Indicial Functions Research Articles

Aeroelastic Response from Indicial Functions with a Finite Element Model of a Suspension Bridge

The present paper describes a comprehensive analysis of the aeroelastic bridge response in time-domain, with a finite element model of the structure. The main focus is on the analysis of flutter instability, accounting for the wind forces generated by the bridge motion, including twisting as well as vertical and horizontal translation, i.e. all three global degrees of freedom. The solution is obtained by direct integration of the equations of motion for the bridge-wind system, with motion-dependent forces approximated from flutter derivatives in terms of rational functions. For the streamlined bridge box-girder investigated, the motion dependent wind forces related to the along-wind response are found to have a limited influence on the flutter velocity. The flutter mode shapes in the time-domain and the frequency domain are consistent, and composed of the three lowest symmetrical vertical modes coupled with the first torsional symmetric mode. The method applied in this study provides detailed response estimates and contributes to an increased understanding of the complex aeroelastic behaviour of long-span bridges.

Nonlinear aerodynamic and energy input properties of a twin-box girder bridge deck section

By means of computational fluid dynamics (CFD), the nonlinear aeroelastic properties of a bridge deck configuration is investigated in this study in terms of amplitude-dependent flutter derivatives and indicial functions. The results are partially compared with experimental results. It shows that the concerned properties exhibit significant dependence on the motion amplitudes. Moreover, based on flutter derivatives, the nonlinear aerodynamic properties can be divided into two groups: the group with torsional amplitudes less or equal than 10°, and the one with amplitudes larger than 10°. Flow patterns around the section of the two groups differ substantially; one group remains an overall streamlined pattern with locally distributed vortices and detached flow, while the other shows fully detached flow with large vortices emerging and developing drastically. Dynamic load coefficients indicate that, as the motion amplitude increases, the smoothness of the hysteresis loops decreases, suggesting irregular fluctuations of the loads resulted from signature turbulence, which becomes progressively prominent. Energy trapping properties derived from indicial functions are expressed in terms of dimensionless coefficients, of which the results indicate there is no possibility of single-DOF flutter, and coupling between vertical and torsional motions is necessary for flutter instability. Moreover, by the analysis of the phase angles involved in coupling, it is indicated that symmetricity of vertical motion has to be consistent with that of torsional motion in the event of a coupled flutter.

Real-time prediction of aeroelastic loads of wind turbine blades in gusty and turbulent wind using an improved load model

Abstract The traditional method of flutter analysis of wind turbine blades employs a mixed-domain (frequency- and time- domain) formulation for self-excited aerodynamic loads, because flutter derivatives are functions of reduced frequencies. However, a time-domain formulation for self-excited and buffeting loads allows the equations of motion to be continuously solved for response time histories of wind turbine blades below flutter speed and captures the transient response in gusty and turbulent winds. The rational function coefficients and buffeting indicial function coefficients that appear in the time-domain formulation were previously extracted for the NREL S830 airfoil section model in a wind tunnel. The objective of the current study is to evaluate how well these experimentally extracted coefficients can predict the aerodynamic loads on a blade in different wind conditions, through a separate set of tests conducted on a much larger (3.3 times) section model of the blade in a larger wind tunnel than originally used. The measured loads (lift and moment) were used to validate simulated loads using the time domain load prediction procedure. To improve the correlation between the measured and predicted loads, an improved loads model was developed using an additional lag term and its performance is presented in smooth, gusty and turbulent wind. An aeroelastic model test of a HAWT blade in a wind tunnel was used for further validation showing the efficacy of the loads model.

An enhanced analytical method for the subsonic indicial lift of two-dimensional aerofoils – with numerical cross-validation

Aerodynamic indicial-admittance functions are generated for the unsteady lift of two-dimensional aerofoils in subsonic flow, using an enhanced theoretical formulation which grants effective synthesis. Both a step-change in the angle of attack and a sharp-edge gust are considered as the flow perturbation. Convenient analytical approximations of the indicial functions are obtained by generalising those available for incompressible flow, taking advantage of acoustic wave theory for the non-circulatory airload and Prandtl–Glauert's scalability rule for the circulatory airload. An explicit formula is proposed for modelling the latter as function of the Mach number in the absence of shock waves, while damped oscillatory terms are introduced for better approximating the former. Indicial-admittance functions are then obtained for the lift development of a wing aerofoil using Euler-based simulations at several Mach numbers, thus providing with sound cross-validation. Appropriate tuning of the analytical expressions is also derived in order to mimic the numerical solutions. The rigor of superposing circulatory and non-circulatory theoretical contributions is thus verified in the light of computational fluid dynamics and all results are addressed with respect to the physical and mathematical assumptions employed, within a consistent framework.

