Journal of Aerospace Engineering

The $2\mathrm{x}\mathrm{x}\mathrm{x}$ series aluminum alloys are preferred as a material for aircraft leading edges because they have high failure strain compared with other aluminum alloys. This parametric study on $2\mathrm{x}\mathrm{x}\mathrm{x}$ series aluminum alloy material parameters was carried out to identify the critical parameters that influence the energy absorption and load transferability during bird strikes. The soft-body impact simulation on an aluminum alloy leading edge was validated with an experiment test. Using commercially available finite-element software, a parametric study has explored the bird impact response of aluminum leading edges with different material parameters. The results showed that static yield limit, elastic modulus, strain-hardening modulus, and strain-hardening exponent are the most influencing material parameters to be considered to improve the crashworthiness performance of the aluminum alloy. Analysis of variance revealed that the most significant material parameters for the performance improvement of the wing leading edge against bird impact are the elastic modulus and static yield limit. After that, considering this material parameters as design variables, a gray relational analysis was carried out to simultaneously reduce reaction force at the spar region and wing center tip deformation. The material parameter values were selected from the optimization study, and a comparison analysis was carried out to study the improvement of the optimized material properties. The optimized aluminum alloy showed up to 25% improvement in energy-absorbing characteristics than a conventional aluminum alloy (AA 2024-T3). Fracture analysis was carried out to find the optimized aluminum alloy’s damage initiation and evolution parameters.

Abstract

The high-speed, low-altitude region in the flight envelope of a wing-in-ground (WIG) effect craft requires attitude angle limitations to protect the aircraft from touching the ground or sea surface, and the safe boundaries of the attitude angles are time-dependent related to the flight height. This paper presents a command governor design for attitude angle protection of a WIG craft. In particular, this command governor adjusts the calculation of attitude angle boundaries and improves the maximal admissible $\lambda \text{-}\mathrm{contractive}$ set to handle the varying constraints. A set-membership state estimator is also employed to address the unmeasured disturbances and noise. The command governor design for pitch angle and bank angle protection is verified by simulation cases of climbing and turning of a WIG craft. The simulation results illustrate that the command governor can predict potential constraint violations and modify the pilot’s commands to handle the varying constraints as well as the contracted constraints affected by the measurement noise.

Practical Applications

The wing-in-ground effect craft is a kind of aircraft flying close to the ground or sea surface. The airframe and the wingtips of these crafts may touch the sea surface during climbing or turning due to the low flight altitude. We use a control method called “command governor” to prevent this dangerous situation and ensure flight safety. The command governor can guarantee the attitude angles of the aircraft within the safe boundaries by predicting potential constraint violations and modifying the pilot’s commands. In particular, we improve the command governor to handle the varying constraints as well as the contracted constraints affected by the wave disturbances and the measurement noise.

In this study, the impacts of the windward shape on the aerodynamic and aero-heating characteristics of supersonic vehicles were investigated using the blunted cone and blunted wedge models. Then, three-dimensional (3D) numerical simulations were performed under three different attack angles (0°, 10°, and 20°) with the Mach number of 10. The prediction accuracy of the adopted approach was validated against a published shock tunnel experiment. The numerical results suggested that the airflow structure, boundary layer parameters, aero-heating, and boundary layer transition were highly affected by the windward shape of supersonic vehicles. Under the same attack angle, the entropy layer thickness for the blunted wedge was found to be obviously greater than that of the blunted cone, whereas the detachment of windward shock waves for the blunted wedge was more obvious than that of the blunted cone. Moreover, the boundary layer parameters of both blunted cone and wedge were found to vary with the position in the flow direction. The commonly used constant assumption for the boundary layer parameters in engineering might be inaccurate and induce undesired errors. Furthermore, the boundary layer transition was found to more easily occur for a blunted wedge due to more significant instability in a streamwise direction. However, cross-flow instability might occur for a blunted cone under nonzero attack angles. In addition, the ratio of $St$ number of the blunted cone to that of the blunted wedge at downstream was inconsistent with the commonly used empirical value of 1.73, especially under large attack angles ($\ge 10°$) because of variable-entropy effects and cross-flow on the blunted cone surface.

