Note: The hardware, single-chip computer program, experimental data, data processing program, filter program, 3D model and modal cloud diagram involved in this article can be replied through the WeChat public account (Amu Lab): "Vibration analysis" or "Vibration reduction design" "obtain.
Multi-rotor aircraft have been widely used in civilian and military markets in recent years due to their excellent maneuverability, stability, and hoverability. The academic and industrial circles paid attention to the state estimation and controller design of multi-rotor aircraft in the early stage. With the marketization of multi-rotor aircraft, research on its aerodynamics, structure and vibration has been increasing.
This paper systematically sorts out the vibration problems of multi-rotor aircraft, and gives an effective and engineering feasible vibration reduction scheme. Figure 1 summarizes the vibration mechanism of a multi-rotor aircraft and the effects of vibration isolators and digital filters.
1 Vibration mechanism
A multi-rotor aircraft is an aircraft in which the thrust generated by multiple rotors overcomes gravity and adjusts attitude displacement. Its rotors often use two-blade rotors with fixed pitch. The common configurations are four-rotor, six-rotor and eight-rotor. This article focuses on the four-rotor configuration, as shown in Figure 2. The local reference system is defined as follows: the x-axis points outward along the arm, the z-axis is upright, and the y-axis is defined according to the right-handed coordinate system.
The vibration mechanism consists of three parts: excitation (vibration source), system and response [8]. For multi-rotor aircraft, the excitation comes from the power system composed of rotors and motors, the system is the frame, and the response of general research is located at the sensor (mainly the inertial measurement unit of the flight controller). They will be introduced separately below.
Excitation: Vibration generated by rotating devices generally comes from dynamic imbalance, and its vibration frequency is equal to the rotation frequency. However, for multi-rotor aircraft, in addition to the fundamental wave component caused by dynamic unbalance, there are also the second harmonic of vibration caused by the periodic aerodynamic force (also called "lift wave" [9]) generated by a single rotor, and the second harmonic caused by multiple The second or more harmonics generated by the mutual coupling of rotor flow fields. NASA conducted a vibration measurement experiment on a quadrotor vehicle [5], and its sampling frequency was as high as 100,000 Hz, thus showing the fine vibration signal spectrum, as shown in Figure 3. In the experiment, the motor rotation frequency is 95Hz. Compared with Figure 3, it can be seen that the fundamental frequency of the vibration signal is the motor rotation frequency, and there are 2, 4, and 8 harmonic components, which verifies the above vibration excitation analysis.
Further analysis of the direction of excitation, the additional dynamic reaction force caused by dynamic imbalance, perpendicular to the z-axis, is mainly the x-axis component and the y-axis component [10]; the lift wave is mainly the z-axis component.
System: The frame of a multi-rotor aircraft is usually composed of engineering plastics or carbon fiber, and different materials have different mechanical properties. The frame of the experimental measurement object used in this article is DJI F450, the main material is engineering plastic PA66-GF33, which can be simplified as an isotropic linear elastic system, and the arm is regarded as an Euler-Bernoulli beam.
Response: According to the stimulus and the system, the characteristics of the response can be inferred, as shown in Table 1. For multi-rotor aircraft, the greatest danger of vibration is to distort the measurement of the inertial measurement unit, which in turn leads to an increase in the state estimation error, which affects the control performance and severely causes the control to diverge. Therefore, this article mainly studies the response of the inertial measurement unit and designs its vibration reduction.
2 Finite element modal analysis
Modal analysis is an important part of vibration analysis. In this section, we will find the position with the smallest amplitude through modal analysis, and provide a reference for the spatial layout of the inertial measurement unit; and give a method to avoid resonance based on the obtained natural frequency.
First use SOLIDWORKS to build a 3D model. Considering that the grid should not be too dense, the three-dimensional model ignores small holes and only retains the key connection holes. In addition, it ignores the influence of electronic components on the modal. The model after establishment is shown in Figure 2. Import the 3D model into ABAQUS for follow-up work.
Divide the grid, as shown in Figure 4. The components are assembled and the surfaces are bound and restricted. Except for the four rotors, the rest are restricted to the degree of freedom of rotation.
