Browse Topic: Mathematical models
This paper deals with the uncertainty estimation of identified frequency and damping trends of whirl flutter modes, obtained by applying system identification methods on experimental data. In particular, two different identification approaches are considered, namely the free-decay analysis by using Matrix Pencil algorithm and the Data-Driven Stochastic Subspace Identification method (SSI), applied to system response to stochastic input. The two approaches lead to as many uncertainty estimation methodologies, both leveraging the bootstrapping statistical process. A full validation procedure is then set up to assess the accuracy of such methods in correctly quantifying the uncertainty of the estimated statistics. To do so, a wing-rotor state-space linear numerical model is used to simulate system response to both dwell and stochastic inputs. The state space numerical system aims to replicate the ATTILA wing-rotor wind-tunnel model, which falls in the framework of Clean Sky 2 European program to investigate the possible occurrence of whirl-flutter instability in tiltrotor configurations. Hence, one of the proposed methodologies is applied to ATTILA experimental data for both modal identification and uncertainty estimation, and the processed flutter trends are reported in a few conditions of particular interest due to the progressive approaching of whirl-flutter condition.
This paper presents a multi-aircraft Markov decision process congestion game to resolve multi-aircraft near midair collisions (NMACs) for small unmanned aerial vehicles (sUAVs). Two key features of this framework are: 1) it leverages the concept of strategic equilibria from game theory to define optimality in multi-aircraft near midair encounters and 2) it extends the existing NMAC metrics to stochastic formulations via the occupancy measure of a Markov decision process. This game-theoretic approach decomposes the classically centralized air traffic control objective to multiple objectives that correspond to each aircraft within the NMAC, and as result, provides an aircraft-centric notion of optimality and safety that is well-suited for distributed conflict resolutions in multi-aircraft NMACs. In addition to modeling multi-aircraft as a game, stochastic metrics that extend the deterministic notions of NMACs are explored. The safety and optimality of the Nash equilibrium multi-aircraft trajectory under a joint NMAC threat is analyzed under different NMAC thresholds and evaluation metrics. Results are simulated numerically for a representative sUAV NMAC geometry.
Enhancing rotor efficiency has been a persistent challenge in the development of micro aerial vehicles (MAV) especially for surveillance and covert operations. This study introduces a new Hybrid Flapping Wing Rotor (Hybrid FWR) configuration inspired by insect's wing flapping mechanics to address the efficiency limitation of traditional rotor designs. Unlike traditional rotary systems that rely solely on rotational motion, the Hybrid FWR combines rotational and flapping motions to significantly enhance lift generation. A comprehensive mathematical model was developed to analyze and predict the optimal aerodynamic performance, demonstrating that the Hybrid FWR configuration achieves a substantial improvement, with a power efficiency increase of up to 2.148-fold compared to conventional micro rotorcraft. Experimental validation was conducted to confirm the theoretical predictions, identifying an optimal hybrid ratio of approximately 0.7, which effectively minimizes aerodynamic resistance during the upstroke phase while maximizing lift during the downstroke. This bio-inspired hybrid approach addresses critical limitations of existing MAV rotors, such as limited operational endurance and range. The findings of this research contribute significantly to the advancement of micro rotorcraft technology, presenting a promising direction for future MAV developments with enhanced flight performance and energy efficiency.
Rotorcrafts frequently operate in environments with severe atmospheric turbulence, for instance transferring people offshore to and from oil rigs as well as operating from and around ships. The presence of high turbulence can deteriorate performance, stability, and controllability of the rotorcraft. Additionally, such challenging conditions also generate loads that both airframe and rotor components must withstand. Following this, it is crucial to consider the impact of these operational atmospheric conditions during rotorcrafts design and development. In this context, numerical models are a fundamental tool to provide an easier and quicker way to explore the operative envelopes of the helicopter compared to performing experimental activities. This paper presents a rotor loads correlation activity between an experimental test designed and carried out by Leonardo Helicopters in which an AW189 helicopter was placed in the wake of a C-27J Spartan aircraft and a multibody structural model built using MSC Adams®. Furthermore, an atmospheric turbulence model is proposed and compared with the wind experimental records with the purpose of defining an analytical tool for the estimation of the wind time histories. The main purpose of the work is to detail the extensive correlation activity and highlight the modeling key ingredients to consider for an accurate load prediction in a turbulent wind environment. The correlation focuses on the effect of the wind during ground operations, considering the rotor at rest or running up/shutting down.