To better understand and predict the flight performance of X-configuration quadrotor aircraft using variable rotor pitch and fixed rotor speed to control the aircraft, a flight performance tool including a rotor model, an aerodynamic interference model, and a new propulsive trim method is derived. The lifting rotors are modeled as circular fixed wings, and generate horseshoe vortices trailed from the retreating and advancing sides to affect the aerodynamics of the other rotors. The equilibrium equations of the aircraft are reduced by using symmetry. An iterative method by separately solving for the aircraft pitch and the collective rotor pitch is proposed to obtain the converged solution of the reduced equilibrium equations. The aerodynamic interference was found to be beneficial for the front rotors at low to medium speeds, because the nearby front rotors induced a larger upwash than the resultant downwash induced by the rear rotors. The interference is harmful to the rear rotors due to larger downwash induced by the rotors right ahead. The rotors can generate nose-up hub pitching moments, which can be used to counter the nose-down fuselage pitching moment, and decrease the thrust difference between the front and rear rotors. The effect of the fuselage pitching moment on the rotor power becomes pronounced at medium to high speeds. The vertical distance between the rotor plane and the center of mass of the aircraft can change the pitching moment acting on the aircraft, and its effect is similar to a change in the fuselage pitching moment. Reducing the fuselage drag can lead to larger rotor induced power due to the stronger aerodynamic interference, but the effect is relatively small, and fuselage drag reductions are desirable.

The cylindrical airbag is widely used in the landing buffer field with a simple structure. It is a three-dimensional closed structure, so the folding model is difficult to establish. Because the modeling process of the existing methods is complex and these methods are difficult to use in the actual folding process, a new direct folding method was proposed in this paper. In the method, the two end faces of the cylindrical airbag are recessed inward, and the cylindrical surface is flattened into a sheetlike structure; then, its finite-element model is established. The folded cylindrical airbag was inflated, and the deployed shape was the same as that of the initial unfolded model after inflation. Moreover, the maximum errors of the equivalent cross-sectional diameter, maximum length, and volume of the inflated airbag were less than 4%, which shows that the method is reasonable and feasible. Based on this method, three finite-element models of cylindrical airbags in different folding states were established, and then the effects of folding mode, inflation rate, and ambient pressure on the deployment process of the folding airbag were studied. In the landing buffer analysis, this method was used to model the folded secondary airbag when establishing the finite-element model for the cushioning dynamic of the combined airbag. The simulation results are in good agreement with the drop test results, which also verifies the effectiveness of this folding method.

Deorbit disposal of constellation satellites at the end of their operational lifetime is considered a necessary task to maintain the safety and sustainability of low Earth orbit (LEO). In LEO region, especially below 800 km, the drag sail device emerges as one of the most promising techniques to accelerate the deorbit process by the advantages of atmospheric drag. In this paper, an attitude-orbit coupled dynamics model is established for satellites equipped with pyramid drag sail devices. The number and length of support booms, the flare angle, and the initial orbital height are not limited, so that the obtained model could describe a number of drag sail systems. Then, by simplifying the attitude dynamics model in the orbital plane and considering circular orbit motions, a key parameter is proposed to evaluate the stability of the drag sail system. With this parameter, the influence of geometries on the attitude stability is analyzed for pyramid drag sails, and further conclusions about the optimal configurations are obtained from the analysis results. Finally, numerical simulations with the attitude-orbit coupled dynamics model are conducted to verify the conclusions drawn from the key parameter, and optimal configurations under three different conditions are obtained as well as simulated with deorbit missions to show the optimality. The results from this work would offer theoretical guidance to the configuration design of drag sail systems in application.

The fault detection of turboshaft engines is very important to ensure the flight safety of helicopters. Because there are few fault data in engine historical operation data, engine fault detection is often regarded as an anomaly detection problem, which is solved by the one-class classification (OCC) method. However, previous studies on fault detection usually ignored the difference between engine data caused by different engine states and operating conditions. Therefore, this paper considers the existence of these differences and introduces transfer learning to solve the problem. In this paper, based on one-class support vector machines (OC-SVM) and transfer learning (TL), an algorithm named OC-SVM-TL is proposed to detect turboshaft engine faults. In this algorithm, the hyperplane of the OC-SVM is transferred from the source domain to the target domain as a knowledge structure to help the target domain to establish a fault detection model with high accuracy. The training process is divided into two steps. The first step is to train the OC-SVM model with the data of the source domain, and the second step is to train the OC-SVM model with the data of the target domain, and the hyperplane difference between the source domain and the target domain is considered in the training process. Finally, the fault detection experiment of turboshaft engine was designed, and the fault detection of turboshaft engine was carried out under different working conditions and different engine states. The experimental results showed that the proposed algorithm has good fault detection performance when the target domain data are few or the amount of target domain data changes.