State-Space Adaptation of Unsteady Lifting Line Theory: Twisting/Flapping Wings of Finite Span

In this paper, a low-order state-space adaptation of the unsteady lifting line model has been analytically derived for a wing of finite aspect ratio, suitable for use in real-time control of wake-dependent forces. Each discretization along the span has between 1–6 states to represent the local unsteady wake effects, rather than remembering the entire wake history which unnecessarily complicates controller design. Sinusoidal perturbations to each system degree of freedom are also avoided. Instead, a state-space model is fit to individual indicial functions for each blade element, allowing the downwash and lift distributions over the span to be arbitrary. The wake geometry is assumed to be quasi steady (no roll up) but with fully unsteady vorticity. The model supports time-varying surge (a nonlinear effect), dihedral, heave, sweep, and twist along the span. Cross-coupling terms are explicitly derived. This state-space model is then validated through comparison with an analytic solution for elliptic wings, an unsteady vortex lattice method, and experiments from the literature.

Airfoil-based piezoelectric energy harvesting by exploiting the pseudoelastic hysteresis of shape memory alloy springs

The modeling and analysis of an electromechanically coupled typical aeroelastic section with shape memory alloy springs for wind energy harvesting is addressed in this paper. An airfoil with two-degrees-of-freedom, namely pitch and plunge, is considered and piezoelectric coupling is added to the plunge degree-of-freedom. A load resistance is assumed in the electrical domain of the problem in order to estimate the electrical power output. Shape memory alloy coil springs are modeled in the pitch degree-of-freedom of the typical section. A nickel–titanium alloy that exhibits pseudoelasticity at room temperature is assumed. The constitutive model for the shape memory alloy is based on classical phenomenological models. The unsteady aerodynamic loads are obtained by Jones’ approximation to Wagner’s indicial function. The resulting nonlinear electroaeroelastic model is cast into a state-space representation and solved with a Runge–Kutta method. The effects of preload values of the shape memory springs and resistive power generation on the aeroelastic behavior of the wind energy harvester are investigated at the flutter boundary and in a post-flutter regime. The nonlinear kinetics of the austenite-to-martensite phase transformation changes the typical linear flutter behavior to stable limit-cycle oscillations over a range of airflow speeds. Such nonlinear aeroelastic behavior introduced by the hysteretic behavior of the SMA springs provides an important source of persistent electrical power.

Experimental Validation of a Flight Simulation Model for Slightly Flexible Aircraft

The paper presents a methodology for modeling the coupled flight dynamics with aeroelasticity in the time domain. The equations of motion are based on linearized mean-axes formulation. Unsteady, incremental aerodynamics due to elastic deformations are modeled in the time domain, applying the strip theory and a indicial function. The methodology is demonstrated with a prototype of the utility aircraft Stemme S15. Comparisons between the simulation and the flight-test data regarding the added aeroelastic dynamics have indicated that the model is suited for the flight control law and aircraft design, and that care must be taken in modeling the control surface lift slope for low-aspect-ratio lifting surfaces.

Improvement of a near wake model for trailing vorticity

A near wake model, originally proposed by Beddoes, is further developed. The purpose of the model is to account for the radially dependent time constants of the fast aerodynamic response and to provide a tip loss correction. It is based on lifting line theory and models the downwash due to roughly the first 90 degrees of rotation. This restriction of the model to the near wake allows for using a computationally efficient indicial function algorithm. The aim of this study is to improve the accuracy of the downwash close to the root and tip of the blade and to decrease the sensitivity of the model to temporal discretization, both regarding numerical stability and quality of the results. The modified near wake model is coupled to an aerodynamics model, which consists of a blade element momentum model with dynamic inflow for the far wake and a 2D shed vorticity model that simulates the unsteady buildup of both lift and circulation in the attached flow region. The near wake model is validated against the test case of a finite wing with constant elliptical bound circulation. An unsteady simulation of the NREL 5 MW rotor shows the functionality of the coupled model.

