Understanding the fixed pitch RPM-controlled rotor modeling for the conceptual design of UAM vehicles

Abstract

Most of the introduced eVTOL models in the UAM market utilize electrically driven rotors with fixed pitch blades, controlled by their rotational speed. This design approach brings new features and challenges that need to be considered during the conceptual design stage, taking new aspects into account, among them the rotor dynamic response. This paper presents an initial parameterization of the rpm-controlled rotors along with the electric motor, based on the conducted literature research. Several isolated rotor variants are modeled and analyzed in terms of performance, and the dynamic response of the rotor rotational speed. The relations between aerodynamics and inertia are discussed, and their effects on the rotor dynamic response are elaborated on.

1 Introduction

Interest in the novel eVTOL concepts has been significantly increasing over the past years. Currently, there are more than 700 registered eVTOL models in the World eVTOL Aircraft Directory of the Vertical Flight Society [1], introduced in various sub-categories for a broad spectrum of use cases.

In contrast to conventional helicopters characterized by a main rotor-tail rotor arrangement, eVTOL developers strive to design their models equipped with multiple thrust generators driven by an electric propulsion system. Multirotor configuration concepts bring significant benefits, compared to single main rotor concepts. Distributing the load on multiple rotors increases the safety in case of a rotor failure event. In addition, the simple architecture of an electric drive train not only enables carbon-neutral transport, but also provides reduced operation and maintenance costs. Duffy et. al. [2] show in a feasibility study that an electrically driven 4-passenger rotorcraft vehicle can provide reductions in seat per mile costs by about 26% compared to its conventional fuel powered counterpart, regardless of the size and number of the rotors.

The rotors developed for eVTOL vehicles are designed based on two control types: the fixed pitch rpm control or the variable pitch control at constant rpm. Depending on the vehicle architecture and operating conditions, both control types are utilized in various eVTOL models. For example, Volocopter employs the fixed pitch rpm-control on the eighteen distributed rotors of the VoloCity [3] configuration. Archer’s Maker [4] is an example of the usage of both control types, with six tilting, variable pitch rotors mounted in front of the wing for vertical and cruise flight, and six non-tilting, fixed pitch propellers to be used mainly for vertical take-off and landing.

Unlike pitch-controlled rotors, rpm-controlled rotors alter the thrust directly by changing the rotational speed of the rotor. Here, the rotational acceleration is controlled by changing the current flux through the electric motor. In its initial application on smaller-scale drones, this approach concept showed good usability. However, applying rpm control on large-scale rotors comes with new challenges and design constraints. Some of these challenges were already addressed by Silva et. al. [5], such as high blade loading during maneuvers and therefore possible blade stall, concerns about the impacts of increasing tip speed on acoustic performance of the vehicle, and high response time of the rotor to achieve appropriate handling qualities.

At the Institute of Flight Systems at DLR, ongoing studies have been concentrated on researching novel multirotor configurations. To this end, initial activities were performed to understand the flight characteristics of a re-implemented passenger-grade quadrotor configuration [6], later with simple modifications [7]. Currently, new configurations have been studied for medical use cases as part of DLR’s Guiding Concept  4 (LK4), which is an internal research project featuring two design scenarios: S1 - A conventional helicopter for primary search and rescue and S2a - a Medical Personnel Deployment Vehicle.¹ Here, the latter scenario is primarily concentrated on transporting an emergency physician to a deployment location using a multirotor configuration. As depicted in Fig. 1, this configuration relies on four rotors for vertical and low-speed forward flight, supported with two propellers for higher cruise speeds. In this concept, the vertical thrust is distributed on four rotors for lift, while two propellers provide horizontal thrust for forward flight. Wings are used to alleviate the loading of the lifting rotors; the magnitude of this unloading depends on the flight speed.

Fig. 1

Concept illustration for LK4S2 - The medical personnel deployment vehicle

Current design processes developed at DLR are well established to study conventional helicopter configurations with pitch-controlled rotors. A benchmark example is the SAR Helicopter sized for LK4-S1 [8]. In order to extend the design capabilities for eVTOL configurations, it is necessary to understand the general design concept of the rpm-controlled rotors in the first place.

The presented study aims to improve the comprehension of the initial modeling of fixed pitch rpm-controlled rotors during the conceptual design stage by summarizing some aspects based on the literature study. It addresses design and performance boundaries and rotor control response characteristics, showing a glimpse of possible design spaces by varying the selected design parameters. For this purpose, rpm-controlled rotor models in various sizes were parameterized along with an electric motor model. The rotor models were analyzed in terms of performance and motor-coupled dynamic response in isolated rotor conditions.

The paper starts with an overview of the methodology of the rotorcraft conceptual design approach at DLR. This is followed by the introduction of the rotor and the electric motor parameterization. Next, the influence of the design parameters on rotor performance is discussed. The dynamic response of the rotor to the rpm-control is then elaborated by relevant quantities as function of design parameters. The paper is concluded with a summary and outlook.

2 Conceptual design approach

In commonly known practice of today’s aircraft design [9, 10], the design process is broken into three major phases: Conceptual design, preliminary design and detailed design.² In the conceptual design phase, studies are conducted to find answers to fundamental questions emerging on a new design, such as What requirements should be considered?, What should the configuration look like?, How much does it weigh?, What does it cost?, What technologies should be used?, etc. [9]. To find the answers, multiple disciplines are synthesized to determine the external configuration of an aircraft with all its dimensions and performance parameters.

