Micro-Cantilever Flow Sensor for Small Aircrafts

Transcription

1 Micro-Cantilever Flow Sensor for Small Aircrafts Mehdi Ghommem 1, Victor M. Calo 1, and Christian G. Claudel 2 1 Center for Numerical Porous Media (NumPor) Applied Mathematics & Computational Science and Earth Sciences & Engineering King Abdullah University of Science and Technology Thuwal , Kingdom of Saudi Arabia 2 Division of Physical Sciences and Engineering King Abdullah University of Science and Technology Thuwal , Kingdom of Saudi Arabia. Abstract We extend the use of cantilever beams as flow sensors for small aircrafts. As such, we propose a novel method to measure the airspeed and the angle of attack at which the air travels across a small flying vehicle. We measure beam deflections and extract information about the surrounding flow. Thus, we couple a nonlinear beam model with a potential flow simulator through a fluid-structure interaction scheme. We use this numerical approach to generate calibration curves that exhibit the trend for the variations of the limit cycle oscillations amplitudes of flexural and torsional vibrations with the air speed and the angle of attack, respectively. The results show smooth variations of these vibrations with the air speed and angle of attack. Key words: Flow sensor, cantilever beam, nonlinear beam model, unsteady vortex lattice method, fluid-structure interactions. 1 Introduction Unmanned air vehicles (UAVs) are flying systems used to perform environmental monitoring, surveillance, and intelligence missions in confined and urban environments. To achieve successfully these missions, these systems must be designed to fulfill some performance requirements such as high maneuverability and capability to overcome gust and operate in bad weather. As such, they need to be connected to reliable sensing mechanisms. In particular, measuring the air speed and angle of attack is fundamental for controlling these flying vehicles. Typical UAVs are much smaller than their manned counterparts as they do not require space and life support for a pilot. To monitor their airspeed and angle of attack, aircrafts rely currently on Pitot tubes and wind vanes, which are bulky, expensive and cannot be installed everywhere as they might perturb the airflow (wings). This is an important limitation for small aircrafts. They also require pressure lines (Pitot tubes) and drilling the aircraft structure (Pitot tubes and angle of attack sensors). As such, there is a need to develop and design miniaturized 1

2 flow sensors that can be easily implemented on small vehicles and provide accurate measurements of the surrounding flow (without perturbing it) as required for controlling and manoeuvring the flights. Flow sensors have been extensively used in many fields such as environmental monitoring, flight control, and medical instrumentation. For many of these applications, they need to be cost-effective, small, light, and have low power consumption. Micromachining has facilitated the production of flow sensors that fulfill these requirements. A variety of miniaturized flow sensors capable of detecting both the flow rate and direction has been proposed and tested (Lee et al., 29, Liu et al., 212, Ma et al., 26, Ma et al., 28, Ma et al., 29, Kim et al., 24, Que et al., 212, Fei et al., 27). In relation to our application of interest, flight control, Que et al. (212) presented a hotfilm flow sensor array which uses a thermal metallic thin-film deposited on a flexible substrate that is mounted on the surface of the air vehicle. In absence of an incoming freestream (zero flow rate), the thermal element reaches a steady-state temperature that corresponds to the equilibrium of the heat transfer process. As a flow travels around the air vehicle, the thermal element undergoes forced convective cooling. As a result, the temperature of the thermal element varies. This variation yields a change in the resistance of the element which is used to measure the flow speed that governs the cooling rate. Thermal flow sensors have not attracted considerable attention due to their significant power requirements and the difficulty to integrate them with other microscale systems (Kim et al., 24). In this work, we propose new integrated airspeed/angle of attack/sideslip sensors (i.e., flow sensors) using micro cantilevers. Micro-cantilevers have been widely employed in many of the current force sensors (Yeh et al., 28, Hsu et al., 27), bio-sensors (Aboelkassem et al., 21), chemical reactions detectors (Changizi et al., 211), and inertial sensors (Bhadbhade et al., 28, Maenaka et al., 1996, Mohite et al., 26, Acar and Shkel, 24, Esmaeili et al., 26). For our application, we measure the airspeed and the angle of attack at which the air travels across a flying vehicle. The idea is to attach a single cantilever beam at a specific location on the vehicle (e.g., wing). Being subjected to an incoming air stream, the beam undergoes flexural and torsional vibrations that are coupled via geometric nonlinearities and aerodynamic loading. Then, one can measure the beam deflections and extract information about the surrounding flow. We generate calibration curves that show the trend for the variations of the bending and torsion amplitudes with the air speed and the angle of attack. Different transduction principles (Aboelkassem et al., 21, Bhadbhade et al., 28, Liu et al., 212) are available and can be used to generate the readout signal of the beam deflections such as capacitive, piezoelectric, piezoresistive, electrodynamic, and optical. The key benefits of this new sensor are the followings: Compactness: this sensor is much more compact than traditional airspeed sensors (Pitot tubes) and angle of attack sensors (wind vanes) Low drag: sensing the angle of attack on very small airplanes is impractical, as the wind vanes dimensions are comparable to the dimensions of the airplane, which causes additional drag or weight imbalance. The Pitot tubes can also induce significant drag. 2

