Aerodynamic modeling and design optimization of a variable-pitch variable-camber piezocomposite rotor

Abstract

Multi-rotor vehicles either use complex mechanisms to control blade pitch angle or use the rotor speed to control the thrust and torque to attain stability, and control. An alternate approach to attain thrust and torque control is by morphing the blades without the use of complex mechanisms, and without varying the rotor speed. This method may require fewer mechanical components, could lead to higher efficiency, and achieve rapid change in thrust. In this context, a variable—camber, variable-pitch hybrid piezocomposite rotor prototype is modeled, optimized, and experimentally tested on a single degree of freedom test setup. The mechanism-free compliant blades are actuated with the Macro-Fiber Composite actuators. First, the baseline aerodynamic model to predict thrust and torque coefficients is presented. The experiments are used to identify the so-called unknown coefficients of an empirical model, and to identify a physics-based correction formula to account for flow induced deformations on the compliant blades. The refined aerodynamic model is used to conduct parametric analyses to predict the thrust and torque coefficients as a function of pitch and the excitation by the piezocomposite actuators. The model is then used for design optimization leading to several optimal designs based on a limited parameter space. Three objective functions are evaluated: Maximizing thrust, minimizing torque, and maximizing thrust-to-torque ratio. The aerodynamic characteristics of these designs are also presented.

Keywords

active rotors piezoelectric actuator active composites design optimization morphing aircraft morphing wing morphing structures MFC

Introduction

For several decades, researchers have explored the potential of smart materials to create actively controlled surfaces on aircraft. In particular, there has been significant interest in applying this concept to rotorcraft blades by altering the camber to improve aerodynamic performance. A propulsion approach known as the Direct-Drive Solid-State Rotor system has been demonstrated in authors’ prior research (Bilgen, 2016a; Bilgen and Alberts, 2016b), in which wireless power generation and data transmission are achieved through the use of rotating electronics (Carneiro Ferreira de Castro and Bilgen, 2019). The progress toward this research by the authors is in close relationship with significant accomplishments in the literature – relevant works are briefly reviewed below.

The Macro-Fiber Composite and adaptive aerospace structures

The Macro-Fiber Composite (MFC) is a piezocomposite device composed of piezoceramic fibers embedded in a thermosetting polymer (High, 2003; Williams, 2004; Williams et al., 2004). The use of single crystal piezoelectric materials to improve the performance of MFCs was explored by Park and Kim (2005). In related work, Mukherjee et al. (2020) constructed a finite element-based approach to simulate how surfaces deform when actuated by MFCs. Bilgen et al. (2010a) demonstrated the viability of MFCs to actively control the camber of an airfoil for a ducted fan aircraft, showing higher mean lift coefficient can be reached. Additionally, the same research group developed a piezocomposite airfoil capable of bi-directional camber variation with MFCs (Bilgen et al., 2010b). Bilgen et al. (2012) examined how varying the spatial configuration of MFC actuators on an airfoil influenced aerodynamic response. Also, Thornburgh et al. (2014) developed an optimized MFC-actuated continuous trailing-edge flap (CTEF) for helicopter rotor control, while Shen et al. (2016) demonstrated improved aerodynamic performance through active airfoil morphing using CTEF systems. Zhang et al. (2021) presented a computational fluid dynamics (CFD) model to evaluate the feasibility of active flow control for airfoils using MFCs. A thorough study on the application of MFC on various aerodynamic surfaces was carried out by Karthik et al. (2018). Multiple experimental studies have been conducted on morphing airfoil concepts, demonstrating their potential for controlling airfoil camber and modifying the aerodynamic response of wings equipped with such airfoils (Bilgen et al., 2011; Moosavian et al., 2017; Pankonien and Inman, 2013; Woods et al., 2016).

Adaptive rotors and active twist rotors

Initial developments in applying piezoelectric materials for active twist rotor were led by Chen and Chopra (1997). Furthermore, subsequent research focused on employing piezoelectric materials for helicopter rotor blades (Barrett et al., 1998; Barrett and Stutts, 1997). Park and Kim (2008) designed and experimentally validated an active twist rotor. Other active twist concepts were developed by Mok (2004) and Grohmann et al. (2006).

Several studies addressed actuator placement strategies and conducted design optimization to determine the optimal configuration of actuators. Kovalovs et al. (2007), and Barkanov et al. (2008) investigated the optimal placement of MFC actuators on rotor and helicopter blades, respectively. Modeling techniques for active twist using MFCs were explored by Glukhikh et al. (2008). An experimental and numerical analysis for an active twist rotor system for a helicopter blade was provided by (Shin, 2005). Henricks et al. (2020) worked on optimizing the rotor design based on twist, taper, and pitch angle.

Motivation, objectives, and contributions

Multi-rotor vehicles traditionally rely on either complex pitch actuation mechanisms or rotor speed variation to control thrust and torque for achieving stability, and control. While extensive research and significant investment have advanced the field, large rotor systems remain costly and mechanically intricate due to the need for pitch actuation and maintenance requirements. In contrast, small rotors typically use fixed blades and require low maintenance but have relatively low responsiveness. An alternative solution is to use morphing rotor blades to adjust thrust and torque without changing rotor speed or relying on complex actuation. This approach could lead to faster response time, improved efficiency, and reduced mechanical complexity.

In the context above, the authors previously demonstrated a practical mechanism-free rotor (Shah et al., 2024; Shah and Bilgen, 2020). The prior publications addressed a fully-mechanism-free rotor; and demonstrated the limitations of such rotor in producing variation in thrust. In prior research, variable-pitch was not considered; hence, aerodynamic response was limited.

The current paper reports authors’ latest research on the so-called hybrid rotor that combines the mechanism-free variable-camber blades with a conventional swash plate-based variable-pitch rotor hub. The goal is to combine the large-but-slow pitch authority of swash plate mechanism with the fast-but-small response pitch authority of piezoelectric materials. The combined system would have the desired authority over a large band to enable both collective and cyclic control.

The design and development of such hybrid rotor system is highly complicated due to the numerous design parameters and the strong coupling between multiple physical domains and subsystems. Achieving optimal performance requires accurate and fast predictive models to drive the design optimization process. Consequently, this paper presents an aerodynamic model, with a static-aeroelastic correction, that has been validated through experimental testing to obtain thrust and torque coefficient of the hybrid rotor. The system combines: (1) semi-compliant mechanism-free blades with piezocomposite actuators and (2) conventional swash plate for pitch actuation of the blades.

