# How to start deriving longitudinal equations of motion for an aircraft?

I hope this is a relevant place for me to ask a math question regarding aircraft design.

I am trying to understand how one would implement a controller to control the pitch angle of an airplane for a small exercise. I understand the control part and its implementation. What I do not grasp is how one acquires the longitudinal equations of motions (which are then used for the control part) which serves as the starting point. What is the starting point or what are the principles used to derive these equations? If I know how to derive these equations for a very simple case, then I know I have to linearize the equations and then apply control theory to it.

For example, how are the left and right hand sides of eq. 4.70 from pp. 164 of the following book book is derived?

I will appreciate a simple explanation of the above case.

Edit:

1. I am attaching two screen shots of two sets of equations from two sources. Links to the books are included below. Both sources state these are longitudinal equations of motion although their general form differ from each other.
2. I think I got to understand one point: these equations were derived considering translation motion on the x and z planes and rotation about the y axis (so stated in the first book) Thereafter, I don't understand the procedure.

1st set of equations from book 1: second set of equations from source 2:

book1: pg. 164 of Morris, Introduction to Aircraft Flight Mechanics: Performance, Static Stability

source 2: pg 3 of this online note

• Not sure, but this document on NASA's website may help. At least they explain their units. – Lnafziger Mar 13 '14 at 3:28
• Please include the equations in your question. I can probably help you but since I don't have the book there is little to start from. – DeltaLima Mar 13 '14 at 7:27
• @user1420 the two things which come to mind are the longitudinal stability formula and the first derivative of longitudinal stability formula, is it one of these? – Thunderstrike Mar 13 '14 at 8:05
• It appears that I'm not able to view the equation from the book link that you've given. Could you provide an image or type it out? – Qantas 94 Heavy Mar 13 '14 at 9:24
• I don't understand why this has a close vote on it, it is quite clearly on topic. – Jay Carr Apr 15 '16 at 4:07

Part 1 (Unfortunately I'm only familiar with #1 and #3 at the moment, not #2)

(Footnote: This might be a bit simpler than your case, but hopefully it you'll be able to fill in the remaining gaps)

From this, you can some the forces up according the direction of the velocity or the lift vector. Doing this horizontally, you get equation 1, and likewise for vertical direction your equation 3.

To make this simpler to handle, we use small angle approximation consider $\cos(0)=1$ and $\sin(0)=0$. This simplifies down to:

$$V:0=T-D$$

$$L:0=mg-L$$

(i.e. thrust equal to drag, lift equal to weight)

Part 2:

This is basically the equations of a kinetic diagram of the Free Body Diagram above, where there can be a change either in airspeed of altitude. What your second equation says is that excess thrust (T-D, Thrust-Drag) can:

• be used to increase altitude: $m\times{g}\times\sin(y)$

and/ or

• be used to increase airspeed: $m\times{v}$

Depends on what you mean by "derive the equations". If you really mean that you want to work your way up to that formulation starting from the basics, well you start from good old Newton:

$$\overrightarrow{F}=m\overrightarrow{A}$$

and the equivalent for moments ($\overrightarrow{}$ indicates vectors).

At this point you need a reference system in which to decompose your vectorial equations (body-fixed, earth-fixed, stability axes: the choice influences which terms you'll be able to simplify later) and a description of your system: which forces are applied on the aircraft? how can I describe them as functions of the aircraft state?

At this point you substitute in your original equation and carry on the computation.

You can consult these lecture slides to see step-by-step how it can be done. Lecture 8 and 9 for the general 6DoF case and lecture 11 for your particular question.