Dynamic stability and buckling of viscoelastic plates and nanobeams subjected to distributed axial forces.

Doctor of Philosophy in Mechanical Engineering. University of KwaZulu-Natal, Durban 2016.Plates and beams are typical examples of structures that must be analyzed and understood. Buckling and vibration represent for such structures a potential source of fatigue and damage. Damage and fatigue are often caused by axial forces. The current research uses differential quadrature method to study the stability of viscoelastic plate subjected to follower forces in one hand, and the Rayleigh-Ritz method to analyze the buckling of Carbone nanotubes subjected to point and axial load in other hand. For plate, the 3D relation of viscoelastic is used to derive the equation of vibration of viscoelastic rectangular plate subjected to follower force. This equation is solved numerically by differential quadrature method, then the dynamic stability analysis is done by plotting the eigenvalues versus the follower force. We employ the Euler Bernoulli beam theory and the nonlocal theory to derive the equation of equilibrium of Carbone nanotubes subjected to point and axial loads. Rayleigh-Ritz method is used to calculate buckling loads, and the effects of equation's parameters on that buckling loads are analysed properly. Frequencies of vibration of viscoelastic plates and critical load obtained by using differential quadrature method are compared to other results with good satisfaction. The same satisfaction is observed when the buckling load values of Carbone nanotubes obtained using the Rayleigh-Ritz methods are compared to those existing in the literature. The cantilever viscoelastic plate undergoes flutter instability only and the delay time appears to influence that instability more than other parameters. The SFSF plate undergoes divergence instability only. The both types of instability are observed CSCS plate subjected to uniformly follower load but the flutter instability disappears in presence of triangular follower load. The values of the mentioned critical loads increase with triangular follower load for all boundary conditions. The aspect ratio has a large influence on the divergence and flutter critical load values and little influence on the instability quality. The laminar friction coefficient of the flowing fluid increases the critical fluid velocity but its effect on the stability of viscoelastic plate behavior is minor. The nonlocal parameter appears to decrease buckling load considerably. Buckling is more sensitive to the magnitude of the tip load for the clamped-free boundary conditions. The application of the present theory to a non-uniform nanocone shows that the buckling loads increases with radius ratio and decreases with small scale constants

Dynamic Stability of a Sandwich Beam Subjected to Parametric Excitation

Vibration control of machines and structures incorporating viscoelastic materials in suitable arrangement is an important aspect of investigation. The use of viscoelastic layers constrained between elastic layers is known to be effective for damping of flexural vibrations of structures over a wide range of frequencies. The energy dissipated in these arrangements is due to shear deformation in the viscoelastic layers, which occurs due to flexural vibration of the structures. Multilayered cantilever sandwich beam like structures can be used in aircrafts and other applications such as robot arms for effective vibration control. These members may experience parametric instability when subjected to time dependant forces. The theory of dynamic stability of elastic systems deals with the study of vibrations induced by pulsating loads that are parametric with respect to certain forms of deformation The purpose of the present work is to investigate the dynamic stability of a three layered sym..

Aeroelastic Analysis of Small-Scale Aircraft

The structural design of flight vehicles is a balancing act between maximizing loading capability while minimizing weight. An engineer must consider not only the classical static structural yielding failure of a vehicle, but a variety of ways in which structural deformations can in turn, affect the loading conditions driving those deformations. Lift redistribution, divergence, and flutter are exactly such dynamic aeroelastic phenomena that must be properly characterized during the design of a vehicle; to do otherwise is to risk catastrophe. Relevant within the university context is the design of small-scale aircraft for student projects and of particular consideration, the DBF competition hosted by AIAA. This work implements a variety of aeroelastic analysis methods: K and PK with Theodorsen aerodynamics via Matlab, NASA EZASE, and the FEMAP NX NASTRAN Aeroelasticity Package. These techniques are applied to a number of baseline test cases in addition to two representative DBF wings. Both wings considered ultimately indicated stability within reasonable flight conditions, although each for a different reason. Analysis results for the Cal Poly 2020 wing, a spar-rib construction emblematic of the collocation design approach, showed that the wing was stable within expected flight regions. The USC 2020 wing model, a composite top spar construction, exhibited unstable behavior, however this was well outside the scope of expected flight conditions. The codebase developed as a part of this work will serve as a foundation for future student teams to perform aeroelastic analyses of their own and support continued aeroelastic research at Cal Poly - SLO

