Phugoid Mode Research Articles

Numerical Modeling, Trim, and Linearization of a Side-by-Side Helicopter in Hovering Conditions

In the present paper, a flight dynamics model is adopted to represent the trim and stability characteristics of a side-by-side helicopter in hovering conditions. This paper develops a numerical representation of the rotorcraft behavior and proposes a set of guidelines for trimming and linearizing the highly coupled rotor dynamics derived by the modeling approach. The trim algorithm presents two nested loops to compute a solution of the steady-state conditions averaged around one blade’s revolution. On the other hand, a 38-state-space linear representation of the helicopter and rotor dynamics is obtained to study the effects of flap, lead–lag, and inflow on the overall stability. The results are compared with an analytical framework developed to validate the rotorcraft stability and compare different modeling approaches. The analysis showed that non-uniform inflow modeling led to a coupled longitudinal inflow–phugoid mode which made the vehicle prone to dangerous instabilities. The flap and lead–lag dynamics introduced damping in the system and can be considered beneficial for rotor dynamics.

Model-based manoeuvre analysis: a path to a new paradigm in aircraft flight dynamics

We propose a closed-form system of nonlinear equations for the pitch plane or longitudinal motions of a fixed-wing aircraft and use it to demonstrate a possible path to the unification of theoretical flight dynamics and practical analysis of aircraft manoeuvres. The derivation of an explicit model free of data tables and interpolated functions is enabled by our use of empirical formulae for lift and drag which agree with experiments. We validate the model by recovering the well-known short period and phugoid modes, and the regions of normal and reversed command. We then use the model to present detailed simulations of two acrobatic manoeuvres, an Immelmann turn and a vertical dive. Providing new quantitative insights into the dynamics of aviation, our model-based manoeuvre analysis has the potential to impact both the academic flight dynamics curriculum and the ground training program for pilots of manned and unmanned aircraft. Possible consequences of future model-centric pilot training may include improved safety standards in general and commercial aviation as well as expedited theoretical course completion in air transport.

The Design of Phugoid Mode Stabilization in Angle of Attack Protection Control Law

To achieve the requirement of maintaining the angle of attack, the design of the angle of attack (AOA) protection control law often causes instability in the pitch attitude and generates phugoid mode oscillations. Conversely, adding a damping ratio of phugoid to suppress the oscillations would reduce the stability of the angle of attack and the maneuverability of the aircraft. Therefore, when designing the AOA protection control law, it is necessary to choose the appropriate damping ratio to allow the aircraft to have a certain balance between maneuverability and stability. At present, people do not pay enough attention to this, but the resulting aviation accidents make this problem urgent to be solved. In this paper, based on the general modeling process, the linear model of a sample aircraft is established. Different gain values are designed according to the different needs of different flight phases for aircraft maneuverability and stability. The simulation shows that in the flight phase, the phugoid mode stabilization control law designed by the eigenstate assignment method (EA) can reduce the phugoid oscillation effectively, while in the landing flare phase when maneuverability is more important, it provides a greater degree of maneuverability.

A Simple, Correct Pedagogical Presentation of Airplane Longitudinal Dynamics

Various shortcomings and inaccuracies in the existing approach and presentation of airplane flight dynamics in pedagogy have been identified. An improved pedagogical presentation is offered which is clean - uses the two timescales in longitudinal dynamics to derive short period and phugoid mode parameters, simple - starts with the longitudinal dynamics equations rather than the complete six degree of freedom formulation, and correct - uses a physically correct model of the dynamic (rate) derivatives and employs the concept of a static residual in deriving the slower (phugoid) mode approximations. The redundant and confusing concepts of dimensional derivatives and static stability are junked. A companion paper presents a similar approach to the lateral-directional dynamics.

