Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad

[Angelfire888]

I am trying to model the affect of free play in a landing gear assembly. However, being unfamilar with Mathcad's differential equation solvers, I am unsure how to get what I want, which is to see the affect of the free play angle on the yaw angle (Y) and lateral tire deformation oscillations. In essence, I am not sure how to handle the differential equations I've obtained through the Mathcad program in order to recreate the baseline model of these oscillations.

Any suggestions on how I could setup the Mathcad program to solve for a 3-equation system that models the landing gear?

The dynamic equations that were derived are:

1) Y'(t) = Y'(t)
2) Y'(t) = c4*Y'(t) + v*Y(t) + c5*y(t)
3) Y''(t) = c2*Y'(t) + c1*Y(t) + c3*y(t)

where c1 = M(Y)/Iz
c2 = k/Iz + K/(v*Iz)
c3 = ((cMα-(e*cFα))*Fz)/(Iz*σ)
c4 = e - a
c5 = -v/σ

Parameter Description Value Unit
v velocity 0…80 m/s
a half contact length 0.1 m
e caster length 0.1 m
Iz moment of inertia 1 kg m2
Fz vertical force 9000 N
c torsional spring rate -100000 Nm/rad
cFα side force derivative 20 1/rad
cMα moment derivative -2 m/rad
k torsional damping constant 0…-50 Nm/rad/s
κ tread width moment constant -270 Nm2/rad
σ = 3a relaxation length 0.3 m

and M(Y) = |(-c*(Y - Yfp)) if Y >= Yfp
|0 if -Yfp <= Y <= Yfp
|(-c*(Y + Yfp)) if Y <= -Yfp

where Yfp is the free play angle (in degrees)

I've attached the following Mathcad file with what I have so far. I am unsure on how to progress. Any help would be much appreciated.

 

[Angelfire888]

Didn't get a chance to finish this thread before my computer froze... but I assume I need to use the 4th order Runge Kutta method?

 

[GregLocock]

I'm not saying you can't do it, but that certainly strikes me as a pretty awesome equation to solve in an analytical fashion. As soon as you include a contact problem then even a numerical solution will bog down.





Cheers

Greg Locock


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[Angelfire888]

I figured I would have to make a lot of assumptions and simplifications since dynamic analysis of shimmy is a real pain due to multiple DOFs. Do you have any ideas how I can tackle these equations though? It's a rheonomic system of differential equations but I'm unfamiliar with how to solve these types of systems, especially (if even possible) in Mathcad. Should I instead look at the effect of free play has on stability as an easier, dirtier way? I'm just trying to determine acceptable maximum allowable freeplay limits of the gear without having to actually go out and test the gear myself.

If I do look at the stability aspect, I assume this (below matrix) is a linearized model of three ordinary differential equations of first order (with state variables as (Y, Y', y) all dependent on time (hence rheonomic condition).

|Y' | = |0 1 0| |Y |
|Y''| |c1 c2 c3| |Y'|
|y' | |v c4 c5| |y |

The characteristic equation is then

λ^3 - (c2 - c5)*λ + (c2*c5 - c1 - c3*c4)*λ + (c1*c5 - v*c3)

Using Routh-Hurwitz criterion, the stability boundaries of the linear model are found using third order polynomial criterion.

Thus, landing gear model is stable when:

-(c2 + c5) > 0
c2*c5 - c1 - c3*c4 > 0
c1*c5 - v*c3 > 0
-(c2 + c5)*(c2*c5 - c1 - c3*c4) > c1*c5 - v*c3

I think I should then take these stability boundaries and see the effect of free play angle (Yfp) on them for different values of velocity (v) and yaw angle (Y) (I will ignore lateral deformation change and assume it is constant to keep things simpler). However, I'm not sure how I should plot and read these stability plots in Mathcad to determine what is the max allowable free play in a gear.

 

[GregLocock]

Well, I build multi body dynamics models for that sort of thing, and for various reasons mathcad would not be the top of my list for tools to use. I would consider using a time based approach, using a fixed, small, time interval, probably of the order of 10^-4 seconds or less. This could be done in mathcad, or excel or preferably matlab.

Obviously one run of the model proves little, you'll need to explore interrelationships between factors. For that I'd design the whole system around a Monte Carlo approach.

Incidentally lash type problems solve much more quickly if you replace the free play plus contact system by a non linear spring with a very low rate in the lash region, and some reasonably graceful polynomial curvature in the contact region, 3rd or 5th order seems popular.

Incidentally so far as quick and dirty goes, yes I'm with you there.




Cheers

Greg Locock


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