Abstract Aeroelasticity is the study of the interaction of aerodynamic, elastic and inertia forces. When flexible structures, such as an airfoil, undergo wind excitation, divergence or flutter instability may arise. We study the dynamics of a two-degree-of-freedom (pitch and plunge) aeroelastic system with cubic structural nonlinearities. The aerodynamic forces are modeled as a piecewise linear function of the effective angle of attack. Stability and bifurcations of equilibria are analyzed. The effect of the structural nonlinearity is investigated. We find border collision, rapid, Hopf, saddle-node and pitchfork bifurcations. Bifurcation diagrams of the system were calculated utilizing MatCont and Mathematica.

Abstract The stability characteristics of a cantilever beam, with and without an intermediate support, subjected to a dynamic terminal moment, is investigated. The moment is assumed to be proportional to the slope of a point along the length of the beam. The proportionally constant, which can be positive or negative, is varied to find the critical stability point. In the absence of intermediate support, stability is lost through divergence when the dynamic moment is proportional to the positive slope, and through flutter when the dynamic moment is proportional to the negative slope. In contrast, the nature of instability switches between divergence and flutter, and between different flutter instability modes while undergoing flutter, in the presence of an intermediate support.

Abstract The stability characteristics of a hinged beam subjected to a dynamic moment is investigated. The moment is proportional to the curvature of the beam at some point along its length. The stability investigations are carried out using a Galerkin approximation, both in the presence and absence of external flow. In the absence of external flow, stability is lost through divergence and flutter depending on the location of the point of measurement of curvature and the sign of the applied moment. In the presence of external flow, additional terms are introduced in the dynamic model. This alters the mechanism of flutter, reduces the value of the parameter at the critical point, and changes the nature of oscillations from standing waves to traveling waves.

Pipes aspirating fluid have applications in the filling and recovery processes for underground caverns — large subterranean cavities used to store hydrocarbons, such as natural gas and oil. This paper deals with the dynamics of a vertical cantilevered flexible pipe, immersed in fluid. Fluid is aspirated from its bottom free end up to the fixed upper end. In this study, the working fluid is assumed to be water. An existing analytical model is used to predict the dynamical behaviour of the aspirating pipe. This model is then discretized with Galerkin’s method, using Euler-Bernoulli eigen-functions for cantilevered beam as comparison functions. Once solved, the model results show a unique kind of flutter comprising three regions, denoted regions 01–03. These regions are delineated by two critical flow velocities, U cf1 and U cf2 . In addition, two frequencies of oscillation, f 1 and f 2 , are found to characterize the aforementioned flutter. The dominant frequency of oscillation changes from f 1 to f 2 as the flow velocity is increased from approximately 3 to 6 m/s — a frequency exchange phenomenon observed and reported here for the first time for this system. The analytical/numerical study was followed by a corresponding experimental study. Experiments were performed on a flexible (Silastic) pipe that was completely submerged in water. The behaviour observed experimentally was similar to the numerical study, as the aspirating fluid velocity was increased from zero to 7 m/s.

One of the important issues in turbomachinery flutter analysis is the intra-row interaction effects. The present work is aimed at a systematic research of the adjacent rows effects on aerodynamic damping. Three models, the isolated rotor, the IGV-rotor and the rotor-stator model, are performed to identify the upstream and downstream stator effects on the rotor blade. It is found that the aerodynamic damping from the stage flutter simulations are quite different from that from isolated rotor. In addition, the mixing-plane method is also applied to calculate the stage flutter characteristics and its accuracy of flutter predictions is compared with the time-marching method. It is indicated that the main difference of aerowork density between MP and TM is in the tip area, and in some cases the result from MP method can be misleading. Furthermore, study with different axial gaps illustrates that there is a nonmonotonic relationship between the rotor blade aerodynamic damping and the gap in the rotor-stator model, while the rotor blade aerodynamic damping monotonically increases with the gap in the IGV-rotor model.

In this study, analysis and results of linear and nonlinear aeroelastic of a cantilever beam subjected to the airflow as a model of a high aspect ratio wing are presented. A third-order nonlinear beam model is used as structural model to take into account the effects of geometric structural nonlinearities. In order to model aerodynamic loads, Wagner state-space model has been used. Galerkin method is implemented to solve dynamic perturbation equations about a nonlinear static equilibrium state. The small perturbation flutter boundary is determined by these perturbation equations. The effect of geometric structural nonlinearity of the beam model on the flutter behavior is significant. As it is observed the system’s response to upper speed of flutter goes to limit cycle oscillations and also the oscillations lose periodicity and become chaotic.