It is important to highlight that the current paper presents a new rotor design by integrating variable pitch into the variable-camber piezocomposite rotor concept. This represents a novel step toward achieving more comprehensive aerodynamic control in terms of amplitude and frequency. The current paper uses theoretical and experimental methods that are fundamentally similar to the authors’ prior research benefiting from prior validation campaigns. However, since the new rotor has unique functionality and geometry, both the theoretical modeling and the experimental setup have critical differences from prior research. The hybrid rotor concept and the associated model is different from authors’ prior research. The results presented are new in terms of theoretical modeling, experiments, system identification, and the consequent design optimization.

Outline of the paper

The following section (2) introduces the Hybrid Rotor concept. The basic propulsion system is described in detail with all the components. Next, in Section 3, the theoretical models used to estimate the performance of the rotor are discussed. Next, Section 4 presents a hybrid prototype, using a carbon fiber blade as rigid section and a stainless-steel sheet as the active trailing section. A single degree of freedom test stand for static thrust tests is presented. Parametric analyses of the hybrid rotor are conducted, and design optimization of the rotor geometry is carried out. Finally, a summary of the results from this research, and a discussion of recommendations for future work is presented.

The hybrid rotor concept and aerodynamic models

This section introduces the proposed hybrid rotor concept, and describes different theoretical models used to analyze aerodynamic performance. Two different theoretical modeling methods are used: an empirical model, and an XROTOR model. The results are compared with each other, compared to the experiments (Section 4), and are used to obtain new designs (Section 5). First, a brief description of the so-called solid-state rotor concept is presented in this section, along with the principal components used in the system. Next, the hybrid rotor concept is presented. Finally, the theoretical models used for parametric analyses and design optimization are presented.

The solid-state rotor concept

The so-called “Solid-State” mechanism-free rotor concept uses MFC actuators to actively control the rotor blade parameters such as camber and angle of attack. This system includes rotating onboard electronics, contactless signal transmission, and piezocomposite actuators that generate uniaxial strain. Figure 1 illustrates the schematic layout of the solid-state rotary hub system (Bilgen and Alberts, 2016). This schematic demonstrates critical components of the system such as the MFC actuators, drive motor, magnet fixture, and the electromagnetic generator.

Figure 1.

The rotary hub schematic with brushless outrunner type drive motor, MFCs, electromagnetic generator, and the permanent magnet fixture (Bilgen and Alberts, 2016).

The electromagnetic generator rotates within a stationary permanent magnet fixture. It is mechanically linked to the drive motor via a hollow drive shaft. The MFC actuators are connected to the onboard high-voltage electronics using the wires running through the hollow drive shaft.

The hybrid rotor concept

The Hybrid Rotor concept, which is the primary contribution of this paper, uses Macro-Fiber Composite actuators to change the shape (e.g. camber, and angle of attack) of the rotor blades. The Hybrid Rotor follows the Solid-State Rotor concept and utilizes similar components to actuate the blades. However, the Hybrid Rotor also utilizes conventional (i.e. mechanical) blade pitch actuations through the use of a swash plate. Figure 2 shows a schematic of the Hybrid Rotor concept. This schematic demonstrates critical components of the system such as the Macro-Fiber Composite actuators, drive motor, hollow rotor shaft, magnet fixture, and the electromagnetic generator. The illustration of the swash plate is omitted.

Figure 2.

An example illustration and implementation of the hybrid rotor concept with drive motor, electromagnetic generator, hollow shaft, and the permanent magnetic fixture. The swash plate is for illustrative purposes only.

Theoretical models

This section presents the theoretical framework developed to model the aerodynamic performance of the hybrid rotor. It begins with a discussion of empirical formulations derived from Blade Element Momentum (BEM) theory, followed by an overview of the modeling approach implemented using XROTOR software.

Empirical model

The empirical model employed in this study is derived from Blade Element Momentum theory (Durand, 1963), which integrates classical momentum theory with blade element theory. In this approach, the propeller blade is discretized into a series of uniformly spaced radial sub sections. Each section is treated as aerodynamically independent, with local flow conditions determining the resulting aerodynamic forces. These forces are then integrated into the overall axial force (thrust) and moment (torque) generated by the rotor. The resulting expressions for thrust and torque are functions of the lift and drag coefficients, fluid density, axial and angular velocities (typically expressed in revolutions per unit time i.e. RPM), as well as the blade’s geometric parameters, including diameter and chord distribution (Durand, 1963), The thrust ( $T$ ) and torque ( $Q$ ) are expressed as follows:

T = f (c_{L}, c_{D}, ρ, V, Ω, D, c)

(1)

Q = f (c_{L}, c_{D}, ρ, V, Ω, D, c)

(2)

where, $c_{L}$ and $c_{D}$ are lift and drag coefficients, $ρ$ is the fluid density, $V$ represents freestream velocity. Furthermore, $D$ is blade diameter, Ω is angular velocity in revolutions per second, and $c$ the rotor effective chord length.

The total thrust and torque generated by the rotor are obtained by integrating the contributions from all radial blade sections. For consistency and to facilitate comparative analysis, thrust and torque are non-dimensionalized. The corresponding non-dimensional coefficients of thrust and torque are defined as:

c_{T} = \frac{T}{ρ Ω^{2} D^{4}},

(3)

c_{Q} = \frac{Q}{ρ Ω^{2} D^{5}} .

(4)

XROTOR model

XROTOR is an open-source computational tool developed by Mark Drela for aerodynamic analysis and design of a wide range of rotor configurations including free-tip, ducted, and windmill rotors (Drela and Youngren, 2003). By providing rotor geometry and relevant flow conditions as inputs, the software computes key aerodynamic performance metrics (Carneiro Ferreira de Castro and Bilgen, 2019). The geometry of the propeller is defined using radial points, each radial point has the span-wise location $(r / R)$ , the chord distribution $(c / R)$ , and the blade angle ( $β_{tip}$ ). The shape of the blade is discretized using “aero sections” defined along the span of the propeller. Aero sections are defined using various aerodynamic properties such as the lift and drag coefficients, moment coefficient, lift curve slope, Reynolds number $(Re)$ , and the critical Mach number. The atmospheric conditions are preloaded in the XROTOR for several different altitudes. Table 1 shows the free stream flow conditions at sea level (altitude = 0 km.)