As the conceptual design phase requires a certain amount of time for the designers to reach the level of maturity before proceeding to the next design phase, one can welcome the fact that tailoring enhanced software tools into design processes would significantly reduce the time spent in the research and development. Therefore, many research organizations and companies have developed their own software tools over the years to conduct their rotorcraft design studies more rapidly. One of the early examples of integrating computer aided design and analysis utilities into rotorcraft conceptual design process can be observed by Hughes Helicopters to achieve rapid and more accurate mass estimations in the 1980s [11]. Popular examples of today’s applications are NDARC by NASA [12, 13] and C.R.E.A.T.I.O.N. by ONERA [14]. At the German Aerospace Center, the multidisciplinary design environment IRIS (Integrated Rotorcraft Initial Sizing) has been continuously developed to study not merely the conventional rotorcraft, but also the novel multirotor configurations in pursuit of providing consistent data for the day by day expanding eVTOL community. As an extensive overview of the conceptual design process in IRIS was already given in [15], a more simplified summary (see Fig. 2) is described in following.

Fig. 2

The process flow of a conceptual design study of IRIS

Before initiating the process, the so called Top-Level Aircraft Requirements (TLARs) are set by the user, covering the general specifications, which the vehicle has to satisfy. The TLARs are classified in three groups: technical requirements, mission requirements and performance requirements. The technical requirements represent the boundaries and conditions for the external configuration (e.g., rotor arrangement, rotor disc loading, rotor solidity, number of blades, etc.). The mission requirements, as the name implies, quantify the operational metrics which the vehicle has to fulfill at lowest as design goal (e.g. range, payload and flight altitude). Based on the mission requirements, the mission profile is laid out. The performance requirements comprise the flight conditions, which the vehicle has to be able to fly without exceeding any structural or propulsion limits within the design mission, defining the operational flight envelope (e.g. minimum climb rates, cruise flight speed, maximum load factor). Depending on the selected configuration or the use case, more requirements can be taken into consideration in addition to the mentioned requirement groups.

In the model initialization block, a first instance of the vehicle configuration is created based on the aforementioned TLARs in a low level of detail. The vehicle is configured using a set of simple rules, derived mostly from statistical data. Here, an initial guesstimate is made about the aircraft size, aerodynamics and propulsion. As no iterative loop is executed at this stage, the focus is on the investigation of the strengths and weaknesses of the configuration before proceeding into the primary sizing stage.

The core task of the primary sizing is to calculate the converged design mass of the vehicle based on the mission and performance requirements. Therefore, the configuration is sequentially tailored to a set of sizing rules, which redetermine the characteristic dimensions of the vehicle. For example, the blade radius and the mean blade chord are two main characteristic dimensions. The two considered sizing rules for determining these dimensions are disc loading and blade loading (see Sect. 3.1 for more detail).³ The former sets the relation between the design thrust and the rotor disc area (see Eq. (2)), where the latter describes how much air each blade has to push through to provide the given design thrust (see Eq. (5)). Since the design thrust is also derived from the design mass - that is iteratively recalculated -, the radius and the mean blade chord are also updated in every iteration step regarding the given disc and blade loading.

Once the sizing rules are applied, the new configuration instance is analyzed to determine the fractions of the new design mass. In IRIS, the design mass is considered in three main fractions. These are the vehicle basic empty mass, the fuel mass and the payload mass. During the analysis, the first two fractions are updated. The basic empty mass is broken down into sub-components and recalculated using statistical regression models, while the fuel fraction is obtained through the performance requirements of the vehicle model resulting from aeromechanical trim analysis at each mission segment. Eventually, the new design mass is reached through the sum of these fractions at each iteration. The sizing iteration is repeated without any human interaction until the design mass eventually converges.

Following the converged design mass, the configuration proceeds to the higher order sizing and analysis. The term "higher order" portrays the increasing level of detail in the sizing and analysis process, which considers more component-wise dimension adjustments utilizing physics based modeling and analysis tools in order to achieve higher accuracy in the design mass prediction. Currently, the only higher order discipline integrated within IRIS is the finite element analysis based structural sizing of the rotorcraft airframe [16]. Unlike in primary sizing, the higher order sizing may require user interaction depending on the complexity of pre- or post-processing the analysis (e.g. preparing the airframe structural topology, selecting the finite element mesh-fineness, etc.). Once the design mass has been updated, the configuration instance is passed back to primary sizing to adjust the characteristic dimensions and to recalculate the design mass fractions which are not grasped in the higher fidelity sizing. The iteration between the primary sizing and the higher order sizing is repeated until the design mass is fully converged in both stages.

The configuration instance with fully converged design mass is assessed by the user in the last stage of the conceptual design process. Here, the user checks, inter alia, the plausibility, advantages and disadvantages of the resulting vehicle model. Usually, this instance is accepted as the final result in the conceptual design phase. However, in certain cases, the resulting vehicle model may not promise the most efficient solution, where some trade-offs in the selected TLARs become unavoidable. Consequently, the user revises the requirements (e.g. by selecting different rotor arrangement or lower disc loading) and re-initiates the conceptual design process afterwards.

