Aerodynamic forces, vibration and resonance

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M

MikeHoulder

Airman
15
1
Jul 24, 2009
Apologies, the only way I know to talk about this is with maths. The objective is to create a reasonably accurate simulation model of my Avro Lancaster. Perhaps for obvious reasons, there is no documentation given with the simulator of these issues. Hence there is the need to create a good mathematical model of the simulation.

First, all joints between components are modelled by point location damped springs. I need to estimate the axial and torque forces on those springs
All airframes have structural limits, say +4g, -2g. Can I use these and the masses of both the whole airframe and the individual components to calculate a reasonable spring strength for a given lateral or angular displacement using Hooke's law. How do I divide the total load under 4g, say, between the various surfaces: wings, tailplanes, fins? Help!!!

Second, damping. I thought initially I could model the spring system as a simple unforced damped spring with 1 degree of freedom; i.e. with the 2nd order differential equation
m.u''(t) + Df.u'(t) + Kf.u(t) = 0 where m is mass, Df is damping coeff, Kf is spring strength coeff, t is time and u is a function giving displacement at time t. It is simple to calculate the critical damping coefficient which is the lowest value eliminating oscillations: Df = sqrt(4.m.Kf).

Doing some experimentation, I found that, at significantly lower values than the critical coefficient for Df, destructive resonant oscillation occurs in the simulator. This could be an artifact caused by arithmetic rounding or discrete state calculation of 4000 times per sec in the simulator. It could also be that the equation is lacking one or more terms. It may need to be:
m.u''(t) + Df.u'(t) + Kf.u(t) = F(t). Help!!!

I had assumed that vibration was not propagated from one component to another. Hence a system with 1 degree of freedom. Unfortunately, vibration is propagated throughout. Hence the system is of multiple degrees of freedom. Help!!!

So I am now looking very grimly at a can of worms. Can anyone, please, sort me out.
Mike
 

Good luck. You are now starting where NASA began in 1963 in the very new world of computerized structural and vibration analysis, joined by increasing sophistication as AeroElastic effects were considered after the Comet disasters.

Today, at places like Lockheed and Northrup, the technology stack to model a.) composites, b.) complete and accurate potential flow aerodynamic models for applied loads in all flight regimes (and stagnation temperatures), to a structural modelling capability 'under the external skin', to a pretty accurate vibration/natural frequency model, to detailed structural anaysis - even including plastic range deformation under 'ultimate loading' - is state of the art.

Your Lancaster airframe is quite simple 'theoretically' because of the design techniques and materials and flight regimes were much simpler - but still very compex when attempting to arrive at natural frequencies and resonance/divergence conditions under both steady state (engine, areo flow) characteristics, no real thermal thresholds and nowhere near transonic flow.

But I do not have much confidence that linear differential model of spring/damper will yield good results.
 
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Thank you very much indeed, drgondog, for an extensive and considered reply. I would dearly love to take a full course in aeronautical engineering; but at my age there just isn't time. I think my problem is somewhat more tractable. I'm not modelling reality at the level of sophistication and complexity you discuss.

The simulator I am using is a 'black box'. I have no documentation on the internal algorithms used. My knowledge of the simulator is confined to experiment and a substantial list of input parameters which again are not documented. I have been assured by one of the technical support team for the simulator that component joints are modelled by point location damped springs. There is an extensive network of users of the software; but, in general, without the ability to solve technical problems at this level of complexity.

The Avro Lancaster model is being designed for use within this simulator. The model should exploit all the functions of the simulator to give the most realistic flight portrayal possible; but at the same time it is limited to those functions and the computer hardware resources in normal use.

If only I had the years to follow the path you outline, drgondog, but I haven't. My objective is very much lower. I would like to create a reasonably accurate maths model of the simulator, not the Lancaster, which would enable me to calculate good values for the simulator input parameters of the model. For my calculations I would use a limited number of values from the real aircraft. I don't know the history of the simulation engine at the heart of the simulator; but, evidently, contact with the originators is either lost or too restricted to obtain a maths model from them. Current practice for the simulator input parameters is gross approximation, copying values from broadly similar models. The specific problem I have needing calculated values is that my Lancaster model is way outside existing user experience, right on the limits of the simulator.

What we have in the simulator is a small number of components; joints between which can be described by both axial and torsional spring stiffnesses and damping factors. In addition the inertial properties of a component can be described in a manner I do not yet understand - something involving tensors I am told. Depending on the component, some aerodynamic properties can also be defined.

