MAE 4283 Exam 2 Review Flashcards

Complete 164-flashcard vocabulary deck covering longitudinal and lateral-directional flight dynamics, approximations, stability criteria, and exam concepts.

Conventional Aircraft Dynamic Modes

phugoid, short period, roll subsidence, spiral, and Dutch roll.

Longitudinal Dynamic Modes

phugoid (long period) and short period.

Lateral-Directional Dynamic Modes

roll subsidence, spiral, and Dutch roll.

Phugoid Mode

primarily involves variations in airspeed and altitude at nearly constant angle of attack.

Short-Period Mode

rapid longitudinal oscillation in pitch attitude and angle of attack

Roll

Rotation around longitudinal axis, wings tip left to right

Spiral

A slow, nonoscillatory lateral-directional mode characterized by divergence or convergence in bank angle and heading through coupled rolling and yawing motions.

Dutch Roll

A coupled yaw-sideslip oscillation accompanied by rolling motion in full three-dimensional aircraft dynamics.

USAF Stability and Control DATCOM

engineering handbook used for predicting and estimating aircraft stability and control derivatives

Static Stability

tendency of an aircraft to return to equilibrium following an initial disturbance.

Dynamic Stability

The time history of the motion following a disturbance, indicating whether the resulting response decays, persists, or grows over time.

Statically Stable but Dynamically Unstable Systems

A condition where an aircraft possesses an initial restoring tendency toward equilibrium, but lacks sufficient damping, resulting in undamped or growing oscillations.

Trim

A reference flight condition characterized by completely balanced translational forces and rotational moments, resulting in zero unintended net acceleration.

Stick-Fixed Dynamics

An analysis condition in which the cockpit controls and aerodynamic control surfaces are assumed to remain completely rigid and fixed at their reference positions.

Six Degrees of Freedom (6-DOF)

three translational motions along body axes and three rotational motions about body axes

Aircraft Dynamic Eigenvalue

A characteristic root of the linearized system matrix whose real part governs exponential growth or decay and whose imaginary part defines the damped oscillation frequency.

Real Part of an Eigenvalue

The modal decay or growth parameter: a negative value indicates exponential decay, a positive value indicates exponential growth, and zero indicates neutral stability.

Imaginary Part of an Eigenvalue

The magnitude represents the damped natural angular oscillation frequency ($\omega_d$) of that specific dynamic mode.

Yaw Damping

An aerodynamic moment generated in response to yaw rate that acts directly to oppose the rotational yawing motion of the vehicle.

Weathercock Stability

The directional static restoring tendency produced primarily by the vertical tail to rotate the aircraft back into alignment with the oncoming relative wind during sideslip.

Dihedral Effect

The rolling moment generated in response to a lateral sideslip angle, standardly parameterized by the aerodynamic stability derivative $C_{l_\beta}$.

Adverse Yaw

The aerodynamic yawing moment directed opposite to the intended turn or commanded roll direction, caused primarily by unequal induced drag across the deflected wings.

Short-Period Damping Significance

Crucial handling-qualities parameter; inadequate short-period damping leads to severe pitch oscillations and excessive pilot corrective workload or pilot-induced oscillations.

Spiral Divergence Tolerance

Slow spiral divergence is acceptable in operational flight because its time to double amplitude is long enough for a pilot or autopilot to correct with minimal workload.

Steady Wings-Level Trim Rates

Mathematical conditions in symmetric, steady, straight, wings-level trim: $p_0 = q_0 = r_0 = 0$ along with zero reference accelerations.

Symmetric Reference Angles

In a reference flight condition devoid of roll and sideslip, the reference bank and sideslip angles vanish: $\phi_0 = 0$ and $\beta_0 = 0$.

Constrained Forward Translation Perturbations

Under a 1-DOF forward translation restriction, all lateral and rotational perturbations vanish: $\Delta v = \Delta w = \Delta p = \Delta q = \Delta r = 0$, along with attitude perturbations.

Constrained Pure Pitching Rates

Under motion constrained purely to pitching, out-of-plane angular rates are eliminated: roll rate $p = 0$ and yaw rate $r = 0$, leaving pitch rate $q$ as the sole rate.

Constrained Pure Rolling Rates

Under pure roll constraint assumptions, pitch and yaw rates vanish identically: $q = r = 0$, retaining only roll rate $p$.

Constrained Pure Yawing Rates

Under pure yaw constraint assumptions, roll and pitch rates vanish identically: $p = q = 0$, retaining only yaw rate $r$.