Numerical simulation of dynamic response of a long-span bridge to assess its vulnerability to non-synoptic wind

Winds generated by non-synoptic events such as those from tornadoes, microbursts or gust fronts, that are non-stationary or transient in nature and extreme in magnitude, can cause major damage to flexible structures. In this paper, a time-domain method is used to simulate the response of a long-span bridge subject to winds generated by a particular type of non-synoptic wind phenomenon such as a microburst to assess the vulnerability of the bridge to such winds. The self-excited or motion-induced and buffeting or turbulence-induced wind loads on the structure were modeled and simulated by Rational Functions and buffeting indicial functions, respectively. Wind from a translating microburst was simulated using empirical relationships that were derived from measurements of a laboratory-simulated microburst and the bridge response calculated to compare it with those induced by an equivalent straight-line wind that is used for structural design. It is shown that microburst induced structural vibration could be larger or smaller than the vibration induced by straight-line wind of equivalent magnitude depending on the relative size of the microburst with respect to the bridge span.

Indicial functions in the aeroelasticity of bridge decks

This paper presents the development of indicial functions (IFs) for two-dimensional bridge deck sections. A new set of IFs predicted for the cross section of the Great Belt Bridge (GBB) are discussed. Two approaches, based on the time- and frequency-domain descriptions, are applied. In time domain, IFs are determined by imposing an instantaneous change to a system state variable, e.g., the angle of attack (AOA) of air flow to the bridge section. In frequency domain, indicial representation is derived from the aeroelastic (or flutter) derivatives typically used to define the self-excited harmonic forces in an eigenvalue problem and generated by imposing a sinusoidal motion to the bridge section and performing a sweep in the frequency range of oscillations of the section. IFs are thus evaluated by exploiting the reciprocal relations that exist between them and the aerodynamic derivatives. To determine the aerodynamic response of a bridge cross section due to a step change in the AOA and to calculate the flutter derivatives from sinusoidal oscillations, the meshless discrete vortex method implemented in DVMFLOW® is adopted. The results from the proposed work can be applied in the development of reduced-order models (ROM) of aerodynamic loads suitable to investigate fluid–structure interaction (FSI) problems associated with practical analysis of wind effects on long-span bridges, including phenomena such as flutter, vortex-induced vibration, buffeting and galloping. The IFs reported in this paper illustrate the importance of flow separation and vortex shedding and their dependence on the magnitude of the change in a state variable such as, in particular, the AOA for the cases reported herein.

Application of indicial function for the flutter analysis of long span suspension bridge during erection

The design of cable supported bridge with long span is challenging due to the sensitivity of the dynamic excitation. The aerodynamic instability caused by fluttering can severely affect the safe operation. An application of indicial function to the flutter analysis in time domain is applied to the Great belt East Bridge for both completed and erection stage. The nonlinear least square method was used to extract the aerodynamic indicial parameters for flutter analysis in time domain. The geometric nonlinearity is considered through the nonlinear dynamic analysis. The results showed the good agreement with the wind tunnel test and the validity of the indicial function as well as the important role of the geometrically nonlinear analysis during deck erection.

Refined optimal passive control of buffeting-induced wind loading of a suspension bridge

Modern design of long suspension bridges must satisfy at the same time spanning very long distances and limiting their response against several external loads, even if of high intensity. Structural Control, with the solutions it provides, can offer a reliable contribution to limit internal forces and deformations in structural elements when extreme events occur. This positive aspect is very interesting when the dimensions of the structure are large. Herein, an updated numerical model of an existing suspension bridge is developed in a commercial finite element work frame, starting from original data. This model is used to reevaluate an optimization procedure for a passive control strategy, already proven effective with a simplified model of the buffeting wind forces. Such optimization procedure, previously implemented with a quasi-steady model of the buffeting excitation, is here reevaluated adopting a more refined version of the wind-structure interaction forces in which wind actions are applied on the towers and the cables considering drag forces only. For the deck a more refined formulation, based on the use of indicial functions, is adopted to reflect coupling with the bridge orientation and motion. It is shown that there is no variation of the previously identified optimal passive configuration.