Then assign material properties to each part, as shown in Table 2. These material properties are approximate to the actual material properties of the F450 frame, but they meet the requirements of the modal analysis of this study.
Then, the modal cloud image can be obtained. Based on the above conclusions, we have analyzed the modal natural frequency cutoff to 355.53 Hz, a total of 50 modal cloud diagrams (see the attachment for the original picture), of which the first 6 orders correspond to rigid body motion and are invalid modals.
Observing these cloud images, the following conclusions are obtained.
(a) In most modes, the rotor and the tripod have the largest deformation due to their slenderness and insufficient material stiffness.
(b) There are only three modes with large deformation of the arm. Observing the vibration shapes of these three modes, it can be seen that the deformation of the middle section of the arm is greater than that of the outer and inner sections, as shown in Figure 5. It can be inferred that the average amplitude of the middle section of the arm is the strongest.
(c) The deformation in the middle of the frame in all modes is very small, indicating that this is an ideal location for the placement of the inertial measurement unit required by the flight controller.
Looking at Figure 6, it can be seen that under the material properties of Table 2, the natural frequency of a certain mode with large rotor deformation is close to the motor speed, indicating that the rotor will produce greater vibration, which will increase noise and reduce the aerodynamics of the rotor. Efficiency, and may form positive feedback in the aeroelastic system, increasing the vibration of the system. In order to avoid such a phenomenon, the elastic modulus of the rotor can be increased to keep the natural frequency of this mode away from the motor speed, as shown in Figure 7.
3 Experimental measurement
After theoretical analysis and finite element analysis, experimental measurement and analysis are required. This section will introduce the experimental measurement system and experimental process.
3.1 Hardware system
This research focuses on engineering applications, so the experimental equipment is commonly used in the UAV field. The sensor for measuring acceleration is the MPU6050 programmable inertial measurement unit, which uses the back-EMF of the electronic governor to measure the motor speed. The single-chip microcomputer is an Arduino Mega 2560 8-bit single-chip microcomputer, so that the quality of the acquired data is in line with the actual engineering and is designed for the subsequent digital filter Standard data is provided. The acceleration sampling frequency of MPU6050 is 1kHz, and the mass is only 1.5g, which has negligible influence on the system mass distribution. For noise reduction considerations, the firmware of MPU6050 supports a low-pass filter bandwidth of up to 260Hz, which shows that the inertial measurement unit filters high-frequency vibration components at the beginning of the design, and the vibration reduction design should focus on the vibration components within the effective bandwidth. In order to restore the real signal spectrum as much as possible, the experiment chooses the highest bandwidth of 260Hz.
The experiment uses a DJI F450 quad-rotor platform, and the flight controller is Ardupilot (its processor performance is equivalent to Arduino Mega 2560). The maximum hovering time of the aircraft is about 10 minutes, which meets the needs of the experiment.
Due to the limited collection rate of the data collection system, the vibration of only one position is measured during each flight. Considering the symmetry of the quadrotor, only the five positions (Arm1 to Arm5) and The center of the frame is measured, and the center of the frame is measured without vibration isolator (Board1) and vibration isolator (Board2). The layout of measuring points and motor numbers are shown in Figure 8.
There are usually six ways to connect the vibration signal measurement sensor to the measured object [11]. We use the adhesive bonding method. The effective bandwidth of this method is about 10000 Hz, which meets the needs of the experiment.
The data acquisition system is shown in Figure 9. The data marked in red are vibration analysis related data. Due to the high frequency of data collection and high requirements for transmission quality, the acceleration and motor speed are written into the SD card in real time during flight experiments, and only the flight information and data collection status are wirelessly sent to the computer in real time.
3.2 Software system
Theoretically, in order to perform discrete Fourier transform, vibration signal acquisition needs to strictly ensure equal sampling intervals [12]. But in this data acquisition system, in order to ensure the reliable recording of experimental data, the single-chip microcomputer will save and close the SD card after every 200 data collected. This operation will result in a data blank of about 10ms. Resampling is required during data processing to change it to standard 1000Hz sampling data.