Fixed-Control Perturbations

In unforced, stick-fixed natural motion, all control-surface deflections and thrust deviations are set to zero: $\Delta \delta_e = \Delta \delta_a = \Delta \delta_r = \Delta \delta_T = 0$.

Unforced State-Space Formulation

In the linear state equation $\mathbf{\dot{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$, unforced natural motion requires setting the control vector $\mathbf{u}(t) = \mathbf{0}$.

Rest-in-Roll Initial Condition

A mathematical starting boundary condition specifying that the roll rate at initial time is zero: $p(0) = 0$.

Neglected Wing-Body Pitch Contribution

The modeling simplification in which designated wing/fuselage pitching-moment stability derivatives (such as $C_{m_{\alpha,\text{wb}}}$) are set to zero.

Neglected Fuselage Yaw Contribution

The directional modeling simplification setting the fuselage static directional stability derivative to zero: $C_{n_{\beta,\text{wf}}} = 0$.

Neglected Vertical Tail Sidewash

The aerodynamic simplification assuming the flow field across the vertical stabilizer has no lateral deflection gradient: $\frac{d\sigma}{d\beta} = 0$.

Neglected Downwash Gradient

The assumption that the wing wake does not alter its downwash angle at the horizontal tail due to changes in angle of attack: $\frac{d\varepsilon}{d\alpha} = 0$.

Neglected Downwash Lag

The assumption that downwash convection delay from wing to horizontal tail is negligible, eliminating the pitch-damping derivative: $C_{m_{\dot{\alpha}}} = 0$.

100% Tail Dynamic Pressure Ratio

The modeling assumption that dynamic pressure at the tail equals freestream dynamic pressure, yielding dynamic pressure ratio $\eta_v = 1$.

Neglected Compressibility on Drag

Setting the Mach-dependent drag derivative $C_{D_u} \approx 0$, which does not remove the baseline $2 C_{D_0}$ dynamic pressure speed term.

Neglected Compressibility on Lift

Setting the Mach-dependent lift derivative $C_{L_u} \approx 0$, which does not remove the baseline $2 C_{L_0}$ dynamic pressure speed term.

Speed-Independent Thrust Derivative

The mathematical modeling assumption that thrust does not vary with forward speed perturbations: $C_{T_u} \approx 0$, while steady trim thrust remains non-zero.

Bilateral Symmetry Inertia Elimination

For an aircraft possessing mass symmetry across the $xz$-plane, the cross products of inertia involving the lateral axis vanish identically: $I_{xy} = I_{yz} = 0$.

Zero Product of Inertia ($I_{xz} = 0$)

An additional mass-distribution simplification that completely eliminates linear dynamic roll-yaw inertial coupling.

Longitudinal State Decoupling

The reduction of aircraft equations of motion by setting lateral-directional perturbation states ($\Delta v$, $\Delta p$, $\Delta r$, and $\Delta \phi$) to zero.

Lateral State Decoupling

The reduction of aircraft equations of motion by setting longitudinal perturbation states ($\Delta u$, $\Delta \alpha$, $\Delta q$, and $\Delta \theta$) to zero.

Phugoid Approximation Assumptions

A 2-DOF dynamic approximation assuming angle-of-attack perturbation $\Delta \alpha \approx 0$ and suppressing fast pitch-rate dynamics.

Short-Period Approximation Assumptions

A 2-DOF dynamic approximation assuming forward airspeed perturbation is negligible ($\Delta u \approx 0$) throughout the high-frequency oscillation.

Steady Pure Roll Condition

Under a sustained constant aileron deflection, steady-state roll rate requires roll acceleration to vanish ($\dot{p} = 0$), settling at a constant $p \neq 0$.

Purely Imaginary Eigenvalue ($\sigma = 0$)

A dynamic condition indicating undamped, sustained harmonic oscillation whose amplitude neither grows nor decays exponentially.

Strictly Negative Real Root ($\sigma < 0$)

A dynamic condition indicating asymptotic stability in which perturbation modal amplitude decays exponentially with time.

Zero Imaginary Part ($\omega = 0$)

A characteristic root condition indicating a purely real, nonoscillatory exponential decay or divergence.

Massless Body Simplification

An inertia estimation technique where a fuselage or body mass is neglected in moment of inertia calculations, retaining only external appendages or fins.

Thin Flat Plate 2D Lift Slope

The theoretical thin-airfoil section lift-curve slope approximation: $a_0 \approx 2\pi\,\text{rad}^{-1}$, requiring aspect ratio corrections for finite wings.