NRC developed a higher-order mathematical model structure of coupled rotor-body flapping dynamics for inflight control applications. The hybrid (rigid body fuselage state and rotating hub rotor state) 8DOF model was developed utilizing explicit measurements from a novel rotor hub state measurement system enabling estimation rotor blade dynamics. The method identified second-order rotor flap dynamics, attitude-rate and rotor flap dynamics response correlation, and response lead of rotor flap dynamics over rigid body dynamics. Reducing implementation resource burdens of past approaches, this novel rotor state measurement and modelling methodology may prove useful in applied development cycles across a spectrum of needs for articulated (helicopter) and non-articulated rotor (tiltrotor, eVTOL) aeromechanics, modelling, monitoring, and operations.
The flow behavior of the two-blade MERIT rotor in hover, focusing on both pre-stall and stall regimes, is investigated through a comprehensive numerical-experimental approach. The study leverages unsteady RANS simulations to compute rotor thrust and power polars and validates them against experimental measurements. Valuable insights are provided into the capabilities of unsteady RANS methods and modern turbulence models for predicting rotor performance across these critical operating conditions. Furthermore, the numerical model incorporates blade deformations by implementing the experimentally measured flap and torsion displacements. A more realistic depiction of the rotor's aerodynamics is provided accounting for the structural deformations of the blades under aerodynamic loads. Highfidelity simulations closely predict the experiments in pre-stall conditions while discrepancies are present when the flow exhibits extended stalled regions. Blade deformations demonstrated to have only a minor effect on thrust and power polars due to the high stiffness of the system.
This paper describes the work performed to determine a 0.999999, 6 nines, reliable fatigue critical component life using field monitored loads. The Tie Bar of the MH-47 is substantiated by Centrifugal Force (CF), which is a direct function of rotor speed, Nr, which is a monitored parameter in the Structural Usage Monitoring System (SUMS). Six nines of reliability has been the Army target for component reliability and it is generally assumed that legacy safe-life methods are near this level of reliability. With monitored loads it is possible to develop a statistical model for loads and determine an actual reliability value. This paper presents multiple methods for the Army's first attempt at establishing a retirement time using an absolute component reliability. Reliability is gained using a reduction of the Endurance Limit and mean and standard deviations of binned loads across multiple aircraft. Most notably fatigue lives can vary widely if the independent variable reliability contributions are assigned arbitrarily.
The parameters of a Pitt-Peters dynamic inflow model for a rotor undergoing collective inputs were extracted from experimental measurements on a hovering rotor. The four-bladed rotor of 2 m diameter featured straight, untwisted blades and a solidity of σ = 0.010. The nominal trim condition was CT /σ = 0.07 at a speed of 840 RPM. The rotor wake was measured using phase-resolved, 2D, 3-component particle image velocimetry (PIV) over a large region of interest (0.84 m x 0.77 m), and the integrated rotor aerodynamic forces were obtained from simultaneous hub loads measurements. The frequency response of rotor inflow to rotor thrust was found by measuring the system response to a stepped-sine collective input, which included frequencies of 0.2, 0.3, 0.4, 0.6, and 0.7/rev. The thrust amplitude increased with input frequency, reaching 27.4% of the steady thrust at the highest input frequency. The inflow amplitude was 4.3% of the steady inflow at 0.2/rev and decreased to 2.0% at 0.7/rev. A first-order transfer function was fit to the discrete frequency response to compute the parameters of the Pitt-Peters dynamic inflow model. The apparent mass term for a rotor undergoing collective inputs was found to be M11 = 0.0288±4.4% and the gain term was found to be ¯L11 = 0.118±6.3%. The results agreed well with examples in literature extracted from high-fidelity numerical models. However, there were differences in the steady-state gain and the frequency at which the inflow magnitude is attenuated. The methodology for extending the present approach to cyclic inflow components is presented.
This work introduces the use of "global" stochastic models to detect and identify rotor failures in multicopters under different operating conditions, turbulence, and uncertainty. The identification of an extended class of time-series models known as Vector-dependent Functionally Pooled AutoRegressive models, which are characterized by parameters that depend on both forward velocity and gross weight, using scalar or vector aircraft response signals under white noise excitation has been described. A concise overview of the residual based statistical decision making schemes for fault detection and identification of rotor failures is provided. The scalar and vector statistical models, along with residual variance and residual uncorrelatedness methods were validated and their effectiveness was assessed by a proof-of-concept application to aircraft flight for healthy and faulty states under severe turbulence and intermediate operating conditions. The results of this study demonstrate the effectiveness of all the proposed residual-based time series methods in terms of prompt rotor fault detection, although the methods based on Vector AutoRegressive models exhibit improved performance compared to their scalar counterparts with respect to their performance in identifying rotor failures in the post-failure controller compensated state.
