flutter

Aeroelastic instability in blades is one of the most important sources of instability in helicopter rotors, and the most critical of these instabilities is flutter. In this paper, in order to investigate the blade flutter and its relationship with the rotor structural parameters, using the Hamilton's principle and considering the Euler-Bernoulli beam theory, the coupled nonlinear partial differential equations governing the rotating elastic blade of a helicopter in the hover flight mode are extracted and converted into a set of ODEs by applying Galerkin method. Then the obtained equations for small perturbations are linearized around the steady state conditions. The effects of the gyroscopic couplings are investigated and assuming the harmonic response, the natural frequencies of the blade in three motion axes are calculated and the relationship between the natural frequency and flutter frequency of the blade with structural and aerodynamic parameters are shown. Using numerical simulation, the results for two types of soft and stiff blades with given characteristics in terms of different parameters such as blade twist angle, pre-cone angle and rotation speed of rotor for the first mode shape are extracted. Finally, the effect of each of the mentioned parameters on the flutter frequency and also, the blade stability region is analyzed. It is shown that by increasing the blade stiffness, the flutter frequency will increase and the system will be stable.

In this paper, the flutter analysis of a CNT-reinforced composite beam carrying an attached mass in the supersonic flow under different boundary conditions is presented. Also, the analysis of the mentioned beam has been investigated taking into account of the first-order shear deformation theory. The aerodynamic Piston theory has been used to estimate the aerodynamic pressure. The equations governing the vibrations of this dynamic system have been determined based on the Hamilton principle. Then, by solving the equations using the generalized differential quadrature method, the natural frequencies of the dynamic system are calculated. In this work, the results have been compared and validated with similar studies. Then, the effects of carbon nanotube reinforcement and the effects of the attached mass on the frequency and stability of the beam have been investigated. The obtained results show that for uniform distribution of CNT the non-dimensional frequency decreases by increasing the ratio of length to thickness for all boundary conditions.

In this paper, a three dimensional model of a box wing configuration is derived by a semi-analytical approach and the aeroelastic behavior is studied. So far, the flutter characteristics have been studied on the typical wing sections or via a whole lot more time and cost in the professional software. The winglet is modeled by two longitudinal and torsional springs and in order to simulate the effect of the winglet on the dynamic behavior, two ends of the springs are placed on the elastic axis of the sections. The governing equations are extracted via Hamilton's principle and in order to apply the aerodynamic forces, Wagner unsteady model is considered. To transform the linear partial integro-differential equations into a set of ordinary differential equations, mathematical techniques are employed. For the purpose of validation, the flutter values of the box wing are obtained by MSC NASTRAN and the proposed numerical procedure. The effects of the sweep angles and the winglet rigidity on the flutter are investigated. The results reveal that increasing the sweep angles and the chord ratio, enhances the flutter speed, remarkably. Furthermore, increasing the torsional rigidity of the winglet is more significant than the longitudinal rigidity on the flutter.

Composite materials are widely used in modern aerospace flight vehicles, especially because of their high specific strength and lightweight than other materials. Since the study of reliability and uncertainty of composite material design variables in aeroelasticity has received less attention, in the present work the reliability of the laminated composite plate due to uncertainty in variables including elastic model, Poisson coefficient, density, thickness, and length is examined. The composite laminated plate is symmetric with different boundary conditions subjected to supersonic airflow. The classical plate theory and the first-order piston theory are utilized to derive the equation of motion. The differential quadrature method has been used to discretize and analyze the aeroelastic equations. The governing equations after discretization are solved by calculating and analyzing eigenvalues and the occurrence of the flutter phenomenon for the laminated composite plate is obtained. To examine the reliability, the distribution of random variables as a normal distribution has been used. Finally, the Monte Carlo simulation method was used for five different boundary conditions to obtain the reliability of the plate flat threshold. According to the presented results, the reliability value of the composite plate flutter threshold for the full hinge boundary condition (SSSS) will be higher than other boundary conditions and the all-bound boundary condition (CCCC) will be lower than other boundary conditions. Also, according to the studies on the condition of the fiber angle of the composite plate, it can be concluded that increasing the angle of the fiber of the composite plate increases the reliability of the occurrence of the filter threshold.