The Application of Finite Element Methods to Aeroelastic Lifting Surface Flutter

Aeroelastic behavior prediction is often confined to analytical or highly computational methods, so I developed a low degree of freedom computational method using structural finite elements and unsteady loading to cover a gap in the literature. Finite elements are readily suitable for determination of the free vibration characteristics of eccentric, elastic structures, and the free vibration characteristics fundamentally determine the aeroelastic behavior. I used Theodorsen’s unsteady strip loading formulation to model the aerodynamic loading on linear elastic structures assuming harmonic motion. I applied Hassig’s ‘p-k’ method to predict the flutter boundary of nonsymmetric, aeroelastic systems. I investigated the application of a quintic interpolation assumed displacement shape to accurately predict higher order characteristic effects compared to linear analytical results. I show that quintic interpolation is especially accurate over cubic interpolation when multi-modal interactions are considered in low degree of freedom flutter behavior for high aspect ratio HALE aircraft wings

Dynamic Stability of Beams Under Parametric Excitation

The present investigation is an attempt to contribute towards the improved understanding of the dynamic stability of beams under parametric excitation. The dynamic stability of ordinary and sandwich beams subjected to longitudinal parametric excitation has been investigated theoretically and experiments have been carried out to validate some of the theoretical findings. The equations of motion have been derived using finite element method. For ordinary beams the instability regions have been established using Floquet’s theory and for sandwich beams modified Hsu’s method proposed by Saito and Otomi has been applied to determine the boundary frequencies of the instability regions. The dynamic stability of a Timoshenko beam with localised damage and having fixed-free, pinned-pinned, fixed-fixed and fixed-pinned boundary conditions has been investigated to study the effects of parameters such as extent of damage, position of damage, static load factor and boundary conditions on its dynamic..

Hydroaeroelasticity

The textbook is devoted to investigate the stability problems for deformable systems streamlined by fluid or gas flow. Special attention is paid to the study of hydrodynamic forces acting on deformable surfaces. The textbook will be intended for engineering students and postgraduate students of higher educational institutions

MODEL UPDATING AND STRUCTURAL HEALTH MONITORING OF HORIZONTAL AXIS WIND TURBINES VIA ADVANCED SPINNING FINITE ELEMENTS AND STOCHASTIC SUBSPACE IDENTIFICATION METHODS

Wind energy has been one of the most growing sectors of the nation’s renewable energy portfolio for the past decade, and the same tendency is being projected for the upcoming years given the aggressive governmental policies for the reduction of fossil fuel dependency. Great technological expectation and outstanding commercial penetration has shown the so called Horizontal Axis Wind Turbines (HAWT) technologies. Given its great acceptance, size evolution of wind turbines over time has increased exponentially. However, safety and economical concerns have emerged as a result of the newly design tendencies for massive scale wind turbine structures presenting high slenderness ratios and complex shapes, typically located in remote areas (e.g. offshore wind farms). In this regard, safety operation requires not only having first-hand information regarding actual structural dynamic conditions under aerodynamic action, but also a deep understanding of the environmental factors in which these multibody rotating structures operate. Given the cyclo-stochastic patterns of the wind loading exerting pressure on a HAWT, a probabilistic framework is appropriate to characterize the risk of failure in terms of resistance and serviceability conditions, at any given time. Furthermore, sources of uncertainty such as material imperfections, buffeting and flutter, aeroelastic damping, gyroscopic effects, turbulence, among others, have pleaded for the use of a more sophisticated mathematical framework that could properly handle all these sources of indetermination. The attainable modeling complexity that arises as a result of these characterizations demands a data-driven experimental validation methodology to calibrate and corroborate the model. For this aim, System Identification (SI) techniques offer a spectrum of well-established numerical methods appropriated for stationary, deterministic, and data-driven numerical schemes, capable of predicting actual dynamic states (eigenrealizations) of traditional time-invariant dynamic systems. As a consequence, it is proposed a modified data-driven SI metric based on the so called Subspace Realization Theory, now adapted for stochastic non-stationary and timevarying systems, as is the case of HAWT’s complex aerodynamics. Simultaneously, this investigation explores the characterization of the turbine loading and response envelopes for critical failure modes of the structural components the wind turbine is made of. In the long run, both aerodynamic framework (theoretical model) and system identification (experimental model) will be merged in a numerical engine formulated as a search algorithm for model updating, also known as Adaptive Simulated Annealing (ASA) process. This iterative engine is based on a set of function minimizations computed by a metric called Modal Assurance Criterion (MAC). In summary, the Thesis is composed of four major parts: (1) development of an analytical aerodynamic framework that predicts interacted wind-structure stochastic loads on wind turbine components; (2) development of a novel tapered-swept-corved Spinning Finite Element (SFE) that includes dampedgyroscopic effects and axial-flexural-torsional coupling; (3) a novel data-driven structural health monitoring (SHM) algorithm via stochastic subspace identification methods; and (4) a numerical search (optimization) engine based on ASA and MAC capable of updating the SFE aerodynamic model

