1. Introduction
With the growing demand for urban air mobility (UAM) vehicles, in particular small unmanned multirotor helicopters, new methodologies and tools are needed to calculate the loads generated by the rotors, in order to predict trajectories and noise emitted during flight. Noise exposure and annoyance from these vehicles, which will be operating in suburban and urban areas with high population density, could limit the success of integrating UAM into the transportation system. Its noise features differ from that of existing rotorcraft: a larger number of rotors will be used to lift UAM vehicles rather than the one or two rotors used on conventional helicopters and tiltrotors and, unlike conventional rotorcrafts, the rotors on some UAM vehicles may operate with variable rotational speed and lower tip Mach numbers, using different propulsors for different functions. Although the types of the noise sources remain the same, their relative importance changes and low noise conditions need to be studied [
1,
2,
3]. A crucial aspect is related to the interactional effects between the rotors and with the airframe, which lead the broadband noise components to assume greater importance [
3,
4,
5]. In addition, electric motors that drive the rotors may be an additional noise source for UAM that are not typically present in conventional rotorcraft [
6]. Typically, a hybrid approach is used for propeller tonal noise calculation. This consists of first modeling the blades’ motion and loading using various methodologies and then feeding an acoustic solver, which often consists of one with a solution to the Ffowcs Williams–Hawkings (FWH) equation [
7].
The typical control strategy of a multirotor drone moves away from the traditional cyclic control of helicopters, introducing a differential control logic thanks to the presence of several electronically piloted rotors: each rotor has a variable rotation speed that guarantees the thrust and the moments necessary to perform flight’s maneuvers. Multirotor helicopters are generally modeled as rigid, six degrees of freedom bodies [
8,
9,
10], with the rotors treated as concentrated loads acting on the fuselage. In the existing literature, there are several ways these loads have been calculated, starting from simple relations as a function of the rotation speed of the rotors [
11,
12,
13]. Such models neglect all side and drag forces, as well as pitching and rolling moments and the assumption could affect the representativeness in conditions far from a hover. At the other extreme, a high fidelity solution is given by computational fluid dynamics (CFD), often used to identify complex aerodynamic interactions [
14,
15]. Unfortunately, these analyses have too expensive computational costs and for this reason models with lower fidelity are preferred. For this purpose, blade element theory (BET) [
16,
17,
18] is a widely used solution and it can also be used to derive closed-form expressions for the forces and moments on a rotor [
19], but only under particular assumptions (linear aerodynamics, uniform inflow, and simple blade geometry). Using BET, a strong discriminant is given by the model used for the induced velocity of the rotors. The first to propose a relationship for the calculation of the induced velocity starting from the total thrust generated by the blades was Glauert [
20], modeling a uniform inflow across the rotor disc, that can only be used for preliminary performance predictions of rotorcraft where detailed knowledge about the inflow distribution is not needed. Since Glauert’s work, multiple models [
21] have been developed to remove the restriction of uniform inflow across the rotor disk, up to obtain increasingly complex models, both stationary [
22] and dynamic ones [
23,
24,
25,
26,
27,
28]. The results obtained from these models are so satisfactory that they are widely used in the literature and in rotorcraft simulation codes [
29,
30,
31].
