall structures. The accurate determination of the nonlinear response of reinforced concrete and reinforced masonry (RC/RM) walls subjected to lateral loading is of uttermost importance for ensuring the safety of the built environment. Analytical models provide a cost efficient and comprehensive tool to study the nonlinear response of RC/RM structures, as compared to experimental tests. Predictive models should capture nonlinear material behavior as well as the geometrically nonlinear response of RC/RM shear wall structures during major seismic events. This thesis outlines the formulation and validation of a nonlinear shell element for the simulation of RC/RM structures. The proposed shell element enhances an existing formulation of a four-node Discrete Kirchhoff shell element through the inclusion of a corotational approach to account for geometric nonlinearities and of nonlinear material models to capture the effect of cracking and crushing in concrete or masonry and the nonlinear hysteretic behavior of reinforcing steel. The analytical results obtained from multiple linear and nonlinear analyses are compared against theoretical solutions and experimental test data. These comparative validation studies show the enhanced shell element can satisfactorily capture the salient features of the response of nonlinear reinforced concrete/masonry shear wall structures including axial-shear-flexure interaction, damage patterns, and in-plane and out-of-plane loading.

Compliant mechanisms are monolithic devices that transfer force and motion by exploiting the elasticity of their members. Nonlinear springs are a class of compliant mechanisms that have a defined nonlinear load-displacement function measured at one point on the mechanism. Various applications benefit from nonlinear springs, including medical devices, MEMS, and commercial products designed for user comfort. Since each nonlinear spring application requires a unique load-displacement function, spring configurations must be custom designed. Research in compliant mechanism synthesis has yet to address a generalized method for designing nonlinear springs. This dissertation presents a generalized nonlinear spring synthesis methodology that (i) generates a planar spring design for any prescribed nonlinear load-displacement function, (ii) synthesizes designs having distributed compliance, and (iii) employs a design parameterization conducive to geometric nonlinearities. Key features of the design parameterization include (i) a branching network of compliant beams used for topology synthesis, (ii) curved beams without sudden changes in cross-section, and (iii) boundary conditions that impose both axial and bending loads on the compliant members and enable large rotations while minimizing bending stresses. To generate nonlinear spring designs, the design parameterization is implemented into a genetic algorithm, where potential spring designs are generated and optimized. Each spring design is analyzed by nonlinear finite element analysis and then evaluated by the objective function for its nonlinear response. To improve optimization performance, the objective function is formulated to exploit scaling rules. Four spring examples each having a unique load-displacement function (J-curve, S-curve, constant-force, and linear), demonstrate the methodology’s effectiveness. Two fabricated designs validate the springs’ nonlinear responses, while demonstrating the applicability of nonlinear springs. The synthesis methodology also works for anisotropic spring designs and has been extended to design compliant mechanisms having prescribed velocity profiles at their output. Other developments include scaling rules for springs, guidelines for arranging nonlinear springs in series and parallel, and physical interpretations of springs’ responses. Results indicate that nonlinear load-displacement responses are generated by altering a spring’s axial stiffness while it deforms. This change in axial stiffness is possible by exploiting geometric nonlinearities and boundary conditions.

This study examines the use of embedded shape memory alloy (SMA)actuators for adaptive control of the themomechanical response of composite structures. Control of static and dynamic responses are demonstrated including thermal buckling, thermal post-buckling, vibration, sonic fatigue, and acoustic transmission. A thermomechanical model is presented for analyzing such shape memory alloy hybrid composite (SMAHC) structures exposed to thermal and mechanical loads. Also presented are (1) fabrication procedures for SMAHC specimens, (2) characterization of the constituent materials for model quantification, (3) development of the test apparatus for conducting static and dynamic experiments on specimens with and without SMA, (4) discussion of the experimental results, and (5) validation of the analytical and numerical tools developed in the study. The constitutive model developed to describe the mechanics of a SMAHC lamina captures the material nonlinearity with temperature of the SMA and matrix material if necessary. It is in a form that is amenable to commercial finite element (FE) code implementation. The model is valid for constrained, restrained, or free recovery configurations with appropriate measurements of fundamental engineering properties. This constitutive model is used along with classical lamination theory and the FE method to formulate the equations of motion for panel-type structures subjected to steady-state thermal and dynamic mechanical loads. Mechanical loads that are considered include acoustic pressure, inertial (base acceleration), and concentrated forces. Four solution types are developed from the governing equations including thermal buckling, thermal post-buckling, dynamic response, and acoustic transmission/radiation. These solution procedures are compared with closed-form and/or other known solutions to benchmark the numerical tools developed in this study. Practical solutions for overcoming fabrication issues and obtaining repeatable specimens are demonstrated. Results from characterization of the SMA constituent are highlighted with regard to their impact on thermomechanical modeling. Results from static and dynamic tests on a SMAHC beam specimen are presented, which demonstrate the enormous control authority of the SMA actuators. Excellent agreement is achieved between the predicted and measured responses including thermal buckling, thermal post-buckling, and dynamic response due to inertial loading. The validated model and thermomechanical analysis tools are used to demonstrate a variety of static and dynamic response behaviors associated with SMAHC structures. Topics of discussion include the fundamental mechanics of SMAHC structures, control of static (thermal buckling and post-buckling) and dynamic responses (vibration, sonic fatigue, and acoustic transmission), and SMAHC design considerations for these applications. The dynamic response performance of a SMAHC panel specimen is compared to conventional response abatement approaches. SMAHCs are shown to have significant advantages for vibration, sonic fatigue, and noise control.