Friction dampers are utilized in turbomachinery to reduce blade vibrations resulting from aeroelastic interactions. In this paper, the microslip friction model is applied to a blade with blade to ground damper and subjected to negative damping. Analysis using the describing function method, also known as the method of harmonic balance, is used to identify the behavior of the system and the maximum negative damping that can be stabilized by such a damper. These results are compared to those for the macroslip friction model.

The paper presents the preliminary design of a novel gripper able to grasp large non-rigid materials that has been conceived to face the challenge of automatic handling tasks in the leather industry. The design has been driven by the requirements to limit production costs and the complexity of the grasping device. A statistical analysis of the different templates sizes has allowed to identify a fixed configuration of the gripping points able to properly pick all the sheets within a great confidence interval. According to the varying shape of the leather templates themselves, that is due to their stacking in plies on the beam, the trajectory of the gripping points has been studied and arranged. Due to the irregular shape of the large sheets that are handled, the edges of the non-rigid materials out of the gripping area might flutter during the transferring phase: a four-bar linkage has been specifically designed, so that the motion of its end-effector prevents unwanted leather creases.

In order to analyze turbine blades vibration caused by flutter, it is necessary to understand both aerodynamic damping and structural damping of high vibration stress. Flutter Vibration mode occurring in rated speed is non-synchronous mode. For measuring non-synchronous mode damping ratio of turbine blades, AC-type electromagnet which can generate high frequency excitation force was developed. Damping ratio characteristics of non-synchronous mode of nodal diameter 12,4 was measured in rotational test. For comparison, synchronous mode of nodal diameter 4 was measured, too. It was concluded as follows. (1) It is possible to excite non-synchronous mode by high frequency excitation electromagnet and calculate damping ratio from measurement resonance curve. (2) Damping ratio of non-synchronous mode ND12,4 was increased by increasing the excitation force. Synchronous mode ND4 is also a similar trend. (3) Nodal diameter 4 damping ratio of non-synchronous mode (Resonant speed=100%) was lower than synchronous mode (Resonant speed=75%).

A linearized parametric continuum model of a long-span suspension bridge is coupled with a nonlinear quasi-steady aerodynamic model giving the aeroelastic partial differential equations of motion reduced to the state-space ordinary differential form by adopting the Galerkin method. Numerical time-domain simulations are performed to investigate the limit cycle oscillations occurring in the range of post-flutter wind speeds. Continuation tools are thus employed to path follow the limit cycles past the flutter speed where the Hopf bifurcation occurs. The stable post-flutter behavior, which can significantly affect the bridge by fatigue, terminate at a fold bifurcation. This result represents an important assessment of the conducted aeroelastic investigations. The stability range of the limit cycle oscillations is evaluated by carrying out sensitivity analyses with respect to the main design parameters, such as the structural damping and the initial wind angle of attack.

Flutter-based micro generators have been successfully demonstrated to power wireless sensors. Since environmental wind speeds vary widely, flutter-based micro generators which are designed to operate within particular range of wind speeds will underperform elsewhere. At low wind speeds, magnets embedded near the ends of the belt will not move the desired distance between the coils, thereby reducing the energy conversion. A broadband flutter-based micro generator will have pick-up coils embedded on several vibrating elements with different dimensions. The coils are particularly concentrated near the point of maximum speed to maximize power output. The variation in fluttering element dimensions allows the microgenerator to generate considerable power at a wide range of wind speeds. In this work, we develop a mathematical model for the flutter-based micro generator, which addresses the wind – structure interaction, induced vibrations and electromagnetic transduction. The model primarily makes use of equations from bridge deck and thin plane analysis of flutter due to their similarities, and they are formulated to provide the velocity. This is later fed into electromagnetic transduction equations to calculate the output power. The model is useful to determine the significant design parameters of a flutter-based micro generator. The dynamic response and power output of a broadband micro generator with coils embedded on a set of cantilever films vibrating with respect to an external permanent magnetic field are calculated.

Modern aircrafts require improved performance and maneuverability while they conduct the missions. The flutter, an aeroelastic phenomenon is one of the important situations that limit the aircraft speed. Furthermore, for aircraft operated at high speed, many uncertainties may exist in its structural and aerodynamics characteristics. Especially, a slight change in the wing structural mode may induce a variation in its aerodynamic force distribution. In this work, an interval-based approach is used to handle the uncertainties associated with the flutter analysis. The set-theoretic representation of uncertainty is motivated by a possible lack of detailed probabilistic information on the distributions of the parameters. The analysis procedure is performed on an aircraft wing structure using finite element idealization and the results have shown the effectiveness and feasibility of the interval method. The order of the aerodynamic, mass and stiffness matrices of the assembled structures is reduced by introducing the first few natural modes of the structure as generalized coordinates. System equivalent reduction expansion process is used for model reduction which uses the generalized inverse and carries information pertaining to the selected modes at the selected set of degrees of freedoms. The system equivalent reduction formulation allows the reduction process to preserve the dynamics of the full system in a reduced set of matrices. Thus the order of the eigenvalue problem in the flutter analysis is reduced to one-third of the corresponding statics problem.