Table 1.

Free stream flow conditions.

Parameter	Variable	Value	Unit
Speed of sound	$a$	340	m/s
Density	$ρ$	1.226	kg/m³
Viscosity	$μ$	1.78E-05	kg/m-s
Kinematic viscosity	$υ$	1.45E-05	m²/s
Air pressure	$P$	1.01E+05	Pa
Air temperature	$T$	287.7	K

In XROTOR, the blade is segmented into four distinct regions, referred to as “aero sections.” Each section is characterized by 13 aerodynamic parameters including coefficients of lift and drag, zero-lift angle of attack $(α_{L = 0})$ , and lift curve slope $({dC}_{L} / d α)$ . These inputs are used by the software to predict aerodynamic performance under specified flight conditions and operating parameters. Performance metrics such as thrust, torque, power, thrust and torque coefficients and efficiency are evaluated as functions of flight speed, blade pitch angle, and zero-lift angle of attack.

Table 2 summarizes the aerodynamic properties required to define an aero section in XROTOR. The minimum coefficient of drag listed in Table 2 is prescribed based on authors’ previous research on piezocomposite variable camber airfoils (Bilgen et al., 2010a).

Table 2.

Two-dimensional aerodynamic parameters.

Parameter	Value	Source/Reference
Zero-lift angle of attack	$α_{L = 0}$	User specified or identified
Lift curve slope	2π	Thin Airfoil Theory (Jones, 1990)
Post stall lift slope	0.1	Thin Airfoil Theory (Jones, 1990)
Maximum $C_{L}$	2.0	Assumed
Minimum $C_{L}$	0.0	Assumed
$C_{L}$ Increment to stall	0.1	Drela and Youngren (2003)
Minimum $C_{D}$	Section 2: 0.013	Section 2: Bilgen et al. (2010a)
Minimum $C_{D}$	Sections 4,5: Identified	Sections 4,5: Identified
Mach number (at 75% span)	0.0234–0.2333	Calculated from angular speed
$C_{L}$ at Minimum $C_{D}$	0.5	Calculated from equation (5)
$C_{L}$ Weighting coefficient	0.004	Drela and Youngren (2003)
Angular speed range	1000 RPM–10,000 RPM	User specified
Reynolds number	0.931E+6–9.319 E+6	Calculated from angular speed
Reynolds number scaling exponent	−0.5	Drela and Youngren (2003)
Pitch moment coefficient	−0.1	Drela and Youngren (2003)
Critical Mach number	0.8	Assumed

Figure 3 presents an illustration of the two-dimensional aerodynamic parameters, namely the two-dimensional lift and drag coefficients, identified above.

Figure 3.

(a) Coefficient of lift ( $c_{L}$ ) versus angle of attack ( $α$ ) plot, with slope of $2 π$ , $α_{c_{L}} = 0$ , $c_{L, α = 0^{°}}$ and lift increment after stall. (b) Coefficient of drag ( $c_{D}$ ) as a function of angle of attack.

Most of the aerodynamic parameters defining the aero sections remain fixed, with the exception of zero-lift angle of attack $(α_{L = 0})$ , $C_{L}$ at minimum $C_{D} (C_{L_{C_{D, \min}}})$ and $C_{D, \min}$ . $C_{L_{C_{D, \min}}}$ is correlated to the zero-lift angle of attack. The post stall lift refers to the rate of change in lift coefficient ( $C_{L}$ ) with respect to angle of attack near the stall region. The $C_{L}$ increment to stall is the last linear part of the $C_{L}$ curve near the stall region. The $C_{L}$ weighting coefficient captures the quadratic dependence of drag on $C_{L}$ , which corresponds to the slope of the $C_{d}$ and $C_{L}^{2}$ curve. The $C_{L_{C_{D, \min}}}$ is calculated based on the zero-lift angle of attack using the following equation:

C_{L} at minimum C_{D} = {dC}_{L} / d α * (- α_{L = 0} \frac{π}{180}) .

(5)

Changes in blade camber induced by the MFC actuators are incorporated into the model through modifications to zero-lift angle of attack ( $α_{L = 0}$ ) and minimum coefficient of drag ( $c_{D, \min}$ .) A Reynolds number scaling exponent of $- 0.5$ is used, and the critical Mach number is set to $M_{cr} = 0.8$ (Drela and Youngren, 2003). The Reynolds number is calculated using equation (6), where $V$ is the tip velocity of the rotor, $c$ is the effective chord length of the rotor, and $ν$ is the kinematic viscosity. The Reynolds number is given by:

Re = \frac{V * c}{ν} .

(6)

A single Reynolds number near $70 % - 75 %$ of the rotor radius is calculated while using blade element analysis (MacNeill and Verstraete, 2017). Both the Reynolds and Mach numbers in this paper are defined using the blade section at $75 %$ radius, where the chord length is $0.0635 m$ ( $2.5 inch$ ) (Deters et al., 2014).

As mentioned previously, the rotor is divided into four aero sections. The properties of each of the aero sections are described in Table 3.

Table 3.

Aero section properties, and their purpose.

Aero section	Distance from center	Purpose	Characteristics
N/A	0 m(0″)	Start: Hub Center – Dead Zone	$α_{L = 0} = 0^{°}$ ${dc}_{L} / d α = 0$
1	0.0254 m(1″)	End: Hub Finish – Dead Zone Start: Flat Section	$α_{L = 0} = 0^{°}$ ${dc}_{L} / d α = 0$
2	0.0889 m(3.5″)	End: Flat Section Start: Tapered & Cambered Section (variable chord)	$α_{L = 0} = 0^{°}$ ${dc}_{L} / d α = 2 π$
3	0.1016 m(4″)	End: Tapered & Cambered Section (variable chord) Start: Cambered Section	$α_{L = 0} : Calculated$ ${dc}_{L} / d α = 2 π$
4	0.2032 m(8″)	End: Cambered Section	$α_{L = 0} : Calculated$ ${dc}_{L} / d α = 2 π$

Two geometric input parameters are considered in the sections next (i.e. blade section camber and blade root pitch angle,) and the coefficient of thrust ( $c_{T}$ ) and torque ( $c_{Q}$ ) and the thrust to torque ratio ( $c_{T} / c_{Q})$ are considered as the performance metrics. A range of rotor speeds is considered in the analyses.