Although the elaborated conceptual design process covers a collective approach for a vast variety of rotorcraft configurations, the parameterization of the design metrics from TLARs up to the constitutive elements of the aeromechanical trim model varies from configuration to configuration, and therefore has a crucial impact on the sizing. Of particular influence is the prediction of the mass of the energy storage (fuel fraction) needed to meet the mission requirements, when it comes to sizing of rpm-controlled eVTOLs. For example, the design’s peak point performances of the electric motors (e.g. due to strong control inputs for avoidance maneuvers at any flight condition) typically are not reached during the design mission profile. However, the required power for peak performance maneuvers still must be taken into account when it comes to determining the size of the motor, and therefore will have an impact on the dimensions of the converged vehicle.

3 System parameterization

Demonstrated in Fig. 3, an rpm-controlled propulsion system is primarily comprised of a rotor group and an electric propulsion group in its core. The rotor blades are attached to the rotor hub with no hinges. The rotor hub is connected with the electric propulsion group via the rotor shaft. The electric propulsion group is constituted of a brushless direct current (BLDC) electric motor, a motor shaft and a gearbox. For the initial approach, additional constituent elements such as power supply, inverter, cooler and actuators are not taken into account in this study. Furthermore, no engine speed controller is involved, in order to analyze the bare motor dynamics.

Fig. 3

Concept sketch of an electrically driven rpm-controlled propulsion system

The electric motor output torque is transmitted from the motor shaft to the rotor shaft through a gearbox system. As electric motors usually have higher nominal rotational speed than the rotors, gearbox systems are utilized for speed transmission. Here, the kinematic relation between the rotational speeds of the rotor and the motor are expressed as

$$\begin{aligned} \omega = r_g \, \Omega \end{aligned}$$

(1)

where $r_g$ is the gearbox ratio, and $\Omega$ and $\omega$ are the angular speeds of the rotor and motor, respectively. In the absence of speed transmission, the gearbox is not included in the propulsion system, so that the gearbox ratio remains identity, i.e., $r_g=1$. In such cases, the electric motor shaft is attached directly to the rotor hub as the rotor shaft.

Taking the basic momentum theory into account, fixed pitch rotors can easily be modeled using standard design parameters provided in the theory books (e.g. Leishman [17] or Prouty [18]). On the other hand, parameterization of the electric propulsion is a relatively new area of research and one of the main topics in the development of eVTOLs. One of the first electric motor performance modeling studies was introduced by McDonald [19] in 2013 for aircraft conceptual design studies. Duffy et. al. [20] performed a survey for propulsion scaling methods based on the data from off-the-shelf motors, inverters and propellers. A comprehensive sizing method of the electric propulsion system was modeled in NDARC [13] based on the aforementioned studies among others. As for the motor dynamics, a characterization approach was introduced by Malpica et. al. [21] to develop motor speed controllers. This method was used by Maser et. al. [22, 23] to investigate the limits of the controllers while optimizing them to achieve the desired handling qualities levels.

The following sections examine the parameterization of the introduced rotor and electric propulsion groups based on the literature study of the aforementioned references.

3.1 Rotor group

The design parameters that have the highest impact on rotor performance are the disc loading - DL, the number of blades - $N_b$, and the rotor solidity - $\sigma$. The disc loading (see Eq. (2)) defines the relation between the weight and the rotor area with the radius - R, while rotor solidity (see Eq. (3)) defines the relation between the blade area and the rotor disc area for the given blade number - $N_b$ and the mean blade chord - $\overline{c}$.

$$\begin{aligned} DL&=\frac{m\,g}{{\pi }\,R^2} \end{aligned}$$

(2)

$$\begin{aligned} {\sigma }&=\frac{{N_b}\, \overline{c}}{{\pi }\, R} \end{aligned}$$

(3)

While the disc loading is associated with the power required to maintain the momentum of the airflow through the rotor disc, the thrust coefficient - $C_T$ strongly depends on the disc loading for a given air density - $\rho$ and the design blade tip speed - $v_{tip}$ as shown in Eq. (4).

$$\begin{aligned} C_T = \frac{T}{\rho \, v_{tip}^2\, \pi \, R^2} = \frac{DL}{\rho \, {v_{tip}}^2} \end{aligned}$$

(4)

In Eq. (4), the presence of the tip speed during the sizing of an rpm-controlled configuration plays a significant role. Unlike the blade pitch-controlled rotors with constant rotational speed, the rpm-controlled rotors have upper and lower design limits in terms of the critical Mach number and blade stall speed. Usually, the design tip speed is set for the hover case by an average value in a certain spectrum, located within the mentioned tip speed limits including a safety margin. Equation (4) also shows that varying the design tip speed must be compensated by the blade area, given that the thrust remains constant.

The thrust coefficient is often weighed by solidity as

$$\begin{aligned} \frac{C_T}{\sigma } = \frac{T}{\rho \, v_{tip}^2 \, N_{b} \, \overline{c} \, R} \end{aligned}$$

(5)

Eq. (5) is also referred to as the blade loading, giving information about how strong the air is forced down by each blade and how much margin is available until the stall. For conventional helicopters, this value ranges as a design parameter between 0.08 and 0.12 [17]. Given a constant blade tip speed and disc loading, varying the blade loading directly alters the blade area through its mean chord.

