4.3 Longitudinal and Lateral-Directional Dynamics

Longitudinal dynamics describe an aircraft's motion in the vertical plane. These equations, derived from Newton's second law, involve pitch angle, angle of attack, and velocity. They're crucial for understanding how an aircraft moves and responds to control inputs.

Lateral-directional dynamics focus on an aircraft's motion in the horizontal plane. These equations involve roll angle, yaw angle, and sideslip angle. Together with longitudinal dynamics, they provide a complete picture of an aircraft's behavior in flight.

Longitudinal Dynamics

Equations of motion derivation

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• Longitudinal equations of motion derived from Newton's second law describe aircraft's motion in vertical plane (pitch axis)
• Involve variables such as pitch angle $\theta$, angle of attack $\alpha$, and velocity $V$ to represent aircraft's orientation and speed
• Equations for force and moment equilibrium along and about aircraft's axes
  • $m(\dot{u} + qw - rv) = X - mg\sin\theta$ represents force equilibrium along x-axis (forward direction)
  • $m(\dot{w} - qu + pv) = Z + mg\cos\theta$ represents force equilibrium along z-axis (upward direction)
  • $I_y\dot{q} = M$ represents moment equilibrium about y-axis (pitch axis)
• Lateral-directional equations of motion describe aircraft's motion in horizontal plane (roll and yaw axes)
• Involve variables such as roll angle $\phi$, yaw angle $\psi$, and sideslip angle $\beta$ to represent aircraft's orientation
• Equations for force and moment equilibrium along and about aircraft's axes
  • $m(\dot{v} + ru - pw) = Y + mg\cos\theta\sin\phi$ represents force equilibrium along y-axis (sideward direction)
  • $I_x\dot{p} - I_{xz}\dot{r} = L$ represents moment equilibrium about x-axis (roll axis)
  • $I_z\dot{r} - I_{xz}\dot{p} = N$ represents moment equilibrium about z-axis (yaw axis)

Short-period and phugoid mode analysis

• Short-period mode is high-frequency oscillation primarily involving changes in angle of attack $\alpha$ and pitch rate $q$
• Heavily damped, quickly converges to steady state due to aircraft's pitch stiffness and damping (elevator control)
• Phugoid mode is low-frequency oscillation involving changes in velocity $V$ and pitch angle $\theta$
• Lightly damped, persists for longer durations due to exchange of kinetic and potential energy (altitude changes)
• Stability and damping of short-period and phugoid modes affect aircraft's longitudinal handling qualities and passenger comfort

Lateral-Directional Dynamics

Roll, spiral, and Dutch roll modes

• Roll mode is heavily damped, first-order mode primarily involving changes in roll rate $p$ and roll angle $\phi$
• Influenced by aircraft's roll damping (wing dihedral) and roll control effectiveness (aileron control)
• Spiral mode is lightly damped, slow convergence or divergence involving coupling between roll angle $\phi$ and yaw angle $\psi$
• Stability depends on aircraft's lateral stability derivative $L_\beta$ (positive for stability, negative for instability)
• Dutch roll mode is oscillatory mode involving coupling between yaw rate $r$, sideslip angle $\beta$, and roll rate $p$
• Frequency and damping influenced by aircraft's directional stability (vertical tail size) and dihedral effect (wing sweep)

Dynamics coupling and handling qualities

• Inertial coupling caused by product of inertia term $I_{xz}$ couples roll and yaw motions, leading to adverse yaw during rolling maneuvers (aileron reversal)
• Kinematic coupling arises from nonlinear terms in equations of motion, coupling longitudinal and lateral-directional variables at high angles of attack or sideslip (stall, spin)
• Coupling can degrade aircraft response and controllability, requiring pilots to provide coordinated inputs to maintain desired flight path (rudder-aileron coordination)
• Excessive coupling can lead to pilot-induced oscillations (PIO) or loss of control, affecting aircraft's handling qualities and safety