Static thrust experiments

This section presents the experimental analysis of the hybrid rotor concept that incorporates classical pitch control with piezocomposite camber control. First, the prototype of the hybrid carbon fiber variable camber rotor previously developed by the 2018–2019 Solid State Rotor Senior Design Project is presented. Next, the benchtop experimental setup is presented. Then the experimental procedure to measure the aerodynamic loads generated by the prototype for a range of MFC voltage control and a range of blade root pitch angles is described followed by the analysis of the experimental results.

Prototype

The prototype fabricated and discussed in this section is a combination of a modified rotor blade from a model scale Blade 360 helicopter, and a stainless-steel sheet bonded with several Macro-Fiber Composite actuators. The prototype was designed and developed by the 2018–2019 Solid State Rotor Senior Design Project advised by the co-author (Onur Bilgen.) The blades consist of two main components: a rigid leading section made from carbon fiber composite, and a flexible trailing section consisting of a stainless-steel sheet bonded with three MFC actuators. The rigid section was adopted from the model-scale Blade 360 helicopter carbon fiber composite rotor blade. The blades from the Blade 360 helicopter were cut to result in a $0.2032 m$ ( $8 inch$ ) tip radius from the original 0.3600 m (∼14 inch) tip radius. The inside of the carbon fiber composite blade was originally filled with rigid foam, which was removed using acetone creating a cavity in the blade. The MFC actuators were bonded to a stainless-steel sheet using 3M DP460 two-part epoxy with conventional vacuum bagging techniques. The stainless-steel sheets used are 0.0508 mm ( $2 * 10^{- 3}$ inch) thick. The stainless-steel sheet with the MFC actuators was slid into the cavity of the carbon fiber blade along with the wiring for the MFCs. Critical geometric properties of the rotor are listed in Table 4.

Table 4.

Geometric properties of the hybrid rotor prototype.

Parameter	Value
Number of blades	2
Tip radius	0.2032 m
Root chord (Minimum chord at hub area)	0.0254 m
Tip chord	0.0635 m
Blade twist	0°

An illustration of the design and the fabricated prototype blades are shown in Figure 4. Most parts of the blade are covered using Kapton tape. The tape has three purposes. First, it protects the MFCs from surface damage; second, it helps prevent the stainless-steel sheets and the piezocomposite actuators to detach from the rigid section of the blade in case of bond failure; and third, it helps to balance the two blades with respect to each other.

Figure 4.

(a) Illustration of the hybrid blade prototype, (b) picture after bonding of Macro-Fiber Composite actuators to a stainless-steel substrate, and (c) final blade prototypes.

Benchtop experimental setup

Benchtop experiments are conducted to measure aerodynamic response to variations of blade active-section camber and blade root pitch angle. These experiments are conducted at specified fixed rotor speeds set by the original flight controller of the Blade 360 helicopter.

The blade prototypes described above are attached to the modified hub of the Blade 360 helicopter. The helicopter body was modified by the 2018–2019 Senior Design Team to have a hollow shaft running from the rotor through the drive gear to the rotating electronics package. A permanent magnet fixture holding four magnets was 3D printed and attached to the bottom section of the modified helicopter body. The electromagnetic generator rotates within these permanent magnets and generates power necessary to power the rotating electronics and eventually the MFC actuators. The wires for the actuators are passed through the hollow shaft from the electronics to the actuators on the two blades.

A single degree of freedom (SDOF) test stand was originally developed by the 2018–2019 Senior Design Team, and fully updated by the author. The original test stand is modified to include the capability of conducting static thrust experiments with the inclusion of a load cell. An illustration of the SDOF test stand is presented in Figure 5.

Figure 5.

An illustration of the single degree of freedom test stand, with the solid-state rotary hub system including rotating electronics and permanent magnet fixture: (a) side view, and (b) top view. The swash plate is intentionally omitted.

The flight controller (including a receiver and a speed control functionality) and the electronic speed controller (ESC) on the helicopter is powered using a BK Precision 1788 programmable DC power supply. A constant 14 V is applied to operate the helicopter. A Spektrum DXe transmitter is used to control the collective pitch of the blades and the throttle (i.e. rotor speed) – this transmitter is also referred to the “main” transmitter throughout the paper. An Endurance RC PCTx pulse-width-modulated (PWM) signal generator is used to control the signal that eventually controls the MFC actuators on the blades. First, the PCTx is used to generate PWM signals to control a separate Spektrum transmitter – this transmitter is referred to as the “auxiliary” transmitter throughout this paper. This transmitter sends radio signals to a receiver on the rotating electronics. Through an on-board microcontroller, the signals from the receiver are interpreted, and a low-voltage DC control signal is generated. This DC signal is used to control the DC-DC converter. The high-voltage output (−500 V to +1500 V) of the converter is connected to the MFC actuators. An Omega LC103B-25 strain gauge-based load cell is used to measure thrust force. The capacity of the load cell is 111.2 N (25 lbf.) The load cell is connected to a NI DAQ 9219 multi-function data acquisition card to measure strain and consequently derive the thrust force. The rotor speeds are measured using a Monarch Pocket Laser Tach 200. An illustration of the signal flow of the experiment is presented in Figure 6.

Figure 6.

An illustration of the test setup with the single degree of freedom test stand, and the modified Blade 360 helicopter with rotating electronics and variable camber blades. The swash plate is intentionally omitted.

The benchtop setup is presented in Figure 7, with the test stand and most of the equipment used during the experiments. It is important to note that the picture shows a compact form of the experiment showing the equipment together. During the experiments, the experiment is set up such that the rotor is away from walls to minimize recirculation. The closest wall to the rotor is the base of the benchtop setup – the distance between the rotor plane and the base is 48 cm. which implies a small ground effect. The NI data acquisition system is not shown in the picture.

Figure 7.

(a) SDOF test setup with the modified Blade 360 helicopter, the main Spektrum DXe transmitter to control the rotor speed and pitch angle, load cell, power supply, and the hybrid rotor system. (b) Closer look at the hybrid rotor prototype, electromagnetic generator, permanent magnet holder, hollow shaft, and wires to the MFCs on the blades.

