Lateral Modes of an Aircraft

Model, analyze and describe the different modes of an aircraft's lateral dynamics

Define a state space model based on a linear model of the lateral dynamics of an aircraft:

In[1]:=

latDynamics=StateSpaceModel



w0+



,-1+

u0

gCos[θ0]

,0,

u0(

)

,0,0,

u0(

)

,0,0,{0,1,Tan[θ0],0,0},{0,0,Sec[θ0],0,0},

u0

,0,

xdr

zdr

xda

zda

,

xdr

zdr

xda

zda

,{0,0},{0,0},

labels

/.

pars

Out[1]=

						rudder	aileron
side slip	-0.00244895	0.	-1.	0.0369392	0	0.000103322	0
roll rate	-4.08858	-0.760711	0.254886	0	0	0.0494535	-0.174933
yaw rate	0.616827	-0.149414	-0.19905	0	0	-0.138089	0.0167267
roll angle	0	1	0	0	0	0	0
yaw angle	0	0	1	0	0	0	0

Its eigenvalues and eigenvectors:

In[2]:=

esys=Eigensystem[Normal[latDynamics]〚1〛]//Chop

Out[2]=

{{-1.16249,0.110116+1.01675,0.110116-1.01675,-0.0199486,0},{{0.0781402,0.752834,0.0667242,-0.647603,-0.0573975},{-0.150927-0.156862,0.667662,-0.139903+0.147137,0.0702939-0.649051,0.128306+0.151494},{-0.150927+0.156862,0.667662,-0.139903-0.147137,0.0702939+0.649051,0.128306-0.151494},{-0.00280804,0.00945661,-0.0175601,-0.474049,0.880268},{0,0,0,0,1.}}}

The modes plotted on the complex plane:

In[3]:=

ComplexListPlotMapThread[Callout,{esys〚1,1;;4〛,{"Roll convergence","Dutch roll 1","Dutch roll 2","Spiral"}}],



Out[3]=

The Dutch roll is an oscillatory mode with a lot of rolling and yawing. Create initial conditions which generate the Dutch roll:

In[4]:=

ics=With[{c=1+},cesys〚2,2〛+Conjugate[c]esys〚2,3〛]//Chop

Out[4]=

{0.0118691,1.33532,-0.574079,1.43869,-0.0463772}

Simulate 20 seconds of the response:

In[5]:=

InputOutputResponse[{latDynamics,ics},0,{t,0,20},"StateResponse"];Plot%,{t,0,20},PlotRangeAll,PlotLegends

labels



Out[5]=

The roll convergence mode is a stable and almost pure rolling motion:

In[6]:=

InputOutputResponse[{latDynamics,esys〚2,1〛},0,{t,0,20},"StateResponse"];Plot%,{t,0,20},PlotRangeAll,PlotLegends

labels



Out[6]=

The spiral mode is mostly a yawing motion with some roll and a very long time constant:

In[7]:=

InputOutputResponse[{latDynamics,esys〚2,4〛},0,{t,0,20},"StateResponse"];Plot%,{t,0,20},PlotRangeAll,PlotLegends

labels



Out[7]=

External Links

B. Etkin and L. D. Reid, Dynamics of flight: Stability and Control. John Wiley & Sons, 1995.

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Contributed by: Wolfram Controls Team

Lateral Modes of an Aircraft

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