In this work, an evaluation of a number of quasi-Newton algorithms and strategies for sparse, symmetric Hessian matrices was performed. It was shown how these quasi-Newton algorithms could be applied to the unconstrained minimization of a nonlinear function as well as a nonlinear least squares approach to solving a system of nonlinear equations. The best of these algorithms were evaluated for a problem with a fairly large number of degrees of freedom with a large load increment. From this study it is concluded that the proposed quasi-Newton method with the double dogleg strategy and an automatic control on Hessian evaluations is the best algorithm for all of the problems considered in this investigation. The algorithm had no difficulty converging to solutions regardless of the size of the model and regardless of the size of the load or time step. The advantage of being able to take large load or time steps may lie in those problems which involve the location of critical points (limit or bifurcation points) of structures with minimal computational effort. All the algorithms which utilized the double dogleg strategy were consistently better able to converge to the solution - a clear validation of the globally convergent property of the double dogleg strategy. Finally, the usefulness of the double dogleg strategy in solving a system of nonlinear equations via the nonlinear least squares approach and in locating multiple equilibrium configurations using deflation speaks for the versatility of the proposed algorithm. In conclusion, the quasi-Newton algorithm proposed in this dissertation is both robust and efficient for small as well as large scale problems of matrices are exploited. Because sparsity and symmetry the algorithm does not place unreasonable demands on core storage requirements. Furthermore, using the deflation technique with tunneling the algorithm can be extremely useful for post-buckling response studies of structures involving many stable and unstable branches. / Ph. D.

This work investigates the dynamics and vibration in gear systems, including spur and helical gear pairs, idler gear trains, and planetary gears. The spur gear pairs are analyzed using a finite element/contact mechanics (FE/CM) model. A hybrid analytical-computational (HAC) model is proposed for nonlinear gear dynamics. The HAC predictions are compared with FE/CM results and available experimental data for validation. Chapter 2 investigates the static and dynamic tooth root strains in spur gear pairs using a finite element/contact mechanics approach. Extensive comparisons with experiments, including those from the literature and new ones, confirm that the finite element/contact mechanics formulation accurately predicts the tooth root strains. The model is then used to investigate the features of the tooth root strain curves as the gears rotate kinematically and the tooth contact conditions change. Tooth profile modifications are shown to strongly affect the shape of the strain curve. The effects of strain gage location on the shape of the static strain curves are investigated. At non-resonant speeds the dynamic tooth root strain curves have similar shapes as the static strain curves. At resonant speeds, however, the dynamic tooth root strain curves are drastically different because large amplitude vibration causes tooth contact loss. There are three types of contact loss nonlinearities: incomplete tooth contact, total contact loss, and tooth skipping, and each of these has a unique strain curve. Results show that different operating speeds with the same dynamic transmission error can have much different dynamic tooth strain. Chapters 3, 4, and 5 develops a hybrid-analytical-computational (HAC) method for nonlinear dynamic response in gear systems. Chapter 3 describes the basic assumptions and procedures of the method, and implemented the method on two-dimensional vibrations in spur gear pairs. Chapters 4 and 5 extends the method to two-dimensional multi-mesh systems and three-dimensional single-mesh systems. Chapter 3 develops a hybrid analytical-computational (HAC) model for nonlinear dynamic response in spur gear pairs. The HAC model is based on an underlying finite element code. The gear translational and rotational vibrations are calculated analytically using a lumped parameter model, while the crucial dynamic mesh force is calculated using a force-deflection function that is generated from a series of static finite element analyses before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses, and included in the force-deflection function. Elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Extensive comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including near resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy. Chapter 4 investigates the in-plane motions in multi-mesh systems, including the idler chain systems and planetary gear systems, using the HAC method that introduced in Chap. 3. The details of how to implement the HAC method into those systems are explained. The force-deflection function for each mesh is generated individually from a series of static finite element analyses before the dynamic calculations. These functions are used to calculated the dynamic mesh force in the analytical dynamic analyses. The good agreement between the FE/CM and HAC results for both the idler chain and planetary gear systems confirms the capability of the HAC model in predicting the in-plane dynamic response for multi-mesh systems. Conventional softening type contact loss nonlinearities are accurately predicted by HAC method for these multi-mesh systems. Chapter 5 investigates the three-dimensional nonlinear dynamic response in helical gear pairs. The gear translational and rotational vibrations in the three-dimensional space are calculated using an analytical model, while the force due to contact is calculated using the force-deflection. The force-deflection is generated individually from a series of static finite element analyses before the dynamic calculations. The effect of twist angle on the gear tooth contact condition and dynamic response are included. The elastic deformations of the gear teeth along the face-width direction are calculated, and validated by comparing with the FE/CM results. / Ph. D.