As mentioned before, the MFC actuators on the blades are controlled wirelessly using a transmitter-receiver pair. The auxiliary transmitter is controlled using the PCTx device, which is controlled using a custom LabVIEW program. In this program, the control for the MFC actuators can be varied in the range of $0$ to $255$ . On the other end of the signal path, the voltage range for the DC-DC converter is $- 500 V$ to $+ 1500 V$ . Table 5 shows the mapping of the control signals between the PCTx device and the MFC actuators.

Table 5.

Mapping between PCTx control value, PWM generated by the PCTx (repeated by the auxiliary transmitter and rotating receiver,) and voltage output of the DC-DC converter.

PCTx control, 0–255 (8-bit)	PWM signals, 1100–1900 $μ s$	Voltage output, −500 V to +1500 V
<95	1100	−500
95	1100	−500
125	1300	0
145	1500	500
165	1700	1000
195	1900	1500
>195	1900	1500

It is noted that input values below and above the specific limits of the generation devices are saturated.

Experimental procedure

A detailed description of the experimental procedure is provided here. After turning the equipment on, a 30-min wait is applied to allow the equipment to reach thermal and electrical equilibrium. Rotor speed control is achieved by manual control with the use of the main DXe transmitter, and a Spektrum AR636A AS3X flight controller on the modified helicopter body. The control of the MFC actuators is described in the previous section. The main transmitter is turned on first, then the helicopter (i.e. ESC, drive motor, and flight controller) is powered. First, the weight of the entire metric (i.e. measured load) parts is recorded to tare the loads during thrust experiments. The signals are recorded for both sweep up and sweep down (for the desired variable under consideration) using an NI data acquisition system, and an in-house LabVIEW code. The main transmitter is used to adjust throttle and collective pitch angle. The auxiliary transmitter is used to control the camber of the blades. After changing the desired variable, a wait time of 30 s is applied to stabilize the surrounding air at the current operating conditions. Following this wait time, signals are measured and recorded for 10 s. Finally, the helicopter is powered off before powering off the main transmitter. Experimental variable sweeps are repeated several times to check for hysteresis, and repeatability.

Three different tests are conducted on the SDOF test stand. First a preliminary thrust test is conducted to understand the influence of throttle and blade root pitch angle. Next, an MFC voltage control sweep is conducted to measure the change in thrust with changing excitation to the MFCs, hence the change in camber. Finally, a third thrust test is conducted to fully understand the influence of blade root pitch angle and blade camber. For the blade root pitch control tests, the blades are subjected to various pitch angles ( $0^{°}$ to $16^{°}$ ). The main transmitter is used to control the throttle and the pitch of the blades. A digital protractor is used to map the pitch angles to the pitch stick position on the transmitter when the rotor was static.

Results – Throttle sweep

First, throttle sweep tests are conducted to measure the thrust generated by the helicopter at different throttle stick positions on the main transmitter. A picture of the throttle stick positions used is shown in Figure 8 along with the results from these tests. Throttle control up and down sweeps are conducted to check for hysteresis.

Figure 8.

(a) Illustration of throttle stick positions on the DXe transmitter and (b) experimentally measured thrust as a function of throttle position for MFC control OFF (−500 VDC) condition.

In Figure 8 the throttle up sweep is marked with black markers, and the throttle down sweep is marked with red markers. The mapping between the throttle stick position, rotor speed (RPM), and the blade tip pitch angle (°) is presented in Table 6.

Table 6.

Relation between the throttle stick position on the main transmitter, measured rotor speed (RPM), and the measured blade tip pitch angle.

Throttle stick position	Rotor speed (RPM)	Blade tip pitch angle (°)
0 (minimum position – zero throttle)	0	−2.1
1 (low throttle and near-zero pitch)	3528	−0.3
2 (nominal throttle, or hover throttle)	4221	2.3
3 (nominal throttle and increased pitch)	4223	4.1
4 (nominal throttle and increased pitch)	4217	7.6

It is noted that the coupling between blade pitch angle and throttle is the standard functionality of the Spektrum AR636A AS3X flight controller. This type of coupling is common in model helicopters that have collective control and is not necessarily related to hybrid rotor concept proposed in this paper. Of course, a coupling between rotor speed, blade root pitch, and blade camber inherently exists, and would be part of the flight control architecture. These control aspects are outside of the scope of this paper.

Results – MFC voltage sweep

MFC voltage control sweep is conducted to measure the loads for varying MFC excitation at a constant throttle level. The throttle stick position on the main DXe transmitter is set such that the blade tip pitch angle of the rotor is approximately $2 . 3^{°}$ . The thrust measured as a function of MFC voltage control is presented in Figure 9. There is a significant increase in the load generated at maximum applied voltage as compared to the minimum voltage applied. At $1500 V$ , the DC-DC converter driving the MFCs reach its maximum excitation, and any increase in the control signal by the PCTx device is saturated at $1500 V$ . It is noted that the camber of the morphing surfaces cannot be measured due to the complexity of such measurement; however, it is clear that significant camber is achieved as observed from the large change in trust.

Figure 9.

Experimentally measured thrust as a function of MFC control voltage at constant angular speed (∼4223 RPM) and pitch angle (∼2.3°). Beyond 1500 V the voltage remains constant and the DC-DC Converters, and the MFCs, reach their maximum actuation.

The MFC control sweeps are repeated multiple times to see if there are any effects on the actuating capabilities of the actuators with repeating the tests. As expected with MFC actuators, hysteresis is observed between the up and down sweeps. The maximum difference in the thrust between up and down sweeps is noticed at MFC control voltage of approximately $500 V$ as expected since it is the center value of the excitation range. The rotor speed is measured to be very consistent, and changing MFC voltage control did not affect the angular speed of the rotor – this is achieved because of the speed control function of the flight controller. The measured rotor speed is 4223 ± 1 RPM for the experimental data presented in Figure 9.

Results – Blade root pitch angle sweep

Figure 10 shows the thrust at various blade root pitch angles for so-called MFC control OFF (−500 V) and control ON (+1500 V) states which are the maximum negative and maximum positive amplitudes of the DC-DC converter. Each sweep is conducted twice, showing very good repeatability between experimental runs.

Figure 10.

