Browsing by Subject "Structural Dynamics"

Dynamic conditions and harmful vibration are prevalent in many engineered systems. Computational optimization methods can usefully guide the design of structures and solid systems to obtain layouts with desired dynamic behaviors, such as minimized or tailored vibration response, while accounting for additional constraints. Due to resonance phenomena in steady-state dynamics, however, objectives used to control dynamic response possess many local minima that may represent poor solutions to a design problem, causing challenges for gradient-based optimization methods. Additionally, designs are typically characterized by high-dimensional parameter spaces, which require costly regularization operations and can lead to slow convergence. This dissertation presents strategies to address these obstacles in dynamic response design problems, seeking inspiration from solution strategies for the related class of medium inversion problems.

We propose an extension of the modified error in constitutive equations (MECE) method, used previously in material identification problems, as a new approach for frequency-domain dynamics topology optimization to mitigate solution dependence upon initial guess and considered loading conditions. The main idea of the MECE strategy is to relax the constitutive relations between stresses and strains and between inertial forces and displacements in the elastodynamic PDE-constraint. An additional penalty-like term measuring the discrepancy in these relations is added to the objective, so that both model error and the original response objective are minimized. The resulting optimization problem features an objective more amenable to minimization, as resonant responses are suppressed in the relaxed elastodynamic model. We show that this approach yields superior designs to conventional gradient-based optimization approaches that solely use a functional of displacements as the objective, while strictly enforcing the constitutive equations. We highlight the merits of our approach in a variety of scenarios for direct frequency response design, considering multiple frequency load cases and different frequency-response control objectives.

We next present an adaptive eigenspace basis (AEB) strategy for effective design dimensionality reduction in topology optimization problems. We construct the density field as a linear combination of eigenfunctions, computed for an elliptic operator defined over the design domain, and solve for the associated eigenfunction coefficients. Restriction to this finite-dimensional eigenspace drastically reduces the design dimension and imposes implicit regularization upon the solution. We furthermore develop a basis adaptation scheme, in which we iteratively recompute the eigenfunction basis to conform to the evolving solution field, enabling further dimension reduction and acceleration of the optimization process. We demonstrate the AEB method on topology optimization problems for static linear elasticity, benchmarking its performance against conventional TO implementations. We further present the formulation of an MECE frequency-response control problem using an AEB design parameterization. We demonstrate the combined methods provide the benefits of both dimensionality reduction and robust performance in multifrequency design scenarios.

We finally present a case study in dynamic design: a PDE-constrained optimization formulation for the design of continuously graded elastic cylinders to achieve passive cloaking from harmonic acoustic excitation, both at single frequencies and over extended bandwidths. We show that elastic parameters in a multilayered, constant-density cylinder may be selected such that the residual between the pressure field from an unobstructed spreading wave in a fluid and the pressure field produced by the cylindrical inclusion is minimized.

The primary focus of this dissertation is the characterization of snap-through buckling of discrete and continuous systems. Snap-through buckling occurs as the consequence of two factors, first the destabilization, or more often the disappearance of, an equilibrium position under the change of a system parameter, and second the existence of another stable equilibrium configuration at a remote location in state space. In this sense snap-through buckling is a global dynamic transition as the result of a local static instability.

In order to better understand the static instabilities that lead to snap-through buckling, the behavior of mechanical systems in the vicinity of various local bifurcations is first investigated. Oscillators with saddle-node, pitchfork, and transcritical bifurcations are shown analytically to exhibit several interesting characteristics, particularly in relation to the system damping ratio. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. The transcritical bifurcation was selected since it may be used to represent generic bifurcation behavior. It is shown that the damping ratio may be used to predict changes in stability with respect to changing system parameters.

Another useful indicator of snap-through is the presence of chaos in the dynamic response of a system. Chaos is usually associated snap-through, as in many systems large amplitude responses are typically necessary to sufficiently engage the nonlinearities that induce chaos. Thus, a pragmatic approach for identifying chaos in experimental (and hence noisy) systems is also developed. The method is applied to multiple experimental systems showing good agreement with identification via Lyapunov exponents.

Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. This phenomenon is investigated using experimental, numerical, and analytical means and the boundaries separating safe (non-snap-through) from unsafe (snap-through) dynamic response in forcing parameter space are obtained for both a discrete and a continuous arch. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many direct applications in both civil engineering, where arches are a canonical structural element, and mechanical engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.