The truncated square pyramid (TSP) folded structure was proposed recently, and its advantages such as easy fabrication, superior energy-absorption capacity, and better impact performance than many other types of core structures have been demonstrated by laboratory tests and numerical simulations. In this paper, a parametric study was conducted to investigate the effects of geometric parameters on the energy-absorption capacity of a TSP unit cell under both quasi-static and dynamic crushing. A novel re-entrant TSP unit cell is proposed by introducing the re-entrant edge at its corner to improve the performance of the originally proposed TSP. The numerical model of TSP unit cell was constructed and validated against the reconstructed TSP structure, and the results matched well. TSP unit cells with different geometric configurations were constructed by varying the open-top edge length $b$, the corner opening angle $\phi$, and the re-entrant edge $z$. The structural response and damage mode of TSP unit cells with different geometric configurations were analyzed. Criteria such as the peak crushing force, average crushing force, uniformity ratio, and energy absorption were compared and used to evaluate the performance of the structure. The TSP unit cells with better performance in terms of energy absorption were selected to investigate further the influences of crushing speeds on their performance. It was found that the corner opening angle $\phi$ of a TSP unit cell has a greater influence on its energy absorption than other geometric parameters. Furthermore, the addition of re-entrant edge at the corner of a TSP unit cell effectively can reduce the peak crushing force, and results in a more uniform crushing process. Lastly, the peak crushing force is affected significantly by the crushing speed when the TSP unit cell has a large edge length $b$, whereas unit cells with small open-top length are less affected. The results provide guidance for the design of a TSP unit cell for high energy absorption against dynamic loads.

It is challenging to set a precise threshold in fault detection and isolation, which helps in reducing false alarms and missed detection rates. In this paper, an adaptive threshold approach is developed for aero-engine fault detection. Based on kernel density estimation (KDE) and backward exponentially mean filtering method, the adaptive threshold setting result for a single steady-state point of aero-engine fault detection is obtained. The flight envelope is reasonably divided, and the fault detection threshold is obtained in each flight subarea. The exponentially weighted moving average (EWMA) method is used to obtain a threshold setting at different performance degradation levels throughout the life of the aero-engine. Then the proposed threshold setting method is utilized to compare the two traditional fixed threshold setting methods and a double threshold-based method. The results show that the proposed adaptive threshold setting method performs better in the fault detection under a single steady-state point. To be specific, the detection time was shortened by 0.44, 0.72, and 0.56 s, and the fault detection rate was increased by 0.46%, 6%, and 0.13%, respectively.

In this paper, an empirical model of surface dielectric barrier discharge (SDBD) plasma actuator with good prediction performance is proposed based on experimental data. Due to the complex electric field and unstable discharge process of asymmetric dielectric barrier discharge (DBD) plasma actuator, a reliable model used to simulate the macro flow is difficult to develop with small calculation costs according to the first principle. Based on summarizing experimental data and improving other semiempirical models, a new empirical thrust estimation model is developed. The model considers six main parameters affecting the actuator thrust, including voltage amplitude, frequency, upper electrode thickness, lower electrode width, and relative permittivity and thickness of the dielectric. The estimated thrust value is inserted into the Navier–Stokes equation as a body-force source term and then the equation is solved by the computational fluid dynamics (CFD) software FLUENT using the finite volume method combined with the Spalart–Allmaras turbulence model. The body-force region is specified by an empirical body-force distribution function based on velocity information. Grid dependence of the model is verified by three meshes with different resolutions. The effectiveness of the empirical model and body-force distribution is verified by different experimental cases. Thus, the model proposed in this paper can properly predict the thrust value and velocity field generated by the DBD plasma actuator with a small computation cost.