The aeroelastic stability of rectangular plates are well-documented in literature for certain sets of boundary conditions. Specifically, wing flutter, panel flutter, and divergence of a plate that is clamped on all sides are well-understood. However, the ongoing push for lighter structures and novel designs have led to a need to understand the aeroelastic behavior of elastic plates for other boundary conditions. One example is NASA’s continuous mold-line link project for reducing the noise generated by commercial transport aircraft during landing; in order to reduce the noise generated by vortex shedding from the trailing edge flap during landing, the project proposes to connect the gap between the trailing edge flap and the rest of the wing with a flexible plate. This paper summarizes the aeroelastic theory, numerical results, and experimental results of a study on the flutter and/or divergence mechanisms of a rectangular plate for different sets of structural boundary conditions. The theory combines a three-dimensional vortex lattice aerodynamic model with a plate structural model to create a high-fidelity frequency domain aeroelastic model. A modular experimental test bed is designed for this study in order to test the different boundary conditions. The test bed is also designed to test different plate thicknesses and sizes with only a small number of modifications. The well-understood boundary conditions are used as test cases to validate the analysis results, and then results of additional configurations that have not been extensively explored are presented. The results of this paper can be used to support the design efforts of projects involving plates or plate-membranes. In addition, the paper adds to the fundamental understanding of plate aeroelasticity and provides a wealth of experimental data for comparison and future validation.

The structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears are given. Planetary gears have only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds. Numerical results verify the critical speed locations. Divergence and flutter instabilities are investigated numerically for each mode type.

In turbo machinery design it is important to avoid vibrations that can destroy the turbine in the last resort. The rotating structure is exposed to periodic excitation forces. Two main types of periodic excitation can be distinguished. Flutter is the effect when mass flow forces couple with a natural vibration mode. The result is a negative damping coefficient and amplitudes will rise up to malfunction of the structure. The engine order excitation is a periodic excitation where the force signal is directly related to the speed of the rotor. A forced response calculation gives information about the blade vibration. Nonlinear coupling, i.e. friction coupling, between blades is used to increase damping of the bladed disk. Dynamic analysis of turbine blades with nonlinear coupling is a complex task and computer simulations are inevitable. Various techniques have been developed to reduce computational effort. The cyclic symmetry approach assumes each blade around the disk to be identical. Thus only one sector of the disk is sufficient to compute the steady state solution of the whole turbine blading. However, it has been observed that mistuning of blades reduces the flutter instability. On the other hand statistical mistuning can lead to dangerously high forced response amplitudes due to mode localization. A compromise is intentional mistuning. The simplest approach is alternate mistuning with every other blade exhibiting identical mechanical properties. This work explains in detail how a turbine bladed disk can be modeled when alternate mistuning is applied intentionally. Cyclic symmetry is used and each sector comprises two blades. This untypical choice of the sector size has significant impact on results of a cyclic modal analysis. Simulation results show the influence of alternate mistuned turbine bladings which are coupled by underplatform damper elements.

A parametric one-dimensional model of suspension bridges is employed to investigate their static and dynamic aeroelastic behavior in response to a gust load and at the onset of flutter. The equilibrium equations are obtained via a direct total Lagrangian formulation where the kinematics for the deck, assumed to be linear, feature the vertical and the chord-wise displacements of the deck mean axis and the torsional rotations of the deck cross sections, while preserving their shape during rotation. The cables elasto-geometric stiffness contribution is obtained by condensing the equilibrium in the longitudinal direction assuming small horizontal displacements and neglecting the cable kinematics along the bridge chord-wise direction. The equations of motion are linearized about the prestressed static aeroelastic configuration and are obtained via an updated Lagrangian formulation. The equations of motion governing the structural dynamics of the bridge are coupled with the incompressible unsteady aero-dynamic model obtained by a set of reduced-order indicial functions developed for the cross section of a suspension bridge, here represented by a rectangular cross-section. The space dependence of the governing equations is treated using the Galerkin approach borrowing as set of trial functions, the eigenbasis of the modal space. The time integration is subsequently performed by using a numerical scheme that includes the modal reduced dynamic aeroelastic Ordinary Differential Equations (ODEs) and the added aerodynamic states also represented in ODE form, the latter being associated with the lag-state formulation pertinent to the unsteady wind-induced loads. The model is suitable to analyze the effect of a time and space non uniform gust load distributed on the bridge span. The obtained aeroelastic system is also suitable to study the onset of flutter and to investigate the sensitivity of the flutter condition on geometrical and aerodynamic parameters. The flutter instability is evaluated using appropriate frequency and time domain characteristics. The parametric continuum model is exploited to perform dynamic aeroelastic flutter analysis and gust response of the Runyang Suspension Bridge over the Yangtze river in China.