Compact and Accurate Phugoid Mode Approximation with Residualization

Literal approximations to the phugoid mode are systematically derived by a process of residualization with fast-slow modal decomposition. Approximations with a static residual alone and another including a dynamic residual as well are obtained. The two sets of approximations are shown to be analytically identical to each other, therefore previous attempts by some researchers to obtain "improved" literal approximations by introducing a dynamic residual are seen to be redundant. Our static residualization yields new expressions for the phugoid mode parameters which include an "inertia" term and a "phase damping" term not generally observed in previous approximations. In fact, the "phase damping" term is seen to be the second-most numerically significant damping effect after the drag and is always found to detract from phugoid damping. A third term involving the pitch damping derivative Cmq is relatively insignificant but interestingly is found in many cases to undamp the phugoid motion. Approximations are derived with and without compressibility effects in the form of the derivative CmMa to highlight the contribution of this derivative in obtaining accurate estimates of the phugoid frequency, even at low speeds. Finally, recommended literal approximations to the phugoid frequency and damping are presented which are compact and whose numerical accuracy is established for ten different combinations of airplanes / flight conditions.

Influence of differential longitudinal cyclic pitch on flight dynamics of coaxial compound helicopter

The Differential Longitudinal Cyclic Pitch (DLCP) in coaxial compound helicopter is found to be useful in mitigating low-speed rotor interactions and improving flight performance. The complex mutual interaction is simulated by a revised rotor aerodynamics model, where an improved Blade Element Momentum Theory (BEMT) is proposed. Comparisons with the rotor inflow distributions and aircraft trim results from literature validate the accuracy of the model. Then, the influence of the DLCP on the flight dynamics of the aircraft is analysed. The trim characteristics indicate that a negative DLCP can reduce collective and differential collective inputs in low speed forward flight, and the negative longitudinal gradient is alleviated. Moreover, a moderate DLCP can reduce the rotor and total power consumption by 4.68% and 2.9%, respectively. As DLCP further increases, the increased propeller power and unbalanced thrust allocation offset the improvement. In high-speed flight, DLCP does not improve the performance except for extra lateral and heading stick displacements. In addition, the tip clearance is degraded throughout the speed envelope due to the differential pitching moment and the higher thrust from the lower rotor. Meanwhile, the changed rotor efficiency and induced velocity alter low-speed dynamic stability and controllability. The pitch and roll subsidences are slightly degraded with the DLCP, while the heave subsidence, dutch roll and phugoid modes are improved. Lastly, the on-axis controllability, including collective, differential collective pitch, longitudinal and lateral cyclic pitches, varies with DLCP due to its effect on rotor efficiency and inflow distribution. In conclusion, a reasonable DLCP is recommended to adjust the rotor interaction and improve aircraft performance, and further to alter the flight dynamics and aerodynamics of aircraft.

Gull dynamic pitch stability is controlled by wing morphing

Birds perform astounding aerial maneuvers by actuating their shoulder, elbow, and wrist joints to morph their wing shape. This maneuverability is desirable for similar-sized uncrewed aerial vehicles (UAVs) and can be analyzed through the lens of dynamic flight stability. Quantifying avian dynamic stability is challenging as it is dictated by aerodynamics and inertia, which must both account for birds' complex and variable morphology. To date, avian dynamic stability across flight conditions remains largely unknown. Here, we fill this gap by quantifying how a gull can use wing morphing to adjust its longitudinal dynamic response. We found that it was necessary to adjust the shoulder angle to achieve trimmed flight and that most trimmed configurations were longitudinally stable except for configurations with high wrist angles. Our results showed that as flight speed increases, the gull could fold or sweep its wings backward to trim. Further, a trimmed gull can use its wing joints to control the frequencies and damping ratios of the longitudinal oscillatory modes. We found a more damped phugoid mode than similar-sized UAVs, possibly reducing speed sensitivity to perturbations, such as gusts. Although most configurations had controllable short-period flying qualities, the heavily damped phugoid mode indicates a sluggish response to control inputs, which may be overcome while maneuvering by morphing into an unstable flight configuration. Our study shows that gulls use their shoulder, wrist, and elbow joints to negotiate trade-offs in stability and control and points the way forward for designing UAVs with avian-like maneuverability.