After the flight experiment is completed, use MATLAB to process the data stored in SD. After reading the data, resampling, amplitude domain analysis (calculate the root mean square error of the data, that is, the average amplitude), frequency domain analysis (calculate the power spectrum) and time-frequency After domain analysis (short-time Fourier transform), data processing is completed.
3.3 Flight experiment
The flow field generated by the rotor near the ground is quite different from the flow field generated by the far ground (called "ground effect") [13], so this study records vibration data in real time during flight, which is different from many studies. Fix the aircraft on the support for vibration test. The flight is carried out indoors (to avoid wind disturbance). When the aircraft is hovering, it starts to collect data, and then landed after the data collection is finished, so as to ensure that the flight status of multiple experiments is approximately the same.
4 Analysis of measurement results
This section analyzes the measurement results from the perspective of amplitude domain, frequency domain and time-frequency domain. Among them, the amplitude domain analysis is used to compare the strength of the three-axis components of the vibration signal at different measuring points, the frequency domain analysis is used to study the spectral characteristics, and the time-frequency domain analysis is used to study the stationarity of the frequency spectrum.
4.1 Amplitude range analysis
Figure 11 lists the average amplitude (calculated with root mean square error) of different measuring points in different directions. After analysis, the following conclusions can be drawn:
(1) The entire machine arm can be regarded as a spatial beam, the y-axis and z-axis directions are transverse bending vibration problems, and the x-axis direction is axial tension and compression vibration problems, and the mechanical difference is obvious. Therefore, the average amplitude distribution of the x-axis is obviously different from the y-axis and z-axis.
(2) The total average amplitude is generally consistent with the results of the finite element modal analysis. The average amplitude of the middle section of the arm is the largest, and the outer and inner sections are small. Although the average amplitude of the center of the rack is not the smallest, it is close to the smallest (Arm5).
(3) The experiment also explored the influence of mass distribution on vibration. The battery, as the heaviest component of the aircraft, is generally located at the geometric center of the horizontal plane of the aircraft. The experiment changed the distance between the battery and the measuring point of Board1 in the z-axis direction, and measured the average amplitude change in the z-axis direction. The result is shown in Figure 12. It can be seen that the mass distribution has a significant impact on vibration, which can be used as one of the basis for the overall design of the multi-rotor aircraft.
4.2 Frequency domain analysis
This analysis is used to verify the spectral characteristics of the response predicted by theoretical analysis in Chapter 1. Figure 13 lists the vibration signal spectrum in the x-axis direction, y-axis direction and z-axis direction of the rack center (see the open source project website for the spectrum of the remaining measurement points). Table 3 records the mean value and variance of the four motor speeds. It can be seen that the speeds of motor 1 and motor 3 are close, about 84r/s, and the speeds of motor 2 and motor 4 are close, about 77r/s. The rotation frequency of the motor 1 and the motor 3 is one fundamental frequency of the vibration spectrum, and the rotation frequency of the motor 2 and the motor 4 is the other fundamental frequency of the vibration spectrum.
By comparing the relationship between the frequency of the main peak of the spectrum and the motor speed, the theoretical speculation is well verified: the fundamental frequency of the vibration signal is the rotation frequency of the motor, the x-axis and y-axis are dominated by the fundamental component, and the harmonics of the z-axis The weight is stronger.
The signal in the z-axis direction shown in Figure 13 has a very high component below 1 Hz, which is different from the x-axis and y-axis. This is a real motion signal, a low-frequency motion caused by the height control of a multi-rotor aircraft. In addition, the attitude control of the aircraft will also cause linear motion at the center of the frame. This motion is also low-frequency. This low-frequency motion signal is more obvious at the outer end of the arm. The gray rectangle at the low frequency in Figure 1(a) is the motion signal described. In the frequency spectrum of Chapter 5, in order to mainly show the change of the vibration signal, the frequency spectrum will be drawn after 1Hz.
4.3 Time-frequency domain analysis
Perform short-time Fourier transform on the collected acceleration signal to get the time spectrum, as shown in Figure 14. It can be seen that the high-brightness spectrum is almost parallel to the horizontal axis with small fluctuations. This is because the motor speed is not constant, there will be small fluctuations, but the overall is stable, which further proves that the fundamental frequency of the vibration signal is the motor speed. One conclusion.