When applying optimization techniques to complex physical systems, using very large numerical models for the solution of a system of parameter dependent partial differential equations (PDEs) is usually intractable. Surrogate models are used to provide an approximation to the high fidelity models while being computationally cheaper to evaluate. Typically, for time dependent nonlinear problems a reduced order model is built using a basis obtained through proper orthogonal decomposition (POD) and Galerkin projection of the system dynamics. In this thesis we present theoretical and numerical results for parameter dependent model reduction techniques. The methods are motivated by the need for surrogate models specifically designed for nonlinear parameter dependent systems. We focus on methods in which the projection basis also depends on the parameter through extrapolation and interpolation. Numerical examples involving 1D Burgers' equation, 2D Navier-Stokes equations and 2D Boussinesq equations are presented. For each model problem comparison to traditional POD reduced order models will also be presented. / Ph. D.

The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus. / Master of Science

The stress-strain state of ferromagnetic plane with a moving crack has been investigated in this study. The model considers a soft magnetic ferroelastic body and incorporates a realistic (nonlinear) susceptibility. A moving crack is present in the body and is propagating in a direction perpendicular to the magnetic field. Assuming that the processes in the moving coordinates are stationary, a Fourier transform method is used to reduce the mixed boundary value problem to the solutions of a pair of dual integral equations yielding to a closed form solution. As a result of this investigation, the magnetoelastic stress intensity factor is obtained and its dependency upon the crack velocity, material constants and nonlinear law of magnetization are highlighted. It has been shown that stress result around the crack essentially depend on external magnetic field, speed of the moving crack, nonlinear law of magnetization, and other physical parameters. The results presented in this work show that when cracked ferromagnetic structure is under the influence of magnetic field it is necessary to take into account the interaction effects between deformation of the body and magnetic field and that such interaction can bring to a new conditions for strengthening the materials. Closed form solutions for the stress-strain state are obtained, graphical representations are supplied and conclusions and prospects for further developments are outlined. / Master of Science

The modified Riks/Wempner method was examined for elastic, geometrically nonlinear space trusses. A computer program, RWCNR, was developed using the Riks/Wempner algorithm. The study includes an examination of three trusses, of which two exhibit bifurcation. A method of branching onto secondary equilibrium paths was studied, and the computer program incorporates this concept. This method of branching was found to work well provided the intersection of the fundamental and secondary equilibrium paths is good. The computer program was found to be very useful and reliable. It successfully traced fundamental equilibrium paths, located critical points (load levels), and branched onto secondary equilibrium paths. The modified Riks/Wempner method was found to be a very reliable method of obtaining equilibrium paths up to, and beyond, limit points, and it could branch onto secondary equilibrium paths. / Master of Science

Control of underactuated mechanical systems is of interest as it allows for control authority over all of a system's degrees of freedom without requiring actuation of the full system. In addition to this, open-loop control of a system provides the advantage of applying to systems with unmeasurable states or where sensor integration is not feasible. Vibrational control is an open-loop control strategy that uses high-frequency, high-amplitude forcing to control underactuated mechanical systems. This thesis is concerned with exploring two underactuated mechanical systems that are controlled using vibrational inputs. The first system, a 3 degrees of freedom (DOFs) 2-link mechanism with 1 actuated DOF which is an example of a vibrational control system with 1 input and 2 unactuated DOFs, is used to review analytical results of stability analysis using the averaged potential. Theoretical and numerical results are presented for the achievable stable configurations of the system and the effects of changing the physical parameters on the achievable stable configurations are studied. The primary contribution of this effort is the development of an experimental apparatus where vibrational control is implemented. The second system is a 4DOF system composed of a 2DOF spherical pendulum supported by an actuated 2DOF cart used to study the effects of multiple vibrational inputs acting on a system. Theoretical and numerical analysis results are presented for three variants of harmonic forcing applied to the two actuated degrees of freedom: 1) identical input waveforms, except for the amplitudes, 2) identical input waveforms, except for the amplitudes and a phase shift, and 3) identical input waveforms, but at different frequencies and amplitudes. The equilibrium sets under open-loop vibrational forcing are determined for all three cases. A general closed-loop vibrational control scheme is presented using proportional feedback of the unactuated coordinates superposed with the zero-mean, $T$-periodic vibrational input. / M.S. / Underactuated mechanical systems are systems where the driven degrees of freedom are fewer than the total degrees of freedom of the system. These systems can be controlled using vibrational control which is an open-loop control strategy that uses high-frequency, high-amplitude forcing to control the states of a system. An open-loop control strategy is one in which there are no measurements of the system states required in the control scheme. This allows for control of systems where sensor integration is not feasible. This thesis is concerned with exploring vibrational control of two underactuated mechanical systems. The stability of the equilibrium sets of these systems is assessed using the averaged potential, which is an energy-like quantity used to determine stability of equilibria of systems with high-frequency inputs. Theoretical and numerical results are presented for both systems and the effects of physical parameters and variants of harmonic forcing on the achievable stable configurations of the systems are studied. The two main contributions of the thesis are the development of an experimental apparatus where vibrational control is physically implemented for one system and the outline of the closed-loop vibrational control scheme.