Cyber Physical Systems couple computational and physical elements, therefore the behavior of geometry (deformations, kinematics), physics and controls needs to be certified using many different tools over a very high dimensional space. Because of the near infinite number of ways such a system can fail meeting its requirements, we developed a Probabilistic Certificate of Correctness (PCC) metric which quantifies the probability of satisfying requirements with consistent statistical confidence. PCC can be implemented as a scalable engineering practice for certifying complex system behavior at every milestone in the product lifecycle. This is achieved by: creating virtual prototypes at different levels of model abstraction and fidelity; capturing and integrating these models into a simulation process flow; verifying requirements in parallel by deploying virtual prototypes across large organizations; reducing certification time proportional to additional computational resources and trading off sizing, modeling accuracy, technology and manufacturing tolerances against requirements and cost. This process is an improvement over the V-cycle because verification and validation happens at every stage of the system engineering process thus reducing rework in the more expensive implementation and physical certification phase. The PCC process is illustrated using the example of “Safe Range” certification for an UAV with active flutter control.

Nonlinear limit cycle oscillations of an aeroelastic energy harvester are exploited for enhanced piezoelectric power generation from aerodynamic flows. Specifically, a flexible beam with piezoelectric laminates is excited by a uniform axial flow field in a manner analogous to a flapping flag such that the system delivers power to an electrical impedance load. Fluid-structure interaction is modeled by augmenting a system of nonlinear equations for an electroelastic beam with a discretized vortex-lattice potential flow model. Experimental results from a prototype aeroelastic energy harvester are also presented. Root mean square electrical power on the order of 2.5 mW was delivered below the flutter boundary of the test apparatus at a comparatively low wind speed of 27 m/s and a chord normalized limit cycle amplitude of 0.33. Moreover, subcritical limit cycles with chord normalized amplitudes of up to 0.46 were observed. Calculations indicate that the system tested here was able to access over 17% of the flow energy to which it was exposed. Methods for designing aeroelastic energy harvesters by exploiting nonlinear aeroelastic phenomena and potential improvements to existing relevant aerodynamic models are also discussed.

We analyse a continuous Cosserat model of a visco-elastic rod subjected to a combination of a conservative load and a follower term in one of the ends. The formalism takes into account the geometric non linearities that appear for large deformations from the straight solution. The resulting equation is a PDE whose solution can be analysed for special cases and the eigenvalues of the linearisation can be computed by a combination of numerical continuation and bifurcation results. We illustrate this method analysing the bifurcations of the straight solution of a rod subjected to a follower force. The dynamics and the bifurcation behaviour of the model are explored as the intensity of force is varied and as the mixture of conservative-follower terms varies continuously from the standard conservative case to the purely follower one. Special attention is paid to the corresponding transition from symmetry breaking pitchfork bifurcation (falling over mode) to the appearance of oscillations in a Hopf like bifurcation (if some material damping is included) or pure reversible Hamiltonian Hopf bifurcation in the absence of damping. After the onset of oscillations a complex dynamical behaviour frequently called flutter instability appears. The study is supplemented with the bifurcation analysis of the two elastically jointed follower pendula model as a simplified model of the continuous problem.

Aeroelastic instabilities of a panel may result in buckling (divergence) or flutter (Hopf bifurcation), when it is acted upon by induced aerodynamic and externally applied loads under supersonic/hypersonic environment in this paper. These instabilities are stabilized using nonlinear bifurcation control with piezoelectric actuation. The center manifold theory is used to extract subsystems which completely capture the bifurcation behavior of the original system near critical parameter values represented by sets of parametrised first-order differential equations with feedback control. The principle of normal form is used to simplify the nonlinear terms of the lower dimensional systems. The proposed controllers, which employ purely nonlinear state feedback, are used to modify the nonlinear characteristics of the post bifurcation limit sets by setting the amplitudes and rates of growth to the desired values. Numerical results show that the resulting closed-loop systems are effectively stabilized at the neighborhood of critical values.