There is an immediate source of vibration probably calculated from engine rpm and propeller characteristics which is propagated throughout the model. I am confident some theoretical model of a damped spring system is applicable; but which?

For interest's sake, here is a graph with two curves. The 'critical Df', critical damping factor, curve gives values calculated with a maths model of a free damped spring with one degree of freedom from the corresponding spring stiffness value. The 'minimum Df' curve gives experimentally derived values for the damping factor below which destructive resonance occurs.



For Df values at or above the critical Df curve corresponding to the spring stiffness chosen, no oscillation occurs (by assumption - I can't test this). For Df values below the minimum Df curve, destructive resonance always occurs. I presume that for Df values between the curves, oscillation occurs but without destructive resonance.

My objective: first, to obtain or create a maths model of the simulator, and then, to obtain data values from the real aircraft to satisfy the maths model.

Many thanks again and my best wishes, drgondog
Mike
 
Mike - Good luck..

I might note that if applied Tensor analysis is performed we are dealing with a higher order representation than magnitude and direction - probably second order Tensor in which three dimensional representations may be expressed and decomposed with matrices to yield the moments, forces and magnitudes on a volume. Frequently used in Performance Anaysis (combined with Stabilty and Control parameters and aerodynamic loads) for all the obvious difficulties in predicting moments about each axis of an elastic ariframe as well as prediction of inertia coupling.

A long winded comment saying "i believe' that a system of springs and dampers coupled with interia properties are the substance of your questions.

I must confess this is stage that I reached in my math disciplines along with applied chaos theory for complex boundary layer separation analysis - but proceeded no farther.

Ok. To model any airframe with these disciplines implies some physical data anlong with force inputs. I would think that the simplest form or model woul be to consider the wing as a coupled beam with several nodes - and each member between the nodes would be a 'stiff dampened spring'. Ditto the fuselage. At each node you have to consider the tension and compression loads along the span as the wing deflects, as well as torsion resistance to wing twist deflections at each joint if the model is 3-D.

The 'joints' have the complex torsion (moment about Y axis) Tensor qualities while the 'members' have axial force vecors along the Y axis. The applied loads in the model would seem to be best modelled as point loads in positive Z axis at each joint. To get inertial responses you must add mass to the rod attributes - then from the applied joint loads (to simulate (poorly) the distributed lift loads - spanwise and chordwise) the Differential equations must be solved to a. balance the forces and moments and b.) to yield the deflections from real life data along the span.

Here are the 'issues' - Do we have a set of spanwise deflection data in plus and minus 'z' for steady state (at rest) and while in steady flight (a deflection curve with data points for each span wise member) as the wing tips translate vertically. Do we have any feel for torsional deflection about the wing spar as a function of span?

I have ZERO idea how any valid assumptions may be made for the torsional spring qualities about the 'Y" axis for the wing in twist - certainly much stiffer at the root than the tip - but how to achieve the data?

Wish I could offer more insight to the problem of developing a high order approximation of a real airframe using dampened springs but I don't have anything to offer except the real issues that immediately come to mind.
 
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Some simple questions;

What is the purpose of the model/simulation? It almost sounds like you're trying to mate a dynamic aircraft model to a pilot to that the aircraft structure will move during manoeuvres. How good do the results need to be?
 
Just a quick one. I'll reply at length in a couple of days or so.
But in relation to tensor analysis, drgondog, do you recognise any of these parameters?

From the new Aerofly 5
<[tmvector3d][InertiaLength][0.270850 0.956110 0.025000]>
<[tmmatrix3d][Inertia][0.030492 0.000000 0.000000 0.000000 0.002466 0.000000 0.000000 0.000000 0.032917]>

From the earler release of the simulator, AFPD:
Inertia = { 60.6669, 0.0000, -0.0001, 0.0000, 0.0000, 7.2266, -0.0000, 0.0000, -0.0001, -0.0000, 66.0247, 0.0000, 0.0000, 0.0000, 0.0000, 14978.6143 }
Kn = 1000
Dn = 10
Dv = 0.2

Many thanks to both of you
Mike
 
No to the actual values. The first set seems to imply an operation to derive the 3D vector in X, Y and Z applied to the Inertial reference, for example. The more I think about the terminology, this first set may be all about spring constants - but I would still have to see the paper.

Candidly I would have to look at the entire model starting with the assumptions, naming conventions and the applied Tensor operations before extractions from the programming.

I have no idea what the 15 value set of data should represent unless a 3x3 and a 2x3 matrix operation is implied.