3 Shorter measurement time constant (for the angle of attack detection) since the inertia of the cantilevers is smaller than the inertia of a wind vane. Short measurement time constant is very important in the context of micro air vehicles, since their inertia is much less than typical larger manned air vehicles. Easiness of integration with other microscale systems: the sensor could be stuck to the surface of the airplane, as opposed to traditional angle of attack sensors that require the drilling of the airframe, potentially weakening it. Small size and low weight: most UAVs size is in the order of few meters (if not centimeters or tens of centimeters) and weigh at most a few kilograms. They are expected to operate in confined spaces and navigate at low altitude over complex terrain. Thus, it is hard to add relatively heavy sensing devices (based on the existing technology) to their small payloads. 2 UAV Flights: Need for New Flow Sensors 2.1 Motivation Air data sensors (airspeed and angle of attack sensors) used in UAVs are typically scaled-down counterparts of sensors used in commercial airplanes, namely Pitot tubes for airspeed measurements and wind vanes for angle of attack (SADS, 212). While Pitot tube scale down easily (the downside being a higher risk of tube blockage and invalid readings), wind vane dimensions are only related to the required angle of attack and aircraft operating speed range, and not to the scale of the aircraft. As such, wind vanes become impractical for small aircrafts. Due to the aforementioned reasons, most UAVs are not equipped with angle of attack sensors. If they are, these sensors lead to a significant degradation in performance, and are not accurate enough for other purposes than stall detection. Thus, the use of UAVs is restricted to missions in which the distance to the ground or to obstacles is large, the weather is good (low turbulence), and the overall stall risk is low. This notably excludes the use of UAVs as low-altitude flood sensing platforms (during inclement weather), or as exploratory vehicles in damaged urban environments following earthquakes. The lack of reliable angle of attack sensors can lead to crashes when margins are low, for instance during slow turns in turbulent conditions. To illustrate the weaknesses of the existing sensors for detecting accurately flow quantities of small aircrafts, we perform a flight test of an UAV developed from a commercial C-17 replica remote controlled plane with four electric ducted fans. The UAV is shown in Figure 1(a). It has a length of 1.8 m, a wingspan of 1.4 m, and a flying weight of 2.2 kg including avionics. Furthermore, it is equipped with a Pitot tube, an altimeter, an inertial measurement unit, and a Global Positioning System (GPS) feeding real time data to a microprocessor. The navigation is performed using Ardupilot Mega 2., an open source autopilot ( based on a A proportional-integralderivative (PID) controller, which feeds commands to the UAV elevators, ailerons and throttle. A description of the complete system is shown in Figure 1(b). As for 3

4 manoeuvring and controlling the UAV flight, a Pitot tube is placed in the nose of the aircraft (see Figure 1(b)). However, the UAV is not equipped with an angle of attack sensor, since the smallest commercially available (DSPM, 213) angle of attack sensor is 4 cm long (about 25 % of the UAV s length) and weighs 272 grams, which corresponds to 6 % of the UAV s useful payload, the maximal take off weight of the UAV being 2.2 kg, for an empty weight of 1.8 kg. 2.2 Need for angle of attack and sideslip sensors The needs for additional air data is illustrated in a crash that took place over a sector of King Abdullah University of Science and Technology (KAUST) campus on January 9, 213. The flight was programmed as a closed circuit of 12 waypoints of 2.5 km, which was to last for about 18 seconds (the takeoff is shown in Figure 1(a), right). This experiment was carried out by Mohamed Abdelkader (PhD student), Mohammad Shaqura (PhD student) and Christian Claudel. During the flight, the aircraft parameters were recorded in an SD card at a sampling rate of 5 Hz, and have been subsequently recovered and analyzed after the crash. The UAV followed the first three waypoints successfully before crashing during a banked turn around the fourth waypoint. The set altitude was around 15 meters above the ground. Turbulence was light to moderate, with peak to peak variations in vertical acceleration on the order of.1 g. Figure 2 illustrates the evolution of some aircraft flight parameters before the crash. We observe that the UAV roll is closely tracking the desired roll commands (lower plot) up to time sample 258. After this time, the UAV fails to track the desired roll angle, becomes inverted (roll exceeding 9 degrees in magnitude), and crashes after a brief inverted flight. The damage of the UAV is consistent with a crash while in inverted flight, since the wheels were intact and the upper fuselage was heavily damaged (the crash occurred beyond visual range, approximately 1 km away from the launch site, and the attitude of the plane while crashing could not be observed directly). The center plot of Figure 2 explains the cause of the crash, in the present case an aerodynamic stall caused by an excessive angle of attack. This plot shows the roll commands sent to the aileron servos (in arbitrary units). The analysis of the lower and center plots of Figure 2 shows that the evolution of the plane in roll is consistent with the roll inputs until time sample 258. After this time, roll slowly increases despite correct counter roll servo commands sent by the autopilot. This inconsistency be caused by two factors: hardware failure, or aerodynamic stall (one of the first consequences of a stall is the loss of aileron control, which can also be followed by aileron reversal). The wings were recovered after the crash and the ailerons were found still attached to the wing, undamaged, with the servos responding normally, which proves that the crash was caused by an aerodynamic stall. The UAV autopilot was the actual cause of the stall, as it tried to maintain the altitude of the UAV following a slight downdraft, during the banked turn. The results of the flight test shown in Figure 2 illustrates the need for accurate angle of attack sensors, and more generally reliable flow sensors for UAV missions, to allow better manoeuvrability and performance (by reducing speed margins), particularly in low and slow operations close to terrain. 4

