Science topic

# Aeroelasticity - Science topic

Aeroelasticity is the science which studies the interactions among inertial, elastic, and aerodynamic forces. It was defined by Arthur Roderick Collar in 1947 as "the study of the mutual interaction that takes place within the triangle of the inertial, elastic, and aerodynamic forces acting on structural members exposed to an airstream, and the influence of this study on design." In more simple terms, it is the same set of conditions causing a flag to flutter in a stiff breeze or a reed to tremble in fast-flowing water. Flutter may occur in any fluid medium.
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Hi, I am modeling a wing with strip theory for flutter analysis. In the case of the doublet lattice method, there is a precondition that the size of the aerodynamic panel should be less than 3 at subsonic speed and an aspect ratio of 1 at transonic speed or higher. Is there such a condition when making a strip in strip theory? Aeroelastic analysis manuals such as MSC nastran or altair, and other papers also specified the prerequisites of the box in relation to DLM, but there was no mention of the strip theory, so I asked. Or as many as I want regardless of the ratio of the strips? Or, just as the strip theory becomes a little more accurate when the aircraft's aspect ratio is 3 or more in flutter analysis, the strip is also modeled with ar 3 or more?? Of course it could be something I can't find. sorry
DYNAMIC STABILITY AND RESPONSE )
A. W. BABISTER
The strip theory has been addressed and not specified by 3 conditions only
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When dealing with the complex nonlinear aeroelastic systems, sometimes we encounter with the supercritical or subcritical Hopf bifurcation. Usually the subcritical bifurcation is more dangerous. Is there an efficient way to predict the bifurcation type of a very complex (can not be studied analytically) nonlinear aeroelastic system?
Numerical experiments are mostly the common way to detect or qualitatively comment on the type of bifurcation, particularly if a pitchfork type bifurcation is super or sub-critical. In higher dimensions this becomes a periodic orbit or a limit cycle in the absence of any external forces (limit cycles are self sustained).
One method could be to observe any form of hysteresis in the observed large amplitude oscillations. If while changing the parameter (eg. α) in one direction; say increasing, there persists a stable fixed point which becomes unstable at αc and a larger limit cycle appears. Now while decreasing the parameter from α > αc (ensure the initial condition is near the larger limit cycle), the larger limit cycle disappears at α=αc and the initial fixed point reappears; then this is supercritical hopf. However if the larger limit cycle is still stable below αc and looses stability at a lower value of αnc jumping to the initial fixed point; this is a signature of sub-critical bifurcation. The larger limit cycle looses stability by means of saddle node type bifurcation for cycles. This also means that a local unstable limit cycle exists between αn and αc located between the fixed point and the larger limit cycle which engulfs the larger limit cycle at the saddle node bifurcation point αn
One method to detect this unstable limit cycle is to use the method of pseudo arc-length continuation or path following algorithm. Packages like COCO exist which use the method of continuation to detect these unstable limit cycles for dynamical systems.
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Hello,
I am studying the performance of 2 wing models (with the same size, constraints, loading and material properties but different material distribution) using Nastran. Wing A is lighter and 24% stiffer than wing B and that is demonstrated in the static analysis (under a distributed load on the top skin).
The two wings have about the same fundamental frequency (34.25Hz and 34Hz). Using the relationship f=sqrt(K/m), the difference between the two frequencies should be significant.
Also, the divergence analysis yields exactly the same displacement for the two wings.
Does any one know what could be causing such results? why the divergence displacement is not smaller for wing A (since it is stiffer) and its frequency is not much higher than wing B?
Rossana Fernandes Can you share the two Nastran models (.bdf)?
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I want to write a code related to the flutter of aeroelastic wing under unsteady load using doublet lattice method (DLM). I want a good reference with numerical examples or a book which learn me coupling doublet lattice method (DLM) with finite element method (FEM) step by step.
Hello Mohsen,
One of my colleagues finished his thesis on this topic (https://commons.erau.edu/edt/440/)
I am sure he would be open to discussions via emails.
Good luck!
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Flutter is a great issue in aerospace design wondering if it the same for high speed automobile design such as formula 1 cars.
