Nonlinear aeroelasticity of high-aspect-ratio wings excited by time-dependent thrust

Effects of engine placement on flutter characteristics of a very flexible high-aspect-ratio wing are investigated using the code NATASHA (Nonlinear Aeroelastic Trim And Stability of HALE Aircraft). Gravity for this class of wings plays an important role in flutter characteristics. In the absence of aerodynamic and gravitational forces and without an engine, the kinetic energy of the first two modes are calculated. Maximum and minimum flutter speed locations coincide with the area of minimum and maximum kinetic energy of the second bending and torsion modes. Time-dependent dynamic behavior of a turboshaft engine (JetCat SP5) is simulated with a transient engine model and the nonlinear aeroelastic response of the wing to the engine’s time-dependent thrust and dynamic excitation is presented. Below the flutter speed, at the wing tip and behind the elastic axis, the impulse engine excitation leads to a stable limit cycle oscillation; and for the ramp kind of excitation, beyond the flutter speed, at 75 % span, behind the elastic axis, it produces chaotic oscillation in the wing. Both the excitations above the flutter speed are stabilized, inboard of the wing.     

Phase locking control in the Circle Map

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.     

Bifurcation in a parametrically excited two-degree-of-freedom nonlinear oscillating system with 1∶2 internal resonance

The nonlinear response of a two-degree-of-freedom nonlinear oscillating system to parametric excitation is examined for the case of 1∶2 internal resonance and, principal parametric resonance with respect to the lower mode. The method of multiple scales is used to derive four first-order autonomous ordinary differential equations for the modulation of the amplitudes and phases. The steadystate solutions of the modulated equations and their stability are investigated. The trivial solutions lose their stability through pitchfork bifurcation giving rise to coupled mode solutions. The Melnikov method is used to study the global bifurcation behavior, the critical parameter is determined at which the dynamical system possesses a Smale horseshoe type of chaos. Project supported by the National Natural Science Foundation of China (19472046)     

Articulated Pipes Conveying Fluid Pulsating with High Frequency

Stability and nonlinear dynamics of two articulated pipes conveying fluid with a high-frequency pulsating component is investigated. The non-autonomous model equations are converted into autonomous equations by approximating the fast excitation terms with slowly varying terms. The downward hanging pipe position will lose stability if the mean flow speed exceeds a certain critical value. Adding a pulsating component to the fluid flow is shown to stabilize the hanging position for high values of the ratio between fluid and pipe-mass, and to marginally destabilize this position for low ratios. An approximate nonlinear solution for small-amplitude flutter oscillations is obtained using a fifth-order multiple scales perturbation method, and large-amplitude oscillations are examined by numerical integration of the autonomous model equations, using a path-following algorithm. The pulsating fluid component is shown to affect the nonlinear behavior of the system, e.g. bifurcation types can change from supercritical to subcritical, creating several coexisting stable solutions and also anti-symmetrical flutter may appear.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Coupled, forced response of an axially moving strip with internal resonance

In this paper, the forced response of a non-linear axially moving strip with coupled transverse and longitudinal motions is studied. In particular, the response of the system is examined in the neighborhood of a 3 : 1 internal resonance between the first two transverse modes. The equations of motion are derived using the Hamilton's Principle and discretized by the Galerkin's method. First, with the longitudinal motion neglected, the forced transverse response is investigated by applying the method of multiple scales to assess the effects of speed and the internal resonance. In general, the speed is shown to affect each mode differently. The internal resonance results in the constant solutions having transition to instability of both a saddle-node type and a Hopf bifurcation. In the region where the Hopf bifurcation occurs, steady-state periodic motion does not exist. Instead the stable motion is amplitude- and phase-modulated. When the coupled system with longitudinal motion is examined with internal resonance, results reveal that the modulated motions disappear. Thus, the presence of the longitudinal motion has a stabilizing effect on the transverse modes in the Hopf bifurcation region. The second longitudinal mode is shown to drift due primarily to a direct excitation of the first transverse mode. Effects of the longitudinal motion on the transverse response are shown to be significant for speeds both away from and close to the critical speed.     