5 (a) UAV (b) UAV sensors Figure 1: UAV testbed. This figure illustrates some of the navigation sensors used by the UAV. The data of the air stream surrounding the vehicle are obtained from the Pitot tube, which was implemented by drilling through the structure of the nose. Other navigation sensors include the IMU, the GPS as well as a magnetometer and an ultrasound ground proximity sensor (not shown here). 5

6 Figure 2: Data recovered after a crash severely damaged a UAV operated at KAUST. The subplots show the last recorded altitudes, roll servo commands, actual roll and target roll. These results demonstrate that the UAV follows the target roll commands up to time 77s before losing roll authority. 2.3 Expected sensor configuration Given their extremely low weight and size, we expect to use multiple flow sensors of the type proposed in this article on each UAV. This combination would allow an accurate sensing of the two angles between the airflow and the aircraft: the angle of attack (usually denoted by α) and the sideslip angle (usually denoted by β). Both angles are very important in the context of UAV control: excessive angle of attack can cause stalls, while excessive sideslip angle in conjunction can lead to loss of control, or to spins if the angle of attack is also excessive). Both angles are a function of the airflow disturbances (turbulence), and thus need to be measured in real time. The sideslip sensors have to be mounted vertically, while the angle of attack sensors have to be mounted horizontally. In order to facilitate the sensing and protrude as much as possible outside of the boundary layer, the sensors should be mounted in the front of the plane, ideally by pairs on each side (angle of attack), and up and down (sideslip). This would allow an accurate angle of attack and sideslip sensing in all configurations, even if one or both of these angles has an extreme value. For instance, if the angle of attack is too high, the upper sideslip sensor will be in the wake of the fuselage and provide incorrect reading. Some processing would of course be necessary to estimate the true α and β angles in all configurations, by determining which sensors are in the wake of the fuselage (and thus provide invalid readings) given the current α and β. This configuration is also used for angle of attack and sideslip sensing using wind vanes in commercial airliners. 6

7 w v z U α φ y x Figure 3: Schematic of the flow sensor based on a vibrating beam. The beam degrees of freedom comprise two flexural (or bending) components, w(s; t) and v(s; t), along the z and y directions, respectively, and torsional component, φ(s; t). 3 Aeroelastic Modeling We present a flow sensor whose principal component is a cantilever beam (see Figure 3). The flow sensor is expected to provide accurate measurements of the airspeed and angle of attack at which the air travels across a small flying vehicle. In particular, our sensor is intended to be more suitable (in comparison with the current technologies) for small vehicles (e.g., UAVs and micro air vehicles). The operating principle is based on attaching a beam to the vehicle and detecting its vibrations to extract information about the surrounding flow. Interacting with the incoming air stream, the beam undergoes two flexural and torsional vibrations that are coupled via geometric and inertia nonlinearities and aerodynamic loading. Combined nonlinear aspects that contribute to unstable responses of this aeroelastic system could be either of the supercritical or subcritical type (Nayfeh et al., 211). In the supercritical instability, the system response is stable to any disturbance below the flutter boundary (Hopf bifurcation). Beyond this boundary, nonlinearities yield limit cycle oscillations (LCOs) whose amplitude increases slowly with increasing flight speed. In the subcritical type, a sudden jump to a large-amplitude LCO takes place at or below the flutter speed, depending on the initial conditions. This type of instability may lead to the failure of the structure and constitutes a catastrophic behavior. From a design standpoint, one needs to avoid such behavior to take place. As for our flow sensor, the goal is to select the geometric and material properties of a vibrating microbeam that undergoes supercritical instability (moderate behavior) and exploit 7