Dear Taiwo,
aero-eleasticity new branch of applied mechanics is interaction between fluid matters and flexible solid bodies typical application in aircraft engineering and which played a major role in structural loads of rotor of turbine blades. Many issues are related with civil engineering and mechanical engineering (compressor, turbine blade etc.).Here the forces like dynamic aero-elastics, static aero-eleastics and flight mechanics forces are interrelated.
Ashish
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I want to have parametric Aeroelastic modeling in MSC Nastran. I want use python to change the BDF file data , then running NASTRAN and extract the simulation response. I also found a package "PyNastran", but I couldn't found a simple tutorial regarding this python package, Can any one suggest me a good reference, tutorial or book.
might be a starting point.
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Can anyone give light on the relation between flutter and Buffeting phenomena apart from being linked by same causal factors (Separation/and shock waves relation of oscillating airfoil)?
Buffetting and flutter are two completely different things.
Flutter is an aerodynamically triggered aeroelastic stability mode oscillation depending on structural stiffness, damping and mass distribution.
Buffetting is an aerodynamic instability of a shock movement due to shock induced separation onset
Thus, flutter is not limited to transonic flow conditions and buffetting is not linked to structrual stiffness of the wing.
Only, in case that the Eigenfrequencies of flutter and buffetting match, you may end up in a buffetting flutter mode
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Can any body suggest me a straightforward way with low computational cost, for obtaining LCO amplitude and frequency for multi-dimensional phase space?
Dear Mohsen,
an old but useful way is to use the Describing Function method. If I am not mistaken Dr. Ronilson Rocha has used this method with chaotic circuits and systems with good success.
Regards.
Luis
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work in the field of aeroelastic stability of bridges and investigate the interaction of flow with a structure at different angles of attack. I'm wondering where this angle comes from and what research is on this topic.
The given article is evaluating the AOA on the box girder of a bridge; it will help you:
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Which modes are controlling the flutter? What the flutter mechanism is and how can the flutter speed be increased? Any hint and examples to the questions above? Thanks
You may find dominent modes (usually the first mode) using centre manifold theories (nonlinear theories)
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Refer the book "Gas Turbine Powerhouse: The Development of the Power Generation Gas Turbine at BBC-ABB-Alstom" by Dietrich Eckardt (available at https://books.google.com/). On page 275 a reference of GT8 rotor damage due to high slenderness ratio is given. I could not find much about such effect in open literature. Does anybody know what is effect of rotor slenderness ratio (L/D) (or rotating body in general-any papers/references)? Few more specific questions:
• What will be the effect of slenderness ratio (other than regular rotor dynamics parameters i.e. critical speed, unbalance response and catenary) on the behavior of steam turbine rotor?
• Is there any allowable limit of slenderness ratio for a steam turbine rotor?
Regards
Naveen Bhatt
It may lead to reduced stiffness of the blade and increase the deformation.
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Dear research community,
I've been using MSC NASTRAN SOL146 as the solver for part of a genetic optimization procedure in MATLAB. The .f06 files are read every iteration and the data is used to perform the calculations of the fitness function. However, after running for a while, suddenly I am facing an error message when creating the .f06, which is related to license authorization problems (USER FATAL MESSAGE 3060). It is random when it happens, could be right after starting the procedure, or after letting the code running for about 12h.
Does anyone has already experienced such thing? In annex follows the screenshot of the .f06 with the error.
I appreciate the attention and look forward to discussing this matter.
Regards,
Victor Gasparetto
Hello Victor,
thanks for the files.
I was thinking about that other users reserve all licences and there are not enough left for you. Thats typically when a bunch of people use nastran. But in this case, there is an other error in the .log-file.
In your case I think, the license server is completly unavailable. For license servers I'm unfortunatly not the expert. Nevertheless, maybe you can check if the license server has network connectivity problems, or other cpu or data intensive jobs "blockade" the server hardware.
Regards, Marinus
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I developed the governing equation of an aeroelastic aircraft wing, I want to verify it with MSC Nastran, Can any one suggest me a simple strain forward reference to learn how model aeroelastic problem in MSC Nastran.
I finally find myself. I share it, may be somebody needs.
The best tutorial is MSC Nastran Documentation, in its package.
I use it, there are many training examples and useful subjects.
I also find a good website.