Michael - to get immersed in this I am going to have to spend way too much time wrapping my head around 40 year old math/engineering reference books - which I do not have time for.
 
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Sorry for the delay in replying, Red Admiral. I've had the builders in with the consequent total paralysis.

First, the simulator is a relatively sophisticated one; but it is aimed at the hobby market. It's primary function is to teach radio control flying from scratch and to allow the planning and practice of aerobatic competition manoeuvres. There are a number of subtle simulation behaviours. But the short answer to your final question "How good do the results need to be?" is that I don't know.

Many models have been created for this simulator. But in general the models are sufficiently similar to allow trial and error methods. The normal approach is to copy the data from a similar model and make a few blind adjustments until reasonable performance is achieved. The normal model has a wingspan of 1.5 metres and a mass of 3 to 5 kg.

I have previously built a model developed from a nice FSX model of an A10-TL2 (with permission) at half scale. That is: a wingspan of 8.76 metres and a mass of 8,000 kg. The procedure of scaling up data values from an existing model failed miserably. For each one of the components of the model, there are at least 124 scalar parameters. Now consider the Lancaster to be modelled at full scale (hopefully): with a wingspan of 31 metres and a maximum mass of 29,000 kg. No way will I get into the right ballpark by trial and error.

Again, the simulation is intended to run on a good, perhaps souped-up, PC computer. But there are limitations on processing power. At some point increasing data numbers will break or slow down the simulator. So I have to aim at the minimum data numbers to give respectable flight characteristics. If the minimum data numbers are too big for the simulator, then all I can do is reduce the scale. I have a 1 to 2.48 scale Lancaster model flying (very badly). So presumeably I will be able to have a model at somewhere between 1 to 1 scale and 1 to 2.48 scale.

Trial and error won't work for setting those 124 parameters for each model component.. The best solution would be full documentation of each parameter from the authors of the simulator. But for one reason or another that solution is not possible. All I can do is create an abstract mathematical model of the simulator with the various pieces of information I have been able to gather together.

That's the important point. My first job is to simulate the simulator. After that I can plug in suitable values for the model to set the required parameters.

I hope that's an adequate answer.
Best wishes
Mike
 
As I understand it, you're trying to accurately model an aircraft in flight? Either simply flying in a straight line or doing some manoeuvres. The structural issues are secondary; they're just part of the simulation?

I'd simply say that you can ignore the structural issues if you're just trying to replicate flight dynamics. Its a level of complication that isn't needed. I use two models for aircraft simulation, one is 2 dof for level flight and instantaneous stuff, the other is pseudo 6 dof for dynamic performance. Neither comes close to structural analysis besides setting g limits and making sure they're not exceeded. Pull together some cl/cd/alpha arrays and an engine deck and you're well on the way to getting a workable 6 dof model. Dynamic responses to pitch/roll/yaw are more second order and can be tweaked to give reasonable results.

Overall it seems like a hellishly complicated way of going about doing things when you consider the paucity of the aero data you're going to be using.

Hope I've understood the question.
 
For AFPD and the new Aerofly 5, each component has at least 124 data values; some of these being combined as vectors. My A10-TL2 model has 21 components. The two main issues determined by some of these data values are: the performance of joints between components and the inertial properties of each component.

Within this simulator, the joints are modelled as damped springs. Thus the structure of the model can have some flexibility and relative movement between components under both static and dynamic loads. 'can have' as it is possible to lock the joints and disable the spring action. But in this case, the model takes on a toy-like appearance in simulation. If the springs of a joint are unlocked, then the spring strengths and damping coefficients become vital. If these values are set incorrectly, the joint explodes and the aircraft disintegrates with considerable violence at some point in the simulation. This is caused by resonance on the spring action. For reasons outlined in my previous post, I need a maths model of the simulator action to be able to set the spring strength and damping coefficients to avoid resonance but retain some flexibility in the joints I wish to be unlocked.

I am beginning to believe that the correct setting of the inertial properties of individual components is even more important for realistic flight simulation. But at this time, this seems a more difficult problem than that of the joints. My A10-TL2 model stays together under almost all conditions now; but its flight characteristics are very poor. The problem is best described as extreme sensitivity to control inputs leading to an immediate high-speed stall. The near-impossibility of recovery from such a stall, even with what should be sufficient height, leads me to think I have cocked up the inertial properties of the major components.