This paper proposes a framework for the design optimization of geometric nonlinearities developed by active elements embedded in prestressable, statically indeterminant, truss-like aerospace structures for the purpose of attenuating their dynamic aeroelastic response under turbulent aerodynamic gust conditions. Dynamic aeroelastic responses are analyzed considering random power spectral density (PSD) gust with a continuous Davenport spectrum (DS) and tuned discrete gust (TDG) with a one-minus-cosine (OMC) wind excitation profiles. A genetic optimization algorithm (GA) is utilized to determine optimal prestress values through active element actuations for the purpose of tuning the geometric stiffness and, therefore, the modal response of the structure when exposed to gust excitations. In addition, a new simplified control metric for comparing active member locations is proposed. A case study is analyzed with this methodology to minimize the pointing error of a simplified antenna structure. Pointing error attenuations of 22.1% and 17.0% were found for the structure under DS mean wind speeds of 889 (349.95) and $\mathrm{2,778}\mathrm{cm}/\mathrm{s}$ ($\mathrm{1,093.61}\mathrm{in.}/\mathrm{s}$), respectively. Using the same two operating cases with the TDG excitation profile resulted in the overall pointing error to be reduced by 36.8% and 37.0%, respectively. The adaptive nature of the presented methodology allows a single actuator layout to mitigate structural response for a variety of load cases, which is a large benefit over many traditionally passive techniques. This paper expands the existing usage of geometric nonlinearities to determine optimal active element location and actuations for given optimization objectives under realistic environmental loading conditions.

Abstract

Heterogeneous constellations are composed of satellites with different structures or satellites carrying payloads with different functions. Different appearances of satellites and changes in attitude lead to different area-to-mass ratios (AMRs), which is an adverse factor in the orbit control of low Earth orbit (LEO). Traditional deployment methods primarily apply to homogeneous constellations, and the overall fuel optimization of the constellation and the trajectory drift difference among heterogeneous satellites are not considered. In the present study, the deployment of microsatellites with different AMRs was explored, so as to deploy the satellites to the target semimajor axis and relative phase in the same plane. Such deployment applies to both heterogeneous and homogeneous constellations in which satellites change the attitude or appearance thereof dynamically and differentially in LEO. By introducing the influence of atmospheric drag and orbit maneuvers into the traditional method, an argument of latitude (AoL) and semimajor axis coordinated control strategy was established. A satellite AoL adjustment amount optimization algorithm based on the Kuhn-Munkres and simulated annealing algorithms was proposed, which enables satellites to complete the target configuration with the fuel close to the expected value. Compared with the traditional method, the cost, average fuel consumption, and fuel consumption balance of each satellite are significantly improved, being beneficial for maintaining the overall fuel balance of the constellation and extending the controllable life of the constellation.

Practical Applications

In the present study, the optimal deployment of heterogeneous microsatellite constellations in circular low Earth orbit was proposed, so as to deploy the satellites with different area-to-mass ratios to the target semimajor axis and relative phase in the same plane. To solve the conflict between fuel demand and availability of some satellites, collaborative reconfiguration was devised in which satellites with sufficient fuel were appropriately used more, helping those with insufficient fuel to save fuel. An AoL and semimajor axis coordinated control strategy was established, and a satellite AoL adjustment amount optimization algorithm based on the Kuhn-Munkres and simulated annealing algorithm was proposed, which enables satellites to complete the target configuration with the fuel close to the expected value. Compared with the traditional method, the cost, average fuel consumption, and fuel consumption balance of each satellite were significantly improved, being beneficial for maintaining the overall fuel balance of the constellation and extending the controllable life of the constellation. The proposed method also applies to the case of satellites changing the attitude or appearance thereof dynamically. Complete methods, formulas, and cases are shown, which can be referenced and adopted in practical projects.

Incremental nonlinear dynamic inversion shows good robustness and low dependence on the model information. In this study, incremental nonlinear dynamic inversion is improved and utilized to solve the six-degrees-of-freedom waypoint tracking problem of fixed-wing unmanned aerial vehicles (UAVs). This research is focused on studying the acquisition of the state derivatives and the processing method of the derivative time delay. A nondelay differentiator design method and an incremental gain method that responds to time delay are developed in this study. Additionally, a universal waypoint navigation method based on incremental nonlinear dynamic inversion is put forward to enable the aircraft to successfully reach its waypoint from any spatial position and under any motion states. Finally, the proposed improvements of the incremental nonlinear dynamic inversion are verified through flight tests.

In this work, an innovative lobe-type rear variable area bypass injector is proposed, which can take into account both mixing exhaust performance to increase thrust at a high Bypass Ratio (BR), and bypass throttling at low BR. In order to explore the aerodynamic characteristics of the variable injector, the unsteady motion characteristics and the congenerous thermal transport effect of three-dimensional wake vortices within the BR range of 0.18 to 0.68 are numerically analyzed. The results demonstrate that the injector can adjust the sidewall of the collapsible lobe based on the injection of bypass flow, and significantly strengthen streamwise vortices induced by lobe peak (SV-LP), which can effectively drive the migration of the heterogeneous fluid element, so as to compensate for the inefficient mixing ability of Kelvin-Helmholtz (K-H) vortices. When the BR is raised to 0.36, the increasing SV-LP will gradually replace the low-disturbance K-H vortices in the dominant position of the quasi-ordered flow field by virtue of a strong entrainment effect. Therefore, the injector can maintain a relatively high thrust augmentation in a wide working envelope.