Exposed Fin Planform Area Rule

Portions of a vertical stabilizer submerged within the fuselage profile are excluded from aerodynamic area, using only exposed wetted planform area.

Radian-Degree Derivative Conversion

When aerodynamic stability derivatives are defined per radian, all input angles expressed in degrees must be converted to radians before numerical substitution.

Neglected Alpha-Dot Derivative Convention

The simplification instructing the analyst to set aerodynamic derivatives with respect to $\dot{\alpha}$ (such as $C_{m_{\dot{\alpha}}}$) to zero, leaving $\alpha$ terms active.

Asymmetric Thrust Decoupling Limitation

An aircraft cannot be assumed laterally and longitudinally decoupled if asymmetric propulsion produces cross-coupling moments even with a symmetric airframe.

Stick-Fixed Damping Presence

Fixing the control surfaces does not eliminate aerodynamic damping, because aerodynamic damping arises from aircraft rotational and translational rates.

Aspect Ratio Formula

The nondimensional geometric ratio of wingspan squared to reference wing planform area: $\text{AR} = \frac{b^2}{S}$.

Wing Taper Ratio (\lambda)

The geometric ratio of wing tip chord to wing root chord: $\lambda = \frac{c_{\text{tip}}}{c_{\text{root}}}$.

Mass in English Engineering Units

The mass calculated by dividing weight in pounds-force by standard gravitational acceleration in feet per second squared: $m = \frac{W}{g}\,[\text{slugs}]$.

Aerodynamic Lift Force Equation

The total dimensional aerodynamic lift defined by dynamic pressure, wing reference area, and lift coefficient: $L = Q S C_L$.

Aerodynamic Drag Force Equation

The total dimensional aerodynamic drag defined by dynamic pressure, wing reference area, and drag coefficient: $D = Q S C_D$.

Dimensional Rolling Moment Relation

The total aerodynamic torque about the body roll axis: $L_{\text{roll}} = Q S b C_l$, where $C_l$ denotes the nondimensional rolling-moment coefficient.

Rolling-Moment Coefficient Equation

The nondimensionalization of rolling moment obtained by dividing dimensional torque by dynamic pressure, wing area, and wingspan: $C_l = \frac{L_{\text{roll}}}{Q S b}$.

Dimensional Pitching Moment Relation

The aerodynamic torque about the body pitch axis defined by dynamic pressure, wing area, and mean aerodynamic chord: $M_{\text{pitch}} = Q S \bar{c} C_m$.

Dimensional Yawing Moment Relation

The aerodynamic torque about the body yaw axis defined by dynamic pressure, wing area, and wingspan: $N_{\text{yaw}} = Q S b C_n$.

Aileron Control Derivative ($C_{l_\delta}$)

The change in rolling-moment coefficient per unit control-surface deflection: $C_{l_\delta} = \frac{\partial C_l}{\partial \delta}$, with deflection measured in radians.

Small-Angle Lift Perturbation Estimate

Linearized lift coefficient estimation near trim: $\Delta C_L \approx C_{L_\alpha} \Delta \alpha$, or $C_L \approx a_0 \alpha$ for symmetric airfoils at zero lift.

Ideal Thin-Airfoil Lift Slope Value

The theoretical two-dimensional potential-flow lift-curve slope of a thin airfoil: $a_0 \approx 2\pi\,\text{rad}^{-1} \approx 0.11\,\text{deg}^{-1}$.

Finite Wing Lift-Curve Slope Correction

The correction converting 2D airfoil slope $a_0$ to finite 3D wing slope: $a \approx \frac{a_0}{1 + \frac{a_0}{\pi e \text{AR}}}$.

Small-Angle Sideslip Approximation

The kinematic relation approximating sideslip angle $\beta$ from lateral perturbation velocity and trim airspeed: $\beta \approx \frac{v}{u_0}$.

Linearized Nondimensional Roll Rate

The standardized dimensionless roll-rate parameter normalized by wingspan and airspeed: $\hat{p} = \frac{p b}{2 V}$.

Linearized Nondimensional Yaw Rate

The standardized dimensionless yaw-rate parameter normalized by wingspan and airspeed: $\hat{r} = \frac{r b}{2 V}$.

Local Incidence from Roll Rate

The spanwise change in local aerodynamic angle of attack induced by roll rate: $|\Delta \alpha(y)| \approx \left|\frac{p y}{V}\right|$.

Strip Theory Roll Damping Derivative ($C_{l_p}$)

The roll damping derivative of a uniform rectangular wing estimated via strip theory: $C_{l_p} \approx -\frac{a}{6}$.

