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
By: MikeHoulder - 8th April 2010 at 21:59
Some experimental numbers
I thought some numbers might be useful. Here is the model I am using for the initial experiments:
This is a very light and simple model aircraft with a wingspan of 120 cm and a total mass of 184 grms.
With all other joints set rigid, I concentrated on the left wing of the model. With selected spring and damping values which slowed the process down to make it visible, destructive resonance was seen. Resonance was restricted to left/right movement of the wing gradually increasing in amplitude to the point of breakage.
This occurred with a spring stiffness (Kf) of 3600 and a damping coeff (Df) of 1.06 on the Fuselage Left wing joint. The KF of 3600 might appear high; but it corresponds, I think, to a maximum structure limit of 4g on the model. With Df at 1.06, destructive resonance occurred during a flight where no control inputs were made. Increasing the Df to 1.07 eliminated resonance under these conditions. Destructive resonance continued to occur above this value if maximum rudder was held; but ceased with a Df of 1.10.
It is possible that the Df unit is 10 Newtons while the Kf unit is 1 Newton. The critical damping coefficient for Kf = 3600 is 51.55 Newtons for this model. Using a value of Df = 5.155 seems to be correct in the simulator. But this is using my existing too simple math model of the damped spring with the formula sqrt(4 . mass . Kf)
Does a critical damping coeff of 51.55 Newtons appear to be reasonably consistent with a spring stiffness of 3600 Newtons?
Anything particularly strange about these numbers?
Mike
By: MikeHoulder - 7th April 2010 at 21:15
Thanks, Tony. That was quick. I thought I could assume a simplification where the vibrations were not propagated. That they are propagated suggests that the simulator deserves respect.
My immediate objective is to get to good Kf and Df values. But there is another problem looming. Some one has mentioned the dreaded words ‘tensor analysis’ in the parametrisation of the inertia values. Know anything about that? I’m clueless.
Mike
By: Rocketeer - 7th April 2010 at 20:35
Interesting…reminds me of my university course…..one obvious thing is that vibration is always propagated between structures….a helicopter is living proof of that……the Merlin has a system to reduce transmitted vibes from the main rotor gearbox to the cabin…..sometime ago i could have helped you easily with your quest….now all that knowledge sits in books at work lol!! You will need to replace each major joint with springs in all three axes (6 springs in toto at each joint)….setting a stiffness (for each spring) could be quite difficult to do. I am sure some others here could help as I am not the only one here with an Aeronautical Engineering degree….I shall read my books!
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April 7, 2010 at 8:21 pm