Thrust as a function of pitch angle for MFC control OFF (−500 V) and ON (+1500 V) conditions.

The rotor speed measured for various pitch angles remained constant for the MFC control OFF and ON states – as previously observed. The rotor speed at $0^{°}$ pitch angle was lower (i.e. 3581 RPM) compared to other pitch angles (i.e. 4223 ± 2 RPM) – also as previously observed. It is noted that these are the result of the functionality of the flight controller and are not related to the system proposed in this paper.

Experimental system identification

A comparison of sweep-up-down averaged experiments (labeled EXP) to the nominal (unmodified) XROTOR model (labeled XRO) for thrust and coefficient of thrust is shown in Figure 11(a) and (b) respectively. The experimental data are presented as markers, and the XROTOR data are presented as solid lines. The XROTOR model is simulated for two conditions: (1) MFC control OFF condition (−500 V) with an initially assumed $α_{L = 0} = - 1 . 677^{°}$ , and (2) MFC control ON condition (+1500 V) with an initially assumed $α_{L = 0} = - 7 . 107^{°}$ . These $α_{L = 0}$ values are identified from experiments conducted at a small-but-non-zero pitch angle (i.e. $4^{°}$ ) to minimize pressure-induced deformation while still having a measurable thrust. It should be noted that the rotor speed recorded in experiments for $0^{°}$ pitch angle is much lower (∼3500 RPM) when compared to other pitch angles (∼4223 RPM.) The experimental results at 0° are not used for model identification and are intentionally not presented in the following plots.

Figure 11.

Comparison of (a) thrust and (b) coefficient of thrust as a function of pitch angle between nominal XROTOR (XRO) model, and the averaged experimental data (EXP). At constant angular speed of 4223 RPM.

Two variations of the XROTOR model are compared to identify the zero-lift angle of attack as a function of pitch angle and rotor speed. System identification is conducted to modify the XROTOR model to account for the flexibility of the trailing (active) sections of the blades. The model variations include: (1) a linear relation of zero-lift angle of attack with the pitch angle, and (2) a power relation of zero-lift angle of attack with the pitch angle. The equations for the linear and the power relations are given by,

α_{L = 0} (L) = (b * pitch + \underset{Pitch = 0}{\underset{︸}{α_{L = 0}}}) + (e^{d * {RPM}^{2}}),

(7)

α_{L = 0} (L) = (b * p i t c h^{c} + \underset{P i t c h = 0}{\underset{︸}{α_{L = 0}}}) + (e^{d * R P M^{2}}) .

(8)

The identified model coefficients for MFC control OFF and ON conditions are presented in the Table 7 for the linear and the power relations. The angular speed is kept constant at $4223 RPM$ to match the average rotor speed observed in the experiments.

Table 7.

Zero-lift angle of attack for zero pitch and coefficients for linear and power equations as a function of pitch and rotor speed squared.

State	Method	$\underset{p i t c h = 0}{\underset{︸}{α_{L = 0}}}$	$b$	$c$	$d$
OFF	Linear	−8.463	0.41	N/A	6.780E-08
OFF	Power	−7.614	0.0679	1.613	6.780E-08
ON	Linear	−14.64	0.41	N/A	6.780E-08
ON	Power	−12.81	0.0679	1.613	6.780E-08

A plot comparing (1) the experimental data (labeled EXP) with the (2) nominal (rigid) XROTOR model (labeled XRO), and with the (3) the modified XROTOR models is presented in Figure 12. The linear relation – flexible blade model is labeled LF, and the power relation – flexible blade model is labeled PF. It is clearly seen that the nominal model with a rigid blade assumption deviates from experimental data at higher pitch angles since the loading increases at those angles.

Figure 12.

Comparison of coefficient of thrust as a function of pitch angle between nominal (rigid) and modified XROTOR models, and averaged experiments. At constant angular speed of 4223 RPM.

The power relation model closely follows the experimental trend of the coefficient of thrust as a function of pitch angle – this is expected because of the well-known quadratic relationship between velocity and pressure. The root mean square errors for the model variations with respect to the experimental data are presented in Table 8.

Table 8.

Coefficient of thrust RMS error between the experiment and the XROTOR models.

State	Nominal (Rigid)	Linear – flexible (LF)	Power – flexible (PF)
OFF	2.845	0.3878	0.1199
ON	3.530	0.3831	0.1294

The root mean square error is calculated using equation (9) where $n$ is the number of data points, $\hat{x_{i}}$ are the experimental values and $x_{i}$ are the analytical values.

x_{RMSE} = \sqrt{\sum_{i = 1}^{n} \frac{(\hat{x_{i}} - x_{i})^{2}}{n}}

(9)

As expected, the best model choice is the power relation as a function of pitch angle, and angular speed squared ( ${RPM}^{2}$ ) after comparing it with the experimental results. This model closely aligns with the experimental results which is also supported by the RMS error analysis presented in Table 8. The resulting equations for MFC control OFF and ON conditions respectively are as follows,

\begin{matrix} OFF α_{L = 0} (L) = (0.0679 * {pitch}^{1.613} + (- 7.614)) \\ + (e^{6.78 * 10^{- 8} * {RPM}^{2}}), \end{matrix}

(10)

\begin{matrix} ON α_{L = 0} (L) = (0.0679 * {pitch}^{1.613} + (- 12.81)) \\ + (e^{6.78 * 10^{- 8} * {RPM}^{2}}) . \end{matrix}

(11)

Single-objective design optimization

A single-objective design optimization is conducted to optimize the geometric parameters of the rotor. Three separate optimizations are considered. The thrust to torque ratio and the coefficient of thrust are the objective function to maximize, and the coefficient of torque is the one to minimize. The chord length, and the zero-lift angle of attack (i.e. camber) are the primary parameters to be optimized. A genetic algorithm optimization procedure is adapted. For this optimization, the blade root pitch angle is assumed to be $0^{°}$ , and the rotor speed is assumed to be $5000 RPM$ . (We note that the experimentally identified model can be used at “any” rotor speed as long as compressibility is negligible.) Table 9 presents the setup of the optimization problem. The geometric properties such as the span and the number of blades listed in Table 4 are kept constant. The aero sections (Table 3) and the two-dimensional aerodynamic properties (Table 2) are also kept constant.

Table 9.