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Dear Enthusiasts,
I hope you are enjoying your research. I have a new topic (well, to me at least) and I want your valuable ideas on it. I want to compute Buffet Forcing Function on an aerodynamic body, like VEga or SLS. Can anyone tell me the simplest way to do it (a) on WT model and (b) on Full scale model.
I have fair idea of computing unsteady pressure values and then computing fluctuations but what after that...? I have a grey area ahead. Please, if someone knows, do help me.
Regards,
Shamoon
Thanks Janet
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Hello everyone,
I would ask for the recommendations for any book in the field of aeroelasticity especially for beginner.
At Warsaw University of Technology we have a 30h Vibrations and Aeroelasticity course and we recommend those books for aeroelasticity:
-Bisplinghoff R.L., Ashley H., Halfman R. L, Aeroelasticity, Addison-Wesley, Cambridge, Massachusets, 1955
-Bisplinghoff R.L., Ashley H., Pronciples of Aeroelasticity, John Wiley, New York, 1962
-Dowell E.H., Curtiss H.C., Scanlan R.H., Sisto F., A modern course in aeroelasticity, Sijthof & Noordhoff, Alpen aan den Rijn, Netherlands, 2004.
-Balakrishnan A.V., Aeroelasticity, Springer, New York, 2012
-Wright J. R., Cooper J.E., Introduction to Aircraft Aeroelasticity and Loads, John Wiley & Sons, Chichester, England, 2007
-Hodges D.H., Pierce G.A, Introduction to structural dynamics and aeroelasticity, Cambridge, NY, 2012
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Dear all,
I want to download some papers from the following International symposium " 14th International Symposium on Unsteady Aerodynamics, Aeroacoustics & Aeroelasticity of Turbomachines (ISUAAAT14) ". I don't know how to download papers as they are not available online and no DOI number. Does anybody know how to download papers from this? thanks in Advance
Thank you Yeasir Mohammad Akib for the help
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Good morning,
I am trying to obtain the aeroelastic divergence speed of a thin plate in MSC Nastran (flight loads). Is there a way of getting the divergence speed through SOL144 using DIVERG card? When I attempt to run SOL144 I obtain the aerodynamic loads. Does anyone have a good reference/tutorial for studying divergence in MSC Nastran?
You can easily get the divergence speed with the SOL 145. I am also working in the field of Dynamic aeroelastic problems (Flutter). In the SOL 145 output file, check KFREQ=0, that indicated divergence.
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Good morning
I would like to perform a ground vibration test (GVT) on a wing model. Can a someone suggest a good material on that (including how to perform it)?
The followings are good sources for ground vibration test (GVT):
I hope it is helpful for you.
All the best.
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Hey,
I am working on the Multi body dynamics of a 300kW VAWT as part of my graduation thesis. I have created a reduced order model of the turbine using SIMPACK (MBD solver from dassault systems). I am facing some difficulties with running modal analysis due to error in modal matrix being singular.
I am reaching out to people who have worked with SIMPACK, hoping to get some help with my model .
I think Anays Rigid Dynamics will give you all the possible options in comparison to any other tool. Try for it to cross verify the results.
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I am trying to perform a numerical analysis on a morphing (folding) winglet and would like to know if there is a "best" tool to perform the analysis. I am currently familiar with Natran Aeroelastic Analysis.
ANSYS
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I am studying the supersonic flutter characterisitcs of rectangular panels. I am new to this domain and have certain fundamental questions that I am unable to answer. I am employing the Piston theory for determining the flutter boundary, i.e. the flutter speed and the corresponding frequency. I have seen several works in which the v-g (velocity-damping) and v-f (velocity-frequency) graphs are plotted and the critical velocity is determined from the v-g curve corresponding to the point where the damping corresponding to a particular mode crosses from negative to positive region. The velocity (is it correct to say free-stream velocity?) is varied from 0 to a particular velocity of interest.
If I am correct, these v-g and v-f curves are plotted for a particular Mach number (M>1 for supersonic) . If this is the case, I am unable to comprehend why the velocity is varied from zero to a higher value? In most of the papers that I have read, M is defined as the free stream Mach number, M=(free-stream velocity/sonic speed). Is the velocity being varied (from vmin=0 to v) while keeping M (>1) constant?