In this paper, stability analysis of pipes conveying fluid is considered. The pipe structure is modelled using the Euler-Bernoulli beam theory and the fluid flow effect is taken into account as a distributed load along the pipe length which contains the inertia, Coriolis and centrifugal forces. In order to achieve different boundary conditions, both ends of the pipe are constrained with two lateral and two bending linear springs. The governing equation of the system is developed using the extended Hamilton’s principle and expressed in the matrix form by applying the method of separation of variables and the Galerkin technique. Then, via the standard eigenvalue analysis method the stability analysis of the system is done and effects of some parameters such as mass flow, gravity and the type of pipes’ boundary condition on the type of instability and the stability margin of the pipe conveying fluids are considered and some conclusions are drawn.

In the present study, an analytical model was presented for investigating the nonlinear aero-elastic response of cracked plate in supersonic flow. In this context, two-dimensional equations of cracked induced isotropic plate were proposed considering pure bending loading and simply support boundary condition. To form this equation, plate modeling based on the classical plate theory (CPT) and the von-Karman nonlinear relations. Also, the Line-Spring Model (LSM) and linear piston theory were considered for crack location and aerodynamic effect, respectively. Applying the Galerkin's method and plate assume modes, the partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). Then, by using Runge-Kutta numerical solution method, these equations (PDEs) were solved and the results were investigated. Eventually, some of effective aero-elastic parameters like flow condition and crack size were prescribed within the limit cycle oscillation (LCO) in flutter status. Results demonstrate that the presence of crack was leading early flutter, increasing the maximum amplitude of the limit cycle oscillations and aero-elastic instability that can ultimately reduce the structural performance.

The main goal of this article is to train a support vector machine in order to determin the boundary of the composite wing aeroelastic instability. Aircraft wing is modeled as a cantilever beam with two degrees of freedom with thrust as a follower force and mass of the engine. For structural modeling of composite wing the layer theory has been used and in the aerodynamic model, the flow has been assumed to be unsteady, subsonic and incompressible. Using the assumed mode method, the wing dynamic equations of the motion have been derived by Lagrange equations. Linear flutter speed according to the eigenvalues of the motion equations has been calculated. The process of flutter speed calculation has been converted to a computer code in which the number of layers, angle of fibers in each layer, the mass of the engine, and the thrust are input variables and the flutter speed is its output. Determination of the instability boundary using this conventional method is time consuming. In this article, a support vector machine has been adopted to reduce the calculation cost. The results indicate that support vector machine can be used in determining the boundary of the wings flutter instability as an accurate and fast tool.

The design and construction of aircraft wings with optimal geometry and physical properties that are highly stable is of high importance to engineers. In this study, with the assumption of uncertainty in system design variables, a robust optimization of the Flutter velocity aeroelastic wing with high-aspect-ratio under the bending-torsion effect is examined with the standard deviation minimization. Therefore, the aeroelastic wing are firstly modeled based on the Euler-Bernoulli cantilever beam model in quasi-steady aerodynamic conditions. After validating the results, in the simulation section, by using the 4th Runge-Kutta numerical solution and the theory of Eigenvalues, the system response time and Flutter velocity are obtained. In the high-aspect-ratio wings, increase the Flutter velocity in the presence of uncertainty in the parameters is important. Therefore, by choosing parameters such as bending and torsional rigidity and mass per unit wing as optimization variables the effect of uncertainty on the design variables and optimized by genetic algorithm. In addition, the values of variables before and after optimization, as well as the rate of improvement of the Flutter velocity are presented in a robust and deterministic optimization. Finally, based on the optimization results, design variables for achieving an appropriate stability structure in terms of the phenomenon Flutter is confirmed.