Optimization details for a piezocomposite variable camber rotor with variable pitch.

Parameter	Value
F (objective functions)	$C_{T}$ , $C_{Q}$ or $C_{T} / C_{Q}$
Function optimizer	GA
Model used	Nominal (rigid) XROTOR model
Objectives	Maximize or minimize the objective function
Generations	100
Population	25
Function tolerance	0.001
Velocity	Zero (static thrust)
Blade span	0.2032 m
Angular speed	5000 RPM
Blade root pitch angle	0°
Blade twist	0°
Chord (min. – max. values)	Design variable (12.5% of span–30% of span)
Zero-lift angle of attack (min. – max. values)	Design variable (−10°− 0°)

The angular speed for this optimization is set to a nominal value of $5000 RPM$ . The span is kept at a fixed length of $0.2032 m (8 inch)$ , and the hub radius is constant at $0.0254 m (1 inch)$ . A comparison of the chord distribution of the three optimized designs is presented in Figure 13.

Figure 13.

Comparison of the chord distribution along the span for all three optimized designs.

A summary of the resulting parameters for the provided objective functions are presented in Table 10. Since a GA is used for optimization as opposed to a gradient-based method, the identified optimal design variables will be close (but not exactly match) the theoretical optimal design. This is expected and GA method is utilized purposefully in this paper for its versatility. Table 10 shows that some of the optimal design variables have reached the limit of the domain – this is also expected given physical limitations on those variables.

Table 10.

Summary of resulting geometric parameters for optimized designs for coefficient of thrust, torque, and thrust to torque ratio.

Objective	Max $c_{T}$	Min $c_{Q}$	Max $c_{T} / c_{Q}$
Function value	0.0896	0.001964	12.65
Tip chord	28.16% span	12.5% span	12.66% span
Zero-lift angle of attack	−9.571°	−0.25°	−8.498°

The optimized rotor geometries for each of the objective functions are analyzed with XROTOR for a range of operational conditions. Blade pitch angle sweep, rotor speed sweep, and the zero-lift angle of attack sweep are conducted, and the resulting performance metrics are presented. First, the pitch angle sweep is conducted, and the performance metrics are presented in Figure 14.

Figure 14.

Performance metrics: (a) thrust coefficient, (b) torque coefficient, and (c) thrust to torque ratio, as a function of pitch angle for three optimum designs.

Next, rotor speed is analyzed using XROTOR for the three designs. The rotor speed is ranged from $200 RPM$ to $10, 000 RPM$ . The blade root pitch angle is set to $10^{°}$ for this analysis. Figure 15 shows the performance metrics as a function of rotor speed. The thrust coefficient ( $c_{T}$ ), torque coefficient ( $c_{Q}$ ), and their ratio ( $c_{T} / c_{Q}$ ) were obtained according to equations (10) and (11), as established previously. When the rotor speed reaches ∼10,000 RPM (corresponding to a tip Mach number of ≈0.6), the values of $c_{T}$ , $c_{Q}$ , and $c_{T} / c_{Q}$ exhibit negligible variation with further increases in angular velocity. This convergence indicates that Reynolds-number effects have become minor and that compressibility effects remain below their onset threshold, confirming the expected RPM-independence of these coefficients in the high-speed regime.

Figure 15.

Performance metrics (a) thrust coefficient, (b) torque coefficient, and (c) thrust to torque ratio, as a function of rotor speed for three optimum designs.

Lastly, the zero-lift angle of attack sweep is conducted to estimate the effect of camber change on the rotor performanceas shown in Figure 16. The optimization for coefficients of torque and thrust to torque ratio resulted in very similar chord distributions, which resulted in very close performance metrics for the two designs. The analyses are conducted for a blade root pitch angle of $10^{°}$ .

Figure 16.

Performance metrics (a) thrust coefficient, (b) torque coefficient, and (c) thrust to torque ratio, as a function of zero-lift angle of attack for three optimum designs.

Conclusions and future work

The key contributions of this paper are the conceptual design, static-aeroelastic modeling, design optimization, and experimental validation of a novel variable camber variable pitch piezocomposite rotor that uses mechanism-free blades for camber control, and a conventional swash plate mechanism for blade root pitch control. The system incorporates a unique mechanism-free active control surface to change the aerodynamic properties of the blade. The system uses electronics on the rotor to power Macro-Fiber Composite actuators which in return changes the camber of the blades. An electromagnetic generator and electronics are connected to the drive motor through a common hollow shaft. The hollow shaft is used for the wires to run from the rotating electronics to the MFC actuators on the blades. The rotating electronics include a radio receiver that receives the control signals from a radio transmitter, which is converted into high voltage with the use of a DC-DC converter.

The proposed hybrid rotor concept is presented in detail. Two theoretical models are discussed that are used to predict the performance of the variable camber rotors. Parametric analyses are conducted to estimate the effects of the parameters on the aerodynamic response of the rotor. The parametric analyses showed that there is a direct effect of the selected geometric parameters on the aerodynamic performance of the rotor. Next, a hybrid piezocomposite variable-camber variable-pitch rotor prototype is presented in detail. A single degree of freedom test stand is developed and used to test the hybrid prototype. Several experiments are conducted on the thrust test stand. For example, a variable MFC voltage control test is conducted, where the throttle position is fixed, and the MFC voltage control is varied from $- 500 V$ to $1500 V$ . For these experiments, hysteresis is noticed for the up and down sweeps, as expected. Next, parametric analyses and the design optimization of the rotor geometry is presented for three different objective functions. Performance evaluation for the three optimized designs is conducted as a function of rotor speed, blade pitch angle, and the zero-lift angle of attack.

The future research can focus on multi-point multi-objective design optimization of the hybrid rotor. Preliminary benchtop tests of the hybrid rotor show promising results in generating higher thrust with full actuation of the MFCs. A more detailed geometric model, and the corresponding aerodynamic model, can be developed, and additional design optimizations can be performed.

Footnotes

ORCID iDs

Mohammad Katibeh

Onur Bilgen

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

Data will be made available upon request.*

References

Barkanov

Gluhih

Kovalov

(2008) Optimal design of the active twist for helicopter rotor blades with C-spar. Mechanics of Advanced Materials and Structures 15: 325–334.

Barrett

Frye

Schliesman

(1998) Design, construction and characterization of a flightworthy piezoelectric solid state adaptive rotor. Smart Materials and Structures 7: 422–431.

