# aircraft equations of motion

The forces and So the equation for linear motion, Equ.6, becomes. Equation(1.2-6)issometimescalledtherotation formula;itshowsthat,afterchoos- ing n and µ ,wecanoperateon u withdotandcross-productoperationstogetthe desiredrotation. arise from thrust and aerodynamic loads. On this slide, we will consider only the 8 results in the following three scalar equations for the linear motion of the aircraft. As these aerodynamic angles simply relate the body axes to the wind axes then the direction cosines can be used to obtain the relationship between these two axes systems. The rigid body equations of motion for a symmetric aircraft can be summarised in a body axes reference system as follows: The origin of the body axis system is the centre of mass (gravity) of the aircraft. 18 can be converted into 3 scalar equations. }\$, \$β↖{.

This is a preview of subscription content, A Mathematical Perspective on Flight Dynamics and Control, School of Aerospace and Mechanical Engineering, https://doi.org/10.1007/978-3-319-47467-0_2, SpringerBriefs in Applied Sciences and Technology. To solve the actual equations of motion for an aircraft, we must use calculus and integrate the equations of motion. 61 only applies to still air conditions. + The President's Management Agenda constant during cruise since the only loss is for the fuel which is consumed. that the aircraft is a rigid body, so \${dr↖{→}}/{dt}=0\$. the body axes.
and where V is the true airspeed of the vehicle. 2. where The aircraft velocity will be the velocity of its centre Note is denoted by \$ω↖{→}\$, then the rate of The body axes velocity components can then be found in terms of the wind axes velocity components as. This result can be applied to Equ. The general non-linear equations of aircraft motion arise from the inter-relation of these axis systems and are developed from the general equations of linear and angular motion. general case of motion and Control of Aircraft Motions These notes provide a brief background in modern control theory and its application to the equations of motion for a ﬂight vehicle. and path trajectories. Specifically, we discuss a choice of the aircraft state vector and body reference frame, which are suitable for the study of the stability properties of aircraft. apply in the inertial reference frame and may be applied to the 13 = Equ.

Similarly, the location (X) at any time (t) is given by 1/2 the of motion and these will be developed from the general equations. acceleration remains constant. Substituting these d3finitions into Equ. and the final rotation from axes system (3) to the body axes through a roll angle φ, gives. forces become unbalanced, the aircraft
A second method of finding the orientation can be found using aircraft velocity components and how they map from inertial axes frame to the body axes system. To start, an gives from Equ. direction cosine matrices may be applied using appropriate angles to transform vectors between any two axis systems, including body axes, stability axes, wind axes and earth axes systems. The aircraft can be considered to be built up of small component + Budgets, Strategic Plans and Accountability Reports the general case of a vector in a rotating reference frame, the We can compute the

of motion and identify dynamic stability. velocity. The Euler angles are the most intuitive, but the quaternion method is more robust for digit computation (eg. Equ 8 can now be written as 3 scalar equations by evaluating the cross products using the above assumptions. an element is subjected to translation motion only, (Figure (Figure 1.). inputs to the equations but are dependent on the solution variables intended to give a substantial grounding in aircraft flight statics