As introduced in Sect. 2, the characteristic dimensions of the rotor blade are determined as a result of the sizing process with respect to the disc and blade loading. The relation between the dimensions of the blade are expressed by the blade aspect ratio - $\Lambda = R / \overline{c}$. Figure 4 shows the resulting blade aspect ratio over the disc loading for a three-bladed and a four-bladed rotor model at a design tip speed of $Ma_{tip}=0.5$ and a design mass of $m={125}\hbox {kg}$; the DL is swept from 100${\hbox {N}/\hbox {m}^2}$ to 350${\hbox {N}/\hbox {m}^2}$, while $C_T/\sigma$ is varied from 0.05 to 0.09. Since the rotor thrust is constant for balancing the weight, the DL sweep directly varies the rotor radius. The radius is determined by rearranging Eq. (2) with respect to R. The blade chord is also varied accordingly to satisfy the selected $C_T/\sigma$ by rearranging Eq. (5) with respect to $\overline{c}$. For constant $C_T/\sigma$ and $N_b$, the blade aspect ratio decreases with increasing DL, which reflects either a shorter blade radius or a longer blade chord. In contrast to disc loading, increasing $N_b$ or $C_T/\sigma$ for constant DL requires a higher blade aspect ratio, meaning higher blade radius or lower blade chord.

Fig. 4

Relationships of the blade aspect ratio and disc loading for various blade loading

Another design relation that has to be taken into account is the rotor mass and inertia, which highly depends on the geometry of the blade. The relation between the blade aerodynamics and inertial properties is expressed by the Lock number - $\gamma$, as given in Eq. (6).

$$\begin{aligned} \gamma = \frac{\rho \, C_{l_\alpha } \, \overline{c} \, R^4}{I_b} \end{aligned}$$

(6)

The Lock number describes the ratio between the aerodynamic forces and gyroscopic forces that are proportional to the flapping mass moment of inertia. This parameter ranges between 4 and 12 in the conventional helicopter main rotors [24]. In general, the flap and lead-lag mass moments of inertia are nearly identical and can therefore be used interchangeably. Thus, the Lock number can be selected as a design parameter for the initial prediction of the flap mass moments of inertia - $I_b$. Since the conventional helicopters are operated with constant rotational speed, the main considerations in selecting $\gamma$ is focused on building a rotor blade that is heavy enough to withstand blade flapping and provide a certain level of autorotation characteristics to the rotor, while not overstressing the rotor hub. However, the rpm-controlled rotors do not feature autorotation capabilities, since the blade pitch angle cannot be controlled to use the kinetic energy stored in the rotors for landing maneuvers. Instead, the rpm-controlled rotors have to be able to respond to control inputs of the pilot resulting with a change in the rotor rpm as quick as possible, which indicates the need of lighter rotor blades. This fact has to be taken into consideration while selecting $\gamma$. As the access on such information of the current eVTOLs under development is still limited, a ballpark figure on selecting the Lock number can be taken from handling qualities studies of NASA conducted on the quadrotor configurations for various passenger capacities, where $\gamma$ is ranging between 6.4 and 12.3 [21].

The rotational inertia of the rotor - $I_{r}$ is computed by adding up the flapping inertia of the individual rotor blades and the hub’s rotational inertia - $I_{hub}$ as follows

$$\begin{aligned} I_{r} = N_{b}\, {I_{b}} + I_{hub} \end{aligned}$$

(7)

Fig. 5 shows the rotor rotational inertia over the disc loading for the previously varied blade loading values. Here, two Lock numbers are selected as $\gamma =6$ and $\gamma =4$, while a representative $25\%$ of the summation of all blade inertias is taken as the hub rotational inertia. The rotor inertia decreases with the disc loading and it is inversely proportional with the blade aspect ratio. Higher rotational inertia lessens the agility of the rotors and of the vehicle, consequently. Slender blades with high aspect ratio, low chord length, and therefore low thickness have a higher inertia relative to a blade with a smaller radius carrying the same load. By keeping the blade loading constant, the reduction in the radius would be compensated by increasing the chord length, which leads to a thicker blade if constant thickness to chord ratio is preserved in a simple structural layout. Since the changes in the design tip speed must be compensated by the blade area, such adjustments would also influence the inertia of the rotor in the same way as a result of blade loading variation.

Fig. 5

Relationships of the rotor inertia with disc loading, blade loading and Lock-number

3.2 Electric propulsion group

The parameterization of the electric propulsion relates to both the performance and dynamic response of the system. The peak torque and mass of the electric motor are scaled based on the maximum available performance. The dynamic behavior of the motor depends on specific electrical parameters as well as on the motor inertia. The following subsections summarize how these parameters are estimated during the model initialization.

3.2.1 Performance scaling

The scaling metrics of the electric propulsion are the maximum available power - $P_{max}$ and the peak torque - $Q_{peak}$. For the primary sizing, the maximum available power to be provided by the motor can be taken as the highest value from the trim power curve in horizontal flight or the highest calculated value from the mission profile. The peak torque - $Q_{peak}$ is the highest torque required to achieve peak point power - $P_{peak}$ (previously mentioned as peak point performances in Sect. 2) at the nominal (or specification) motor speed - $\omega _{spec}$, given as

$$\begin{aligned} Q_{peak} = \frac{P_{peak}}{\omega _{spec}} \end{aligned}$$

(8)

Knowing that the peak power is greater than the maximum available power, a rough value for $P_{peak}$ can be selected as a multiple of $P_{max}$, since it is unlikely to accurately predict $P_{peak}$ during the early sizing stages. Hence, the peak torque can be estimated from the regression trends at a base rotational speed - $\omega _{base}$, which is a rated value of the nominal motor speed such that

$$\begin{aligned} \omega _{base}=\frac{P_{max}}{P_{peak}} \omega _{spec} \end{aligned}$$