A hybrid method consists of a quasi-Newton method and a homotopy method for locating multiple equilibrium configurations has been proposed recently. The hybrid method combined the efficiency of a quasi-Newton method capable of locating stable and unstable equilibrium solutions with a robust homotopy method capable of tracking equilibrium paths with turning points and exploiting sparsity of the Jacobian matrix at the same time. A quasi-Newton method in conjunction with a deflation technique is proposed here as an alternative to the hybrid method. The proposed method not only exploits sparsity and symmetry, but also represents an improvement in efficiency. Limit points and nearby equilibrium solutions, either stable or unstable, can be accurately located with the use of a modified pseudoinverse based on the singular value decomposition. This pseudoinverse modification destroys the Jacobian matrix sparsity, but is invoked only rarely (at limit arid bifurcation points where the Jacobian matrix is singular). / M.S.

The L2 gain of a nonlinear time-dependent system measures the maximal gain in the transfer of energy from admissible input signals to the output signals, in which both the input and output signals are measured with the L2 norm. For general nonlinear systems, obtaining a sharp estimate of the L2 gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of L2 gains for systems with quadratic nonlinearity. The approach utilizes a recently developed method that solves a class of Hamilton-Jacobi-Bellman equations via a Taylor series-based approximation, which is scalable to high-dimensional problems given the utilization of linear tensor systems. The ideas are demonstrated through a few concrete examples that include a one-dimensional problem with an explicit energy function and several Galerkin approximations of the viscous Burgers equation. / Master of Science / With nonlinear systems that are of the form of input-output models, questions often arise as to how to measure the energy that passes through such systems and determine strategies to look for specific signals that allow the designer freedom to explore certain system behaviors. The energy comes in the form of a signal. For general nonlinear systems, obtaining a sharp estimate of such energy gain is challenging both theoretically and numerically. In this thesis, we explore a computationally efficient way to obtain numerical estimations of these gains for systems with quadratic nonlinearity. The approach combines fundamental theoretical understandings established in the literature with scalable software recently developed in approximating the solution of the underlying partial differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In this approach, the energy gain is linked to a single scalar parameter in the HJB equation. Roughly speaking, the energy gain is the lower bound of this scalar parameter above which the HJB equation always admits a non-negative solution. Thus, it boils down to approximating the HJB solution using the software while changing this scalar parameter. We will present the theoretical foundation of the approach and illustrate the foundation through several academic examples ranging from low to relatively high dimensions.

Underwater gliders are robust and long endurance ocean sampling platforms that are increasingly being deployed in coastal regions. This new environment is characterized by shallow waters and significant currents that can challenge the mobility of these efficient (but traditionally slow moving) vehicles. This dissertation aims to improve the performance of shallow water underwater gliders through path planning. The path planning problem is formulated for a dynamic particle (or "kinematic car") model. The objective is to identify the path which satisfies specified boundary conditions and minimizes a particular cost. Several cost functions are considered. The problem is addressed using optimal control theory. The length scales of interest for path planning are within a few turn radii. First, an approach is developed for planning minimum-time paths, for a fixed speed glider, that are sub-optimal but are guaranteed to be feasible in the presence of unknown time-varying currents. Next the minimum-time problem for a glider with speed controls, that may vary between the stall speed and the maximum speed, is solved. Last, optimal paths that minimize change in depth (equivalently, maximize range) are investigated. Recognizing that path planning alone cannot overcome all of the challenges associated with significant currents and shallow waters, the design of a novel underwater glider with improved capabilities is explored. A glider with a pneumatic buoyancy engine (allowing large, rapid buoyancy changes) and a cylindrical moving mass mechanism (generating large pitch and roll moments) is designed, manufactured, and tested to demonstrate potential improvements in speed and maneuverability. / Ph. D.