This paper develops a composite adaptive control law for the stabilization and parameter identification of a two-degree-of-freedom aeroelastic system, which is equipped with leading- and trailing-edge flaps. The multiple-input multiple-output (MIMO) model represents the plunge and pitch dynamics of a prototypical wing section. It is assumed that all model parameters, except the signs of the principal minors of the input matrix, are unknown. The composite adaptive control system includes a control module and a composite parameter identifier for the estimation of parameters. The parameter estimation law is synthesized using information on the tracking error, as well as on a model prediction error. In addition, unlike works published in the literature for aeroelastic systems, the novelty of this paper lies in the inclusion of a regressor-dependent matrix integral in the parameter adaptation law. By the Lyapunov analysis, it is shown that the origin of the closed-loop system is stable. Interestingly, the regressor matrix’s integral component contributes a negative semidefinite nonincreasing quadratic function of the parameter error in the derivative of the Lyapunov function. This, in turn, enhances the stability and parameter convergence properties of the closed-loop system. Simulation results show that this composite adaptive control system accomplishes simultaneous suppression of limit cycle oscillations (LCOs) and identification of parameters despite gust input, unmodeled actuator dynamics, and time-varying nonlinearity.

Aerodynamic effects on small aircraft flying over tall buildings are studied. Flow is modeled using unsteady Navier–Stokes equations. The effects of buildings on the stability of aircraft are studied by time-accurately coupling stability equations with the flow equations. Responses are computed at varying dynamic pressures. The present work provides a road map for designers to use high-fidelity simulations for the safe design of air taxis in urban environments where flying closer to buildings is unavoidable.

This paper presents a kinetic particle simulation campaign of plasma and dust environments related to excavation and construction activities at the lunar terminator regions. The electrostatic field caused by the local plasma environment was resolved by a fully kinetic finite-difference (FD) particle-in-cell (PIC) code. Trajectories of lofted charged dust grains were traced in the obtained electric field as well as lunar gravity. Two surface terrain scenarios, one convex and the other concave, were considered, each with three cases of dust generation locations with respect to the surface feature. Results show that under average solar wind conditions, lofted charged dust grains are generally concentrated within several meters from the surface near the location of origination, but some dust grains can be lofted as high as about 80 m from lunar surface. A slight amount of dust can even reach over 100 m and migrate further due to greater dust charge. Based on these results, preferred locations to perform excavation and construction activities are recommended.

This study presents a new method for aerodynamic optimization of a civil aircraft fuselage. Typically, the fuselage is designed for minimum drag force at cruise conditions, which is essential for determining the flight cost. Evolutionary optimization methods, as a powerful tool, require the accurate computations of flows around several hundreds of candidate geometries, which in turn are too costly and time-consuming. One strategy is to use two-dimensional simulations that reduce the computational cost by at least one order of magnitude. This approach was applied in the current work, and the results were obtained with the aim of minimizing the drag coefficient and maximizing the lift coefficient. A genetic algorithm (GA) was used for shape optimization, and the objective function evaluation was carried out using computational fluid dynamics (CFD). A fuselage shape parameterization method is presented that accounts for important flow features. The 150-seat aircraft model designed in Amirkabir University of Technology was considered as a reference body. In the first step, the optimization is carried out for the single objective of minimizing the drag coefficient. Then, the multiobjective optimization is performed with the simultaneous drag coefficient minimization and lift coefficient maximization. Finally, the two-dimensional (2D) side view of the optimum fuselages is used for generation of the corresponding three-dimensional (3D) geometry. The results show that such 3D optimum fuselage reduced the drag coefficient by about 11% while increasing the lift coefficient by 29% in comparison with the original geometry.

Accurate prediction of the aerodynamic forces induced by free-stream disturbance has been a challenge for flight safety. In this paper, a novel data-based approach to model the online unsteady and nonlinear response of aircraft, i.e., aerodynamic drag and lift coefficients from inflow disturbance, which can be measured in practical flights by Light Detection and Ranging (LiDAR), is established and tested. Numerical simulations with the NACA 0012 airfoil were performed to collect samples in order to train a neural network. Each sample consists of the time series of the inflow disturbance and the aerodynamic coefficients, both transformed to the Fourier space to reduce training cost and the degree of overfitting. The impacts of the number of samples and their distributions on the prediction were analyzed. Four inflow profiles with increasing complexity were tested. Various hyperparameters were first investigated, and it was found that the neural network trained with activation function TanH and optimized scaled conjugate gradient algorithm had the best performance. Finally, neural networks for both drag and lift coefficients were trained with a total of 1,300 randomly distributed samples, and a mean error of 2.1% was achieved in the test.