(9)

An example $Q_{peak}$ trend over $P_{max}$ for various $\omega _{base}$ rates of high thrust-to-weight motors is illustrated in Fig. 12, where the data and trend lines are taken from the NDARC documentation [13].⁴

3.2.2 Motor-specific parameters

The torque moment - $Q_S$ acting on the drive shaft of an equivalent DC-motor at an arbitrary stationary rotational speed is given as

$$\begin{aligned} Q_S = r_g \, K_m \, i_a - r_g^2 \, B\,\Omega \end{aligned}$$

(10)

The first term in Eq. (10) is the output torque of the electric propulsion with $r_g$ - gearbox ratio, $K_m$ - motor torque constant and $i_a$ - motor current, while the second term is the friction torque with B representing the proportionality factor related to the viscous losses of the electric motor. Neglecting the armature inductance, Eq. (10) can be reformulated as introduced in [21]

$$\begin{aligned} Q_S = \frac{c\, {K_e} r_g}{R_a} V_a -\frac{c\, {K_e}^2 {r_g}^2}{R_a} \Omega - {r_g}^2 \, B\,\Omega \end{aligned}$$

(11)

Equation (11) basically extends the first term in Eq. (10) such that the back-electromotive force (EMF) is taken into account as a function of $\Omega$. Here, the additional terms are the back-EMF constant - $K_e$, the equivalent resistance - $R_a$, armature voltage - $V_a$, and the proportionality constant- c. Eq. (10) is no longer calculated by the motor current in Eq. (11) but rather by the supply voltage. Furthermore, taking the relation between the back-EMF constant and the motor torque constant as $K_m=c\, K_e$, both parameters hold identical values [21]. The back-EMF constant is parameterized as

$$\begin{aligned} K_e = \frac{\eta \, \Bigl \lceil {\frac{\sqrt{P_{max}}}{V_{ref}}} \Bigr \rceil \, V_{ref}}{\omega _{spec}} \end{aligned}$$

(12)

where $\eta$ is the motor efficiency and $V_{ref}$ is the reference voltage of the power supply [21]. In Eq. (12), the ceiling function ($\lceil \, \rceil$) in the numerator approximates the number of the power supply units to be serially connected to provide the required electric tension. This approximation can be considered as a numerical proportion of the square root from the maximum available power to the reference voltage [21]. Here, a lithium-ion battery cell with a reference voltage of $V_{ref}={4.2}{\hbox {V}}$ [13] can be considered as a typical power supply unit. Looking at performance maps of commercially available motors [20], the highest efficiency is achieved at a certain speed and torque, normally around $\eta ={95}{\%}$, which can be assumed as the starting value during the initial parameterization. Based on the back-EMF constant, the equivalent armature resistance is calculated as [21]

$$\begin{aligned} R_a = \frac{1-\eta }{\eta }\frac{{\omega _{spec}}^2}{P_{max}}{K_e}^2 \end{aligned}$$

(13)

3.2.3 Mass & inertia

The rotational inertia and mass of the motor are essential for determining the dynamic properties of the coupled rotor-motor system. Based on current design trends for high torque-to-weight motors, a mass estimation can be made using the regression model given by [25]

$$\begin{aligned} W_{m} = 0.5382\, {Q_{peak}}^{0.8129} \end{aligned}$$

(14)

where $W_m$ is the approximated overall mass of the electric propulsion in lb and $Q_{peak}$ is the peak torque of the motor in ft-lb. Note that the additional weight due to the thermal management system and engine speed controller is neglected in Eq. (14). The motor mass is determined in SI by converting $W_m$ from lb to kg as follows

$$\begin{aligned} m_{m} \bigl [{\hbox {kg}}\bigr ]= W_m \bigl [{\hbox {lb}}\bigr ] \, 0.4536\biggl [\frac{{\hbox {kg}}}{{\hbox {lb}}}\biggr ] \end{aligned}$$

(15)

The trend of $m_m$ over $Q_{peak}$ as formulated in Eqs.(14) and (15) can be observed in Fig. 13.

The rotational inertia of the motor - J can only be approximated, since there is no accessible data currently available in the literature. Therefore, [21] suggests estimating J based on an equivalent cylindrical geometry with

$$\begin{aligned} J = \frac{1}{2}m_{m}\biggl (\frac{d_{m}}{2}\biggr )^2f_{m} \end{aligned}$$

(16)

where $m_{m}$ is the mass, $d_{m}$ the external diameter and $f_{m}$ is the inertia factor of the motor. The inertia trends resulting from Eq. (16) as a function of $m_m$ for various $d_m$ are demonstrated in Fig. 14.

4 Performance

The performance of the rotor is quantified by the rotor shaft power - P. Similar to the thrust coefficient, the power is also written in a coefficient-based formulation as follows

$$\begin{aligned} P = C_P \, \rho \, \pi \, R^2 \bigl (\Omega R\bigr )^3 \end{aligned}$$

(17)

The power coefficient - $C_P$ of an overall rotorcraft configuration can be approximated using momentum theory relations for stationary level flight as shown in Eq. (18) [17].

$$\begin{aligned} {C_P} = \frac{\kappa {C_T}^2}{2\sqrt{\mu ^2+\lambda ^2}}+\frac{\sigma C_{d0}}{8}\left( 1+K \mu ^2\right) + \frac{1}{2} \frac{f_{dx}}{\pi R^2} \mu ^3 \end{aligned}$$

(18)

Here, $C_P$ excludes propulsive efficiency and other losses. The first term represents the induced power, the second the profile power and the third the parasite power of the overall configuration at an advance ratio - ${\mu }=v~ \cos {\alpha } / v_{tip}$ and an inflow ratio - $\lambda = (v_i + v~ \sin {\alpha }) / v_{tip}$. Based on this approach, the power calculation is taken into consideration separately for hover and forward flight.

