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The figure shows input and output variables for both longitudinal and lateral stability transfer functions.

I completely understand what each term stands for but I was wondering what would be the best way to physically interpret the relations. I am having trouble visualizing the relationships.

For your reference:

  • $\eta$ = elevator control
  • $\epsilon$ = thrust pertubations
  • $\xi$= alieron control
  • $\zeta$ = rudder control

  • $u,v,w$ = translation velocity in $x,y,z$ axis respectively.

  • $p,q,r$ = angular rate in $x,y,z$ axis respectively.

  • $\phi,\theta, \psi$ = roll, pitch and yaw respectively

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The object of the game is find an expression for the forces, X,Y,Z; and the moments L,M,N on the airframe expressed as a function of the state vector {x,y,z, phi,theta,psi, u,v,w, p,q,r} and the control vector {aileron,elevator,rudder,flap,thrust}. This is done by first picking a steady trim state and linearizing the dynamic system about this trim state.

The dotted line in the block suggests that a common separation of variables scheme is being used that simplifies the influence of the controls on the forces and moments. For this scheme to be valid, you need geometric symmetry, aerodynamic symmetry (v,p,r,phi = 0), and negligible onboard angular momentum (engines). Then you can construct a pair of functions where the rate of change of {u,w,q,theta} depends only on {u,w,q,theta} and {thrust,flap,elevator}(called the longitudinal dynamics subset); the rate of change of {v,p,r} depends only on {v,p,r,phi} and {aileron,rudder}; {phi} only on {v,p,r,phi} (together, the lateral dynamics subset); and the position update (x,y,z,psi) depends on all eight prior terms but not the control vector (the navigation subset).

See Flight Vehicle Aerodynamics by Mark Drela, chapter 9. I lifted this info from that book. It also lists expressions for all the elements in the matrices. Those expressions (there is a whole raft of them) is where the physical interpretation comes from in terms of plugging in the vehicle characteristics and seeing how things shake out.

For instance, the lateral dynamics subset has a total of 24 elements. 7 are zero, one is purely geometrical, and one is a constant, leaving 15 that are dependent on the vehicle properties.

See also Dynamics of flight - Aircraft Stability and Control by Etkin

This PDF from Stanford by Roberto Bunge appears to have the same development, but I haven't been through it all the way. Stanford Aircraft Flight Dynamics PDF

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