# How to calculate the variation of Z force with elevator deflection (Zδe) in Nelson for pitching dynamics?

I have been trying to model the pitching dynamics of a General Aircraft (Nelson, "Flight Stability and Control"). I need to use the Zδe which uses CZδe parameter in the control matrix B, but Nelson provides no method to find it. The state space equations are:

There are a couple ways you can estimate control derivatives ($$C_{L_{\delta e}}$$ is a control derivative). You can then transform the coefficient from stability-axis to body-axis for use in the state space equations of motion.

Elevator is basically a full-span or part-span plain flap. Knowing that, you have several tools at your disposal:

1. USAF DATCOM: It contains semi-analytical equations to estimate 2D flap lift increment, and corrections to 3D with aspect ratio and taper ratio. Not the easiest to read resource, but it is free and forms the backbone of many conceptual design software and books.

2. ESDU: Items 70011 and 74011 contains semi-empirical relationships for estimating full-span 3D flap control derivative. ESDU contains step-by-step examples and is the authoritative source for all things aerospace in conceptual design and validation. Unfortunately, it isn't free.

Before going further, I should mention that tail experiences downwash from the wing. So converting from the tail stability-axis to aircraft body-axis requires converting the total tail incidence angle ($$\alpha_t = \alpha+i_t-\epsilon$$). Of course, you'll need to find downwash itself, which would be the topic of another question.

1. Vortex-Lattice Method: AVL is a reliable and free VLM, and is the go-to tool for conceptual low subsonic aerodynamics. Good thing about it is you don't necessarily need to estimate downwash; it directly outputs the control derivatives in aircraft stability-axis.

For conventional layout, all three tools should give you similar answers. Otherwise, you may not know the true value until you have access to wind-tunnel.

The variation of Z-force with elevator deflection is simply the component in Z-direction of the lift force developed by the elevator. It is a function of:

• Local Angle of Attack.
• Elevator deflection angle.
• Trim tab or stabiliser deflection.
• Elevator surface area.
• Local dynamic pressure at the elevator.