Frequency- and time-domain methods related to flutter instability problem

Bridge flutter phenomenon presents an important criterion of instability, which should be considered in the bridge design phase. This paper presents different bridge flutter methods which can be used to solve the flutter problem. Most commonly used frequency-domain approach is based on the formulation of aero elastic forces with frequency dependent coefficients called flutter derivatives. The critical wind speed, as the main critical condition for the onset of flutter is obtained based on these aero elastic forces. Aero elastic forces can be also expressed in the time-domain, using so-called indicial functions. These functions are usually determined from the corresponding flutter derivatives. In situations when fluid-memory effects tend to become small the quasi-steady theory can be used as an approximation of aero elastic forces. A numerical example related to the typical bridge cross-section follows presented approaches. .

Numerical Study of Flutter of a Two-Dimensional Aeroelastic System

Numerical Study of Flutter of a Two-Dimensional Aeroelastic System: This paper deals with the problem of the aeroelastic stability of a typical aerofoil section with two degrees of freedom induced by the unsteady aerodynamic loads. A method is presented to model the unsteady lift and pitching moment acting on a two-dimensional typical aerofoil section, operating under attached flow conditions in an incompressible flow. Starting from suitable generalisations and approximations to aerodynamic indicial functions, the unsteady loads due to an arbitrary forcing are represented in a state-space form. From the resulting equations of motion, the flutter speed is computed through stability analysis of a linear state-space system.

Indicial functions and flutter derivatives: A generalized approach to the motion-related wind loads

This paper presents a general time-domain description of the loads acting on a moving cylindrical body immersed in a two-dimensional low-speed flow, aiming to consistently extend the framework of thin airfoil theory to mildly bluff sections, such as those usually employed for decks of modern long-span bridges. In order to systematically accommodate typical features of bluff-body aerodynamics, the classical Theodorsen and Wagner results are reorganized within a unified dimensionless approach, and generalized preserving their main formal structure. Accordingly, circulatory and non-circulatory contributions are separately described and superimposed, and generalized downwash-related terms are introduced. The strong duality between time-domain and frequency-domain representations is focused, and direct relationships between proper Wagner-like indicial functions and Theodorsen-like circulatory functions are deduced. Thereby, following the Scanlan formulation for bridge deck sections, flutter derivatives are represented by superimposing circulatory and non-circulatory effects, resulting in a frequency-domain description fully consistent with the Theodorsen's theory.The model is based on few parameters that can be estimated by simplified strategies and, when applicable, by asymptotic relationships. An identification procedure involving few experiments or numerical simulations is proposed and numerically implemented. Simulation results obtained in the case of a flat thin plate and of a closed box section, similar to the cross-section of the Great Belt East Bridge, are successfully compared with theoretical and/or numerical solutions, highlighting effectiveness and soundness of the presented identification strategy.

Transonic Aerodynamic Load Modeling of X-31 Aircraft Pitching Motions

The generation of reduced-order models for computing the unsteady and nonlinear aerodynamic loads on an aircraft from pitching motions in the transonic speed range is described. The models considered are based on Duhamel’s superposition integral using indicial (step) response functions, Volterra theory using nonlinear kernels, radial basis functions, and a surrogate-based recurrence framework, both using time-history simulations of a training maneuver(s). Results are reported for the X-31 configuration with a sharp leading-edge cranked delta wing geometry, including canard/wing vortex interactions. The validity of the various models studied was assessed by comparison of the model outputs with time-accurate computational-fluid-dynamics simulations of new maneuvers. Overall, the reduced-order models were found to produce accurate results, although a nonlinear model based on indicial functions yielded the best accuracy among all models. This model, along with a time-dependent surrogate approach, helped to produce accurate predictions for a wide range of motions in the transonic speed range.

