Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
(OP)
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 (y) 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.
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.





RE: Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
RE: Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
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.
RE: Nonlinear Model of Landing Gear Freeplay using Runge-Kutta in Mathcad
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
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?