8 this type of instability for sensing purposes. Furthermore, our study aims at designing a microbeam so that the flutter speed (at which the beam starts to oscillate) is inside the operating range of variations of the airspeed as discussed in Section Structural model The structural model used in this work is a nonlinear displacement-based beam model proposed by Freno and Cizmas (211). This model is derived for inextensional and non-uniform cantilevered beams with a straight elastic axis and without neither overly complex cross-sections nor significant warping. The beam degrees of freedom comprise two flexural (or bending) components, w(s, t) and v(s, t), along the z and y directions, respectively, and torsional component φ(s, t) (see Figure 3). The equations of motion are obtained using the Galerkin method and include up to third-order non-linearities that account for flexural-flexural-torsional coupling. The displacements w and v and torsion φ are approximated by w(s, t) = l w i (t)w i (s), v(s, t) = i=1 m v i (t)v i (s), φ(s, t) = i=1 n φ i (t)φ i (s) (1) where the shape functions, W i s, V i s, and Φ i s correspond to bending and torsional motions, respectively, and are derived from the uncoupled linear equations. The governing equations of the time-varying coefficients w i s, v i s, and φ i s are given by where ) (M L + M NL ẅ v φ + + ) (C L + C NL ) (K L + K NL ẇ v φ w v φ i=1 = Q (2) w = ( w 1 w l ) T, v = ( v1 v m ) T, and φ = ( φ1 φ n ) T and the mass M, damping C, and stiffness K matrices are obtained from numerical integration of the shape functions and beam characteristics such as cross-section, mass distribution, mass moment of inertia, and stiffness over the beam length. Detailed derivation along with the analytical expressions of these matrices are provided in (Freno and Cizmas 211). The linear structural damping matrix C L is considered as proportional to the stiffness matrix (Rayleigh damping). Q is a vector of external forces and moments that may arise due to excitation, inertia (in presence of rigid-body motion), or aerodynamic loading. The aerodynamic forces and moments are usually computed from the pressure distribution over the beam which is the manifestation of the interactions between the beam and the incoming air stream. For the sake of subsequent analyses, we let the time coefficients be represented by X = ( w 1 w l v 1 v m φ 1 φ n ), (3) 8

9 introduce the vector Y = ( X Ẋ ) T, (4) and write the equations of motion in first-order differential form as Ẏ = ( Ẋ ) (M L + M NL ) (Q 1 (C L + C NL ) X (K L + K NL ) X ) (5) or Ẏ = F(Y) (6) To verify the implementation of the beam model, we compare the natural frequencies and forced response of a tapered beam (cross-section is linearly varying along the beam elastic axis) with those obtained by Freno and Cizmas (211) through the commercial finite element software package Abaqus. The tapered beam is assumed homogeneous with a mass density of 271 kg/m 3, a Youngs modulus of 7 GPa, and a modulus of rigidity of 26 GPa. The beam is 1-m long and consists of a 1 m.5 m cross-section at the fixed end and a.5 m.5 m cross-section at the free end. More details on the material and geometric properties are given in (Freno and Cizmas 211). The frequencies corresponding to the first five vibrational modes obtained from the finite element analysis by Abaqus based on a mesh size of are compared with those obtained from the linear contribution to the beam model using five shape functions for each of the three independent displacements. The corresponding results are presented in Table 1. The frequency response predicted by the beam model agrees well with the finite element response as can be inferred from the small values of the absolute errors. Table 1: First five frequencies of tapered beam (flexural modes). Comparison with finite element analysis. Mode Finite element analysis Beam model Error Abaqus current (Hz) (Hz) (%) Next, we consider the forced response induced by a time-varying force applied at the free end. The transient displacement along the y-axis is computed and compared with that obtained by Freno and Cizmas (211) from the nonlinear beam model, considered 9

10 v mode modes modes t 1.8 Figure 4: Beam displacement along y-axis at s = L for different numbers of modes. in this study, and the finite element analysis using Abaqus. Results of y-displacement at the beam tip obtained for varying numbers of modes are shown in Figure 4. As expected due to the inherent nonlinearity, increasing the number of modes affects substantially the prediction of the beam response. The use of five mode shapes for each of the three independent displacements yields convergence of the beam displacement. In Figure 5, we plot the y-displacements at the points located along the elastic axis at s = 3 L, 4 L, 1 2 L, 1 4L, where L is the beam length. Five mode shapes are considered. The results are obtained from the linear and nonlinear settings of each of the models. As would be expected, we observe significant differences between the linear and nonlinear responses (in terms of frequency and amplitude). Except a slight discrepancy in the frequency that can be seen for long times, we remark a good agreement between the results obtained from the nonlinear model and those obtained from the finite element analysis using Abaqus. These results show the capability of the current model to capture the nonlinear aspects of the beam vibrations and reproduce results of higher fidelity solvers. 3.2 Aerodynamic model A potential flow solver based on the unsteady vortex lattice method (UVLM) (Ghommem et al., 212, Ghommem et al., 212, Stanford and Beran 21) is used for the prediction of the unsteady aerodynamic forces and moments. The unsteady vortex lattice method computes the loads generated by pressure differences across the beam surface resulting from acceleration- and circulation-based phenomena. This accounts for unsteady effects such as added mass forces, the growth of bound circulation, and the wake. UVLM applies only to ideal fluids, incompressible, inviscid, and irrotational flows where the 1

