International Journal of Aerospace Engineering

Volume 2018 (2018), Article ID 9632942, 10 pages

https://doi.org/10.1155/2018/9632942

## Propeller Force-Constant Modeling for Multirotor UAVs from Experimental Estimation of Inflow Velocity

Department of Aerospace and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA

Correspondence should be addressed to Gaurang Gupta; ude.cu.liam@ggatpug

Received 11 July 2017; Revised 6 November 2017; Accepted 6 December 2017; Published 10 April 2018

Academic Editor: Angel Velazquez

Copyright © 2018 Gaurang Gupta and Shaaban Abdallah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Design and simulation of an unmanned aerial vehicle (UAV) highly depends on the thrust produced by a motor-propeller combination. The aim of this paper is to model a generalized mathematical relationship between the motor RPM and the corresponding thrust generated for the preliminary design process of low Reynold’s number applications. A method is developed to determine a generalized mathematical model which relates inflow velocity to coefficient of thrust using experimental data from 291 motor-propeller data points, comprising of input RPM and corresponding output thrust. Using this relationship, the Force Constant is calculated, which defines each Thrust-RPM mathematical model. In the first part, expression of the inflow ratio obtained from Blade Element and Momentum Theory (BEMT) is approximated to a simplified form. In the later part, the proposed mathematical model is validated against two new sets of pairs of motor-propeller combinations. A special note in the Appendix talks about the application of this mathematical model. The computed results are found to be in good agreement with the experimental data.

#### 1. Introduction

The last few decades have shown an increase in the usage and development of UAVs. Most of today’s UAVs used for reconnaissance, surveillance, and disaster management missions are powered using an electrical propulsion system which has shown significant improved propulsion efficiency and noise reduction over conventional combustion engine systems. UAVs have proven themselves very useful during the time of disaster management, where they hover in one place to securely drop medical supplies or get visuals of places [1, 2].

Energy density is defined as the amount of energy that a substance or component can store or transform per unit mass of itself. Fuel, which can be burnt, has an energy density of the order of magnitude three to four times more than that of solid-state fuel cell storage like a lithium polymer (LiPo) battery [3, 4]. But the downside of using an engine is that it is highly inefficient and creates a lot of pollution. As studies suggest that the brushless direct-current (BLDC) motors are highly efficient, their usage has become very common in electric UAVs. As the propulsion system of an electric UAV consists of batteries, electronic speed controllers (ESC), motors, propellers, and so forth, a general survey shows that the weight of the propulsion system can account for approximately 50% of the total weight of the entire system as shown in [5–7]. Examples can also be taken from [8–10]. Thus, the optimization of the propulsion system of a UAV becomes a very crucial aspect of UAV design. There can be many other components in the propulsion system but the battery, motor, and propeller have a far more significant impact on the overall system. The other important aspect is to stabilize the UAV. Most of the general control methods are based on thrust force and the angular velocity of the motor assuming a simple parabolic relation. The performance and characteristics of the vehicle depend on the strong interaction between them. The purpose of this paper is not to present an optimization method or an optimal study but rather to study the interaction between these components and generalize them. The results of this paper have a significant application scope in the preliminary design process of any UAV (small or medium sized). This will help save time and drive the costs of the project down by eliminating the necessity of numerous amounts of initial bench tests using different propellers and motors to determine the best combination of propulsion system for the vehicle.

The method structured in this paper starts with estimating the inflow velocity which is simplified using approximations. There exists an alternative approach to estimate the inflow velocity. Commonly known as the * thumb rule*, the geometric characteristics of the propeller are considered at radius of the propeller from the center. The major parameters considered are the twist angle, the width of the propeller, and the airfoil section at that point. All these parameters will have to be calculated or measured from the actual propeller as these specifications are not available in off-the-shelf specifications. *The advantage of the method discussed in this paper is that it uses only the parameters which exist in off-the-shelf available specifications like propeller diameter and pitch.*

The first part of the paper discusses the development method and techniques utilized to simplify the conventional relation between the thrust generated by a propeller and its given RPM which is derived from very famous theories, namely the Blade Element and Momentum Theory. The next section discusses the assumptions undertaken to simply the estimation of inflow velocity which is the most important component in estimating the thrust generated by a propeller, using the existing relation. Secondly, *the method used in the paper ensures that the output thrust is a function of only those parameters or a propeller that are available in any off-the-shelf propeller and not the geometric parameters which are used to design a propeller*. The later sections demonstrate the validation of the proposed model and the estimates of error induced due to simplification of the original coefficient of thrust versus RPM relation.

#### 2. Propeller Static Thrust—RPM Modeling

Most of the propeller designs are based on the work of Betz mentioned in [11, 12]. The design principle is based on optimizing the propeller’s geometry for a certain specific operating condition such that the power required for that operation is minimized or can also be understood as maximizing the thrust generated for the given power. The thrust estimation model for a propeller in this study is based on the very famous Blade Element and Momentum Theory (BEMT). This theory helps in estimating the aerodynamic loads developed on the propeller, which can be used to estimate the thrust generated at a given RPM.

##### 2.1. Thrust Model

First, we estimate using the axial momentum theory [13, 14]. The major assumptions are as follows: (i) no rotational motion is imparted to the flow by the propeller disk; (ii) the Mach number is small, so the fluid can be assumed to be incompressible; and (iii) the flow is steady as the propeller is assumed to be a thin disk (of cross-section area *A*), through which air passes, and the induced velocity is assumed to be constant at all points which lie on the same radius.

A simplified model of a propeller stream tube is shown in Figure 1. The disk is assumed to be uniformly loaded, and the velocity of air across the rotor disk is , which is assumed to be uniform across the disk and has the same magnitude before and after the disk [15]. On the rotor disk (Figure 2), at radius *r* from the center, consider a ring with infinitesimal thickness *dr*. It is assumed that this elemental area of the disk is uniformly loaded. Thus, the elemental thrust coefficient, , that is, the nondimensional form of thrust for this elemental area, is
where inflow ratio is defined as the ratio of inflow velocity *v*_{1} to tip velocity Ω*R*; similarly, is defined as the ratio of freestream velocity *V* to tip velocity Ω*R* and as *r*/*R*. Detailed derivation to obtain is explained in Appendix A.