Component-wise models for static, dynamic and aeroelastic analyses of metallic and composite aerospace structures

In the framework of structural mechanics, the classical beam theories that are commonly adopted in many applications may be affected by inconsistencies, because they are not able to foresee higher-order phenomena, such as elastic bending/shear couplings, restrained torsional warping and 3D strain effects. Depending on the problem, those limitations can be overcome by using more complex and computationally expensive 2D and 3D models or, alternatively, by adopting refined beam models, to which many scientists have dedicated their research over the last century. % One of the latest contributions to the development of advanced models, including variable kinematic beam theories, is the Carrera Unified Formulation (CUF), which is the main subject of the research discussed in this thesis. According to CUF, the 3D displacement field can be expressed as an arbitrary expansion of the generalized displacements. Depending on the choice of the polynomials employed in the expansion, various classes of beam models can be implemented. In this work, for instance, Taylor-like and Lagrange polynomials are adopted. The former choice leads to the so-called TE (Taylor Expansion) beam models, whereas LE (Lagrange Expansion) beam models with only pure displacement variables are obtained by interpolating the problem unknowns by Lagrange polynomials. The strength of CUF lies in the fact that, independently of the choice of the polynomials, the governing equations are written in terms of fundamental nuclei, which are invariant with the theory class and order. In this thesis, both strong and weak form governing equations for arbitrarily refined CUF models are derived. Subsequently, exact closed-form and approximate solutions are sought. Exact solutions of any beam model with arbitrary boundary conditions are found by formulating a frequency-dependant Dynamic Stiffness (DS) matrix and by using the Wittrick-Williams algorithm to carry out the resulting transcendental eigenvalue problem for free vibration analysis. Conversely, a linear eigenvalue problem is also derived by approximating the strong form governing equations by Radial Basis Functions (RBFs). On the other hand, weak form solutions are discussed by Finite Element Method (FEM), which still deserves important attentions due to its versatility and numerical efficiency. The various problems of the mechanics are addressed, including static, free vibration and dynamic response problems. Based on CUF and the proposed numerical methods, advanced methodologies for the analysis of complex structures, such as aircraft structures and civil engineering constructions, are developed. Those advanced techniques make use of the Component-Wise (CW) and the Multi-Line approaches. The CW method exploits the natural capability of the LE CUF beam models to be assembled at the cross-section level. This characteristic allows the analyst to use only CUF beam elements to model each component (e.g., stringers, panels and ribs) of the structure and purely physical surfaces are employed to construct the mathematical models. In the ML framework, on the other hand, each component of the structure is modelled via TE beam elements of arbitrary order. Compatibility of displacements between two or more components is then enforced through the Lagrange multipliers method. The second part of this thesis deals with aeroelasticity. In particular, the Vortex (VLM) and the Doublet Lattice Methods (DLM) are employed and extended to CUF to develop aeroelastic models. VLM is used to model the steady contribution in the aerodynamic model, whereas DLM provides the unsteady contribution in the frequency domain. The infinite plate spline approach is adopted for the mesh-to-mesh transformation. Finally, the g-method is described as an effective means for the formulation of the flutter stability problem. Particular attention is given to the extension of this methodology to exact DS solutions of CUF beams. Simplified, discrete, dynamic gust response analysis by refined beam models is also discussed. In this work, vertical gusts and one-minus-cosine idealization is addressed. Accordingly, gust loads in terms of time-dependent load factors are formulated. Subsequently, the mode superposition method is briefly introduced in order to solve the linear dynamic response problem in the time domain by using both weak and strong form solutions of CUF models. In the final part of the work, extensions of 1D CUF models for Fluid-Dynamics problems are carried out. CUF approximation of laminar, incompressible, Stokes flows with constant viscosity was introduced in a recent thesis work and it is here extended to the hierarchical p-version of FEM, which makes use of Legendre-like polynomials to interpolate the generalized unknowns along the 1D computational domain. Finally, the structural, aeroelastic and fluid-dynamics formulations are validated by discussing some selected results. In particular, regarding structures, the efficiency of the various numerical approaches when applied to CUF is investigated and simple to complex problems are considered, including metallic and composite wings. The aeroelastic analyses show that classical beam models are not adequate for the flutter detection, and at least a third-order beam model is required. Contrarily, classical beam models can be quite accurate in dynamic gust response analysis if no coupling phenomena occur, i.e. when the response is dominated by only pure bending modes. Regarding fluid-dynamics, it is demonstrated that CUF models can reproduce the results by finite volume codes for both simple Poiseuille and complex non-axisymmetric fluids in cylinders. In general, the capability of the proposed CUF models to provide accurate results with very low computational efforts is firmly highlighted. Similar analyses are possible only by using 3D models, which usually require a number of degrees of freedom that is some two order of magnitude higher

