Longitudinal Dynamic Stability: Short Period & Phugoid

Start before the classroom omissions

EQ-L03-01 preserves the descriptor model so that an acceleration derivative is not mistaken for an ordinary state derivative. Its second row contains (1−$Z_{\dot{w}}$)$\Delta w$̇. Its third contains $\Delta q$̇−$M_{\dot{w}}$$\Delta w$̇. Thus the acceleration variables are coupled before solving for the state rates. This is an AE426 rearrangement of the cited textbook equations; the source does not print this exact E/F/H organization.

AE426-DERIVED · Nelson §4.4.1 p.149 Eq.4.50; Yechout §7.3.2 p.356 Eq.7.26. Trim-aligned, zero-wind symmetric reference; $X_q$ and $X_{\dot{w}}$ omitted. This reorganized descriptor form is course algebra, not a verbatim textbook matrix. Require E nonsingular before solving.

Let d=1−$Z_{\dot{w}}$ and require d≠0. Solve the vertical row first, then substitute its entire right side into the moment row. For each state column j, A₂ⱼ=F₂ⱼ/d and A₃ⱼ=F₃ⱼ+$M_{\dot{w}}$A₂ⱼ. Apply the same operation to every input column. This ordering explains why the classroom pitch row contains combined quantities. Adding $M_{\dot{w}}$ again to an imported solved row would count the same effect twice.

The nonlevel descriptor retains −g cos${\theta }_0$ in the forward row and −g sin${\theta }_0$ in the vertical row. Setting ${\theta }_0$=0 removes only the latter. The companion crosswalk CW-07 also retains general reference w₀ and $X_q$; its forward-rate coupling is $X_q$−w₀. EQ-L03-01 is already specialized to aligned reference axes and an x-force model without $X_q$ or $X_{\dot{w}}$. None of these omissions follows merely from the word longitudinal.

NELSON-EXPLICIT · notation aligned · Nelson p.149 Eqs.4.51–4.54; Yechout pp.356–357 corroborates. ${\theta }_0$=0, $Z_q$=$Z_{\dot{w}}$=0; $M_{\dot{w}}$ retained. Third-row entries already contain the acceleration-derivative elimination. B(2,2) is Z_δT, correcting Nelson Eq.4.54. Controls remain symbolic.

Definitions, normalization and units

For example, $X_u$=(∂X/∂u)₀/m and $M_q$=(∂M/∂q)₀/$I_y$, with the other independent variables held fixed at the declared trim. The course raw derivatives Dᵡ_u and Dᴹ_q retain force/moment units; $X_u$ and $M_q$ here are acceleration-equation derivatives. Nelson p.150 Eq.4.55 prints a force-symbol numerator for $M_u$; the appropriate numerator is ∂M/∂u. Do not divide a normalized imported derivative by mass or inertia a second time.

Radians are dimensionless in SI algebra, but the radian metadata remains useful: it prevents a degree-valued input from being interpreted as a radian-valued state. A matrix entry has the dimensions of its output rate divided by its input state. The $U_0$ entry therefore does not have inverse-second units. Zero entries retain the unit type of their positions.

Transform aerodynamic angles and controls consistently

AE426-DERIVED · Course notation §8.3; Nelson p.154 Eqs.4.72–4.74; Yechout p.356. Only the aligned, zero-wind case gives $\Delta \alpha$=$\Delta w$/$U_0$ and a diagonal T_α. Transform B and output matrices consistently; instantaneous wind axes are not a fixed body frame.

Differentiate atan2(w_A,u_A) at the reference to obtain the general angle differential. At nonzero reference angle of attack it includes a $\Delta u$_A term. With wind, obtain air-relative velocity by subtracting wind expressed in body axes; even a constant navigation-frame wind has changing body components when attitude changes. If a constant nonsingular state transformation T is used, transform A to TAT⁻¹, B to TB and C to CT⁻¹. A time-varying T adds ṪT⁻¹ and needs a new model declaration.

The symbolic input order is [Δδ_e, Δδ_T]ᵀ. Nelson Eq.4.54 prints X_δT in B(2,2); its own force equations require Z_δT. The corrected symbolic matrix is retained, with no numerical control model inferred. The course positive-surface convention remains unresolved and would need an explicit sketch for forced-control calculations. It does not affect the free-response eigenproblem.

The literal matrix is the numerical authority

NELSON-EXPLICIT (US) / AE426-DERIVED (SI) · Nelson p.158 Example4.3; DS-A-LON-N4.3 / DS-LON-NELSON43-v1. Category A VERIFIED TEXTBOOK AIRCRAFT DATA: unnamed general-aviation airplane. Use this literal matrix; do not reconstruct it from separately rounded coefficients. State order is [u,w,q,θ]: source units [ft/s,ft/s,rad/s,rad], SI units [m/s,m/s,rad/s,rad]; A entries retain the declared row/column unit pattern. SI third-row literals retain the approved displayed precision.

Nelson pp.155–161 supplies the general-aviation example and source condition: sea level, Mach0.158, $U_0$=176 ft/s, CG29.5% MAC, pitch inertia3000 slug·ft² and mean chord5.7 ft. Converted geometry provides context for the mode-shape scaling, while the literal displayed A defines the numerical example. Reconstructing A from separately rounded aerodynamic coefficients would introduce another matrix version.

AE426-DERIVED · Course notation §§8.5,12 and source Nelson p.158 matrix. Only the two velocity coordinates change units. Source g=32.2 ft/s² gives 9.81456 m/s². Never multiply the entire matrix by 0.3048.

The unit transformation changes velocity coordinates by the exact foot-to-metre factor and leaves rate and angle coordinates unchanged. It scales the gravity and kinematic columns differently from the moment-row velocity columns. The source 32.2 ft/s² gravity value converts to 9.81456 m/s²; substituting a different standard gravity value changes the reference model. The approved SI moment-row decimals are displayed to ten places, so their final-place rounding is tracked separately from the exact source-coordinate transformation.

Check the determinant and its coefficients

NELSON-EXPLICIT · Nelson pp.150–151 Eqs.4.56–4.60. Constant-coefficient free response; a nonzero eigenvector is required. The four-state determinant is quartic.

A practical calculation forms the characteristic polynomial from the frozen matrix, computes its roots, groups conjugates and verifies each root. The λ³ coefficient must equal −tr(A), the λ⁰ coefficient must equal det(A), and the sum/product identities provide independent transcription checks. Nelson p.158 prints 5.05 as the cubic coefficient of a rounded polynomial, whereas its displayed matrix gives 5.013. Using the printed polynomial as a second input silently changes the problem. This lesson computes from the matrix and retains the discrepancy honestly.

All table values are generated by the shared calculation kernel. For an upper-half-plane root, first use its real and imaginary parts to obtain ${\omega }_n$; then divide −σ by that radius for ζ. Obtain the cycle time from ${\omega }_d$ and the decay measures from σ. Rounding is a display operation after calculation, not an intermediate arithmetic step. Cycles to half amplitude are t₁/₂/T and are therefore based on the damped period.

Verify vectors as well as roots

A small polynomial residual alone does not verify an eigenvector. Compute Av−λv in the original SI coordinates and report a relative residual using a declared norm, for example ‖Av−λv‖/(‖A‖‖v‖). Check the conjugate vector and repeat the root calculation after the SI/source similarity transformation. These tests distinguish numerical correctness from a physically good approximation: a large phugoid-reduction error can coexist with tiny full-model residuals.

Secondary Laplace view, after the eigenproblem

AE426-DERIVED · Yechout pp.357–358 Eqs.7.28–7.31 supports the longitudinal input/output analysis. Transfer function uses zero initial conditions. Particular channels may hide modes through excitation, observation or cancellation; no unverified B or control sign enters the free plots.

Transforming the state equation gives (sI−A)ΔX(s)=$\Delta \mathbf{x}$(0)+BΔU(s). Thus the initial-condition response remains even when the perturbation input is zero. Only the zero-initial-condition input/output term defines G(s). For an appropriate minimal realization its poles coincide with the internal eigenvalues, but a chosen channel may omit a mode through lack of excitation, lack of observation or cancellation. A numerator and its residues affect what an output looks like; a common denominator does not promise equal visible modal content.

Separate force, moment and kinematics

Before reducing anything, the vertical force couples $\Delta w$ to $\Delta q$ through $U_0$, while the primitive moment equation contains $M_w$$\Delta w$, $M_{\dot{w}}$$\Delta w$̇ and $M_q$$\Delta q$ as well as the speed contribution $M_u$$\Delta u$. In aligned coordinates the corresponding angle form is $M_{\alpha }$$\Delta \alpha$+$M_{\dot{\alpha }}$$\Delta \alpha$̇+$M_q$$\Delta q$. The static restoring derivative and the rate-related terms have different physical roles and different units; the latter must not be folded into a static-stability statement.

A conventional negative $M_{\alpha }$ creates nose-down acceleration for a positive $\Delta \alpha$ when the other terms are held fixed. A conventional negative $M_q$ opposes positive q. The $M_{\dot{\alpha }}$ contribution depends on the changing aerodynamic angle and enters the solved pitch row after substituting the vertical equation. Its effect cannot be read by looking only at the uncombined $M_q$ symbol. Evaluate the combined trace when discussing damping.

A complete fast mode still uses four coordinates

AE426-DERIVED · Eigenproblem EQ-L03-04 and L02 modal development. One real contribution per conjugate pair; c_k absorbs the factor two. Do not also add its conjugate and double the response. A general modal sum requires a complete basis.

AE426-INSTRUCTIONAL-SYNTHESIS · Verified matrix and EQ-L03-08; original course excitation. All components follow the same SI eigenvector with θ=1. This is a deliberate small mathematical initial condition, not a measured elevator-release experiment.

For the upper fast root, the last matrix row imposes v_q=λv_θ. With v_θ=1, the pitch-rate component is therefore complex and nonzero. The vertical row simultaneously imposes a relationship among v_u, v_w and v_q. The resulting velocity direction need not move in phase with the fuselage attitude. Plotting only |v| would discard precisely the relative phases that determine those snapshots.

A pure-mode initial state is deliberately constructed as 0.01Re(v_SP), after the stated θ normalization. It is a valid free response of the textbook matrix, not a fabricated aircraft dataset and not a measured elevator release. An arbitrary initial pitch-angle displacement with all other states zero usually excites both conjugate pairs. The distinction explains why a measured pitch history can contain a rapid early transient followed by a slow tail.

Why a constrained-pitch analogy is limited

A one-degree-of-freedom pitch analogy constrains quantities that the free aircraft can change. It can illustrate restoring stiffness and pitch inertia, but it removes vertical-force coupling and may omit an acceleration-derivative contribution. Its frequency estimate is therefore conditional. The principal lesson reduction keeps both α and q, allowing the vertical-force equation to contribute to the fast eigenvalues.

YECHOUT-EXPLICIT · Yechout p.363 Eq.7.37. A further estimate requiring the displayed inequality. Hold the other quantities fixed for sensitivity claims; recheck near an aft-CG/static-stability boundary.

The further stiffness/inertia estimate requires the moment-force product to be small compared with −$M_{\alpha }$. When that inequality holds, a more-negative $C_{\text{mα}}$ at fixed dynamic pressure, geometry and inertia tends to raise the estimated frequency, while greater $I_y$ tends to lower it. Real changes to a tail or CG may alter several derivatives and trim together; isolated trends are not a validated redesigned aircraft.

Read the short time window honestly

The displayed fast-mode solution uses the frozen SI matrix and small amplitude. Translational quantities are scaled by $U_0$, angle is in radians, and pitch rate carries its own scale and unit. A small $\Delta u$/$U_0$ compared with the other declared coordinates is evidence about this mode shape. It does not make a raw m/s entry numerically comparable to a radian entry, nor does rapid disappearance of this mode guarantee that the full aircraft has no slower residual motion.

Derive the height output from the declared state

AE426-INSTRUCTIONAL-SYNTHESIS / AE426-DERIVED · Nelson Fig.4.10 p.148; Yechout pp.359–360 Fig.7.14. Small-angle, longitudinal, no-wind output relations; h is positive up and is derived, not a fifth eigenproblem state. Thrust and drag make the energy exchange nonconservative.

The small-angle relation uses the same reference frame and zero-wind convention as the force model. In level trim, vertical inertial velocity positive upward is approximately $U_0$θ−w to first order, hence Δḣ=$U_0$$\Delta \theta$−$\Delta w$. Since $\Delta w$=$U_0$$\Delta \alpha$ in the declared aligned coordinates, this is $U_0$($\Delta \theta$−$\Delta \alpha$). Products such as $\Delta u$$\Delta \theta$ are second order and are omitted. An initial height offset affects the plotted path but does not alter any of the four natural roots.

For a single eigenmode with nonzero λ, a compatible complex height amplitude is ($U_0$v_θ−v_w)/λ. The real response follows by using the same complex modal coefficient and exponential as the state. If height is instead numerically integrated from an arbitrary h(0), retain that integration constant. An augmented position-state realization would add a kinematic integrator; that extra zero root must be identified separately and is not a third longitudinal oscillatory mode.

What energy exchange does and does not say

Expand kinetic energy mV²/2 about V=$U_0$ and potential energy mgh about the reference height. Their first-order changes are m$U_0$$\Delta u$ and mgΔh. The energy interpretation uses the reference speed because quadratic perturbation terms are omitted. The absolute energy of the translating reference aircraft is not zero merely because its perturbation energy is zero.

In a simple point-mass interpretation the mechanical-energy rate contains the along-path propulsive power minus drag power. The zero-loss limit makes kinetic/potential exchange especially clear. In the actual local powered or unpowered model, changes in drag, thrust and speed modify that budget. Damping comes from the net work over the motion, not from gravity destroying energy. The state-equation eigenproblem supplies the quantitative answer; a qualitative energy picture alone cannot recover its damping ratio.

Phase and the long time window

The full-model PH eigenvector makes $\Delta \alpha$ much smaller than the slow speed/attitude pattern under the declared scaling, while preserving a nonzero phase difference. Speed extrema, attitude extrema and height extrema need not occur simultaneously. For a pure complex mode, the time signal is the real projection of each complex component; a shared envelope multiplies differently phased sinusoids. A path drawing intended as quantitative must therefore be generated from those synchronized outputs, rather than assigning arbitrary phase labels.

The kernel computes the PH period and envelope half-time from the slow full root. One cycle does not remove a fixed fraction independent of the root; its envelope ratio is $e^{\sigma T}$. On the long time axis the much faster short-period mode has essentially disappeared, while the phugoid can remain visible. Weak damping is common in this conventional example, but the name phugoid neither enforces stability nor supplies a universal acceptable time range.

Powered-flight caution

YECHOUT-EXPLICIT · notation aligned · Yechout pp.364–365 Eqs.7.39,7.41. Use total $X_u$ once. Add a separate propulsive derivative only when the imported aerodynamic derivative excludes it; thrust-command units require their own definition.

Yechout separates the propulsive speed derivative in the slow x-force balance. When $X_u$ already represents the total derivative, add nothing further. When a source supplies aerodynamic and propulsive pieces separately, sum them before evaluating damping. The lift-to-drag limit introduced later neglects the propulsive speed contribution and additional coefficient-speed terms. It cannot replace the total-derivative calculation indiscriminately for a powered aircraft.

Compute, scale and normalize in a declared order

AE426-INSTRUCTIONAL-SYNTHESIS · Comparable physical interpretation: Nelson p.161 Table4.5/Fig.4.12. The displayed vectors are recomputed from the frozen SI matrix. Common v_θ=1 normalization and the stated scales fix a display convention; amplitudes are not participation percentages or energy fractions.

Solve Av=λv in SI coordinates. Divide the entire complex vector by its θ entry, including the common phase. This makes that entry exactly real and equal to one. Then apply the diagonal display transformation D_z=diag(1/$U_0$,1/$U_0$,c̄/(2$U_0$),1). The resulting complex components share a common reference. Because the last display scale is one, θ remains the normalization reference after this transformation.

A displayed component magnitude is |${\mathbf{v}}_z$,i| and its phase relative to θ is arg(${\mathbf{v}}_z$,i/${\mathbf{v}}_z$,θ). If a component is zero, its phase is undefined rather than a meaningful zero angle. Multiplying the unnormalized vector by any common nonzero complex number changes neither these final ratios nor the physical trajectory when its modal coefficient is compensated inversely. Scaling separate components independently would instead redefine the coordinates.

Reconstruct the real state without double counting

AE426-DERIVED · Eigenproblem EQ-L03-04 and L02 modal development. One real contribution per conjugate pair; c_k absorbs the factor two. Do not also add its conjugate and double the response. A general modal sum requires a complete basis.

One real contribution per conjugate pair is Re(c_kv_k$e^{\lambda \_kt}$), where c_k absorbs the factor two used in an all-complex eigenvector sum. Alternatively, use c_kv_k$e^{\lambda \_kt}$+conjugate(c_kv_k$e^{\lambda \_kt}$). These are compatible conventions after coefficient conversion, but they must not be added together. The reference pure-mode construction uses the first convention with real ε=0.01 and v_θ=1.

AE426-INSTRUCTIONAL-SYNTHESIS · Verified matrix and EQ-L03-08; original course excitation. All components follow the same SI eigenvector with θ=1. This is a deliberate small mathematical initial condition, not a measured elevator-release experiment.

The explicitly defined mixed example sums the two pure real contributions at the same ε, so its initial state is 0.01Re(v_SP+v_PH) and its initial θ perturbation is 0.02 rad. This is a stated mathematical superposition, not a measured disturbance or a claim that all state-only releases excite the modes equally. The explorer prints its initial condition and state units before interpreting the response.

A general initial condition has modal coefficients

Place the four complex eigenvectors into a matrix V with conjugate columns. When V is nonsingular, solve Vc=$\Delta \mathbf{x}$(0) for c. For a real initial state the coefficients of conjugate eigenvectors are conjugates, so the reconstructed state remains real. A robust implementation also evaluates $e^{\mathbf{A}t}$$\Delta \mathbf{x}$(0) as an independent calculation. Agreement checks initial-state recovery and time response without relying on the same modal algebra twice.

The matrix exponential remains meaningful for defective matrices even when the ordinary eigenvector expansion does not. The present frozen model has distinct roots, so that complication does not block its modal reconstruction. Nevertheless, root separation and a well-conditioned eigenvector basis are numerical properties to check rather than universal consequences of having four states.

Interpret time-scale separation with the chosen coordinates

Compare damped frequencies to assess cycle separation and envelope times to assess how long the modes persist. These are different ratios. A rapidly decaying fast oscillation can disappear before completing much of a cycle, while the slow mode can persist for many seconds. Use the same declared coordinate scale for both modes when comparing their shapes. Neither raw eigenvector magnitudes nor a table normalized separately by each component supplies formal left/right eigenvector participation factors. The course dashboard deliberately reports mode-shape amplitudes and phases only.

The redrawn phase representation rotates at the computed ${\omega }_d$ while its amplitude envelope uses σ. Nelson p.161 Fig.4.12 labels the short-period rotation near2.5, close to its decay-rate magnitude rather than the example’s imaginary part. The present original representation uses the recomputed oscillation frequency and keeps rotation separate from radial decay.

Derive the two equations before the solved matrix

Under $\Delta u$≈0, the vertical-force row is $\Delta w$̇=$Z_w$$\Delta w$+$U_0$$\Delta q$. Divide by $U_0$ and use $\Delta w$=$U_0$$\Delta \alpha$ to obtain the angle equation. The primitive moment balance is $\Delta q$̇=$M_{\alpha }$$\Delta \alpha$+$M_{\dot{\alpha }}$$\Delta \alpha$̇+$M_q$$\Delta q$. The derivative conversions are $Z_{\alpha }$=$U_0$$Z_w$, $M_{\alpha }$=$U_0$$M_w$ and $M_{\dot{\alpha }}$=$U_0$$M_{\dot{w}}$. These follow from a constant reference transformation, with the other independent variables held fixed.

BOTH / CROSS-CORROBORATED · Nelson pp.154–155 Eqs.4.71,4.75; Yechout pp.361–363 Eqs.7.32–7.33. Neglect fast-time $\Delta u$ and remove the x-force equation; the earlier $Z_q$/$Z_{\dot{w}}$ omissions remain. θ is recoverable from θ̇=q and is not zero.

Expand and identify trace and determinant

Write a=$Z_{\alpha }$/$U_0$. The determinant is (λ−a)(λ−$M_q$−$M_{\dot{\alpha }}$)−($M_{\alpha }$+$M_{\dot{\alpha }}$a). Expanding the product gives λ²−(a+$M_q$+$M_{\dot{\alpha }}$)λ+a$M_q$+a$M_{\dot{\alpha }}$−$M_{\alpha }$−$M_{\dot{\alpha }}$a. The last two acceleration-derivative products cancel exactly. What remains in the constant term is a$M_q$−$M_{\alpha }$; the coefficient of λ still contains $M_{\dot{\alpha }}$.

BOTH / CROSS-CORROBORATED · Nelson p.154 Eq.4.78; Yechout p.362 Eq.7.33. $M_{\dot{\alpha }}$ cancels from the determinant, while remaining in the trace and damping coefficient.

BOTH / CROSS-CORROBORATED · Nelson p.155 Eqs.4.80–4.81; Yechout p.363 Eqs.7.35–7.36. Positive determinant and an appropriate discriminant are needed for the conventional oscillatory interpretation. Retain the leading minus in ζ; Nelson p.159 omits it in a worked substitution.

A positive determinant gives a real positive natural-frequency scale. An oscillatory pair additionally needs trace²−4det<0. Stability requires the trace to be negative for this two-state case with positive determinant; a derivative name does not enforce that sign. Nelson p.159 loses the leading minus in a worked damping substitution. The governing p.155 formula and Yechout Eq.7.36 support the signed expression used here.

Recover outputs without adding feedback

Integrate θ̇=q using a declared initial θ. For a pure nonzero eigenvalue, its compatible attitude amplitude is v_q/λ. For a general reduced initial state, integration may retain an arbitrary constant attitude component that the isolated α/q model does not feed back into speed. That reconstructed attitude is an output of a fast approximation, not a complete four-state long-time prediction. The ignored speed/gravity loop limits its use after the fast transient.

Retain $Z_q$ and $Z_{\dot{w}}$ when they matter

The descriptor calculation shows how the principal approximation changes if the source omissions are not appropriate. Let d=1−$Z_{\dot{w}}$, a=$Z_{\alpha }$/($U_0$d), and b=($U_0$+$Z_q$)/($U_0$d). Then α̇=aα+bq and q̇=($M_{\alpha }$+$M_{\dot{\alpha }}$a)α+($M_q$+$M_{\dot{\alpha }}$b)q. Its determinant is a$M_q$−b$M_{\alpha }$, while its trace is a+$M_q$+$M_{\dot{\alpha }}$b. This is a generalization of the same force/moment logic, not the frozen dataset used by the comparison tool.

Laplace and Cramer-rule extension

For symbolic elevator forcing, define b_α as the verified vertical-force input derivative divided by $U_0$ and b_q as the corresponding solved pitching derivative, including $M_{\dot{\alpha }}$b_α. Keep them symbolic because the numerical lesson has no approved control-sign fixture. With zero initial conditions, Cramer’s rule gives α/δ=[(s−a₂₂)b_α+a₁₂b_q]/P_SP(s) and q/δ=[a₂₁b_α+(s−a₁₁)b_q]/P_SP(s). The θ channel adds division by s through θ̇=q, so its apparent extra kinematic factor must be interpreted in the declared reduced realization.

Preserve the matrix rounding contract

For numerical SP verification, extract the specified frozen matrix’s w/q submatrix, then transform w to α if desired. Its entries already include the source elimination and rounding. Rebuilding them from Nelson p.157’s separately rounded $M_w$, $M_{\dot{w}}$ and $M_q$ produces a slightly different model. The extra stiffness/inertia estimate additionally drops $M_q$$Z_{\alpha }$/$U_0$ from frequency squared and needs the explicit magnitude test in EQ-L03-13; it is not the main approximation.