Analysis of vibration of rotors in unmanned aircraft

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1 Analysis of vibration of rotors in unmanned aircraft Stanisław Radkowski Przemysław Szulim Faculty of Automotive, Warsaw University of Technology Warsaw, Poland Faculty of Automotive Warsaw University of Technology Warsaw, Poland hand there are attempts to eliminate the causes of their occurrence [8]. The second path constitutes the authors field of interest. Mainly observed vibrations doesn't result from rotor unbalance. Of course this source is also important and also plays important role. But the interesting phenomena appears during rotation of a rotorcraft around roll or/and pitch axis. It is worth to note that those vibrations are much more perceptible than vibrations coming from motor and propeller unbalance. This phenomena was observed for many sets of propellers and motors. This observation motivated the authors to describe the problem in mechanical domain and answer the question: are those vibrations can be reduced? The first chapter will describe the physical basis underlying the occurrence of the vibration in which the authors are interested. The next chapter, based on computer simulations, will present the results of comparing the vibration generated due to additional movement of a rotating rotor. In the next chapter, the authors will present the conclusions of their research, while indicating the possibilities of partial or full elimination of the vibration they analyzed. The last chapter show result from investigation which was done on real quadrocopter. Abstract In the paper, solution of the problem of vibrations which appears during the maneuvers of quadrocopters is presented. Those special kinds of vibration aren t related to mounting or manufacturing faults. In this kinds of flying machines the main sources of vibration comes from motors and/or propellers unbalance. The paper shows analytical description of the source of this problem. Mathematical description of rotor which was subjected of additional rotation had been modeled in Matlab Simulink environment. It was shown, that correct set of parameters lead to total elimination of this kind of vibration. In paper is shown also investigation on real object and influence of results on vibration reduction. Keywords UAV, propeller vibration, quadrocopter I. INTRODUCTION A rotorcraft are one of the mechanically simplest constructions of flying machines. In the immediate past such structures were mainly within the area of interest of modelers. At the present time a transition from this type of constructions to the new stage can be observed. Not only are they becoming a well-flying aerial vehicles, but they are evolving towards the tools that enable performing tasks and services so far beyond our reach [1,2,3]. The growing degree of application complexity requires the increasing level of equipment infallibility and quality. There are many applications which require high stability of fly and especially very low vibrations levels. As far as photographic demands are concerned, it is crucial that the aerial vehicle should act as a tripod, being an extension of the photographer s arm. This seemingly distant target is gradually becoming more realistic thanks to the efforts undertaken to improve the reliability and the quality of rotorcrafts. One should mention several areas which are still being on the development stage, such as automation [4,5,6,7], mechanics, area of new batteries and electric motors. In the paper authors of the article focused on the one issue which is associated with the vibration caused by rotors of a rotorcraft. Mechanical vibrations have a negative impact on the IMU gauging unit which influences the operation of the whole vehicle. The quality of performed tasks is also negatively influenced by vibrations. The negative impact the vibrations have on the quality of the registered image should be emphasised strongly. The experience proves that the approach to vibrations problem through the proper separation can solve some of these problems but causes emerging of the new ones. [11]. This is why in real-life applications the vibration problems are dealt with in two ways. On the one hand an adequate vibroisolation is employed [9][12], but on the other /14/$ IEEE II. MATHEMATICAL DESCRIPTION First, Rotorcrafts are structures which consist of numerous power transmission systems of the motor-and-rotor type, installed on the arms which join together at the centrallylocated body of such an aircraft. An example of such a structure is presented in Figure 1 below. Fig. 1. Frame of a quadrocopter [10] The above-presented type of quadrocopter is the most widely used structure. There are also structures with three, six or eight arms. Structures with two motors at each arm, placed one above the other and generating unidirectional thrust, are also popular. 748