Barrett

Stutts

(1997) Design and testing of a 1/12th-scale solid state adaptive rotor. Smart Materials and Structures 6: 491–497.

Bilgen

Alberts

(2016a) Solid-state rotor: A demonstration. In: 24th AIAA/AHS Adaptive Structures Conference, San Diego, CA.

Bilgen

Alberts

(2016b) A direct-drive piezocomposite solid-state rotor with contactless power and data transmission. In: SMASIS 2016, Stowe, VT.

Bilgen

De Marqui

Kochersberger

, et al. (2011) Macro-fiber composite actuators for flow control of a variable camber airfoil. Journal of Intelligent Material Systems and Structures 22: 81–91.

Bilgen

Friswell

Kochersberger

, et al. (2012) Surface actuated variable-camber and variable-twist morphing wings using piezocomposites. In: 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Denver, CO.

Bilgen

Kochersberger

Inman

, et al. (2010a) Macro-fiber composite actuated simply supported thin airfoils. Smart Materials and Structures 19: 055010.

Bilgen

Kochersberger

Inman

, et al. (2010b) Novel, bidirectional, variable-camber airfoil via macro-fiber composite actuators. Journal of Aircraft 47: 303–314.

10.

Carneiro Ferreira de Castro

Bilgen

(2019) A piezocomposite solid-state rotor: Theoretical analysis of thrust and efficiency metrics. In: ASME 2019 Conference on smart materials, adaptive structures and intelligent systems, p.V001T04A006.

11.

Chen

Chopra

(1997) Wind tunnel test of a smart rotor model with individual blade twist control. Journal of Intelligent Material Systems and Structures 8: 414–425.

12.

Deters

Ananda Krishnan

Selig

(2014) Reynolds number effects on the performance of small-scale propellers. In: 32nd AIAA applied aerodynamics conference, p.2151.

13.

Drela

Youngren

(2003) XROTOR User Guide. Massachusetts Institute of Technology. Available at: http://web.mit.edu/drela/Public/web/xrotor/xrotor_doc.txt

14.

Durand

(1963) Aerodynamic Theory: A General Review of Progress. P. Smith.

15.

Glukhikh

Barkanov

Kovalev

, et al. (2008) Design of helicopter rotor blades with actuators made of a piezomacrofiber composite. Mechanics of Composite Materials 44: 57–64.

16.

Grohmann

Maucher

Jänker

(2006) Actuation concepts for morphing helicopter rotor blades. In: 25th International congress of the aeronautical sciences, Hamburg, Germany, pp.3–8.

17.

Henricks

Wang

Zhuang

(2020) Small-scale rotor design variables and their effects on aerodynamic and aeroacoustic performance of a hovering rotor. Journal of Fluids Engineering 142: 081209.

18.

High

(2003) Method of Fabricating NASA-Standard Macro-Fiber Composite Piezoelectric Actuators, National Aeronautics and Space Administration. Langley Research Center.

19.

Jones

(1990) Wing theory. Princeton University Press.

20.

Karthik

Guru Prasath

Swathi

(2018) Surface morphing using macro fiber composites. Materials Today Proceedings 5: 12863–12871.

21.

Kovalovs

Barkanov

Gluhihs

(2007) Numerical optimization of helicopter rotor blade design for active twist control. Aviation 11: 3–9.

22.

MacNeill

Verstraete

(2017) Blade element momentum theory extended to model low Reynolds number propeller performance. Aeronautical Journal 121: 835–857.

23.

Mok

(2004) Design optimization of active twist rotor blades. Doctor of Philosophy, The Univerisyt of Michigan.

24.

Moosavian

Chae

Pankonien

, et al. (2017) A parametric study on a bio-inspired continuously morphing trailing edge. Bioinspiration, Biomimetics, and Bioreplication 2017, 1016204.

25.

Mukherjee

Faruque Ali

Arockiarajan

(2020) Modeling of integrated shape memory alloy and macro-fiber composite actuated trailing edge. Smart Materials and Structures 29: 085005.

26.

Pankonien

Inman

(2013) Experimental testing of spanwise morphing trailing edge concept. In: Active and Passive Smart Structures and Integrated Systems 2013, 868815. International Society for Optics and Photonics.

27.

Park

J-S

Kim

J-H

(2005) Analytical development of single crystal macro fiber composite actuators for active twist rotor blades. Smart Materials and Structures 14: 745–753.

28.

Park

J-S

Kim

J-H

(2008) Design and aeroelastic analysis of active twist rotor blades incorporating single crystal macro fiber composite actuators. Composites Part B Engineering 39: 1011–1025.

29.

Shah

Bilgen

(2020) A piezocomposite actuated propeller with an onboard power generator. In: ASME 2020 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, p.V001T04A002.

30.

Shah

Katibeh

Bilgen

(2024) Aerodynamic modeling and design optimization of a variable-camber piezocomposite rotor. AIAA Journal 62: 3391–3402.

31.

Shen

Liu

Thornburgh

, et al. (2016) Design and optimisation of an aerofoil with active continuous trailing-edge flap. Aeronautical Journal 120: 1468–1486.

32.

Shin

(2005) Integral twist actuation of helicolpter rotor blades for vibration reduction. Doctor of Philosophy, Massachusetts Institute of Technology.

33.

Thornburgh

Kreshock

Wilbur

, et al. (2014) Continuous trailing-edge flaps for primary flight control of a helicopter main rotor. In American Helicopter Society (AHS) Annual Forum (No. NF1676L-17681).

34.

Williams

(2004) Nonlinear mechanical and actuation characterization of piezoceramic fiber composites. Doctor of Philosophy, Virginia Polytechnic Institute and State University.

35.

Williams

Inman

Schultz

, et al. (2004) Nonlinear tensile and shear behavior of macro fiber composite actuators. Journal of Composite Materials 38: 855–869.

36.

Woods

BKS

Parsons

Coles

, et al. (2016) Morphing elastically lofted transition for active camber control surfaces. Aerospace Science and Technology 55: 439–448.

37.

Zhang

Nie

Gao

, et al. (2021) Conceptual design and numerical studies of active flow control aerofoil based on shape-memory alloy and macro fibre composites. Aeronautical Journal 125: 830–846.