Any help in clearing these misconceptions would be greatly appericiated.
Thanks
Thank you very much for this resource, Dr. Pany!
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I'm particularly interested in predicting the flutter speed of a 3D aircraft using FEM for the structural analysis and panel methods for the aerodynamic force. I would like to find some theory/applications concerning modeling flexible structures by mode expansions in a panel method code in order to express the aerodynamic force by {F}=[A]{q}, where {q} are the generalized displacements and [A] is the generalized aerodynamic matrix. This matrix is a function of the "reduced frequency" k.
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I need to calculate the forces caused by vortex shedding(or vortex induced vibrations) on a tapered cantilever structure under wind loading.
I did some reading and some researchers suggests to calculate the response using some empirical equations to include the motion induced forces due to lock in and negative damping as I understood.
I would look on line for the Strouhal number of similar structures at similar Reynolds numbers to see what is already been done. When in doubt 0.22 is a good first guess at the Strouhal number. From there I would set up an excel spreadsheet with different heights above the base plane, then do a 1/7th power law for the boundary layer effect and calculate the wind velocity in 10 or so different points up the height. From there you should be able to get the vortex shedding frequency as a function of wind speed and height to compare to your structural resonances. Figuring out the unsteady lift force is more problematic from first principles but I bet there is a lot out there in the published technical literature. Good luck
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I would like to ask if results of the research presented on the ASA Meeting "Aeroelastic analysis of a closing reed of the mouth organ
harmonica)" heve been publicated somewhere? I am currently working on analysis of the sound generation in free reed instruments also featuring 3D numerical simulations. I would be very grateful for any help. Thank you in advance.
В данный задачи дифференциальный оператор будет несамосопряженным, методы решения для соответствующих спектральных задач немногочисленны.
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Hi All,
I wanna use the mentioned formulation for computing noise radiation from a fluid solid interaction system. I faced with some difficulty in my mind due to lack of enough knowledge, so I would like to have your support; Furthermore, any collaboration would be so pleasant.
Sincerely,
Hamed
@ Rashid Naseer,
Hi, I just clarified my plan for you through private messaging.
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Hello everyone!
Please I need a reference that could be used to build an aeroelastic model for an aircraft. The model should be simple, but yet accurate. It might be consisted of a simple aerodynamic model (lifting line, lifting surface, etc.) and a simple structural model (beam theory, etc.). The reference should give complete details and hands-on on how to build the model (not only theoretical derivations). The model should be fully implementable on MATLAB. It also needs to be finally represented in a state-space model that accounts for the ordinary rigid-body states, in addition to the flexible degrees of freedom. The results of the analytical model will be validated with real flight test data.
The aircraft to be modeled is a single-seated sailplane that has a wing span of 18 m, wing chord of 0.7 m, wing area of 11 m^2 and mass of 450 kg. The wing has low sweep and twist angles, and also a low tapering ratio with a considerable dihedral angle. The aircraft is constructed of composite materials (carbon fiber) and has a T-tail and ordinary control surfaces (aileron, elevator and rudder). The flight envelope is mainly characterized by altitudes up to 3 km and speeds from 100 to 160 km/h TAS.
The possibility of a mutual cooperation that will result in a published paper is also welcomed.
Thank you!
Hi, I am not a specialist of this area but I have a colleague working in AIRBUS Toulouse that certainly can give you some elements. You can contact me directly and I will give you the email of this engineer.
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Hello,
I have condensed a Finite element model from 8 nodes with 6 dof at each node into 2 nodes with 6 dof at each node using a static condensation method (or Guyan reduction). Now i would like to translate this nodal data (represented in stiffness matrix) into a 3D beam element. So for that i need the beam element properties (like Moment of inertia (Iyy and Izz), Polar moment of inertia (J), Shear modulus (G), Young's Modulus (E), Cross sectional Area and length (L)) from the 12 by 12 stiffness matrix.
What makes it difficult is that the model which i have reduced is made of different material at each sides hence the model does not have a constant Young's Modulus (E) or shear modulus (G).
Can anyone help me in finding out the beam element properties using the beam element stiffness matrix? Well we all know how to do the opposite.