4.1 Hover

The rotor performance at hover is decisive in the sizing of the rotor disc. At this state ($\mu =0$), the disc loading is directly related to the induced power, while the solidity affects the profile power. The effective drag area of the fuselage assembly - $f_{dx}$ and the correction factor for the asymmetric velocity distribution on the rotor - K have no effect here. Hence, Eq. (18) simplifies to

$$\begin{aligned} C_P = \kappa \frac{ {C_T}^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d0}}{8} \end{aligned}$$

(19)

One of the metrics defining the power efficiency at hover is the Figure of Merit - FM, which gives the relation between the ideal power and the total power as shown in Eq. (20).

$$\begin{aligned} FM = \frac{{C_T}^{3/2}/\sqrt{2}}{C_P} = \frac{{C_T}^{3/2}/\sqrt{2}}{\kappa \frac{ {C_T}^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d0}}{8}} \end{aligned}$$

(20)

Given the loss factor associated with the inflow - $\kappa$ and the mean blade airfoil drag coefficient - $C_{d0}$, FM shows how close the total power gets to the ideal power by varying the rotor design parameters. Considering a constant rotor solidity, applying the right hand side of Eq. (4) on Eq. (20) shows that the rotor disc efficiency is directly correlated with DL. The same applies for the rotor solidity, when DL is taken constant. These correlations are depicted in Fig. 6 for various DL and $C_T/\sigma$. Here, it is important to note that the Figure of Merit, without any additional parameters, may misdirect the judgement about efficiency in hover, if the $\kappa$ and $C_{d0}$ in the denominator of Eq. (20) are not consistently selected.

Fig. 6

Figure of Merit for various disc loading (DL) and blade loading ($C_T/\sigma$)

4.2 Forward flight

In stationary forward flight, the rotor system must compensate the aerodynamic drag while maintaining the flight condition. The methodologies for computing the required power of a single swashplate controlled rotor, or of an rpm-controlled rotor do not fundamentally differ. The merely difference is the computation of the advance ratio $\mu$ and the inflow ratio $\lambda$ as the rpm of the rotor changes dependent on the flight condition. Additionally, the advance ratio influences the profile power of the rotors, due to the asymmetric velocity distribution on the rotor disc. This influence is formulated by the correction factor $1+K\mu ^2$ in Eq. (18), where an average value of $K = {4.65}$ is usually taken for $\mu <{0.5}$ [17].

Nine rotor variants were computed for DL and $C_T/\sigma$ combinations for forward flight in wind tunnel mode. The rotor variants were modeled using the blade element method with a 3-state Pitt-Peters dynamic inflow model [26]. The trim equilibria were found using the Newton–Rhapson method. An effective drag area of $f_{dx}={0.23}{\hbox {m}^{2}}$ was applied for all variants. The results for rotor thrust - T, rotor shaft tilt angle - $\theta$, rotor rotational speed - $\Omega$, $C_T$, and P over the horizontal speed - v ranging from 0m/s to 50m/s are illustrated in Fig. 7.

Fig. 7

Trim results over horizontal flight speed for different disc loading and blade loading variants

Since the design mass and the drag area are common, all rotor variants produce the identical amount of thrust within the speed spectrum. The thrust remains constant up to about $v={15}{\hbox {m}/\hbox {s}}$ and then gradually increases. To balance the drag force in forward flight, the rotor tilts towards the flight direction resulting in a negative $\theta$, whose magnitude increases with the horizontal speed. Regardless of the disc loading and blade loading, all $\theta$ lines are quite similar to one another. Looking at $\Omega$, the higher the DL gets, the faster the rotor spins, since the blade pitch cannot be adjusted. The changes in the blade loading do not show a significant effect on the $\Omega$ until 40m/s. The rotors spin slower up to $v={15}{\hbox {m}/\hbox {s}}$, as the blade angle of attack increases due to tilting of the shaft. From this point on, the rotors begin to spin faster, since the generated thrust is increased. The associated $C_T$ changes with respect to forward flight, showing that $C_T$ increases initially up to $v={15}{\hbox {m/s}}$ for all DL and $C_T/\sigma$ cases, mostly due to the reduction in $\Omega$. This occurs because the rotor maintains the thrust by increasing $C_T$ up to ${15}{\hbox {m/s}}$. As the rotor spins up faster, $C_T$ starts to drop. Note that the blade loading reaches its maximum value around $v={15}{\hbox {m/s}}$ along with $C_T$, indicating the critical flight speed for a possible blade stall. While increasing DL shifts this maximum to higher speeds, the maximum blade loading represents a constraint in the rotor design already in hover, in order to provide a blade stall margin in the low-speed forward flight. A theoretical upper limit of $C_T$ can be determined by the relation between the average lift coefficient (assumed to be constant along the blade span) and the blade loading - $\overline{C}_l=6\, C_T / \sigma$ [18]. Taking a maximum airfoil lift coefficient of 1.2, the maximum $\overline{C}_l$ yields a $C_T/\sigma$ verge of 0.2, which is considered as a reasonable upper limit [18]. Other test results show that this limit is concentrated about 0.17 when taking the margins for transient maneuvers and steady turns [18]. The power first decreases with increasing speeds up to about $v={15}{\hbox {m/s}}$, then increases again for all cases. While the blade loading has a small effect on the power in the low speed region, substantial effects can be expected at higher speeds. On the other hand, higher disc loading results in an overall higher rotor power required.