A Novel Design Approach for Low Speed Recovery of High Performance Fighter Aircraft

In this paper, a novel design approach for low-speed recovery of a high-performance fighter aircraft is presented. It is shown that the phugoid mode has an important bearing on the problem of low-speed departure. Based on the analysis of the phugoid mode trajectories, a novel low-speed protection algorithm is presented in this paper. The proposed low speed recovery is achieved in three phases. The first phase consists of detecting the incipient departure followed in the second phase by the application of suitable recovery controls and finally the third phase ends with the transfer of controls to the pilot. The design of the first and the third phase consist of choosing the correct trigger conditions which ensures safe recovery of the aircraft in all conditions. The proposed Automatic low speed recovery is triggered when the aircraft trajectory crosses a fixed boundary in the region spanned by the dynamic pressure and its rate of decrease. It is observed that this boundary is approximately a straight line, implying that it is equivalent to a forward prediction in time to indicate when the aircraft will reach the lowest controllable airspeed. This Automatic Low Speed Recovery with Forward Prediction (ALSR-FP) algorithm is found to be simpler than other existing design methods and effective in preventing low speed departure for a variety of pilot inputs that result in the aircraft losing airspeed leading to stall. In the second phase control inputs are chosen to align the velocity vector to the direction of local gravity. The recovery phase is considered complete after the aircraft reaches the dynamic pressure which is approximately 10 % higher than the minimum dynamic pressure for control. Performance of the ALSR-FP is demonstrated using the high-performance fighter aircraft ADMIRE model which has a delta wing configuration, canards and multiple redundant controls. It is also shown that the proposed algorithm can be easily implemented on board for any other fighter and civil aircraft.

Derivation of control inputs for deep stall recovery using nonlinear frequency analysis

AbstractAlthough the problem of locked-in deep stall is well documented over many years, there currently exists no consistent procedure that can guarantee recovery. Past studies have suggested that it might be possible to rock the aircraft in pitch to destabilise the statically stable deep stall trim point, thereby gaining enough momentum to push the nose down. However, the methods used in these studies are either of preliminary or empirical nature and cannot guarantee recovery. In this paper, we use bifurcation analysis to derive a recovery manoeuvre, specifically by assessing the aircraft’s nonlinear frequency response under an elevator forcing. The ensuing nonlinear Bode plot detects unstable (divergent) solutions near resonance that contribute to a successful deep stall recovery. Moreover, the nonlinear resonant frequency is slightly lower than the result obtained using linear analysis, and time simulation shows that relying on the linear result does not lead to a successful recovery. It is also found that at the high angles of attack associated with deep stall, the frequency separation between the short period and phugoid mode is significantly reduced, leading to only one visible peak in the frequency response. This feature is also reflected in the time-domain step response.

Numerical estimation of longitudinal damping derivatives of a flying wing micro aerial vehicle

Longitudinal damping derivatives, [Formula: see text], of an aerial vehicle is important from an aerodynamic stability point of view. Experimental calculation of longitudinal damping derivatives using wind tunnel is not a cost-effective method; therefore, researchers have developed numerical solutions as an alternative. In this research, the longitudinal damping derivatives of a flying wing micro aerial vehicle (FWMAV) were calculated using numerical simulations by adopting pull-up maneuver and forced harmonic motion in pitch axis. Pull-up maneuver with four steady rotational rates was simulated to obtain pitch rate derivative, C mq. Combined derivative, [Formula: see text], was obtained by simulating forced harmonic motion of FWMAV around a mean angle of attack of 0° with amplitude of oscillation of ± 3° using four reduced frequencies (0.02, 0.03, 0.04, and 0.05). Unstructured surface and volume mesh was used in a spherical domain engulfed inside a large cuboid domain for moving reference frame strategy. Reynolds number taking mean aerodynamic chord as a reference length was 2.33 × 105. Spalart–Allmaras turbulence model was used. Pitch rate derivative, combined derivative, and acceleration derivative were found as − 0.03/rad, − 7.39/rad, and − 7.36/rad, respectively, by the use of a phase method at a reduced frequency of 0.03. During flight dynamic analysis, it was found that [Formula: see text] has a significant contribution on damping in short period mode with no effect on Phugoid mode. The research concluded that for tailless configurations, acceleration derivative [Formula: see text] can exist and can provide necessary damping in the longitudinal flight mode.