The aim of this work is to derive and experimentally validate a model for both the aeropropulsive and aeroacoustic characterization of a quadcopter drone, evaluating the effects of induced velocity modeling. For this purpose, BET and quasi-stationary aerodynamics are combined with the Peters and He finite state wake model [
23,
24], to feed both the flight dynamics of the drone and the aeroacoustic solver. The proposed methodology is formulated to be compatible with a dynamic system simulation, by developing an easily generalizable model: by modeling the individual rotors and the airframe separately, it allows the study to be extended to other configurations of rotary wing aircraft, not only quadcopter drones but also helicopters and innovative configurations such as those being designed for UAM, with rotors arranged both vertically and horizontally or even tiltable. This way of proceeding is common to other works [
1,
9] but, at least for the part of flight dynamics, in the literature it is difficult to find comparisons with experimental analyses that establish the differences obtained varying the number of the inflow states. The rotor model is customized to the design of the drone blades used in the experimental analysis by using a laser scanner 3D for the geometry and the XFOIL program for the profile aerodynamic coefficients. The forces and moments calculated in this way are stored in lookup tables, feeding a trim tool developed by Aerospace Italian Research Center (CIRA). In this way it is possible to obtain the values of the rotor commands (the four rotational speeds) and the attitude of the aircraft needed to maintain steady flight conditions, to be compared with the experimental results. For the aeroacoustic part, propeller noise is usually divided into a tonal and a broadband component: the latter is closely linked to the presence of turbulence [
3,
4,
5] and is not taken into account in this work. The model is based on the hypothesis of compact noise sources, modelled as monopoles (thickness noise) and dipoles (loading noise) [
7,
32,
33,
34,
35], following the Farassat 1A formulation [
36,
37,
38,
39,
40]. Given accurate blade geometry and blade motion, thickness noise is accurately predicted. For loading noise, the values of the aerodynamic forces acting on the blades are saved at any instant of time, allowing the calculation of the acoustic pressure they generate at any microphone point. The pressure contributions are algebraically added together, taking into account the different arrival times, since each point is at a different distance from the microphone.
The main part of this work concerns the experimental–numerical comparisons for the validation of the models. For the flight dynamics part, the values of the rotor commands and the drone attitudes calculated numerically by the trim tool are compared with those recorded in the experimental analyses, for three different flight conditions (hovering, axial climb, and forward flight) and three different levels of complexity of the Peters and He wake (1, 15, and 45 inflow states models). For the aeroacoustics part, the comparisons are made on the noise spectra, in particular for the tonal noise in correspondence with the blade pass frequency (BPF) (i.e., the number of revolutions per second of the rotor multiplied by the number of blades), as has been done several times in the literature [
6,
41,
42,
43]. In this case the numerical results are obtained considering two levels of complexity of the Peters and He wake (15 and 45 inflow states models), for three different flight conditions: for the first two (hovering and forward flight) the commands and assets of the drone are taken by the experimental acquisitions, in order to evaluate separately the quality of the aeroacoustic solver; for the third one (hovering), the solver takes the numerical data from the flight dynamics simulation, so as to allow a global evaluation of the entire model.
The article is organized to briefly present the theoretical framework used to create the model, so as to give more attention to experimental–numerical comparisons, the results of which are discussed in the conclusions. The appendices contain useful information for the reader to reproduce the results.
2. Simulation Model
The model developed in this work can be divided into two parts: one for the isolated rotors and the latter for the entire body of the aircraft. The simulation is applied to a small quadrotor drone developed by Intel (total weight of 1450 g) and a 3D scanner Hexagon Absolute Arm 7-Axis has been used to obtain the chord and pitch values of the blades (see
Appendix A). To get the aerodynamic coefficients of the blade profile, we used the XFOIL program, loading the section shape at the 75% radial station. According to the operating conditions and geometric properties of the blade, we used a Reynolds number
and five different Mach numbers, from 0 to 0.4. Under the conditions in which the program showed convergence issues (at high angles of attack), the coefficient’s values have been connected with the experimental ones of a NACA 4412 profile, at the same Reynolds number.