5 Dynamic response

Depending on the maneuver type, demands on rpm-controlled rotors’ actuators may vary. For example, Maser et. al. [23] show that maneuvers along the heave and yaw axes demand higher actuator usage than those along the roll and pitch axes, emphasizing the significance of electric motor modeling for rpm-controlled rotors. Therefore, the propulsion system has to ensure the desired response of the rotor during the most critical maneuvers.

To determine the response characteristics of the propulsion system, the moment equilibrium about the rotor is taken into account as follows

$$\begin{aligned} \frac{\textrm{d}\Omega }{\textrm{d}t} = \frac{Q_A + Q_S}{I_r + J r_g^2} \end{aligned}$$

(21)

where $Q_A$ is the aerodynamic torque acting on the rotor, $Q_S$ is the motor shaft torque introduced in Eqs. (10) and (11), and the terms in denominator are the rotor inertia - $I_r$ and the inertia of the electric propulsion group - J coupled with the square of the gearbox ratio - ${r_g}^2$ (see Eq. (1)).

The dynamic response of an rpm-controlled rotor during its rotational speed transition is identified by the time constant - $T_c$. Pavel [27] shows the direct impact of $T_c$ on an rpm-controlled quadrotor response time and therefore on the overall handling qualities. Given the rotor and motor parameters, $T_c$ can be analytically obtained for hover through the design parameters or through a non-linear time simulation for an arbitrary flight condition. Following sections show both approaches in more detail.

5.1 Analytical approach

Linearizing Eq. (21) with respect to the rotational speed, Eq. (22) describes $T_c$ and its relationship with associated parameters [21].

$$\begin{aligned} \begin{aligned} T_c&= -\frac{1}{\frac{\partial }{\partial \Omega }\Bigl ( \frac{\textrm{d}\Omega }{\textrm{d}t} \Bigr )}\\&=\frac{I_r + J {r_g}^2}{\frac{c {K_e}^2 {r_g}^2}{R_a} - \frac{\partial Q_A}{\partial \Omega } + B {r_g}^2 - \frac{c K_e r_g}{R_a} \frac{\partial V_a}{\partial \Omega } } \end{aligned} \end{aligned}$$

(22)

With the aerodynamic torque derivative - $\partial Q_A / \partial \Omega$ highly depending on the flight condition, a general analytical solution that covers all scenarios is not possible. Nevertheless, $\partial Q_A / \partial \Omega$ can be approximated for hover cases using the design parameters and performance metrics - introduced in Sect. 3.1 and Sect. 4.1 - as given in Eq. (23) [23].

$$\begin{aligned} \frac{\partial Q_A}{\partial \Omega } = \frac{2P}{\Omega ^2} = \frac{2 \pi R^4}{\sqrt{2 \rho }} \frac{\sqrt{DL^3}}{{v_{tip}}^2 {FM}} \end{aligned}$$

(23)

Fig. 8 shows the effect of the disc loading and blade loading on $\partial Q_A / \partial \Omega$. Although the aerodynamic torque derivative exhibits a generally increasing trend against the blade loading, it is more sensitive to changes in DL, than to $C_T/\sigma$, since Eq. (23) is proportional to $R^4$.

Fig. 8

$\partial Q_A / \partial \Omega$ trend for various disc loading and blade loading during hover at sea-level

Applying the right hand side of Eq. (23) in Eq. (22) offers insights for selecting a plausible gearbox ratio. Assuming that $\partial V_a / \partial \Omega$ is negligibly small, the three other terms in the denominator of Eq. (22) must be positive, so that sufficient torque is achieved to reach higher rotational speeds, i.e. $T_c>0$. Assuming the sum of these three terms are greater than 0, rearranging them with respect to $r_g$ shows in Eq. (24) a first relation for determining the lower bound of the gearbox ratio based on the design parameters.

$$\begin{aligned} r_g \overset{!}{>}\ \sqrt{\frac{2 \pi R^4}{\sqrt{2 \rho }} \frac{\sqrt{DL^3}}{{v_{tip}}^2 {FM}} \, \frac{1}{c\, \frac{{K_e}^2}{R_a}+B}} \end{aligned}$$

(24)

Although increasing the $r_g$ helps reducing $T_c$, the selection of ${r_g}$ should be realized with care, as it raises the inertia of the electric propulsion system quadratically, which in turn increases the mass of the propulsion system.

In following, the influence of the rotor design parameters on $T_c$ are highlighted. For this purpose, a propulsion system to lift a design weight of $m\,g = {1.225}{\hbox {kN}}$ at a design tip speed of $Ma_{tip}={0.5}$ is varied in terms of DL and $C_T/\sigma$ under the mean sea-level conditions in hover mode. A common electric propulsion parameterization is considered with $r_g = {4}$, $J{r_g}^2 = {0.473}{\hbox {kg}\, \hbox {m}^2}$, $K_e = {1.2}{\hbox {Vs}}$, $R_a = {0.62}{\hbox {ohms}}$, $B = {0.15}{\hbox {Nm}\,\hbox {s}}$. Figure 9 shows the effects of DL and $C_T/\sigma$ on the rotor response $T_c$.