5 Vibration damping design
According to the analysis of the vibration mechanism, the key to the design of vibration reduction is to filter the vibration signal with the fundamental frequency of the motor rotation frequency and harmonic components. The following will introduce the design of the vibration isolator and the design of the digital filter respectively, and it is proved through experiments that the average amplitude of the vibration signal is greatly reduced after the vibration signal passes through the vibration isolator and the digital filter.
5.1 Vibration isolator design
Vibration isolators are elastic elements and damping elements that connect equipment and foundation to reduce and eliminate the vibration force transmitted from the equipment to the foundation and the vibration transmitted from the foundation to the equipment [14]. In this question, the foundation is the middle of the rack and the equipment is an inertial measurement unit.
The amplitude-frequency characteristic curve of the vibration isolator is shown in Figure 15. In order to effectively reduce vibration, the natural frequency of the vibration isolator should be lower than the excitation fundamental frequency, that is, the rotation frequency of the motor, under the condition of constant motor speed. But the natural frequency should not be too small, because the real motion signal has aperiodic terms, which will cover the entire frequency spectrum in the frequency domain (see the gray part of the frequency spectrum in Figure 1(a)), and the natural frequency is too low It will cause a large number of frequency band distortions, leading to increased observation errors [15]. Comprehensive consideration, we take the natural frequency 
.
Among vibration isolation materials, rubber is the most widely used. It can be made into products of various shapes and various hardnesses, and has good elasticity, sufficient strength and can absorb part of the energy to quickly suppress impact and vibration [16].
The selection of rubber material is mainly determined by the static elastic modulus and hardness, and the determination of the static elastic modulus is related to the natural frequency, shape and temperature.
Assuming that the shock absorber works at room temperature 
, check the "Vibration Isolation Design Manual" [16] to get the temperature influence coefficient 
; choose 
natural rubber with Shore hardness , and its static elastic modulus 
. The rubber pad is designed in a ring shape, as shown in Figure 16. Since the pressure bearing area of the vibration damping block should be satisfied 
, assuming the outer diameter 
and inner diameter 
of the torus, the area ratio of the torus to the cylindrical surface is:
(6)
The vertical shape factor is:
(7)
Check the "Vibration Isolation Design Manual" [16], the shape influence coefficient is 
.
The required bearing area of the damping block is:
(8)
The actual bearing area of the damping block is:
(9)
Because 
, it meets the design requirements.
In order to 
produce close vibration isolation effects in three directions, the annular rubber pad is in the form of an oblique cylinder. Its axis is along the diagonal of the cube.
Therefore, the material of the ring rubber pad of the vibration isolator is selected to be natural rubber with Shore hardness 
and static elastic modulus 
; height 
, outer diameter 
, and inner diameter 
; the installation method is inclined installation. The physical map is shown in Figure 18.
As shown in Figure 19, after installing the vibration isolator, 
the fundamental power peaks in the three directions are respectively attenuated to approximately 74.3%, 5.5%, and 6.6% of the original; and 
the harmonics in the directions are effectively suppressed.
As shown in Figure 20, the 
directional average amplitude and total strength after installing the vibration isolator are 73.2%, 26.0%, 20.0% and 48.3% of those without vibration isolator. The vibration isolator effectively reduces the vibration transmission between the foundation and the equipment, and the vibration suppression effect of the direction and direction is particularly obvious.
5.2 Digital filter design
In the multi-rotor aircraft industry, low-order IIR low-pass filters, such as second-order Butterworth low-pass filters, are often used for vibration and noise. Such a filter is simple in design and fast in calculation, but there are two disadvantages. The first is that it causes a large phase lag, which may lead to overshoot or even divergence of attitude control; the second is that the real motion signal has non-periodic terms, and the frequency domain will be covered with the entire frequency spectrum. The introduction makes this part of the signal distorted. The use of band stop filters can solve these two problems.
As analyzed above, the vibration interference of multi-rotor aircraft is concentrated in several narrow bandwidth frequency bands, and the center frequency of the frequency band is a multiple of the motor rotation frequency. Using a band-stop filter can accurately remove the signal interference of these narrow bands and reserve other useful signal bands.