11 Current simulations Freno & Cizmas, International Journal of Non-Linear Mechanics, Figure 5: Beam displacement along y-axis at s = L, 4 L, 1 2 L, 1 4L: Comparison of results obtained from our current simulations (upper plot) with those obtained by Freno and Cizmas (211) (lower plot). 11

12 Freestream direction Γ 1,1 Γ 1,2 Γ 1,3 Γ 1,4 Γ 2,1 Γ 2,2 Γ 2,3 Γ 2,4 Z Collocation point b 2,2 c 2,2 X Y Vortex ring with a circulation Γ 2,2 Figure 6: Aerodynamic mesh: vortex lattice sheets that model the boundary layer at the beam surface. separation lines are known a priori. Thus, UVLM requires the fluid to leave the beam smoothly at the trailing edge through imposing the Kutta condition (Preidikman and Mook 2, Hall et al., 21, Wang, 24) and does not cover the cases of flow separation at the leading-edge and extreme situations where strong beam-wake interactions take place. In spite of these restrictions, the use of UVLM remains adequate for the application of our interest (Stanford and Beran 211, Preidikman and Mook 2, Hall et al., 21, Wang, 24). The UVLM solver proceeds as follows (Ghommem et al., 212, Ghommem et al., 212, Ghommem et al., 213): The beam surface is discretized into a lattice of vortex rings. Each vortex ring consists of four short straight vortex segments, with a collocation point placed at its center (see Figure 6). A no-penetration condition is imposed at the collocation points. The normal component of the velocity due to beam-beam interactions, wake-beam interactions, and free-stream velocities is assumed to vanish at each collocation point. Using the Biot-Savart law to compute velocities in terms vorticity circulations Γ yields a linear system of equations that can be expressed as: A be be Γ be = A wa be Γ wa + V n (7) where A be be and A wa be are beam-beam and wake-beam influence matrices, respectively. The vector V n collects the normal component of the velocity at each collocation point due to the beam motion and the incoming freestream. The vectors Γ be and Γ wa stand for the circulations of the vortex elements on the beam and wake, respectively (Ghommem et al., 212). 12

13 Vorticity is introduced to the wake by shedding vortex segments from the trailing edge. These vortices are moved with the fluid particle velocity and their individual circulation remains constant (i.e., Γwa t+ t = Γ t wa). The wake elements have been truncated in the flowfield and load computation and only those which are located within 1 b are accounted for, where b is the beam thickness. This truncation was observed to speed up significantly the simulation while introducing negligible loss in the solution accuracy. To evaluate the aerodynamic loads, we follow the approach given by Katz and Plotkin (Katz and Plotkin 21). First, we compute the local lift and drag contributions of each element (see Figure 6). Then, we transform the elemental lift and drag to the inertial frame. We sum them and we resolve the total force into three components. The lift and drag of the i th row and j th column of the aerodynamic mesh (see Figure 6) are given by (Katz and Plotkin 21, Stanford and Beran 21) and L i,j = ρ b i,j [ D i,j = ρ b i,j [ V i,j (Γ i,j be ( Γ i,j Γi 1,j be ) + c i,j t V i,j ind (Γi,j be ( Γ i,j Γi 1,j be ) + c i,j t be + Γi 1,j be 2 be + Γi 1,j be 2 )] cos(α i,j )(8) ) ] sin(α i,j ).(9) where V i,j is the velocity vector of element (i, j) due to beam deformation (aeroelastic response), ρ is the fluid density, c i,j and b i,j are the chord and span of each element (i, j). V ind is the velocity induced by the streamwise vortex lines of the wing and the wake. V ind is computed by summing the velocity induced by all the segments along the chordwise direction of the bound vortices. At the first row (i.e., m = 1) the m 1 terms are omitted from Equations (8) and (9). α i,j is the angle of attack relative to the freestream direction of element (i, j) and is computed by resolving the velocity V i,j along the tangent and outward normal of each element and then computing the arctangent of the ratio of the two components. The aerodynamic quantities defined in Equations (8) and (9) act along the local lift and drag vectors of each element which are transformed to the fixed inertial frame and summed to obtain the total aerodynamic forces and moments applied on the beam resulting from its interaction with the incoming freestream. Further information concerning the UVLM-based model, as well as verification studies, can be found in (Ghommem et al., 212, Ghommem et al., 212). We express the forcing vector as F aero = where F z and F x are the aerodynamic forces defined in the inertial frame acting along the z- and x-axis directions, respectively, and M is the aerodynamic moment about the elastic axis of the beam. The right-hand side of Equation (2) is F z F x M 13