Aeroelastic modelling and analyses of a short wing-propeller configuration

Rotorcraft designs with the ability to fly as an aircraft but take off and land vertically as helicopters were first introduced in the early 1950s. However, the development of such concept stopped after a few failed attempts. Similar configurations, in which wings are equipped with additional propulsive propellers, has re-emerged in the past ten years for the compound helicopter and tilt-rotor aircraft. Those designs are aimed at introducing a new way of providing propulsive thrust, increasing the cruise speed, while retaining hovering advantages. However, the still open issue of high level of vibration resulting from fluid-structure interaction is exacerbated in the case of those rotorcrafts, with even more critical effects on the fatigue life, maintenance costs, on-board instrumental efficiency and comfort. Being able to model and predict the complex aeroelastic behaviour associated with the wing-propeller system is extremely important for achieving an optimised design. This project is an initial attempt to apply comprehensive analysis onto such compound aero-structures. Aiming at providing sufficient information of dynamic responses in the time-domain for the preliminary design stages, low-fidelity aeroelastic models are reviewed and basic structural and aerodynamic theories are introduced as basic building blocks. With this intention, a numerical computational approach is developed for characterising aeroelastic behaviour of a short-wing/propeller configuration on small rotorcrafts. The computational tool consists of many components and the integration of them. Firstly, a two-dimensional aerodynamic model is developed based on thin aerofoil theories, including Wagner's, K\"{u}ssner's, Theodorsen's and Sears' unsteady models. The combinations of those theories are investigated for different output. The Wagner's and K\"{u}ssner's theories are found to be the most suitable approach for obtaining time-domain dynamic responses, which is one of the main objective outcomes for this thesis. Theodorsen's model, formulated in the frequency domain, is better suited for instability analyses and Sears' model is ideal for dynamic steady-state response solutions. At the same time, propeller inflow is simplified as several velocity components in its axial and vertical directions based on the general characteristic of propeller slipstream. Hence, the propeller effects on the wing structure can be taken into account by the two-dimensional aerodynamic model. As for the mathematical representation of a short wing structure, beam theories are reviewed. Considering flapping, torsion and lead-lag motion, governing equations based on different beam theories are derived. In order to evaluate different structural effects and form a simple, yet realistic, representation, modal analyses based on different theories are compared. To perform efficient modal analyses, a numerical tool based on the transfer matrix method is introduced and validated. Structural coupling between flapping and torsion is examined and coupled modes are introduced. With the aid of modal analyses results, structural coupling, rotary inertia and shear deformation are found to contribute the most towards the structural modal behaviour. Hence, coupled flapping-torsion motion characterised by Euler-Bernoulli beam theory and separate lead-lag motion defined as a Timoshenko beam are found to be the most appropriate. The integration of aerodynamic and structural models allows further investigation of aeroelastic stability boundary and time-domain dynamic responses under forced conditions. A numerical procedure is developed, featuring coupled modes, which has been proved to be accurate enough when validated against experimental results. Instability, both statically and dynamically, is then studied and flight condition defined. To obtain solutions of aeroelastic response under a defined flight condition, solution strategy and procedures for convergence analyses are specified. A case study on the short wing under propeller inflow is presented and time-domain dynamic responses are obtained. Collecting all modular solvers developed for the aeroelastic model, the tool was further exploited to demonstrate its capability in analysing phenomena involving different structural, flight and propeller conditions in preliminary design stages. It was found that, apart from the aeroelastic instability, the wingspan and propeller operational speed needs to be designed carefully in order to keep deformation at a minimal level