The 3D beam element stiffness matrix definition and Finite element model descriptions are given in the attached .doc. The finite element model shown in figure 1 is before condensation, then the model is condensed into 2 nodes and the condensed stiffness matrix is given by Table 1.
Any help will be appreciated.
Thank You,
Paul Thomas
You can try the general-purpose multiscale constitutive modeling which can take a block of material and homogenization it to a beam. The code can be freely launched in the cloud at https://cdmhub.org/resources/scstandard
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I am trying to do flutter analysis in ANSYS workbench with 2 way system coupling. However, there is a point am concerning. The problem is that, with the Transient structural module of ANSYS I will transfer only bending (flexural) data to the Fluent but flutter needs torsional deflection at the same time. How can I design my project? Thanks from now for your advices.
Regards

better go for aero elastic analysis in NASTRAN it is the easy way to find out flutter analysis.
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What is the best way to solve wing flutter problem using Finite element modeling approach with Euler-beam structural model and Theonderson's theorem of unsteady aerodynamics as the aerodynamic model?
I am trying to do flutter analysis in ANSYS workbench with 2 way system coupling. However, there is a point am concerning. The problem is that, with the Transient structural module of ANSYS I will transfer only bending (flexural) data to the Fluent but flutter needs torsional deflection at the same time. How can I design my project? Thanks from now for your suggestions.
Regards
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Dear all
i want to simulate a problem same as attached figure in Comsol, but modeling of virtual plunging/pitching spring is my concern,
could be anybody help me?
The Best
My guts feeling is that it has something to do with the flight dynamics. Not really want you to model an airfoil with spring, but just want you to determine its stiffness.
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hello everyone,
I faced problem to compile my udf as shown below,
I dont know what caused the problem in my udf codes until fluent unable to compile it.
I also try to compiled other udf and it goes smooth without any error.I think this is not the problem with compiler or fluent as I already setup the VS2012 and Fluent 16 according to system required.
It maybe something wrong in my udf codes,but I dont know to track at which line caused the error.
Can anybody help me to solve this problem.
#include "udf.h"
#include "dynamesh_tools.h"
static real v_prev = 0.0;
DEFINE_CG_MOTION(osc, dt, vel, omega, time, dtime)
{
Domain *d = Get_Domain(1);
real x_cg[3], force[3], moment[3];
real accl, dv;
real mass = 0.2;
real wn = 12.0;
real k = mass * wn*wn;
real c = 2 * mass*wn*0.00;
int i;
NV_S(vel, = , 0.0);
NV_S(omega, = , 0.0);
for (i = 0; i<3; i++)
x_cg[i] = DT_CG(dt)[i];
Compute_Force_And_Moment(d, t, x_cg, force, moment, TRUE);
force[1] += -k*x_cg[1] - c*vel[1];
accl = force[1] / mass;
dv = accl*dtime;
v_prev += dv;
vel[1] = v_prev;
printf("Computed force: %g \n", force[1]);
printf("Velocity: %g\n", vel[1]);
}
DEFINE_CG_MOTION(osc_inert, dt, vel, omega, time, dtime)
{
vel[0] = 0.0;
vel[1] = v_prev;
}
Hi Khairil,
What type of error did you get during compiling? I checked your code and compiled it in one sample, it was compiled successfully without any errors on my computer; the screenshot is attached. Maybe you need to update your compiler.
Regards,
Siamak
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Hi everyone,
I am doing aeroelasticity simulation for 2D airfoil.I'm trying to do a simulation of a wing that is both pluging and pitching.
I have created 2 types of mesh using GAMBIT sotware
1)structured mesh (consist of square grid)
2)unstructured mesh consist of triangles shape grid.
I run the simulation in fluent by compiling the UDF .c file for both type of mesh.
everything goes well for unstructured mesh.when I click "preview mesh motion" button under dynamic mesh I can see the movement of the airfoil.but the quality for this type of mesh is very low.
for structured mesh,fluent gives me an error "Update-Dynamic-Mesh failed". Negative cell volume detected" for the same model.I enable the 3 methods (layering,smoothing and remeshing) under "mesh method"
can anybody help me on this.
I experienced the same problem while moving particular zone. Problem is remeshing, remeshing happening only in triangular zone that may be fluent limitation.