2.1. Rotor Model
The forces and moments produced by each rotor are determined using BET and the Peters and He finite state model [
16,
17,
18,
23,
24]. This theory calculates the forces generated by a blade element located at each radial and azimuthal position, and then integrates over the span of the rotor, and sums over the blades. In a conventional rotor, the flapping hinge located near the root of the blade prevents much of the moment loading from being transferred to the aircraft. A stiff rotor, such as those on a typical quadrotor drone, however, readily transmit these moments, and therefore must be accounted for in the analysis. Each rotor has a rotational velocity, and a direction that is either clockwise (CW) or counterclockwise (CCW). The azimuthal angle of a rotor blade (
) is defined to be zero at the aft of the rotor disk, and increases in the direction of rotation. The four rotors are assumed to be identical in their pitch and chord distributions, except for the fact that half are designed to spin clockwise and the other half counterclockwise. On any single blade element, the angle of attack is determined by the blade pitch and the velocity components of relative wind measured in a frame of reference jointed with the blade (
x positive from root to tip,
y positive from leading edge to trailing edge and
z positive down). All the velocity components depend on the linear and angular velocity of the body and on the atmospheric wind: to calculate the y-component of velocity the main contribution is due to the rotational speed of the rotor; the z-component instead is influenced by the local induced velocity generated by the rotor. The three components are reported below, calculated in the blade reference system:
where
u,
v,
w,
p,
q, and
r are the components of the linear and angular velocity of the aircraft,
,
, and
are the coordinates of hub center respect to body axes,
is the radial distance from the hub axis,
is the induced velocity and the signs at the top are valid for a counterclockwise (CCW) rotor, while those at the bottom are for a clockwise (CW) one.
Peters and He Inflow Model
The Peters and He inflow model is a dynamic wake model based on Euler equations, with first-order dynamics in the inflow states [
23,
24]. The induced inflow distribution
is expressed azimuthally by a Fourier series and radially by Legendre functions, as showed in Equation (
4).
where the induced flow distribution at the rotor plane can be truncated at a desired harmonic,
N, and, for each harmonic, a specific number of radial shape function,
, can be chosen. The radial expansion functions
have the following form:
where
In general, the model is truncated in a way that is consistent with blade structural dynamics modeling in both azimuthal harmonic and radial variations. The truncation of the harmonic number depends on the frequency range of the specific problem of interest. Authors have established a table for a proper choice of the number of the inflow radial shape functions based on mathematical consistency of the highest polynomial power of
r for the radial shape function related to each harmonic [
29,
30]. An important feature of the Peters and He inflow model is that the derivatives of the inflow can be defined as a function of the lift distribution.
In this work we will examine three different state models (1 state 0 × 0, 15 states 4 × 4, 45 states 8 × 8), indicated as
where
m is the number of harmonic and
n is the highest power for radial variation. The code written in Matlab for an isolated rotor (i.e., a rotor having a fixed hub, on which wind can blow from any direction) allows obtaining useful information along the rotor blade. First of all, the blades are discretized into 25 radial stations, equally spaced, in the center of which the aerodynamic computational points (ACP) are positioned. After assigning the rotational velocity of the rotor and the wind vector, at each instant of time, starting from zero initial conditions, the Peters and He matrices are calculated and the value of the wake states are obtained (see
Appendix B).
2.2. Drone Model
For the flight dynamic model, short-term atmospheric fights at reduced speed assumptions can be considered valid [
44]. The aircraft dynamics are completely defined by 12 scalar variables, in particular: position and velocity of the center of mass G with respect to an inertial reference system; angular velocity with respect to the center of mass and attitude angles of a reference system jointed with the aircraft with respect to the inertial reference system. By projecting the conservation equations of momentum and angular momentum in the body reference system we have the classic formulation of the equation of motion of a rigid body and the system is completed by the kinematic attitude equations [
16,
17,
18]. The airloads acting on the drone fuselage along the three body axis have been modeled as those provided by an equivalent flat plate area. The equivalent areas have been computed from CAD files of the drone 3D scanning.
The quadrotor helicopter uses two sets of two counter-rotating rotors to achieve trim, with adjacent rotors spinning in opposite directions. The speed of each rotor can be individually controlled to change its thrust and torque, each of which can be calculated in the way described in the previous section. The aircraft subject of the simulation model is inherently unstable and requires a control system to stabilize its dynamics. A classic control logic for the four rotor speeds acts on four axis channels: the first one provides a baseline rotational speed for all the rotors, used to balance the drone’s weight; a “differential pitch RPM” increases the rotational speed on the front rotor and decreasing the speed on the aft rotors causing a nose up pitching moment; similarly, “differential roll RPM” creates a roll right moment by increasing the speed on the left rotors and decreasing the speed on the right ones; lastly, “differential yaw RPM” increases the speed of the rotors spinning counterclockwise and decreases the speed of the rotors spinning clockwise inducing a nose-right yawing torque on the aircraft.