In this research, nonlinear transverse vibrations of a fluid conveying microtube under parametric magnetic axial resonance condition is studied. For this purpose, nonlinear governing equations of transverse motion of beam-like microtube are derived using Reddy’s first-order shear deformation theory with considering the effect of fluid viscosity and fluid centripetal acceleration. In this model, nonlinear terms of Hetenyi foundation and nonlinear geometric terms of the Von-Karman theory under magnetic excitations in the presence of fluid flow beyond the flutter instability is considered. In the following, the effects of foundation parameters on the linear flutter specifications of fluid conveying magnetizable microtubes are studied. Then, the nonlinear system behavior for fluid flow velocities more than critical velocity corresponding to the coupling of the first and second vibration modes is studied using multiple scales method. Nonlinear response curves in velocities above critical velocity are obtained and effects of variations of various system parameters including flow velocity, amplitude, and frequency of the magnetic field, Hetenyi foundation stiffness constants, viscosity, and dimensions ratio on the nonlinear response of the system are investigated. Some results indicate that increasing the values of shear stiffness parameter of the Hetenyi foundation has an unstable effect so that with its increasing, the flutter instability occurs at lower frequencies.

In this study aeroelastic instability of laminated beam-plates with two overlapping delaminations with equal length that is subjected to supersonic ‎flow is investigated analytically. The classical laminate plate theory with small deflection assumptions with quasi-steady aerodynamic theory is ‎utilized to derive the governing equations of the system. Additionally, it is assumed that the delaminations parts are forced to vibrate with each other. ‎In this case to predict critical flutter pressure, the delaminated composite beam-plates are divided into five parts. The boundary conditions at the ‎ends of beam-plates and at the ends of each delamination region are employed to predict flutter pressure. The effect of different parameters such as ‎delaminations length, axial position, their position through thickness of delaminated beam-plate and different stacking sequence are considered on ‎flutter pressure of beam-plate with two overlapping delaminations. According to the results, flutter pressure of delaminated beam-plate with two ‎overlapping, equal length delaminations in comparison to beam-plate with single delamination is reduced 20 through 40 percentages approximately. ‎Additionally, it has been shown, as far as two delaminations are closer to the mid-plane of the beam-plate, the aeroelastic behavior will change ‎further. ‎

In this study, with the assumption of uncertainty in system design variables, a robust Optimization of the Flutter velocity aeroelastic wing with high-aspect-ratio under the bending-torsion effect is examined . Therefore, the aerospace wings are firstly modeled based on the Euler-Bernoulli cantilever beam model in quasi-steady aerodynamic conditions. After validating the results, in the simulation section, by using the Runge-Kutta numerical solution and the theory of Eigenvalues, the system response time and Flutter velocity are obtained. Therefore, by choosing parameters such as bending and torsional rigidity and mass per unit wing as optimization variables the effect of uncertainty on the design variables and optimized by genetic algorithm. In addition, the values of variables before and after optimization, as well as the rate of improvement of the Flutter velocity are presented in a robust and deterministic optimization. Finally, based on the optimization results, design variables for achieving an appropriate stability structure in terms of the phenomenon Flutter is confirmed.

In this paper, active flutter control of a swept wing with an engine is carried out. The aircraft wing is considered as a uniform swept cantilever beam carrying an engine. The piezoelectric layers are attached to the wing to control the vibrations. To simulate aerodynamic loads, the Theodorsen model is used. The equations of motion are determined via Hamilton’s variational principle and are transformed to a set of ordinary differential equations through the assumed mode method. Lyapunov controller is used to control the system. Effects of design parameters like engine trust, location and mass and wing sweep angle, are evaluated on the flutter speed, and the control system has been applied at the flutter situation. Results show that the control system can substantially suppress the vibration in investigated cases. According to the results, the length of the piezoelectric layers affects the speed of the flutter and the flutter speed increases by increasing the length of these layers. Also, according to the influence of the Lyapunov gains on the performance of the system, it is necessary to select these values carefully to control system has the best performance for the different values of the system parameters.