14 given then by Q aero i = Q aero l+i = Q aero l+m+i = L L L W i F z ds for i = 1,, l V i F x ds for i = 1,, m Φ i M ds for i = 1,, n (1) The above integrals are computed numerically using the Simpson s 1 3 rule. 3.3 Aeroelastic coupling Aeroelastic coupling is performed with an iterative scheme that accounts for the interaction between the aerodynamic loads and the vibrations of the beam. The procedure is based on Hamming s fourth-order predictor-corrector method (Preidikman and Mook 2, Hall et al., 21, Wang, 24, Ghommem et al., 212). This method requires the solution to be known at three previous time steps and the current one. Thus, different schemes are considered to generate the solution at the first three time steps. For the first time step, Euler and modified Euler methods are used as predictorcorrector schemes. For the second time step, Adams-Bash forth two-step predictor and Adams-Moulton two-step corrector schemes are used. For the third time step, Adams- Bashforth three-step predictor and Adams-Moulton three-step corrector schemes are used. All of the above predictor-corrector schemes are based on a combination of explicit and implicit techniques to solve a system of first-order differential equations and are detailed below. An illustration of the algorithmic flow of the aeroelastic coupling is presented in Figure 7. To proceed with the numerical integration procedure, we let t be the time step size and introduce the following variables t j = j t Y j = Y(t j ) Ẏ j = Ẏ(t j) F j = F(Y(t j )) The numerical procedure includes the following steps: At t = t, we use the initial conditions to evaluate the right-hand side Ẏ = F = F(Y ) At t = t 1, we convect the vorticity at the trailing-edge to its new position based on the state of the system at t = t and use the geometry information to compute the pressure distribution over the beam via the unsteady vortex lattice method. The predicted solution, Y 1 p, is computed using the forward Euler method Y 1 p = Y + t F 14

15 Time marching loop Aeroelastic subiteration Predictor-corrector scheme Deformation of the aerodynamic surface grid UVLM solver aerodynamic forces and moment Beam solver Displacements and rotation NO Convergence YES UVLM convect wake Figure 7: Flow chart of the numerical procedure. The predicted solution is corrected using the modified Euler method where k is the iteration number and Y 1 k+1 = Y + t 2 (F1 k + F ) F 1 k = F(Y 1 k ) Y 1 1 = Y 1 p The previous correction is repeatedly applied until is less than a prescribed tolerance ɛ. If e 1 > ɛ, then we set and keep correcting the solution using e 1 = Y 1 k+1 Y1 k Y 1 k+1 = Y 1 k F 1 k = Ẏ1 k+1 = Ẏ1 k If e 1 < ɛ, then we set Y 1 k+1 = Y + t 2 (F1 k + F ) Y 1 = Y 1 k+1 and compute the solution at t = t 2. Ẏ 1 = Ẏ1 k+1 = F 1 = F(Y 1 ) 15

16 At t = t 2, we convect the vorticity at the trailing-edge to its new position based on the state of the system at t = t 1, update the vortex rings of the wake, and use the geometry information to compute the pressure distribution over the beam via the unsteady vortex lattice method. The predicted solution, Y 2 p, is computed using the Adams-Bashforth two-step predictor method Y 2 p = Y 1 + t 2 (3F1 F ) The predicted solution is corrected using the Adams-Moulton two-step predictor method where Y 2 k+1 = Y1 + t 12 (5F2 k + 8F1 F ) F 2 k = F(Y 2 k ) Y 2 1 = Y 2 p The previous update is repeatedly applied until is less than a prescribed tolerance ɛ. If e 2 > ɛ, then we set e 2 = Y 2 k+1 Y2 k and keep correcting the solution using If e 2 < ɛ, then we set Y 2 k+1 = Y 2 k F 2 k = Ẏ2 k+1 = Ẏ2 k Y 2 k+1 = Y1 + t 12 (5F2 k + 8F1 F ) and compute the solution at t = t 3 Y 2 = Y 2 k+1 Ẏ 2 = Ẏ2 k+1 At t = t 3, we convect the vorticity at the trailing-edge to its new position based on the state of the system at t = t 2 and update the vortex rings of the wake, and use the geometry information to compute the pressure distribution over the beam via the unsteady vortex lattice method. The predicted solution, Y 3 p, is computed using the Adams-Bashforth three-step predictor method Y 3 p = Y 2 + t 12 (23F2 6F 1 + 5F ) 16