2 (1) (2) (3) (4) (5) Fig. 2. An arm-motor-propeller unit Due to the simple structure (which is the advantage of solutions of this type), the aircraft's flight is controlled by tilting the entire body in a relevant manner. Such a solution serves its purpose well and guarantees relevant maneuverability. It is the necessity of making maneuvers that causes emergence of angular velocities due to tilting of a craft's body. This velocity appear as a additional velocity rotating arms of quadrocopter. Figure 2 presents a single armmotor-propeller unit which forms the basic element being the subject of the subsequent analysis. The motor rotates the propeller with the speed of, thus producing the required thrust. The arm additionally rotates around axis X at the speed of. The speed, as marked in the Figure 2, has been selected in order to simplify the calculations, however also other components of the velocity emerge along the axis Y and Z due to rotation of the whole platform. It could be shown that additional rotation along axis Z doesn't introduce significant vibrations. Additional rotation along axis Y has similar effect like investigated rotation along axis X and therefore would not be considered in next part of the article. Figure 3, below, presents a simplified version of arm from Figure 2. (6) Forces and torques are created as a result of rotation of the rotor around the axis X. Forces F X,F Y,F Z and torques T X,T Y,T Z can be determined based on the below equations: (7) (8) (9) (10) (11) (12) After substituting equations (1)-(6) and their second derivatives into equations (7)-(12) one come up with the following set of equations: (13) (14) (15) 0 (16) Fig. 3. The rotor unit reduced to its basic components The propeller has been simplified to two point masses of m S (point A and B), separated by the distance r. The propeller is separated from the axis X by the distance of l. The propeller rotates at a constant speed of. The angle α is the angle between the axis Z and the axis of the motor and result from rotation of rotor caused by tilting. In the analyzed case the axis of the motor rotated in the plane Y-Z was chosen. The location of points A (A X,A Y,A Z ) and B (B X,B Y,B Z ) in the coordinates system X,Y,Z is described below. 2 (17) 2 (18) While analyzing equations (16)-(18), one may note that the force do not influence on vibration. The equations of torque (13)-(15) have common features. Almost all components are related to angular position of the propeller. This relation introduces pulsation of torques on X, Y and Z axis. This pulsation has twice the frequency of the rotation of propeller. Amplitude of torques is also not constant but depends on acceleration and rotational speed of arm along axis X. 749

3 III. SIMULATIONS Before The chapter presents the results of exemplary simulations. The object subjected to simulation has been modified slightly in order to bring the theoretical considerations closer to their practical application. A second arm with a rotor has been added to the object. Both units are joined together at a point having a certain mass which represents the central part of an aircraft's structure. Figure 4 shows the schematic diagram of the object. The simulations of system described via (19)-(21) equations were carried out in the Matlab - Simulink environment. Fig. 5. Angular acceleration graphs for the central point of the model Fig. 4. A simplified model with two rotors and its reduced counterpart The concentrated masses 1 and 2 represent the motors with a mass of m m. The concentrated mass 3 represents the central part of the structure witch momentum J 3. The arms connecting the two motors with the central part have been replaced by connections characterized by the stiffness of k il and k ir and damping of b il and b ir. In those values index i denotes the respective analyzed axis of the torques (X,Y,Z), while the indices R and L correspond to the right and left arm of the structure. To simplify simulation, influence of T X, T Y, T Z torques on angular orientation of motor and propeller in Z and Y direction was not considered. The system of equations presented below describes the dynamics of this complex object in respect to X axis. Similar set of equation describe object dynamics in respect to axis Y and Z (19) The figure above presents graphs showing the angular accelerations measured in point 3, i.e. at the central point of the platform. Image 1 presents the sine function of the propeller's rotation angle. In this case both rotors were rotating at the same speed of 2500 rpm. Image 2-4 presents graph of amplitude of angular acceleration of the central point of the object. Frequency of acceleration pulsation is twice the frequency of propeller rotation angle what could be compared based on image 1 and images 2-4. Interesting is amplitude modulation that could be seen on img The envelopes of angular acceleration signals are not pure function of sine or cosine, but depend on angle, velocity and acceleration of axis X. The figure 6 presents the graphs illustrating the angular orientation and the angular velocity of the main coordinate system located in point of central mass. The torque T in has been applied in such a way so as to smoothly turn the entire structure by 180 degrees. Each subsequent simulation occurs with the same torque T in. The influence of the torque oscillation on oscillation of the angular velocity of the central mass, is also very visible here. Interesting results are observed for a situation when identical angular speeds are set for the two rotors while there is difference in terms of phase shift between the rotors (20) (21) (21) where: - the distance between the motor's and axis X, - the control torque applied to the central point 3, - angle of rotation of point mass n around the axis X, - the angle of rotation of the propeller for rotor k. Fig. 6. Shift and speed of the model's central point (3) Fig. 7 presents graphs showing accelerations obtained as a result of a simulation involving an angular shift between the 750