The point is that there are two very real issues in creating a model for this simulator. Solving them produces what I believe is a substantially better simulation. I regret deeply the lack of detailed documentation of the simulator. But I feel the only option I have is this gruesome and painful task of creating my own mathematical model of the simulator.
Mike
 
There is some repetition here. I apologise. But perhaps it is clearer and I need to set the scene for my problem which is told at the bottom.

I believe that AFPD/Aerofly is a serious simulator with predictive ability. That is, it has enough physics and engineering theory built into the software to be able to model a significant amount of real-life aerodynamic and structural properties. For instance, it may be able to answer questions to some extent such as:
Will the wings of a an r/c model design remain attached under high g manoeouvres?
Could structural failure occur when an Avro Lancaster performed an extreme 'corkscrew'
manoeouvre to avoid attack by a night fighter?

Of course, Aerofly does not cost 500,000 euros and does not have several hundred support engineers. But it provides the possibility at least to enter this area of engineering at a low cost and with limited resources.

My project is to build a full-scale simulator model of the Avro Lancaster which has a very high level of detail and which can start to examine aerodynamic and structural characteristics.

Any AFPD/Aerofy model consists of a limited number of components such as wings, tailplanes, chassis, wheels etc. One critical issue is that of defining the joints between components. These can be set as rigid. But for my purposes the joints should use the optional representation supplied within AFPD/Aerofly as damped springs using spring stiffness (K) and damping coefficient values (D). Anyone who has built an AFPD model knows the consequence of incorrect values for these two parameters: exploding joints and violent disintegration of the model. The usual solution to this problem is to copy values from an existing similar model. In my case, there are no existing similar models. My model has a wingspan of 31.09 metres and a weight of 29,000 kg when existing models normally range around 1.5 metres wingspan and weights of 5 kg.

My immediate objective is to derive as much as possible values for these parameters from the real aircraft. This requires the creation of a mathematical model of a damped spring system which corresponds to the actual software of AFPD/Aerofly. Access to the software code itself is not possible and for good reasons there is no documentation. So the mathematical model I create cannot be exactly the same as the maths of the software. The task is to simulate the simulator itself.

AFPD/Aerofly itself is not a professional level simulator. With limited computational resources it cannot calculate all the forces acting and their interactions. So I assume each joint is simulated by a damped spring system with one degree of freedom. There is some evidence that AFPD/Aerofly calculates forces, both linear and rotational, velocity and position of the joint in one unified maths operation about all three axes using Tensor Analysis. While I might be forced very reluctantly to study Tensor Analysis of which I now know nothing, my plan for now is to analyse all these cases independently; that is: linear and rotational forces about each axis separately, linear separately from rotational.

Consider the joint between the fuselage and wing using only linear forces in the vertical axis. Without external forces other than gravity, the spring of the joint is stretched downwards under the weight of the wing and is at rest; the amount of displacement of the spring and the wing together is determined by the stiffness of the spring. If the wing now is displaced up or down by an external momentary force, the spring will oscillate until it returns to rest under the action of the damper.



There are only three forces acting here: the linear inertia of the mass of the wing, the stiffness of the spring against the momentary displacement of the wing (K) and the damping force against the movement of the spring (D).

The displacement of the wing as it oscillates up and down during the following time can be described by the following equation:
M . u''(t) + D . u'(t) + K . u(t) = 0
where M is the mass of the wing, D is the damping coefficient and K is the spring stiffness. The function u(t) gives the vertical displacement of the wing against time. This is a second order homogeneous differential equation. The term u''(t), displacement/distance differentiated twice, gives acceleration. The term u'(t), displacement/distance differentiated once, gives velocity. For solution methods, see Paul Dawkins' Online Maths Notes at Pauls Online Math Notes. The equation has value 0 as no external force is applied. The energy of the system is that given by the initial momentary displacement and after that swapped between the spring and the wing until it is exhausted by the damper.

With simulation, there must be continuous forces external to the component joint unless the both the model and its engine are stationary. There are aerodynamic forces on the wing and vibration from the engine which 'force' the behaviour of the spring/damper system. At this stage of the work, aerodynamic forces are ignored. The only external force to be considered is that caused by the vibration of the engine. The engine vibration is a periodic function against time. And in this case the equation becomes:
M . u''(t) + D . u'(t) + K . u(t) = X . cos(w . t)
where X is an unknown constant derived from the engine vibration whose value will be found approximately by experiment and where w (omega) is the frequency of the vibration. This is now a non-homogeneous second order differential equation. Solution methods are given in the above reference.