17 The predicted solution is corrected using the Adams-Moulton Three-step Predictor Method Y 3 k+1 = Y2 + t 24 (9F3 k + 19F2 5F 1 + F ) where F 3 k = F(Y 3 k ) Y 3 1 = Y 3 p The previous update is repeatedly applied until is less than a prescribed tolerance ɛ. If e 3 > ɛ, then we set and keep correcting the solution using If e 3 < ɛ, then we set e 3 = Y 3 k+1 Y3 k Y 3 k+1 = Y 3 k F 3 k = Ẏ3 k+1 = Ẏ3 k Y 3 k+1 = Y2 + t 24 (9F3 k + 19F2 5F 1 + F ) and compute the solution at t = t 4 Y 3 = Y 3 k+1 Ẏ 3 = Ẏ3 k+1 For t = t j = t 4, t 5, t 6..., we convect the vorticity at the trailing-edge to its new position based on the state of the system at t = t j 1 and update the vortex rings of the wake, and use the geometry information at aeroelastic subiteration to compute the pressure distribution over the beam via the unsteady vortex lattice method and then evaluate the aerodynamic forces and moments needed to determine the right-hand side of Equation (2). The predicted solution, Y j p, is computed using the Hammings Fourth-Order Modified Predictor-Corrector Method Y j p = Y j t(2fj 1 F j 2 + 2F j 3 ) The predicted solution is modified using the local truncation error from the previous time step where Y j 1 = Yj p ej 1 e j 1 = Y j 1 k+1 Yj 1 k 17

18 The modified predicted solution is corrected using the correction equation Y j k+1 = 1 8 [ ] 9Y j 1 Y j t(f j k + 2Fj 1 F j 2 ) where F j k = F(Y j k ) Y j 1 = Y j p The previous solution is repeatedly applied until is less than a prescribed tolerance ɛ. e j = Y j k+1 Yj k The local truncation error is estimated for use in the current and next time step The final solution at step j is e j = (Yj k+1 Yj 1 p ) Y j = Y j k+1 ej To calculate the solution at the next time step, we set Y j 4 = Y j 3 Y j 3 = Y j 2 Y j 2 = Y j 1 Y j 1 = Y j e j 1 = e j and repeat previous steps as much as desired. Stopping conditions are a limit on the number of subiterations N s or a maximum value for the error between two successive solutions within one iteration ɛ. In our simulations, N s is set equal to 2 and ɛ is equal to 1 6. During the subiterations, we do not recalculate the position of the wake. We convect the vorticity at the trailing-edge into the wake and update the wake position only once the solution converges within an iteration. 4 Results and Discussion We consider a beam made of silicon and with a uniform square cross section. Material and geometry properties are presented in Table 2. The center mass of the crosssections of the undeformed beam is assumed to lie on the elastic axis. The first natural 18

19 frequencies of the vibrating beam are given in Table 3. From a practical standpoint, the values of these frequencies are important when selecting the proper transducers to measure the beam oscillations and then extract the flight characteristics in terms of airspeed and angle of attack. Following our work on resolution independent UVLM simulation (Ghommem et al., 213), particular care was taken when selecting the mesh and time step sizes. The wing is discretized into 8 panels along the beam length and 6 chordwise panels, providing 48 vortex rings for the UVLM solver. A small time step of t = s is selected and the tolerance on the norm of the residual vectors ɛ is taken equal to 1 6. Under this setting, the aeroelastic solutions were observed to converge in less than 5 subiterations of the fluid-structure interaction scheme. Table 2: Flow sensor specifications. Beam Mass Young s Rigidity dimensions density modulus modulus (Polysilicon material) L(m) b(m) h(m) ρ (kg/m 3 ) E (GPa) G (GPa) Table 3: First five natural frequencies of the flow sensor (flexural modes). Mode Natural frequency (khz) We show in Figure 8 the time histories of the flexural and torsional vibrations of the beam at the free end (s = L) as result of the interactions with the incompressible flow. The airspeed U considered in these simulations is equal to 15 m/s. After a transient phase, the beam deflections achieve bounded oscillations, referred as limit cycle oscillations (LCO). These moderate oscillations, obtained beyond the flutter boundary (at which the beam starts to undergo nondecaying oscillations), are the manifestation of Hopf bifurcation and correspond to supercritical instability (moderate behavior). Considering longer beam (higher aspect ratio) yielded subcritical instability (i.e., large LCO amplitudes that may lead to the structure failure). As such, one needs to carefully select the beam dimensions for the intended application. In Figure 9, we plot the variations of the LCO amplitude of the z-axis displacement w at the beam tip (i.e., s = L) with the airspeed. The airspeed is varied between 19

20 x w L (m) t (s) 2 x w L (m) t (s) (a) Displacement along z-axis w L at the beam free end x v L (m) t (s) 1.5 x v L (m) t (s) (b) Displacement along y-axis v L at the beam free end 8 x 1-5 (rad) x t (s) (rad) t (s) (c) Torsion φ at the beam free end Figure 8: Limit cycle oscillations (LCO): flexural and torsional deformations at the beam free end (s = L). The airspeed U is taken equal to 15 m/s and the angle of attack α is set equal to. 2