Nonlinear aeroelastic analysis of a two-dimensional wing with control surface in supersonic flow

Based on the piston theory of supersonic flow and the energy method, a two dimensional wing with a control surface in supersonic flow is theoretically modeled, in which the cubic stiffness in the torsional direction of the control surface is considered. An approximate method of the cha- otic response analysis of the nonlinear aeroelastic system is studied, the main idea of which is that under the condi- tion of stable limit cycle flutter of the aeroelastic system, the vibrations in the plunging and pitching of the wing can approximately be considered to be simple harmonic excita- tion to the control surface. The motion of the control surface can approximately be modeled by a nonlinear oscillation of one-degree-of-freedom. The range of the chaotic response of the aeroelastic system is approximately determined by means of the chaotic response of the nonlinear oscillator. The rich dynamic behaviors of the control surface are represented as bifurcation diagrams, phase-plane portraits and PS diagrams. The theoretical analysis is verified by the numerical results.     

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Direct flutter and limit cycle computations of highly flexible wings for efficient analysis and optimization

The usefulness of flutter as a design metric is diluted for wings with destabilizing (softening) nonlinearities, as a stable high-amplitude limit cycle (subcritical) may exist for flight speeds well below the flutter point. It is thus desired to design aeroelastic structures such that the post-flutter behavior is as benign (i.e., supercritical) as possible, among the other constraints commonly considered in the optimization process. In order to account for these metrics in an accurate and efficient manner, direct tools are utilized to first locate the Hopf-point (flutter speed), and then to obtain a nonlinear perturbation solution via the method of multiple scales. The latter scheme provides a scalar variable whose sign and magnitude dictate the nature of the limit cycle. The accuracy of these methods is demonstrated with a high-aspect-ratio highly flexible wing, modeled with nonlinear beam finite elements and the ONERA dynamic stall tool. Stiffness and inertial design variables are allowed to vary spatially throughout the wing, in order to conduct gradient-based optimization of the limit cycle under flutter and mass constraints. The resulting wing structure demonstrates strongly supercritical behavior, as well as several design conflicts between linear (flutter) and nonlinear (limit cycles) sensitivities, which are not present in the uniform baseline wing.     

Numerical study of the dependence of the critical flutter speed and time of a plate on rheological parameters

The nonlinear flutter of some aircraft elements is modeled. A viscoelastic model is used. Numerical algorithms for solving integro-differential equations are developed. The critical flutter speed and time for a viscoelastic plate are determined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 97–104, June 2008.     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。     

Convective inertia effects in wall-bounded thin film flows

We apply the boundary layer equations to inertial flow in wall bounded films that might be characterized as ‘thin’, say ɛ ≤ 0.1 where ɛ is the ratio of the characteristic lengths, yet to which the lubrication approximation of Reynolds no longer applies. Two particular flow geometries are investigated, nominally parallel plates and nominally inclined plates, both with and without spatially periodic perturbation of the stationary plate. A Galerkin-B spline formulation of the governing equations is employed, and we rely on parametric continuation to obtain solutions at higher values of the Reynolds number. In particular, we are able to demonstrate that the boundary layer equations yield accurate results for a wide range of Reynolds numbers when the aspect ratio is less than 1/10. We also find that in both nominally parallel and nominally inclined geometries the sign of the inertial force correction is determined by the film contour in the neighborhood of the exit, this result might have implications in the design of MEMS devices.     

Swimming performance of a biomimetic compliant fish-like robot

Digital particle image velocimetry and fluorescent dye visualization are used to characterize the performance of fish-like swimming robots. During nominal swimming, these robots produce a ‘V’-shaped double wake, with two reverse-Kármán streets in the far wake. The Reynolds number based on swimming speed and body length is approximately 7500, and the Strouhal number based on flapping frequency, flapping amplitude, and swimming speed is 0.86. It is found that swimming speed scales with the strength and geometry of a composite wake, which is constructed by freezing each vortex at the location of its centroid at the time of shedding. Specifically, we find that swimming speed scales linearly with vortex circulation. Also, swimming speed scales linearly with flapping frequency and the width of the composite wake. The thrust produced by the swimming robot is estimated using a simple vortex dynamics model, and we find satisfactory agreement between this estimate and measurements made during static load tests.     