Fig. 9

$T_c$ trend for various disc loading and blade loading during hover at sea-level

The trend of $T_c$ exhibits a steep drop for low DL levels, converging towards a finite value and thus becomes shallower for higher DL levels. In terms of design characteristics, this trend indicates that rotors with lower disc loading have more sluggish dynamic response compared to those with higher disc loading, for a given constant thrust value. The rapid reduction of the time constant for the disc loading in lower regions is due to the fact that R quadratically decreases with DL, leading to a drastic fall of $I_r$ (see Fig. 5) and $\partial Q_A / \partial \Omega$ (see Fig. 8). Being located at the numerator of Eq. (22), $I_r$ is linearly correlated with the $T_c$. As $\partial Q_A / \partial \Omega$ decreases, the influence of the other terms in the denominator of Eq. (22) exhibit more dominating effects on $T_c$.

5.2 Non-linear time simulation

To give a better perspective for the reflections of the changes in $T_c$ on the rotor response, non-linear time simulations are computed for six rotor variants with the previously stated parameters at hover in wind tunnel mode. In each calculation, the rotational speed is increased with a step input from the initial trim state - $\Omega (t={0}{\hbox {s}})$ to a new steady state - $\Omega _{ss}$. The response of the rotational speed is expressed in the form of feature scaling as shown in Eq. (25).

$$\begin{aligned} {\Delta \Omega }' = \frac{\Omega (t) - \Omega (t={0}{\hbox {s}})}{\Omega _{ss} - \Omega (t={0}{\hbox {s}})} \end{aligned}$$

(25)

In this form, the time constant $T_c$ corresponds to the duration at $\Delta \Omega '=0.632$, such that

$$\begin{aligned} T_c = t_{\Delta \Omega '=0.632} \end{aligned}$$

(26)

Figure 10 shows the changes in the normalized rotational speed in time due to a step input at a constant $C_T/\sigma =0.07$. The full transition of the ${100}{\hbox {N/m}^{2}}$ variant takes an entire second to complete; the exponential decrease in this duration can be observed for the ${200}{\hbox {N/m}^{2}}$ and ${300}{\hbox {N/m}^{2}}$ variants, confirming the trend shown in Fig. 9.

Fig. 10

Time simulation with various DL at $C_T/\sigma ={0.07}{}$

The effect of blade loading $C_T/\sigma$ on the rotational speed response at a constant $DL={225}{\hbox {N/m}^{2}}$ is shown in Fig. 11. Also here, the transition duration is not very sensitive to blade loading, allowing this relationship to be neglected for the selected disc loading.

Fig. 11

Time simulation with various $C_T/\sigma$ at $DL={225}{{N/m}^{2}}$

6 Conclusions

A study was carried out to investigate the modeling of fixed pitch rpm-controlled rotors for passenger-grade UAM vehicles during the conceptual design phase. The modeling concept was implemented as a propulsion system consisting of a rotor group and an electric propulsion group. Multiple rotors were modeled by varying the conventional rotor design parameters, while the methods for the parameterization of the electric propulsion group were presented based on literature study. Eventually, the effects of the design parameters on the rotor performance and dynamic response were investigated. Based on these investigations, following conclusions are noted:

• A basic methodology is currently established for the rpm-controlled rotor modeling in the conceptual design phase of UAM configurations.

• Dynamic response properties of the rpm-controlled rotors can be analytically estimated using design parameters for the hover state.

• There is an upper boundary for the selection of the blade loading, which is constrained by the blade stall.

• Reducing the blade inertia lowers the time constant, quickening the rotor response with respect to the rpm change.

• High disc loading rotors respond quicker to rpm-control inputs than rotors with lower disc loading.

• There is a lower boundary for the selection of the gearbox ratio, below which the motor output torque cannot overcome the aerodynamic torque and other dissipative moments.

This paper presents an initial investigation of the conceptual design activities for the novel eVTOL configurations at DLR. Based on this study, the model initialization and sizing capabilities in IRIS will be extended to design and optimize rotorcraft configurations with rpm-controlled rotors. To achieve a well-tempered sizing of the overall vehicle configuration along with sub-systems such as the propulsion system, energy storage, and structure, the dynamic behavior of the rotors must also be taken into account to some extent in the early stages. Therefore, the goal is to build up the sizing procedures, such that not only the trim performance, but also the vehicles’ flight dynamics and response characteristics are taken as metrics. In addition, optimization processes will  be integrated into rotor parameterization to achieve quick and reliable sizing of power-efficient fixed pitch rotor blades. These processes will be used to conduct the design study of the introduced eVTOL multirotor configuration - LK4S2a: the Medical Personnel Deployment Vehicle.

Appendix A Electric motor scaling

See Figs. 12, 13 and 14.

Fig. 12

Peak torque regression over maximum available power for high thrust-to-weight motors with $Q/W >{3.5}{\hbox {lb-ft/lb}}$ or $\approx {1}{\hbox {Nm/N}}$ [13]

Fig. 13

Motor mass regression model in kg over motor peak torque in lb-ft [25]

Fig. 14

Rotational inertia of the electric motors as a function of motor peak torque