21 5 x w L (m) 1 x 1 3 w L (m) w L (m) 1 x t(s) t(s) 5 x t(s) U(m/s) w L (m) Figure 9: Calibration curve: variations of the amplitude of LCO of the z-axis displacement w at the beam tip with the airspeed. The angle of attack α is set equal to. and 23 m/s. This interval can be expanded depending on the operating range of the flying vehicle to which the beam is attached. We observe a decaying response below the flutter speed (U f = 4 m/s). At this speed, the aeroelastic system undergoes a supercritical Hopf bifurcation as indicated by the oscillatory deflection of the beam. As the airspeed increases, limit cycle oscillations (resulting from the system s nonlinearity) take place and their amplitude increases smoothly in a nonlinear fashion. As such, to fulfill the goal of our flow sensor, we propose to measure these vibrations and extract the airspeed using the calibration curve shown in Figure 9. In practical situations, different transduction principles can be used to generate the readout signal of the beam deflections such as capacitive (Aboelkassem et al., 21, Bhadbhade et al., 28) and piezoelectric (Liu et al., 212). As for detecting the angle of attack, at which the air stream hits the vibrating beam, we analyze the torsional deflection of the beam. In Figure 1, we show the power spectrum of the torsional deflection φ at the beam tip for different values of the angle of attack α. For α, we observe a peak at the the zero frequency. This indicates that the beam oscillates around a static equilibrium position. The presence of multiple peaks at many frequencies is the manifestation of the system s nonlinearity. In Figure 11, we plot the variations of the amplitude of the static deflection at the beam tip with the angle of attack α. The angle of attack is varied between -1 and +1. Small aircrafts are usually designed to operate within this range in order to avoid undesirable aerodynamic effects such as stall which could be accompanied with a large loss of the lift force. Furthermore, the UVLM solver considered here to simulate the aeroelastic behavior of the beam does not capture flow separation at the leading-edge caused by high angles of attack. The static deflection increases linearly with the angle of attack α and switches from negative to positive values as the value 21

22 power spectrum = = 4 = 1 P s f x 1 5 Figure 1: Power spectrum of the deflection of LCO (torsional motion at the beam free end) for different values of the angle of attack α (in degrees). The x-axis refers to the frequency f (in Hz). Results are obtained for an airspeed U equal to 15 m/s. of α crosses zero. Therefore, the static deflection can be used as a detector of the angle of attack. From a practical standpoint, one might increase the deflection angles and then improve the sensitivity of the flow sensor to variations in the angle of attack by selecting properly the geometrical properties of the beam. For instance, varying the shape of the beam (length and cross-section) affects the torsion constant (Roark et al. 22) and then enables increase in the deflection angles. The smooth variations of the amplitude of LCO of the flexural vibrations with the air speed and static deflection of the torsional vibrations with the angle of attack is quite encouraging because it indicates that sensing these motions can produce easy measurement of these aerodynamic quantities. 5 Conclusions We model and analyze a flow sensor consisting of a nonlinear vibrating cantilever microbeam subjected to an incoming air stream. Based on the proposed system, we propose a new sensor to measure the airspeed and the angle of attack at which the air travels across a small flying vehicle. This approach exploits Hopf bifurcation, which is the manifestation of the nonlinearity of the vibrating system, to detect the flow characteristics. The idea is to attach the beam to the vehicle and measure its deflections in order to extract information about the surrounding flow. As such, we generate calibration curves that exhibit the trend for the variations of the LCO amplitudes of flexural and torsional vibrations with the airspeed and the angle of attack, respectively. The results show smooth variations of these vibrations with the air speed and angle of attack. This finding is quite encouraging because it indicates that sensing the beam 22

23 6 x s (rad) Figure 11: Calibration curve: variations of the static deflection of LCO (torsional motion at the beam free end) with the angle of attack α (in degrees). Results are obtained for an airspeed U equal to 15 m/s. vibrations can produce easy measurement of the fundamental aerodynamic quantities (air speed and angle of attack) as required for flight control and manoeuvrability. Building the new sensing device along with testing it on a real-life UAV constitute our future research topic. Acknowledgement Christian Claudel gratefully acknowledges the help of Mohamed Abdelkader, Mohammad Shaqura, Yanning Li, Marlon Diaz, Hongkui Li, Farhan Abdulrahim and Shayma Alhuwaider for their help setting up the UAV system and conducting the flight tests 1. References Aboelkassem Y, Nayfeh A, and Ghommem M (21) Bio-mass Sensor Using an Electrostatically Actuated Microcantilever in a Vaccum Microchannel. Microsystem Technologies 16: Acar C and Shkel AM (24) Structural design and experimental characterization of torsional micromachined gyroscopes with non-resonant drive mode. Journal of Micromechanics and Microengineering 14(15): This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. 23

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26 Wang Z (24) Time-domain simulations of aerodynamic forces on three-dimensional configurations, unstable aeroelastic responses, and control by neural network systems, PhD Dissertation, Virginia Tech, Blacksburg,