So solution is
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Hello,
I am currently developing a code to study wing flutter. The structural code (which is based on the Ritz method) and the aerodynamic code (which is based on the Doublet lattice method) work well separately. I need to combine them. Does anyone know a good book or paper I could follow to help me with this part of the research?
Thank you,
Thank you for your suggestion. It looks like a good paper.
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When designing the control systems, I always confused about how to get the bandwidth of the systems. Nearly all the textbooks says like :"In the systems where the low-frequency magnitude is 0 dB on the Bode diagram, the bandwidth is measured at -3dB frequency." But almost all of the systems I meet in practice do not satisfy the conditions that the low-frequency magnitude is 0 dB on the Bode diagram. The low-frequency is always above or lower than 0 dB. What's more, the open-loop cut-off frequency on the Bode diagram is always connected with the bandwidth of close-loop. In this situations, if there is a cut-off frequency, the low-frequency will not be 0dB.
In wiki, it says the bandwidth is the range of frequency which is less than 3dB below the maximum value. While in a ppt created by Dr Grant Covic for paper: Systems & Control 1988: Document 3 of 3, it says:"Some type 0 systems do not have 0dB low frequency gains, hence the bandwidth is normally taken over the frequency range that has gain > –3dB above the designed passband gain." So more confused. So I want to ask your opinions of how to calculate the bandwidth when the low-frequency is above or lower than 0 dB? And if there are some books or articles talking about this? Thank you very much for your help.
For your maximum value (for example 34 dB or 15 dB, or -5 dB) you must define the value 0dB. From this maximum value (34dB, 15dB, -5dB) then decrease value 3 dB and you will get the value on level -3 dB, it means (31dB or 12dB or -8dB). On these values (31dB or 12dB or -8dB) then you define searched frequency bandwidth.
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Hi I am doing aeroelastic simulation on 2D airfoil.the method im trying to use is 2 way Fluid structural interaction..i have done the meshing in the ICEM-CFD.I faced problem to export out the 2D meshing into static structural but the static structural does not recognize my geometry.it shows the question mark on the geometry.
Hi Khairil, I had the similar problem in simulating 2D 2 way-FSI problem using FLUENT. Have you found out how to come up with the problem?
Regards,
Siamak
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Why is the investigation of the wing divergence consider only the torsion equation? is it true to get the divergence equation by setting all the time dependent terms to zero in the wing governing equations of motion ?
As Andrea Da Ronch has stated the divergence is a static aeroelastic phenomenon and from a mathematical point of view it is described by an eigenvalues problem (as happens for all instability phenomena).
Moreover, if you are working with a straight wing (in other word the elastic axis of the wing structure is perpendicular to the asympototic flow direction) you must refer only to the omogeneous torsional equilibrium equation, on the other hand, if you are working with a swept wing, bending and torsion equations are coupled (the coupling effect depends on the definition of the angle of attack along the wing span) ... in this case you must solve a bending-torsion eigenvalue problem.
for more details see: Aeroelasticity – R. L. Bisplinghoff, H. Ashley, R. L. Halfman, 1996
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In static bending tests, I use normal and tangential forces to get tip defle ction without converting them to lift an drag.
Is it incorrect?
If you know normal and tangential forces (to the rotor disc? To the section?) you can directly use them (be careful to express forces in the same reference of the structural equations, there are only geometric relations). But often you don't know normal and tangential forces and instead you can evaluate drag and lift with some kind of model. It is possibile to pass from lift and drag to normal and tangential, but it requires not only geometric relations but also the knowledge of inflow (induced by wind and wake) over the disc. Here you can find some useful informations:
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Hi Guys
My core research topic is Aeroelasticity in Fixed wing aircraft and I'm  in Literature review stage.As things getting rolled I also need to know the current trends and scenarios in Aeroelasticity in wings.And also it would be very useful if someone shares books on Aeroelasticity.
Introduction to Aircraft Aeroelasticity and Loads by JR Wright (AIAA) and
A Modern Course in Aeroelasticity by Earl Dowell are good books
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can anyone suggests me of how to choose a typical section from low aspect ratio trapezoidal wing (not high aspect ratio) for aeroelastic analysis and how to get the bending and torsion stiffnesses of this low aspect ratio wing?