UAV generalized longitudinal model for autopilot controller designs

PurposeThis paper aims to overcome the main obstacle to compare the merits of the different control strategies for fixed-wing unmanned aerial vehicles (UAVs) to assess autopilot performances. Up to now, the published studies of control strategies have been carried out over disperse models, thus being complicated, if not impossible, to compare the merits of each proposal. The authors present a worked benchmark for autopilots studies, consisting of generalized models obtained by merging UAVs’ parameters gathered from selected literature (journals) with other parameters directly obtained by the authors to include some relevant UAVs whose models are not provided in the literature. To obtain them it has been used a dedicated software (from U.S. Air Force).Design/methodology/approachThe proposed models have been constructed by averaging both the main aircraft defining parameters (model derivatives) and pole-zero locations of longitudinal transfer functions. The suitability of the used methodologies has been checked from their capability to fit the short period and the phugoid modes. Previous analytical model arrangement has been required to match a uniform set of parameters, as the inner state variables are neither the same along the different published models nor between the additional models the authors have here contributed. Besides, moving models between the space state representation and transfer function is not just a simple averaging process, as neither the parameters nor the model orders are the same in the different published works. So, the junction of the models to a common set of parameters requires some residual’s computation and transient responses assessment (even Fourier analysis has been included to preserve the dominance of the phugoid) to keep the main properties of the models. The least mean squares technique has been used to have better fittings between SISO model parameters with state–space ones.FindingsBoth the SISO (Laplace) and state-space models for the longitudinal transfer function of an “averaged” fixed-wing UAV are proposed.Research limitations/implicationsMore complicated situations, such as strong wind conditions, need another kind of models, usually based on finite element method simulation. These particular models apply fluid dynamics to study aerostructural aircraft aspects, such as flutter and other aerolastic aspects, the behavior under icing conditions or other distributed parameter problems. Even some models aim to control other aspects than the autopilot, such as the trajectory prediction. However, these models are not the most suitable for the basic UAV autopilot design (early design), so they are outside the objective of this paper. Obviously, the here-considered UAVs are not all the existing ones, but the number is large enough to consider the result as a reliable and realistic representation. The presented study may be seen as a stepping stone, allowing to include other UAVs in future works.Practical implicationsThe proposed models can be used as benchmarks, or as a previous step to produce improved benchmarks, in order to have a common and realistic scenario the compare the benefits of the different control actions in UAV autopilots continuously presented in the published research.Originality/valueA work with the scope of the presented one, merging model parameters from literature with other (often referred in papers and websites) whose parameters have been obtained by the authors has been never published.

On the Handling Qualities of Two Flying Wing Aircraft Configurations

The coupling of the longitudinal and lateral stability modes of an aeroplane is considered in two cases: (i) weak coupling, when the changes in the frequency and damping of the phugoid, short period, dutch roll, and helical modes are small, i.e., the square of the deviation is negligible compared to the square of the uncoupled value; (ii) strong coupling, when the coupled values may differ significantly from the uncoupled values. This allows a comparison of three values for the frequency and damping of each mode: (i) exact, i.e., fully coupled; (ii) with the approximation of weak coupling; (iii) with the assumption of decoupling. The comparison of these three values allows an assessment of the importance of coupling effects. The method is applied to two flying wing designs, concerning all modes in a total of eighteen flight conditions. It turns out that lateral-longitudinal coupling is small in all cases, and thus classical handling qualities criteria can be applied. The handling qualities are considered for all modes, namely the phugoid, short period, dutch roll, spiral, and roll modes. Additional focus is given to the pitch axis, considering the control anticipation parameter (CAP). The latter relates to the two kinds of manouever points, where damping vanishes, that are calculated for minimum speed, take-off, and initial and final cruise conditions. The conclusion compares two flying wings designs (the “long narrow” and “short wide” fuselage concepts) not only from the point of view of flight stability, but also from other viewpoints.