In the present study, the flutter and aeroelastic response of mistuned bladed disks to the engine order excitation are studied with the aim of determining the effects of disk structural properties and also establishing an efficient method of analysis. For modeling the solid-fluid interaction, the Whitehead’s incompressible, two dimensional cascade theory is used. The structure is also modeled, using a 4 degrees of freedom lumped mass-spring system, which accounts for the bending and torsional deformation of the blade and the disk. This model would enable us to study the effect of structural coupling of adjacent sections as well as the disk flexibility. The solution is based on expansion of the mistuned-blade response in terms of the traveling-wave modes of a tuned bladed disk. The adopted method would be appropriate for determining the aeroelastic response, since the aerodynamic loads are available only for each individual traveling-wave mode. The obtained solution is used to study the effects of disk flexibility on the aeroelastic instability, variations of natural frequencies with different numbers of nodal diameters, and the sensitivity of the vibration amplitude response to the mistuning. Furthermore, the effects of mistuning in blades torsional frequencies and the mistuning in engine order excitation is considered. Parametric studies show that for disks with a lower bending stiffness, the mistuning can significantly influence the aeroelastic behavior such that the for a certain amount of the natural frequency, the disk response could be increased more than 8 times due to the presence of mistuning.

In this study, we derived the rotating airfoil system of equation considering Loewy aerodynamics. To this end, we define the local coordinate system on airfoil and reference coordinate on the hub. We define the free air velocity vector and the airfoil rotating speed vector according to the reference coordinate. So, the Kinetic and Potential energies are derived based on linear stiffness and linear damping according to the Hamiltonian principle. Wakes behind the rotating blades form into the helix. Therefore, we the equation of motion with Loewy aerodynamic which compensates the wake effects. Stability analysis is performed by the well-known P-K method. Flutter speed and stability boundary are estimated. Comparing the results of stability analysis and the reference validates the applied method. Furthermore, we proposed the PID Control to suppress the flutter speed. the PID controller input and command. The desired time and error tolerance are selected to design PID controller. Unit step response shows that pitch angle response is under-damped. However, step response tracks input well. Besides, disturbance rejection by considering the gain from input to output to remain below the gain value is analyzed.

In this paper, we study and introduce the drag force reduction in flexible structures. Firstly, a review of the inspiration of this phenomenon by nature is presented, and thenthe amount of drag force reduction in the flexible structure is measured experimentally. The obtained results show that this reduction can be up to 50% of the total drag force. In addition, influential factors on this phenomenon have been investigated.

In this research, behavior of a cylindrical sandwich panel with the flexible core in different boundary conditions, under the influence of follower force are studied. For this purpose, using the theory of the first-order shear deformation, and the principle of Hamilton, the governing equations have been extracted and by using generalized differential quadrature method (GDQM)are solved. The obtained results indicate that by considering simplicity of present method compared to the other methods in the analysis of stability problems of the cylindrical sandwich panel, the present results have the acceptable accuracy.Also, the results review show that the flutter phenomenon occurs in the free and clamp boundary conditions and for other boundary conditions only the divergence phenomenon or static buckling occurs.Another significant result is that increasing the number of composite layers, makes the flutter phenomenon in the cylindrical sandwich panel with the flexible core occurs later

In this paper, the flutter of an adaptive wing with adjustment of spar position is studied. Despite of two-dimensional models which was used in earlier research on this subject, in this study, more realistic model of the wing with adjustment of spar position contains sweep angle and unsteady aerodynamic loadings is employed. Two uniform spars which can move in chordwise direction are considered along the wing. The wing bending and torsion equations of motion have been derived by Hamilton’s principle. In order to use this principle, kinetic energy, strain energy and virtual work of forces have been obtained for spars and wing separately and then their sum embedded in the Hamilton’s principle. To simulate the aeroelastic loading on the wing Peter’s unsteady aeroelastic model is used. Assumed mode method has been used to discretized the aeroelastic governing equations and the numerical results has been validated with previous published papers, which good agreement has been reported. To review and presentation of results, four different types of motion for two spars have been utilized. In each type of motion, initial and final situation of spars are considered based on their velocity. Comparison of adaptive wing with simple wing in the same conditions shows that flutter velocity and frequency increases for adaptive wing. Finally, effects of different design parameters on the aeroelastic behavior of adaptive wing have been evaluated. Results indicate that these parameters can influence the stability region of such wings, significantly. Results indicates that increasing the thickness of the wing skins and spars reduces the wing flutter speed.