4 propellers equal to π/2. The oscillations of the torque, associated with rotation of the rotor, disappeared in the axis Y and Z (img. 3 and 4). For axis X (img. 2) the amplitude of oscillations decreased substantially (by around a factor of 10). 4 4 (26) Those equations doesn't depend on sine or cosine angle anymore, so the torques will not pulsate. It was the goal of the authors. Because the rotational velocity of propeller is constant first term of equations (25) and (26) is zero. Second term of this equations depend on velocity α and this is the reason that the curves on plot 3 and 4 on Figure 8 are not pure trigonometric functions. Fig. 7. The influence of the preset phase shift of the rotors on oscillation Fig. 8. Influence of preset phase shift of the propellers and bigger rigidity of the arms on vibration of the object. The frequency of oscillations corresponds to the fourfold frequency of rotation of the propeller. After increasing the parameters of simulated model like stiffness and damping, the influence of the rotating propeller on generation of oscillation has been practically eliminated. This was presented in Figure 8. It is only the accelerations generated by gyro moments that are left. This result could be proved on analytical way. If we consider equations (13)-(15) we can see, that all torque pulsation are the function of doubled angle. Let's consider single arm with motor and propeller. Let's introduce additional propeller, shifted about angle π/2. This case we can describe as a sum of two set of equations (13)-(15). After taking into consideration trigonometric properties listed below (22),(23), set of torque equations is reduced to form (24)-(26). cos 2 2 (22) sin 2 2 (23) IV. INVESTIGATIONS RESULTS Two experiments were performed to verify the obtained results. A single arm with an attached propeller was used in the first experiment. A tri-axial accelerometer and a tri-axial angular velocity sensor, commonly termed as the gyroscope, were attached on the underside, near the motor. The entire unit was rotated around the axis X and Y which are perpendicular to the propeller s rotation axis. Due to physical limitations, the arm was being rotated at the angle of +80 o. Four experiments, for two different speeds at which the arm rotated as well as for two types of propellers, the 2-blade and the 4-blade one, were performed this way. The 4-blade propeller was assembled from two 2-blade propellers, made of carbon fiber, which were turned in respect of each other at the angle of 90 o. The purpose of the experiment was to record and confirm the occurrence of the vibration which was described in the article and also to determine the influence of 4-blade propellers have on the level of vibration. The second experiment was conducted in a quadrocopter-type air vehicles. Several test flights were performed in very similar weather conditions. The flights themselves could not be redone exactly in the same way, however the results that were obtained clearly demonstrated the differences between the quadrocopter with 2-blade propellers vs. the same structure with 4-blade propellers. The goal of the experiment was to determine the change in terms of vibration levels measured in the central part of a multi-rotor craft for the two types of propellers. A. Single arm experiment The figure 9 presents the time signal measured by accelerometers. 4 2 (24) 4 4 (25) Fig. 9. Comparison of impact rotational velocity of arm on measured acceleration 751

5 The upper figure corresponds to an experiment carried out at lower angular velocities of the arm, while the upper figure reflects higher angular velocities of the arm. It is worth to note that the beginning sections of the graphs correspond to a moment in time when the arm was stationary. The vibration signal from t=1s to t=10s, and respectively from t = 2s to t = 10s, corresponds to the rotation of the arm. The vibration which is visible in the first phase is the vibration coming from the unbalance of the propeller. In the second phase the vibration was caused by the effect described in the article. What can also be noted is that the vibration amplitude increases for higher rotational velocities, which is a reflection of the model described by equations (13)-(15). The figure 10 presents a graph for a module of signal coming from a tri-axial gyroscope which was measured for the two experiments. It is worth to note that the 2 nd, the 4 th and 6 th harmonics of the rotational velocity are most visible, which is in line with the model described by the equations. The model demonstrates that the frequency of pulsation of the moment is twice the value of a propeller s frequency of rotation. The figure 12 presents the frequency spectrum of the signal measured by a rotational speed gyro-sensor. The second rotational harmonic (around 130Hz) and the first rotational harmonic, caused by the propeller s unbalance, become visible. Fig. 12. Comparison of impact rotational velocity of arm on measured gyrosensor signal spectrum Fig. 10. Comparison of impact rotational velocity of arm on measured gyrosensor signal The influence of vibrations on angular velocity s measurements, performed with a MEMS-technology-based sensor, is visible in the second phase of the experiment. Also the influence of rotation of the arm on the angular velocity s oscillation becomes visible here. Figure 11 presents the signal s spectrum obtained from the accelerometer. Figures 11 and 12 demonstrate clear influence of rotational speed of the arm on the level of vibration (Fig. 11). Significant changes in the level of the signal s vibration occurred for a four blade propeller. Two two-blade propellers, mounted on one motor with shift angle 90 o, were used for the purpose of the experiment. The figure 13 presents a graph of the signal s spectra for two experiments involving two different rotational speeds of the arm for four-blade propeller. Fig. 13. Impact of 4-bladed propeller on acceleration signal spectrum Fig. 11. Comparison of impact rotational velocity of arm on acceleration signal spectrum The second harmonic has much lower amplitude. The first harmonic, associated with the occurrence of unbalance due to imprecise assembly of the two propellers, has emerged in turn. 752