An Investigation of the Longitudinal Dynamic Stability of a Box-Wing Aircraft

This study was undertaken to identify the longitudinal dynamic characteristics of the Box-Wing Aircraft. Research on the Box Wing aircraft gained traction following increasing demand for an environmental friendly aircraft that would reduce noise and pollutant emissions. The Box-Wing Aircraft is renowned for its potential aerodynamics advantages over conventional aircraft. However, due to its departure from conventional configuration, the stability dynamics of the Box Wing has not been fully explored. This study investigates the longitudinal dynamic stability characteristics of a Box Wing Aircraft at the conceptual design level using the J2 Aircraft Dynamics Software. The study integrates empirical mass estimation methods and aerodynamic data generated from a vortex lattice tool into J2 Aircraft Dynamics Software Suite. This was done to assess the Short Period Oscillation and Phugoid responses of a Box Wing Aircraft and a conventional cantilever wing aircraft. For a 2s step input, the short period oscillation induced on the Box Wing was damped out after 19s resulting in damping ratio of 0.68 and the undamped natural frequency is 1.8 <i>rad/s</i>. In the Phugoid mode, the Box Wing Aircraft gives a damping ratio of 0.006 and undamped natural frequency of 0.07 <i>rad/s.</i> On the other hand, the SPO induced on the reference conventional aircraft settles after 14s producing a damping ratio of 0.76 and undamped natural frequency of 3.12 <i>rad/s</i>. The damping ratio and undamped natural frequency for the Phugoid mode of the reference conventional aircraft were however 0.001 and 0.07 <i>rad/s</i> respectively. These results indicate that the reference conventional cantilever wing aircraft possesses acceptable longitudinal dynamic stability characteristics while the Box Wing Aircraft would require stability augmentation systems to improve its flying and handling qualities.

Dynamic Response Analysis of a High Glide Ratio Parachute System

This paper is concerned with the dynamic stability study of a gliding parachute-payload system along its gliding path. To scrutinize the respective dynamic response characteristics after releasing from high altitude, a modified multi-body model is developed. In the stability analysis procedure, the yawing motion of the payload is considered in system dynamics, which in turn creates a state-dependent matrix in the stability analysis and makes the linearization algorithm more cumbersome. To solve the problem, a unified Jacobian-based symbolic differentiation algorithm is implemented and the dynamics is linearized about various operating points along gliding segment of a typical planned trajectory. Based on results, the system has short period and phugoid modes in longitudinal channel just like an aircraft. In addition to dutch roll mode, the system has a low frequency coupled roll-spiral mode in lateral-directional channel which is a result of effective canopy anhedral angle. It is shown, the coupled mode can be decomposed into two distinct roll and spiral modes for small anhedral angles. Based on results, as the parachute descends, both the period and damping ratio for the short period mode were increased by 18 and 30%, respectively. For the phugoid mode the period of oscillations is decreased by 20% and the damping ratio, almost remains constant. For the lateral-directional channel,. As the parachute descends, the dutch roll mode is destabilized whereas the other modes are stabilized. Furthermore, from a practical point of view, lengthening the suspension lines stabilizes the coupled roll-spiral mode whereas destabilizes the other modes.

Pole placement and LQR implementation on longitudinal altitute holding control of wing in surface effect vehicle

The longitudinal altitude holding control system (LAHCS) of wing in surface effect (WiSE) vehicle has been developed using Simulink/Matlab. The LAHCS is designed to maintain the altitude of the vehicle stands at 1 m above the surface, with a maximum allowable deviation of 0.5 m. The purpose is to gain an additional lift generated by the surface effect to increase the aerodynamic performance. This control system is investigated using two approaches, i.e., the pole placement and the linear quadratic regulator (LQR) methods. Originally, the system shows an unstable response on the phugoid mode, indicated by the positive value of its Eigen. After the pole placement method is applied, the system is stable and capable of maintaining the reference command altitude. This method produces 0.27 of the maximum altitude deviation when the disturbance, represented by the doublet input elevator ±5° is applied. Moreover, the time needed for the system to reach the steady-state response of altitude is around 2.2 seconds. In comparison, the LQR method is also applied to the system with the same scenario. Although the settling time response is quite similar to the previous result, its maximum altitude deviation is significantly reduced by around 80 %. In conclusion, both of the methods used to design the LAHCS are capable of maintaining the altitude of the WiSE vehicle always below its maximum deviation tolerance.

Effects of propeller flow on the longitudinal and lateral dynamics and model couplings of a fixed-wing micro air vehicle

This paper analyzes the effects of propeller flow on the linear coupled longitudinal and lateral dynamics of a 150 mm wingspan fixed wing micro air vehicle (MAV). The effects propeller flow on the lift, drag, pitching moment and side force is obtained through wind tunnel tests. The aerodynamic forces and moments are modeled as a function of angle of attack, sideslip angle, control surface deflection and propeller rotation per minute. The nonlinear six degrees of freedom model is linearized about straight and constant altitude flight conditions for different trim airspeed to obtain linear coupled longitudinal and lateral state space model. The eigenvalues and eigenvectors of linear coupled longitudinal and lateral state space model are compared with and without propeller flow effects. The variation in the natural frequencies and damping ratios of short period mode, phugoid mode and Dutch roll mode are analyzed for various trim airspeed. An increase in the natural frequency is observed for phugoid mode and Dutch roll mode with propeller effects. The stability of the spiral mode is enhanced by the propeller flow and also the response of the roll subsidence mode is faster with propeller effects. Detailed analysis of eigenvalues and eigenvectors shows the importance of incorporating propeller flow in analyzing the dynamics of the MAV.

