Nano and viscoelastic Beck's column on elastic foundation

Beck's type column on Winkler type foundation is the subject of the present analysis. Instead of the Bernoulli-Euler model describing the rod, two generalized models will be adopted: Eringen non-local model corresponding to nano-rods and viscoelastic model of fractional Kelvin-Voigt type. The analysis shows that for nano-rod, the Herrmann-Smith paradox holds while for viscoelastic rod it does not

Interior feedback stabilization of wave equations with dynamic boundary delay

In this paper we consider an interior stabilization problem for the wave equation with dynamic boundary delay.We prove some stability results under the choice of damping operator. The proof of the main result is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent

Generalized Kelvin–Voigt damping models for geometrically nonlinear beams

Strain-rate-based damping is investigated in the strong form of the intrinsic equations of three-dimensional geometrically exact beams. Kelvin–Voigt damping, often limited in the literature to linear or two-dimensional beam models, is generalized to the three-dimensional case, including rigid-body motions. The result is an elegant infinite-dimensional description of geometrically exact beams that facilitates theoretical analysis and sets the baseline for any chosen numerical implementation. In particular, the dissipation rates and equilibrium points of the system are derived for the most general case and for one in which a first-order approximation of the resulting damping terms is taken. Finally, numerical examples are given that validate the resulting model against a nonlinear damped Euler–Bernoulli beam (where detail is given on how an equivalent description using our intrinsic formulation is obtained) and support the analytical results of energy decay rates and equilibrium solutions caused by damping. Throughout the paper, the relevance of damping higher-order terms, arising from the geometrically exact description, to the accurate prediction of its effect on the dynamics of highly flexible structures is highlighted

Validation of Spatial Hysteresis as the Internal Damping Mechanism in an Unmanned Micro Aerial Vehicle Model

Unmanned aerial vehicles have been an area of interest in both research and industry for the past several decades. Advancements in technology have allowed such aircraft to decrease in size. UAVs are less expensive than traditional aircraft and are less restricted in where they can fly due to their compact size, leading to shifts in the way infrastructure, agriculture, and transportation surveillance and operations are handled. However, small aerial vehicles and flexible, composite ones are more susceptible to crashes. This has led to an increased interest in methods to control such aircraft. In order to accurately model for a composite, flexible-wing aircraft, there is need for a more complex framework which takes into account the non-linear, spatially varying components associated with the frame. Throughout this project, the application under consideration is the internal damping coefficients. In order to compare damping mechanisms, experiments were conducted in which a time history of the displacement at the tip of a cantilevered beam was measured. The optimal parameters were found for each model using a least squares cost equation for comparison with the measured data. These damping parameters were then incorporated into the generalized beam equation so that performance could be evaluated. This process was repeated for a variety of models. This project builds upon previous studies on spatial hysteresis, a non-local internal form of damping. Spatial hysteresis damping was proposed as a damping model for large, flexible, composite space structures. This method was first proposed by H.T. Banks and D.J. Inman for large space structures constructed of graphite epoxy composite materials (see, for example, [2], [9], [10]). These structures, due to their use in spacecraft, were much more rigid than the materials in which we are primarily interested. Spatial hysteresis was not adopted on a large scale because it is computationally expensive. In the 1980s, when the model was proposed, it was extremely time consuming to incorporate spatial hysteresis using the current technology. Spatial hysteresis involves a kernel function and additional integration variable [8], [11], [15]. However, as computer processing power has increased, so has the potential to incorporate spatial hysteresis into the partial differential equation for a cantilevered beam. Due to this, recent research has proposed that such a damping model could also be used for composite, flexible wing UAVs ([16], [18], [19], [3]). These projects found that, by incorporating spatial hysteresis damping into an Euler Bernoulli beam model for a micro aerial vehicle (MAV), the aircraft was controlled more effectively in flight than by using Kelvin Voigt damping alone. This work expands the field in that it merges two research areas: the experimental work done on space structures and theoretical work on applying spatial hysteresis to UAV models. It allows a theoretical form of internal damping to be experimentally validated for use in mathematical models of MAVs. This is significant because, by having a more complete understanding of composite, flexible wing materials, UAV development is more able to address control issues accurately and efficiently. With the boom of the UAV industry, there is a clear need for a mathematical model which accurately describes the materials used. This project aims to address that need

Backstepping-Based Exponential Stabilization of Timoshenko Beam with Prescribed Decay Rate

This is an open access article under the CC BY-NC-ND license.In this paper, we present a rapid boundary stabilization of a Timoshenko beam with anti-damping and anti-stiffness at the uncontrolled boundary, by using PDE backstepping. We introduce a transformation to map the Timoshenko beam states into a (2+2) × (2+2) hyperbolic PIDE-ODE system. Then backstepping is applied to obtain a control law guaranteeing closed-loop stability of the origin in the H1 sense. Arbitrarily rapid stabilization can be achieved by adjusting control parameters. Finally, a numerical simulation shows that the proposed controller can rapidly stabilize the Timoshenko beam. This result extends a previous work which considered a slender Timoshenko beam with Kelvin-Voigt damping, allowing destabilizing boundary conditions at the uncontrolled boundary and attaining an arbitrarily rapid convergence rate

Analysis of a mathematical model for the heave motion of a micro aerial vehicle with flexible wings having non-local damping effects

In this work we analyze a one dimensional model for a flexible wing micro aerial vehicle which can undergo heaving motion. The vehicle is modeled with a non-local type of internal damping known as spatial hysteresis as well as viscous external damping. We present a rigorous theoretical analysis of the model proving that the linearly approximated system is well-posed and the first order feedback system operators generate exponentially stable C0–semigroups. Furthermore, we present numerical simulations of control designs used on the linearly approximated model to control the associated nonlinear model in two different strategies. The first strategy used to control the system is a target tracking strategy. The second strategy used in this work is morphing the system to a target state over time. The controllers used in this work include Linear Quadratic Regulator, Linear Quadratic Gaussian, and central control. In light of the theory of this work we have incorporated the appropriate Riccati equation solutions into the control design for a system with a mode problem (i.e. zero eigenvalue for stiffness operator). This work remains consistent with the literature that concerns multiple component structures with a mode problem

Large Deflections of Inextensible Cantilevers: Modeling, Theory, and Simulation

A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam's inextensibility---local arc length preservation---rather than traditional extensible effects attributed to fully restricted boundary conditions. Enforcing inextensibility leads to: nonlinear stiffness terms, which appear as quasilinear and semilinear effects, as well as nonlinear inertia effects, appearing as nonlocal terms that make the beam implicit in the acceleration. In this paper we discuss the derivation of the equations of motion via Hamilton's principle with a Lagrange multiplier to enforce the effective inextensibility constraint. We then provide the functional framework for weak and strong solutions before presenting novel results on the existence and uniqueness of strong solutions. A distinguishing feature is that the two types of nonlinear terms prevent independent challenges: the quasilinear nature of the stiffness forces higher topologies for solutions, while the nonlocal inertia requires the consideration of Kelvin-Voigt type damping to close estimates. Finally, a modal approach is used to produce mathematically-oriented numerical simulations that provide insight to the features and limitations of the inextensible model