The equations of motion of the quadrotor helicopter are solved for the four control inputs defined above as well as the pitch and roll attitude needed to maintain steady flight conditions. A trim tool developed by CIRA has been used and, to take into account the wake models, different lookup tables have been created: by varying the intensity and the direction of the wind on an isolated rotor at various rotational speeds, using the model described above, the average forces and moments in a rotation period have been calculated, for the three different Peters and He models.
2.3. Aeroacoustics Solver
Two main sources of sound may be associated with a moving propeller blade: the displacement of fluid by the moving body leading to thickness noise, and the moving lift force distribution leading to loading noise. A description of the loading noise is obtained by representing the propeller blade force by an equivalent distribution of point forces, followed by a summation of the respective sound fields. In a similar way, the thickness noise is obtained by discretizing the propeller blade volume by an equivalent collection of volumes, followed by a summation of the respective sound fields. Because the acoustics considered here are likely linear, the effects of all sources may be summed by the observer independently. However, the effect that a body has on the acoustic field from another body (i.e., fuselage and ground) is neglected in that instance. Effectively, the sound propagates to the observer as if no other bodies are present. Aeroacoustics results are obtained in post processing using the numerical implementation of Farassat 1A formulation of the Ffowcs Williams–Hawkings equation, for loading and thickness noise, simplifying the integral considering constant values at each blade station (ACP) [
7,
32,
35,
36,
37,
38,
39,
40].
During the dynamic simulation, the vectors of the aerodynamic forces and the inertial positions and velocities of the blade points are stored at all times, and at the end, these can be used to calculate the total acoustic pressure generated by the four rotors in any microphone position in the inertial space. It is important to underline that the atmospheric wind effects have been modeled only through the load oscillations (calculated by the rotor model), while the convection effect of the mean flow is not taken into account in the aeroacoustic equations used.
4. Conclusions
In the present work, an integrated code for the aeropropulsive and aeroacoustics characterization of a quadcopter drone has been developed and validated through experimental data. The impact of the induced velocity on the attitude and commands in flight conditions has been evaluated using BET and the Peters and He dynamic wake model. It has been found that the 45 states of the Peters and He dynamic model give good adherence to the experimental results obtained for flight dynamics, in particular for the prediction of angular velocities and attitude in trim conditions. On the other hand, models with a few states in atmospheric wind conditions show quite high errors even in simple conditions such as hovering. For the aeroacoustics part, the 15 and 45 states models do not show evident differences between them, and both are able to correctly predict the tonal noise values up to the second BPF since the interactional effects of aerodynamics have not been modeled. In general is possible to observe a good agreement for the first BPF of each rotor and an underestimation for the second one. Since the proposed model is generalizable, the same procedure exhibited can also be applied to other types of rotary-wing aircraft, although new experimental–numerical comparisons should be made to evaluate the effect of the number of wake states. In order to obtain better correlations in the future, more attention will have to be paid to the airloads acting on the fuselage, through CFD simulation and using the drone 3D scanner, to correctly predict the moments acting on the fuselage in forward flight; the effects of elasticity of the blades can be considered to more accurately estimate the angles of attack of the blade sections; models for broadband noise can be added in the aeroacoustics model as well as models of unsteady aerodynamics, instead of only considering tonal noise in the OSPL estimate; new experimental tests may be carried out, in the absence of external disturbances, reducing as much as possible the effect of wind and background noise to obtain better comparisons and more easily generalizable analyses.
In conclusion, the integrated model developed has shown satisfactory results and, if appropriately improved, it is useful to evaluate the impact of design strategies on noise emitted by propellers, such as increasing the number of blades, optimizing the blade airfoil and planform shapes, modifying the rotor-airframe and inter-rotor distances and positions, manipulating the rotor phasing, etc. Furthermore, it is suitable for planning low-noise trajectory for drones in complex scenarios such as the urban one, both in the tactical phase, based on assets and commands acquired online during the flights, and in the strategic phase on the basis of simulated data.