Aeroelastic Stability and Response of Composite Swept Wings in Subsonic Flow Using Indicial Aerodynamics

In this study, the aeroelastic stability and response of an aircraft swept composite wing in subsonic compressible flow are investigated. The composite wing was modeled as an anisotropic thin-walled composite beam with the circumferentially asymmetric stiffness structural configuration to establish proper coupling between bending and torsion. Also, the structural model consists of a number of nonclassical effects, such as transverse shear, material anisotropy, warping inhibition, nonuniform torsional model, and rotary inertia. The finite state form of the unsteady aerodynamic loads have been modeled based on the indicial aerodynamic theory and strip theory in the subsonic compressible flow. Novel Mach dependent exponential approximations of the indicial aerodynamic functions have been developed. The extended Galerkin’s method was used to construct the mass, stiffness, and damping matrices of the nonconservative aeroelastic system. Eigen analysis of the system was performed to obtain the aeroelastic instability (divergence and flutter) boundaries. Also, solving the equations of motion in the time domain leads to the aeroelastic response of wing in different flight speeds. The obtained results are compared with the available results in the literature, which reveals an excellent agreement. The numerical results obtained in this article seek to clarify the effects of geometrical and material couplings and flight Mach number on the aeroelastic instability and response of composite wings in subsonic compressible flow.

Time-Domain Aeroelastic Loads and Response of Flexible Bridges in Gusty Wind: Prediction and Experimental Validation

Flexible bridges could experience vortex-induced and buffeting excitations below their design wind speeds and divergent self-excited oscillations (flutter) beyond a critical wind speed. For regular straight-line winds that are stationary or weakly stationary, a frequency domain formulation is routinely used for flutter analysis of bridges. However, when evaluating performance of flexible bridges subjected to gusty winds that are transient in nature, the frequency-domain formulation is not applicable, and hence time-domain methods should be used. In this paper, time-domain formulations were used to predict aeroelastic loads acting on a rigid bridge deck section model subject to stationary and gusty straight-line winds based on the knowledge of upstream wind speed and model displacement measurements. In this procedure, rational functions and indicial functions are used to formulate self-excited forces and buffeting forces, respectively. The functions used here were recently obtained from wind tunnel tests performed on a streamlined bridge deck section model with a smaller geometric scale. The results of the validation using a larger section model of this bridge deck subject to a ramp-type gust are presented here.

Challenges in the aerodynamics modeling of an oscillating and translating airfoil at large incidence angles

The challenges in the modeling of the unsteady lift of a two-dimensional NACA 0012 airfoil oscillating and translating at large incidence angles are investigated in this paper. Forced oscillation motions with two reduced frequency values of 0.1 and 1.0 were used. The CFD results show that a hysteresis loop was developed in the lift variation with angle of attack; shapes of the loops change significantly with reduced frequency value. A dynamic stall vortex was identified in the pitching and plunging motions with k=0.1. This vortex delays the onset of flow separation over the upper surface to a higher incidence than would occur in steady conditions. The unsteady lift of the very fast motion is enhanced significantly due to the formation of a compression wave on the lower surface and an expansion wave on the upper surface. The results show clear limitations of unsteady aerodynamic theories for modeling motions at high incidences. Also, the CFD solution of indicial functions show a large oscillation at angles of attack beyond the stall angle, but the predictions of a model based on Radial Basis Function matches CFD values quite well.

Effect of thrust on the aeroelastic instability of a composite swept wing with two engines in subsonic compressible flow

This paper aims to investigate aeroelastic stability boundary of subsonic wings under the effect of thrust of two engines. The wing structure is modeled as a tapered composite box-beam. Moreover, an indicial function based model is used to calculate the unsteady lift and moment distribution along the wing span in subsonic compressible flow. The two jet engines mounted on the wing are modeled as concentrated masses and the effect of thrust of each engine is applied as a follower force. Using Hamilton's principle along with Galerkin's method, the governing equations of motion are derived, then the obtained equations are solved in frequency domain using the K-method and the aeroelastic instability conditions are determined. The flutter analysis results of four example wings are compared with the experimental and analytical results in the literature and good agreements are achieved which validate the present model. Furthermore, based on several case studies on a reference wing, some attempts are performed to analyze the effect of thrust on the stability margin of the wing and some conclusions are outlined.