Deflected Plate Roll Control Derivative

The rolling-moment effectiveness for symmetric opposite-deflection incidence plates via strip theory: $C_{l_\delta} \approx \frac{a}{4}$.

Centerline Moment of Inertia for Thin Flat Plate

The mass moment of inertia of a uniform thin flat plate about its central roll axis: $I_x \approx \frac{m b^2}{12}$.

Wing Couple Roll Torque

The net aerodynamic rolling moment computed by integrating or summing spanwise lift increments multiplied by their moment arms to the roll axis.

Plate Material Mass Relation

The structural mass of an aerodynamic surface computed from its material volume and density: $m = \rho_{\text{material}} \times \text{volume}$.

Pound-Mass to Slug Conversion

The standard conversion between gravitational mass and engineering slug units: $1\,\text{slug} \approx 32.174\,\text{lbm}$.

Newton's Law for Pure Yaw Rotation

The angular equation of motion relating yaw acceleration to total external aerodynamic yaw moment: $I_z \ddot{\psi} = \Sigma N_{\text{aero}}$.

Yawing-Moment Coefficient Taylor Expansion

The linear expansion of yawing-moment coefficient with respect to sideslip and yaw rate: $\Delta C_n \approx C_{n_\beta} \Delta \beta + C_{n_r} \left(\frac{b \Delta r}{2 V}\right)$.

Fixed-Pivot Yaw Kinematic Identities

The geometric small-angle relations linking body angles to wind angles in a pure-yaw pivot test: $\beta \approx -\psi$ and $r \approx \dot{\psi}$.

Second-Order Pure-Yaw Governing ODE

The second-order ordinary differential equation governing rotational yaw response: $I_z \ddot{\psi} + c_\psi \dot{\psi} + k_\psi \psi = N_{\text{input}}$.

Aerodynamic Yaw Restoring Stiffness ($k_\psi$)

The effective rotational spring stiffness in pure yaw arising from directional stability: $k_\psi \approx Q S b_{\text{ref}} C_{n_\beta}$.

Aerodynamic Yaw Damping Constant ($c_\psi$)

The effective rotational damping coefficient in pure yaw motion: $c_\psi \approx -\frac{Q S b_{\text{ref}}^2 C_{n_r}}{2 V}$.

Pivot-Constrained Aerodynamic Kinematic Linkages

Kinematic equalities for single-degree-of-freedom pivot fixtures: in pure pitch $\Delta \alpha \approx \Delta \theta$, and in pure yaw $\Delta \beta \approx -\Delta \psi$.

Constant Input Static Equilibrium Method

The mathematical procedure of setting all state time derivatives to zero ($\mathbf{\dot{x}} = \mathbf{0}$) to solve for steady-state equilibrium.

Velocity-Perturbation Angular Conversions

Small-angle approximations relating translational velocities to aerodynamic angles: $\Delta \alpha \approx \frac{\Delta w}{u_0}$ and $\Delta \beta \approx \frac{\Delta v}{u_0}$.

Airspeed-Mach Relationship

The fundamental kinematic relation between freestream velocity, Mach number, and local speed of sound: $V = M a$.

Speed Sensitivity from Mach Gradient

The conversion between Mach-derivative and speed-derivative at constant atmospheric conditions: $C_u = M \left(\frac{\partial C}{\partial M}\right)$.

Spanwise Strip Aerodynamic Force

The incremental aerodynamic lift generated by a spanwise strip of chord $c$ and width $dy$: $dF \approx Q c a \Delta \alpha\,dy$.

Initial Roll Acceleration Formula

The instantaneous angular acceleration in roll following an abrupt step control input from rest: $\dot{p}(0) = L_\delta \delta$.

Linearized Dihedral Stability Sign Convention

The standard sign requirement for stabilizing dihedral effect in lateral dynamics: $L_\beta < 0$ (or $C_{l_\beta} < 0$).

Yaw Damping Sign Convention

The standard physical sign indicating stabilizing aerodynamic resistance to yaw rate: $N_r < 0$ (or $C_{n_r} < 0$).

Directional Stability Sign Convention

The standard sign requirement for positive weathercock/directional static stability: $N_\beta > 0$ (or $C_{n_\beta} > 0$).

Roll-Due-to-Yaw-Rate Sign ($L_r$)

The typical sign convention where positive yaw rate increases advancing wing speed, generating positive roll: $L_r > 0$.

Approximate Spiral Eigenvalue Root

The simplified analytical formula for the real spiral pole: $\lambda_{\text{spiral}} \approx \frac{L_\beta N_r - L_r N_\beta}{L_\beta}$.