Now assume there is no damping and the vibration frequency is at or close to the natural frequency of the spring (= sqrt( K / M) ). In this case the middle term D . u'(t) disappears from the equation. But now the solution to this equation includes a term t . sin(w . t). The solution is an expression giving the value of u(t).As t, the time, increases, the value of u(t) also increases and doesn't stop increasing with time until the forces caused by the increasing displacement break the spring explosively and the Lancaster disintegrates. This is destructive resonance.



Now we come explicitly to my problem.

This is a problem of my lack of knowledge. If there is damping, the middle term above always exists in the equation and, consequently, I have been unable to set an equation with damping where the solution contains that dangerous term t . sin(w . t). Hence there is no resonance. Mathematically, it seems that the use of damping avoids resonance altogether. We have this situation.



After 4 seconds, the trace shows the engine vibration which remains constant.

But in the simulator, engine vibration causes destructive resonance even if there is damping of, perhaps, a low value. I can see it happening.

My maths model no longer matches the simulator. Unless I receive some help or am lucky with my searches in Internet, I have hit a brick wall.

Mike (very hopefully)
 
Mike - in 'real life' the wing has a natural frequency and specific forces act on that wing - both steady state and transient.

The 'steady state' loads are the Normal aerodynamic loads imposed on the wing due to stable and un accelerated flight, as well as the In PLane engine thrust.

The transient forces include asymmetrical Loads imposed in manuevers and are complex in that there are various inputs from rudder, ailerons, elevator which of themselves are not equal or in equilbrium - causing various accelerations - which in my opinion is driving the 'Spring/Damper' solution approach you are experimenting with.

Additionally - there are various harmonic inpiuts to your semi rigid structure (elastic) including wake turbulence aft of boundary layer separation which is definitely a non linear input, as well as the vortex behind each engine as well as the vibration of the engines - all on cantilever structure. This latter input is one other area of investigation in which a spring damper solution may be useful.

Having said all this, in my opinion trying to model a RC aircraft to predict structural integrity is hitting a nail with a hammer - and I personally have serious doubts regarding the approach for two reasons.

One you may spend the rest of your like trying to model a system with non linear (and largely unpredictible or definable) 'connections' as well as non linear (and unpredictible without very sophisticated modelling and knowledge of the 'real' variables and data) to address the non stable flight environment (like a cork screw)..

The second reason is thatan R/C model should be very straight forward to model with saftey in mind with classic knowledge of airframe structures and Strength of materials (assuming balsa as your primary structural material).

In my opinion studying Tensors is only useful in this experience to attempt to reverse engineer the model you have presented above - You sure don't need tensor analysis to analyze the airframe structure of either the Lancaster RC model or the real thing... but it would be useful if you decided to model a Stability/Flight Dynamics model.

Just my thoughts..
 
Drgondog, I'm delaying somewhat my analysis and reply to your excellent assistance.
The support team of the simulator have published a collection of examples to illustrate the functioning of the simulator in regard to component joints and harmonic vibration. By trying to match the theory to the simulator functioning I will understand the theory rather better.
See my post - it's quite lengthy now with all the contributions - at
AeroFly 5
titled "Calculating TMD Joint Values in both AFPD and Aerofly 5".

My post and the replies are in English - thank God, my bad Spanish is the limit for me - but the majority of other posts in the forum are in German.

I enclose a copy of a couple of the replies and documentation of the 10 experiments offered by Marc Borchers. This documentation is hidden within the .TMC files of Marc's zip file - at the end of the replies so far. It is displayed within the simulator; but I have extracted it to show you what is going on.

I would offer to capture videos for you of the simulator operation with these experiments; but my attempts so far have been very poor quality.

My very best wishes to you
Mike
 

Attachments

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Mike - Marc Bourcher's explanations were thorough, concise and correct.

I admit being puzzled by this selection of a modelling approach to determine the structural integrity of the wing/body combination for your model Lancaster. The spring/damper linear equation approach could be useful to predict aeroelastic responses to applied loads from a vibration mode point of view - but I am skeptical that it will be useful in this particular requirement to 'model a corkscrew mauever' or achieve realistic deflections and/or stresses.

It still seems that the actual applied loads can be determined relatively easily 'by hand' and you could then proceed to analyze the stresses to be applied to a balsa (or other material) spar.

The wing would be treated as a beam with a distributed (start with semi elliptical lift loading profile) aero loading. Conservatively you would then look at the spar at the fuselage joint to size the required area properties of the spar based on your design (i.e. box, H).
 
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