Design of active flutter suppression and wind-tunnel tests of a wing model involving a control delay

In this study, a delayed controller was designed for active flutter suppression of a three-dimensional wing model. The design of controller can be divided into two steps. At the first step, a short time delay was artificially introduced into the control loop and the dynamic equations of the aeroelastic system with delayed control were converted into a set of delay-free state-space equations by using a state transformation. At the second step, the control law was synthesized by using the theory of optimal control for the delay-free state-space equations. To demonstrate the performance of the delayed controller, the margin of time delay was studied. The numerical results showed that the delayed controller had good robustness with respect to the time delay. Moreover, the delayed controller was digitally implemented and tested for the three-dimensional wing model in NH-2 subsonic wind-tunnel. The experimental results illustrated that the critical flow speed of flutter instability of the wing model could be effectively increased from 36.5 m/s to 39 m/s.     

Nonlinear Dynamics of a Flexible Spinning Disc Coupled to a Precompressed Spring

This paper analyses the nonlinear transverse vibrations of a rotating, clamped-free, flexible disc coupled to a precompressed spring. This is representative of a large class of loadings in rotating disc systems such as air jet and electromagnetic excitation commonly used in experiments. Such a loading induces a simultaneous critical speed resonance and parametric instability. The disc is modelled as a Von Kármán plate, and the equations of motion are discretised by a Galerkin projection onto a pair of 1:1 internally resonant modes. The large amplitude wave motions and their stabilities are studied using the averaging method and via numerical continuation techniques. The analysis is carried out in a co-rotating as well as a ground-fixed frame. Numerical simulations are used to verify the above analyses. The response predicted by these analyses is substantially different from that arising from a critical speed resonance or of a parametric instability alone. As many as five equilibrium solutions can coexist at supercritical speed. Two distinct regimes of large amplitude response appear to exist depending on the relationship between the strength of the parametric excitation and the damping. The existence of these regimes underscores the subtle competition between critical speed resonance and parametric instability that is likely to be observed in experiments near critical speed in such systems.Contributed by Prof. A.K. Bajaj.     

Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.     

FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID (Ⅱ)

Based on the nonlinear mathematical model of motion of a horizontally cantilevered rigid pipe conveying fluid, the 3:1 internal resonance induced by the minimum critical velocity is studied in details. With the detuning parameters of internal and primary resonances and the amplitude of the external disturbing excitation varying, the flow in the neighborhood of the critical flow velocity yields that some nonlinearly dynamical behaviors occur in the system such as mode exchange, saddle-node, Hopf and co-dimension 2 bifurcations. Correspondingly, the periodic motion losses its stability by jumping or flutter, and more complicated motions occur in the pipe under consideration.The good agreement between the analytical analysis and the numerical simulation for several parameters ensures the validity and accuracy of the present analysis.     

FLOW-INDUCED INTERNAL RESONANCES AND MODE EXCHANGE IN HORIZONTAL CANTILEVERED PIPE CONVEYING FLUID （Ⅱ）

Based on the nonlinear mathematical model of motion of a horizontally can-tilevered rigid pipe conveying fluid, the 3:1 internal resonance induced by the minimum critical velocity is studied in details. With the detuning parameters of internal and primary resonances and the amplitude of the external disturbing excitation varying, the flow in the neighborhood of the critical flow velocity yields that some nonlinearly dynamical behaviors occur in the system such as mode exchange, saddle-node, Hopf and co-dimension 2 bifurcations. Correspondingly, the periodic motion losses its stability by jumping or flutter, and more complicated motions occur in the pipe under consideration. The good agreement between the analytical analysis and the numerical simulation for several parameters ensures the validity and accuracy of the present analysis.     

Turbocharger rotor dynamics with foundation excitation

To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed.