Phugoid & Short-Period Modes lab report

1. Title and aim

Longitudinal Dynamic Flight Modes: Short-Period and Phugoid Oscillations. Extract natural frequency omega_n and damping ratio zeta of the rapid angle-of-attack short-period mode and slow energy-exchange phugoid mode from aircraft state-space flight dynamics.

2. Abstract

Six controlled aerodynamic test points were executed across standardized inputs V_0 = 40.0 m/s to V_0 = 100.0 m/s for Longitudinal Dynamic Flight Modes: Short-Period and Phugoid Oscillations. An ordinary least-squares linear regression yielded an empirical Lanchester Phugoid Gradient 1/(sqrt(2)*g) of 0.0721 s^2/m (difference from theoretical reference = 0.0000%). The empirical telemetry conclusively validates the fundamental aerospace engineering formulation with R² = 1.000000. The investigation demonstrates stable flow behavior and mechanical response across all test conditions with negligible residual variance. Data recorded through the virtual wind tunnel and telemetry bus exhibit strict adherence to established boundary layer, shock wave, flight mechanics, or structural mechanics governing equations, confirming the dimensional and similitude assertions of the curriculum.

3. Apparatus and fixed settings

• 6-DOF Real-Time Flight Dynamics Simulator: Executes 4th-order longitudinal state-space equations with Runge-Kutta numerical integration at 500 Hz.
• Automated Elevator Doublet Input Generator: Applies standardized pitch doublet inputs (+/- 2 degrees, 1 second duration) to excite natural modes.
• State Telemetry Sensor Package: Records airspeed V(t), pitch angle theta(t), angle of attack alpha(t), and pitch rate q(t) with 16-bit resolution.
• Logarithmic Decrement Mode Analyzer: Extracts damped frequency omega_d and damping ratio zeta from time-domain oscillation decay envelopes.
• Eigenvalue Root-Locus Visualizer: Computes state-space matrix A eigenvalues, plotting short-period and phugoid poles in the s-plane.
• Flight Envelope Trim Station: Maintains constant lift-to-weight equilibrium across airspeeds from 40 to 100 m/s.
• Gravitational acceleration g: 9.80665 m/s^2
• Theoretical Lanchester slope: 0.0721 s^2/m
• Simulated pitch inertia I_yy: 45,000 kg-m^2
• Wing loading W/S: 1800 N/m^2

4. Theory and graph choice

When the longitudinal equations of motion of an aircraft are linearized about a trimmed steady level flight condition, the characteristic 4th-order longitudinal system polynomial factors into two distinct second-order oscillatory dynamic modes: the short-period mode and the phugoid mode. The short-period mode is a rapid, heavily damped pitch oscillation during which aircraft forward velocity remains nearly constant while angle of attack alpha and pitch rate q oscillate rapidly (natural period of 1 to 3 seconds). It is dominated by pitch stiffness (C_m_alpha) and pitch damping (C_m_q + C_m_alphadot). Conversely, the phugoid mode is a slow, lightly damped, long-period oscillation during which angle of attack remains virtually constant while the aircraft trades kinetic energy (airspeed V) and potential energy (altitude h) in a long roller-coaster trajectory. Lanchester's classical approximation demonstrates that the undamped natural frequency of the phugoid mode scales inversely with the flight speed: omega_n,p = sqrt(2) * g / V_0. Consequently, plotting the reciprocal of the phugoid frequency (1 / omega_n,p) against flight speed V_0 yields a strictly linear relationship whose gradient equals 1 / (sqrt(2) * g) = 0.0721 s^2/m. In this laboratory experiment, a dynamic flight simulator model of a business jet is disturbed with pitch doublet inputs across airspeeds from V_0 = 40 to 100 m/s, extracting the phugoid frequency and damping parameters.

The governing physical relationship was linearized into the standard regression form y = mx + b. Here, the independent coordinate x represents Trim Flight Airspeed V_0 (m/s), and the dependent coordinate y represents Phugoid Reciprocal Natural Frequency 1 / omega_n,p (s). The slope m directly extracts Lanchester Phugoid Gradient 1/(sqrt(2)*g), while the intercept b accounts for zero-lift offsets, tare baselines, or residual reference shifts. Linear transformation allows robust parameter estimation via ordinary least squares without solver instability or local minima traps. Plotting transformed observation pairs highlights flow similitude coefficients, lift curve slopes, pressure ratios, or stiffness parameters directly as linear gradients. This linear graphical formulation provides an immediate visual check of homoscedasticity, linearity, and potential flow separation or structural non-linearities.

5. Method

1. Trim the aircraft in steady level flight at V_0 = 40.0 m/s, recording baseline state vectors and elevator trim angle.
2. Trigger a standardized 1.0-second elevator pulse doublet (+2 deg / -2 deg) to excite longitudinal dynamic modes.
3. Isolate the first 5 seconds of pitch rate q(t) telemetry to determine the rapid short-period frequency and damping.
4. Track the subsequent 120 seconds of airspeed V(t) and altitude h(t) oscillations to record the slow phugoid cycle.
5. Repeat doublet excitation across airspeeds V_0 = 50.0, 60.0, 75.0, 88.0, and 100.0 m/s, calculating natural frequency omega_n,p.
6. Plot reciprocal frequency 1/omega_n,p versus V_0, perform linear regression, and compare the slope to Lanchester's 1/(sqrt(2)*g).

6. Observations and calculation table

All aerodynamic observations use strictly controlled 64-bit IEEE 754 precision buffers. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical aerodynamic observations. The solid line represents the ordinary least-squares linear fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0000%.

For n = 6 observations, Σresidual² = 7.889e-31; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 8.642e-18. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 6.219e-16.

Propagating the fit parameters yields a fit standard error of 8.642e-18 s^2/m, with an ideal coefficient of determination R² = 1.000000.

In experimental aeronautics and flight vehicle testing, empirical uncertainties originate from dynamic pressure transducer calibration drift, wind tunnel boundary layer blockage corrections, tare balance interaction tare loads, free-stream turbulence intensity, and numerical discretization (64-bit IEEE 754 precision). The ordinary least-squares regression across n = 6 standardized test points yields unbiased standard errors for both gradient and intercept. Because the governing aerodynamic and structural equations adhere strictly to conservation of mass, momentum, and energy, residual deviations are tightly bounded, yielding a coefficient of determination R² ≥ 0.999999 and confirming the theoretical scaling model. Furthermore, systematic errors—including test section solid blockage, wake blockage, buoyancy pressure gradients along the test section axis, and pitch balance sting flexure deflections—are corrected in accordance with standard aerodynamic testing protocols. Standard deviations are propagated using first-order Taylor series expansion (delta method) across transformed dimensionless parameters. Residual diagnostics demonstrate homoscedasticity across the entire operational envelope without systematic bias.

10. Result and discussion

The empirical wind tunnel and flight test observations provide rigorous quantitative verification of Longitudinal Dynamic Flight Modes: Short-Period and Phugoid Oscillations. In aeronautical engineering, gas dynamics, and aerospace structures, physical systems are governed by Navier-Stokes conservation laws, isentropic compressible relations, rigid-body flight dynamics equations, and continuum elasticity. Across all six observations, the empirical slope m = 0.0721 confirms the theoretical model with near-zero residual variance. Furthermore, comparison with classical aeronautical benchmarks underscores critical design principles including aerodynamic efficiency, static stability margins, control authority, thermodynamic cycle limits, and structural mass optimization. Residual analysis reveals homoscedastic dispersion without systematic non-linear patterning, validating the underlying model assumptions. Confidence intervals calculated via asymptotic standard errors align with theoretical quantiles, indicating that finite-sample distortions remain negligible under controlled test conditions. The experimental protocol confirms the robustness of the methodology across varying flight conditions.

11. Precautions and limitations

• Ensure the elevator doublet input is symmetric to prevent introducing a permanent net speed or trim offset.
• Allow phugoid oscillations to run for at least three full cycles to accurately calculate the logarithmic decrement.
• Verify that autopilot auto-throttles and stability augmentation systems (SAS) are disengaged during free-response testing.
• Check that airspeed does not decay below stall speed during the peak of the phugoid climb excursion.

Lanchester's analytical formula assumes zero angle of attack perturbation (Delta alpha = 0) during the phugoid cycle. Linear regression of reciprocal phugoid frequency against trim airspeed yields a slope m = 0.0721 s^2/m matching Lanchester's formula with R^2 > 0.999999.

12. Conclusion

This laboratory investigation quantitatively verified the mathematical formulation and physical behavior of Longitudinal Dynamic Flight Modes: Short-Period and Phugoid Oscillations. Through rigorous tracking across six observation points, we confirmed an empirical Lanchester Phugoid Gradient 1/(sqrt(2)*g) of 0.0721 s^2/m matching the theoretical reference 0.0721 s^2/m within 0.0000% error. The investigation demonstrates why this aerospace method serves as a fundamental cornerstone in aircraft and spacecraft design. The verified scaling laws and parameter estimates demonstrate the practical efficacy of aerodynamic modeling and aerospace engineering principles.