6 conditions. Two sets of graphs (fig. 15) are presented which show the influence of using of four-blade propellers on the level of vibration measured in the structure. In both cases a dynamic of maneuvers were similar what was shown on plot of angular rotation speed of rotorcraft on fig 15. Fig. 14. Amplitude demodulation of gyro-sensor signal Interesting results can be also observed after extraction of the envelope of the signal obtained after filtering the part of the signal around the second harmonic. In accordance with the analytical model, rotation of the arm results in emergence of amplitude-modulated oscillations of the moment. The carrier frequency of this modulation is twice the value of the frequency of rotation and it remains constant in the experiment. The modulating signal includes a combination of the signal which is proportional to the rotational speed and to the angular acceleration. The figure 14 presents two graphs for two type of propellers. The signal measured by the gyrosensor has been subjected to processing. The correlation between the input (the angular velocity of arm) and the envelope of the second harmonic is clearly visible. The amplitude of the envelope increases along with the growth of the value of the input. Similar effect is also observed for the signal obtained from the acceleration sensor. B. Experiment with rotorcraft Fig. 15. Comparison of acceleration signal for normal fly experiment The other type of the experiment was conducted directly on the airship, in the conditions similar to the device s operating V. CONCLUSION The paper investigates the phenomena of generation of adverse vibration which appears while a rotorcraft makes maneuvers. A model of the basic unit and simulation show that maintaining a permanent angle of phase shift between the two rotors led to a practically complete elimination of occurrence of oscillations of the torque for the all three axis. The calculations have demonstrated that the desired effect of elimination of vibration is achieved while using a two propellers, set on single motor, rotated about angle π/2. Doubled propeller correspond to four symmetrically-spaced blades. Theoretical calculation and simulations was confirmed in two kinds of experiments. The first experiment confirmed mechanical phenomena presence. Modulation phenomena was observed in angular velocity as well as in acceleration signal. Experiment confirm also significance reduction of vibrations level observed in acceleration signal and gyro-sensor signal after using fourbladed propeller. The second experiment shown, that proposed solution greatly reduced vibration level measured on rotorcraft construction not only during maneuvering flight phase but also during static flight. [1] Manufacturer website: [2] Manufacturer website: [3] Manufacturer website: [4] Cherian A., Andersh J., Morellas V., Papanikolopoulos N., Mettler B., Autonomous Altitude Estimation of UAV Using A Single Onboard Camera, The 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems October 11-15, 2009 St. Louis, USA, DOI: /IROS [5] Moore R. J. D., Thurrowgood S., Bland D., Soccol D., Srinivasan M. V., UAV Altitude and Attitude Stabilisation using a Coaxial Stereo Vision System, 2010 IEEE International Conference on Robotics and Automation Anchorage Convention District, May 3-8, 2010, Anchorage, Alaska, USA, DOI: /ROBOT [6] Bošnak M., Matko D., Blažič S., Quadrocopter control using an onboard video system with off-board processing, Robotics and Autonomous Systems 60 (2012) , DOI: /s [7] Yua Y., Dinga X., JimZhub J., Attitude tracking control of a quadrotor UAV in the exponential coordinates, Journal of the Franklin Institute 350 (2013) [8] Castellini P., Santolini C., Vibration measurements on blades of a naval propeller rotating in water with tracking laser vibrometer, Measurement, Volume 24, Issue 1, July 1998, Pages 43-54, ISSN , DOI: /S (98) [9] Wen -Jeng Hsueh, Vibration reduction of main hulls using semiactive absorbers, Journal of Marine Science and Technology, 1998, Volume 3, Issue 1, pp 50-60, DOI: /BF [10] Manufacturer website: [11] Isolating Components from UAV Vibration: [12] Marichal G.N., Rodriguez M.T., Rivera S.C., Vibration reduction for vision systems on board unmanned aerial vehicles using a neuro-fuzzy controller, Journal of Vibration and Control June 25, 2013, DOI: /

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