In this paper, a dynamic tube nonlinear model predictive control (NMPC) scheme is developed to solve the powered-descent guidance (PDG) problem in the presence of bounded disturbances and no-fly zones. First, owing to its simplicity and robustness, time-varying boundary layer sliding mode control (SMC) is used as an ancillary controller to compensate for external disturbances. Consequently, the tube geometry dynamics can be established as first-order differential equations to calculate the robust control invariant tube. Hence, the nominal trajectory and tube geometry are optimized simultaneously to improve the control performance by augmenting the tube geometry dynamics in open-loop MPC optimization. Moreover, the tube can be shrunk to reduce the conservativeness and enhance optimization feasibility by exploiting the tube geometry and tracking error dynamics. In addition, a constraint-tightening method is employed to ensure that the PDG problem satisfies all the constraints while accounting for the uncertainty caused by the disturbance. Finally, numerical case studies and Monte Carlo simulations validate the effectiveness and performance of the proposed strategy.

During the practical applications, the existence of measurement biases brings a great challenge to the attitude control design because they can lead to the mismatched disturbances in the spacecraft attitude kinematics. In this paper, an adaptive dynamic surface control (DSC) scheme is proposed for the attitude tracking of spacecraft under measurement biases. The inertia uncertainties, external disturbances, and actuator saturation are also included. A key idea of the proposed adaptive DSC scheme is that the adaptive updating laws are utilized in each step of the control design to estimate the mismatched and matched disturbances, respectively. The uniform ultimate boundedness of the whole closed-loop system is theoretically proved. Finally, the effectiveness and advantages of the proposed control scheme are verified through simulations and comparisons.

Due to its significant capability for energy and environmental sustainability, the hybrid laminar flow control (HLFC) shows excellent technical appeal for civil aircraft. To use computational tools to speed up the HLFC design process, it is crucial to accurately predict the transition location and reveal the coupling mechanism of suction control and pressure gradient. We carry out HLFC wing glove flight experiments under different flight conditions. More than 40% chord laminar flow region is maintained for some flight conditions. We then perform numerical simulations based on the $e^N$ method. The good agreement between the deterministic simulation and experimental data indicates that the $e^N$-based method using the critical $N$ factor from natural laminar flow (NLF) can capture Tollmien–Schlichting (TS) instabilities for HLFC under similar conditions. For the HLFC simulation, the suction velocity is determined using an algebraic model as a boundary-layer condition, which is verified by the test data. We further consider geometric uncertainties to the laminar-to-turbulent transition prediction. We conclude that as long as TS instabilities are fully suppressed in the leading edge region, the variation of stochastic solutions about predicted transition locations is less than 8% chord for most flight conditions. Besides, experimental results locate in the given confidence intervals. For this wing glove test, both deterministic and uncertainty transition prediction results of the HLFC wing by using the critical $N$ factor of TS waves calibrated throught NLF experiment agree with HLFC experiment well.

In this paper, the Lamb wave based structural health monitoring for aluminum bolted joints with multiple-site fatigue damages was investigated experimentally. In the experiments, the Lamb waves were generated and received by piezoelectric transducers. Three specimens were prepared for fatigue testing and monitored by the Lamb wave signals. Due to the complex geometry of the specimen, multiple fatigue cracks were observed at different bolt holes during the fatigue testing. The signals before and after the initiation of the fatigue cracks were compared to calculate the damage index (DI). In general cases, DI can serve an indicator to map the fatigue crack size. In the experiments, it was found that the phase shifts between the baseline signal before fatigue crack initiates, $S_b(t)$, and the signal with fatigue cracks, $S(t)$, can result in an overestimated DI. To solve this problem, the envelope damage index (EDI) was introduced to monitor the fatigue cracks. The envelope curves of $S_b$ and $S$ are considered in the calculation of EDI. In the experiments, three specimens were prepared for fatigue testing. Crack-size observations of the specimens indicated that the EDI is a better parameter than DI for monitoring the fatigue crack size and location.