Regarding low speed airfoils, you should refer to the NACA series for laminar flow airfoils developed in the 1940s or use one of those referred to in NACA TN 2502.
Regarding modeling of the structure, a finite element method would seem best to determine the torsional and bending stiffness.
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I am not a specialist of transonic aerodynamics and I try to understand the difference between buffet and buzz phenomenon.
1) I wonder if the buffet is the periodic variation of lift due to shock wave oscillation and buzz is the name of the aeroelastic answer of the buffet excitation. Is that right ?
2) What is the link beetween the shock wave oscillation and the boundary layer.
3 Is it possible to have buffet without boundaly layer ?
Thank you
Best regards
If you considered a spring, it will be like considering the elasticity of the structure. In that case based on your BCs an ICs, you could observe damping or diverging or self-sustained response (flutter). But as @Fulvio_Sartor said buzz is related only to ailerons vibration.
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Consider a clamped-free composite wing with a sweep angle. I have attached a file below which shows the potential energy expression. I have a problem applying the lagrange equation on the potential energy (the term which contains both bending and torsion) in order to find the stiffness matrix (I'm using Ritz Assumed Modes method: 4 modes for bending and 4 modes for torsion).
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In the case you are investigating the beam flutter subjected to aerodynamic forces, you will get to variables in the characteristics equation (critical  air speed and critical frequency), in the case of the absence of aerodynamic forces and the coupling is caused by an external force ( a follower force) what is the procedure of determining the critical force from the characteristics equation that includes two variables (the force and the frequency). It is also highly appreciated you could recommend a reference.
Could you send the characteristics equation as pdf- or png-file ?
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Is bending of wind turbine blade tips at higher wind speeds good or bad when compared to a very stiff blade, especially for small wind turbines?
Does it affect the power output? Is there a maximum ratio for tip deflection
Some concepts exist for flexible/aeroelastic blades, but I can't find any solid references.
Tilting the rotor maybe a solution to ensure that no collisions will happened due to deflection of the blade.
However, I think this is increasing the costs of design and manufacturing (not sure).
Another point, you should make CFD to determine the aerodynamics loads on the blade when rotor is tilted and un tilted then calculate the deflection of the blade due to these loads.
If you find the deflection is so dangerous then you should think of modifying the tip by inserting some webs beside the main spar of your blade.
Another option, Try to design different types of the spar I mean:
1- use box girder.
2-one shear web.
3-two shear webs
4-two shear webs with additionally web at the trailing edge.
Try and check the deflections.
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See above
Use Fluid Structure Interaction Method in ANSYS workbench.
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I'm trying to construct the 12 x 12 beam element stiffness matrix from a section constitutive matrix (6 x 6 with shear stiffnesses, axial stiffness, bending stiffnesses and torsional stiffness on the diagonal). I can get a 6 x 6 beam element matrix as would be used in a multibody formulation using the method described here:
Malcolm, David J., and Daniel L. Laird. "Modeling of blades as equivalent beams for aeroelastic analysis." ASME 2003 Wind Energy Symposium. American Society of Mechanical Engineers, 2003.
However, I want to use the 6 x 6 matrix that comes out of that process in a conventional finite element model which needs a 12 x 12 matrix. Is there a simple method of converting between the two? I've found information for a simple stiffness matrix but mine is fully populated as it's for a composite beam with lots of material couplings.
Peter, the constitutive matrix (6x6) gives you the relation between stresses and strains. Writing these strains as a function of nodal displacements implies the definition of the interpolation matrix (6x12). After replacing this discrete version of strains (discrete because it is a function of nodal displacements) in the variational equilibrium equations you get the desired 12x12 stiffness matrix. Please check this for reference: https://www.researchgate.net/publication/257583385_A_composite_beam_finite_element_for_multibody_dynamics_Application_to_large_wind_turbine_modeling?ev=prf_pub.
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I have derived the equations of motion for a system and would like to investigate its stability, the equations are inhomogeneous. The trend is to assume sinusoidal motion ( alpha = alpha0 * exp ( jwt) where w is the frequency, and the other degree of freedom the same), but in the case of nonhomogeneous equations the term exp ( jwt) will not be cancelled out and the flutter determinant can not be reached.