The present paper designs and validates a finite element bird model in order to develop a useful tool for the numerical simulation of an aeronautical bird strike event. The bird is simulated in Abaqus/Explicit environment using the Smoothed-Particle Hydrodynamics (SPH) technique. This formulation is a common approach to the problem, as noticed in many published works that investigate the bird strike problem using other solvers such as LS-Dyna. They are taken as a starting point of the present work in the definition of the bird model initial geometry and equation of state. A comparison and a correlation between some experimental tests and their simulations was conduct, in order to develop and validate the bird numerical model. In particular, impacts against rigid targets, such as plates and wedges, and against deformable ones are evaluated. The result is a validated SPH FE bird model, which can be involved in the next bird strike analysis being a reliable numerical tool.
This paper describes an experimental-numerical technique for evaluating the full 6 × 6 stiffness matrices for beams based on measured strains using digital image correlation (DIC). The general formulation makes the method well suited for isotropic beams with simple cross-sectional configurations or beams made of anisotropic materials with complex geometries, as typically exhibited in composite rotor blades. A 2-D finite element code, SectionBuilder, is used to generate a finite element mesh of the cross-section and evaluate the warping field, which is then combined with the experimental strain data to calculate the stiffness matrix. A detailed error analysis is performed to allow for the propagation of the experimental errors into the stiffness calculation and provide an uncertainty quantification for use in comprehensive analysis codes. Experimental results are presented for an isotropic beam and two composite rotor blades. Overall, the stiffness properties from the experimental measurements and numerical models showed good agreement and the experimental measurements were able to capture all the expected non-zero stiffness components.
This paper presents an object-oriented, equation-based framework for multi-engineering modeling of a quadrotor UAV, which includes the rigid body dynamics, simplified aerodynamics, gyroscopic effects, electrical power system and battery losses, and DC motor dynamics. An open-source drone modeling library is introduced by explaining the mathematical models and multi-domain components used to model the drone. Animation and visualization techniques for the drone using CAD models are also introduced and explained. The proposed drone model is simulated under different flight scenarios using motor and power system models with different levels of detail, aiming to provide better means for design and understanding, of multi-engineering aspects of UAVs. This model provides a foundation for future UAV open-source model development, electrified power propulsion design, visualization and interaction, and system identification.
The use of formation flight to achieve aerodynamic benefit as applied to rotorcraft is, unlike its fixed-wing counterpart, an unproven principle. This document presents a proof-of-concept of rotorcraft formation flight through a numerical research study, supported by results from an independent wind-tunnel experiment. In both cases, two helicopters are placed in an echelon formation aligned on the advancing side of the main rotor, though they do not simulate directly comparable flight conditions. The vertical and lateral alignment is varied in order to observe the achievable reductions in main rotor power required during cruise flight. The wind-tunnel experiment data yields an estimated maximum total power reduction for the secondary aircraft of 24%, while the numerical models yield reductions between 20% and 34% dependent on flight velocity. Both experiments predict a higher potential for aerodynamic benefit than observed for fixed-wing formations, which is contributed to the asymmetric upwash profile in the rotor wake. Optimal lateral alignment of both experimental and numerical results is found to feature overlap of the rotor disk areas due to circular area effects. Experimental data shows an optimal vertical alignment of the secondary rotorcraft below the primary, due to wake displacement. This is not present in the numerical simulations as a result of the applied leader wake modeling.
This document addresses measurement uncertainty and consumer risk as they relate to AS8879 thread inspection. It describes the rationale, theory and methodology used to generate the technical content of the AS5870. The document describes how to calculate measurement consumer risk. It documents all of the calculation methods which industry employs today to calculate what is commonly called measurement uncertainty (Appendices A, B, C, D, E and F). These, in turn, are used to calculate measurement uncertainty ratios which are required inputs to calculate measurement consumer risk. Users of this document can apply the information described herein for the evaluation of the capability of their measurements based on the measurement consumer risk. It involves the analysis of the measurement (product) distribution and biases of both the product and measurement system distributions. It protects the consumer from the worst case distribution results.
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