Longitudinal Flight Modes Natural Frequencies of Glider Type Small UAVs

Geometry and mass of an aircraft are important factors in flight mechanics and in the calculations of stability and natural frequencies of its flight modes, which are of great importance in controller design process. In this paper, comparison between the formulae of the large aircraft applied on small UAVs scale, XFLR5 as a numerical program and experimental data obtained from flight tests is made to investigate their accuracy. A method to validate natural frequencies for UAVs is presented. It is found that for the phugoid mode, methods of Roskam (exact), Ostoslavsky, and XFLR5 estimate the frequency within range of the experimental results while the methods of Roskam (approximation) and Hull overestimate it and should not be applied in case of small UAVs. Considering the short-period mode, all methods can predict the actual frequency with acceptable accuracy.

Neural Partial Differentiation Based Nonlinear Parameter Estimation from Noisy Flight Data

This paper focuses on the application of neural partial differentiation (NPD) approach to estimate the longitudinal parameters of an aircraft HFB 320 from noisy flight data. By exciting both short period and phugoid modes of an aircraft with thrust variation, the aircraft system dynamics becomes highly nonlinear and aerodynamic parameters appears nonlinear to the state trajectories of velocity, AOA, pitch rate and pitch angle. This paper highlights the application of NPD for such a class of nonlinear dynamics; previously it was used only for the estimation of parameter appearing linear to the states. The extracted the nonlinear longitudinal parameters of HFB 320 aircraft are compared with the parameters estimated by using adaptive Unscented Kalman Filter (UKF) approach. Finally, the estimation results are validated by comparing with flight data and the responses obtained from the estimates by adaptive UKF.

A numerical study into the longitudinal dynamic stability of the tailless aircraft

PurposeThe purpose of this research is a study into a mathematical approach of a tailless aircraft dynamic stability analysis. This research is focused on investigation of influence of elevons (elevator) on stability derivatives and consequently on the aircraft longitudinal dynamic stability. The main research question is to determine whether this impact should be taken into account on the conceptual and preliminary stage of the analysis of the longitudinal dynamic stability.Design/methodology/approachAerodynamic coefficients and longitudinal stability derivatives were computed by Panukl (panel methods). The analysis of the dynamic stability of the tailless aircraft was made by the Matlab code and SDSA package.FindingsThe main result of the research is a comparison of the dynamic stability of the tailless aircraft for different approaches, with and without the impact of elevator deflection on the trim drag and stability derivatives.Research limitations/implicationsThis paper presents research that mostly should be considered on the preliminary stage of aircraft design and dynamic stability analysis. The impact of elevons deflection on the aircraft moment of inertia has been omitted.Practical implicationsThe results of this research will be useful for the further design of small tailless unmanned aerial vehicles (UAVs).Originality/valueThis research reveals that in case of the analysis of small tailless UAVs, the impact of elevons deflection on stability derivatives is bigger than the impact of a Mach number. This impact should be taken into consideration, especially for a phugoid mode.

Modeling and Closed Loop Flight Testing of a Fixed Wing Micro Air Vehicle.

This paper presents the nonlinear six degrees of freedom dynamic modeling of a fixed wing micro air vehicle. The static derivatives of the micro air vehicle are obtained through the wind tunnel testing. The propeller effects on the lift, drag, pitching moment and side force are quantified through wind tunnel testing. The dynamic derivatives are obtained through empirical relations available in the literature. The trim conditions are computed for a straight and constant altitude flight condition. The linearized longitudinal and lateral state space models are obtained about trim conditions. The variations in short period mode, phugoid mode, Dutch roll mode, roll subsidence mode and spiral mode with respect to different trim operating conditions is presented. A stabilizing static output feedback controller is designed using the obtained model. Successful closed loop flight trials are conducted with the static output feedback controller.