In the current multi-rotor aircraft industry, notch filters (notch filters for short) have become a hot spot. The notch filter usually refers to the second-order band rejection filter, and its amplitude-frequency characteristic curve is shown in Figure 21(b). Its design simplicity and operation speed are equivalent to the second-order low-pass filter, which is suitable for engineering applications. However, since the notch filter is only second-order, the frequency band that can be eliminated is too narrow. Comparing Table 3 and Figure 13, due to the limited accuracy of the motor speed control of the multi-rotor aircraft and the accuracy of the sensor measuring speed, the center frequency of the fundamental frequency band of vibration interference has a certain deviation from the measured motor speed. In actual engineering, the true fundamental vibration frequency cannot be accurately obtained, and it can only be approximated by the measured rotational speed. The error of this approximation is large, so it is necessary to use a band stop filter with a wider bandwidth.
In this study, after the vibration signal is low-pass filtered by the vibration isolator, only two fundamental frequency components generated by two groups of motors with different speeds are left. To this end, only need to design a band stop filter containing these two fundamental frequency components. What this article uses is the fourth-order Chebyshev type II band stop filter, and its design parameters are shown in Table 4. For comparison, two notch filters (second-order Chebyshev type II band stop filter) were designed at the same time, and the parameters are also listed in Table 4. The center frequency of the band-stop filter is the average value of the average speed of the four motors, and the center frequency of the notch filter is the average value of the speed of the No. 1 motor and the No. 3 motor and the average value of the No. 2 motor and the No. 4 motor. The following will compare the effects of a fourth-order band rejection filter and two notch filters in series.
Figure 21 compares the amplitude response of the designed fourth-order band stop filter and the notch filter. It can be seen that the passbands of the two are very flat without attenuation, and the stopband attenuation also meets the design requirements. The only difference is the bandwidth. The bandwidth of the notch filter is too narrow. Once the center frequency deviates from the actual vibration fundamental frequency, the filtering effect will be reduced sharply. Figure 22 proves the above conclusion. For the signal in the z-axis direction, the fundamental frequency component of the vibration passing through the fourth-order band rejection filter is completely filtered, while the fundamental frequency component of the vibration passing through the two notch filters still remains, the x-axis and y The same is true for the axis. Figure 23 shows the comparison of the average amplitude, which shows that under the experimental conditions, the fourth-order band stop filter in the x-axis direction is equivalent to the two notch filters, while the fourth-order band stop filter in the y-axis direction and the z-axis direction All the devices have better damping effect.
When a multi-rotor aircraft is flying, the motor speeds are different in different flight conditions. The rotation frequency of the motor, as the center frequency of the stop band of the band stop filter, is the core design parameter. Therefore, it is necessary to dynamically adjust this parameter and design it as an adaptive filter. In actual use, the motor speed can be obtained by an electronic speed controller with speed feedback function or other principle speed sensors, and sent to the flight controller, which adjusts the band-stop filter parameters in real time accordingly. Figure 1 shows this process.
to sum up:
In this paper, theoretical analysis is used to speculate that the vibration response takes the motor rotation frequency as the fundamental frequency and is accompanied by high-order harmonic components. The experiment proves this speculation, which lays the foundation for the design of vibration isolators and digital filters. The finite element modal analysis shows that the middle of the frame is the place with the weakest vibration, and it is suitable to install the inertial measurement unit of the flight controller. The subsequent experimental measurements proved this conclusion. The designed experimental measurement device is derived from commonly used drone sensors and single-chip microcomputers, and the quality of the data obtained is consistent with the actual engineering, so that the designed digital filter can be directly deployed. Time-frequency domain analysis shows that the vibration signal is stable when hovering, and further proves that the fundamental frequency of the vibration signal is the motor speed.
In terms of mechanical filtering, this paper designs a rubber vibration isolator for multi-rotor aircraft, and the effectiveness of its vibration reduction is proved through experiments.
In terms of digital filtering, this article compares low-pass filters, notch filters and fourth-order band-stop filters, and through experiments proves that the fourth-order band-stop filters are more suitable for filtering the vibration signals of multi-rotor aircraft.
The entire vibration reduction process is as follows: First, the vibration isolator removes the second and above harmonic components and most of the fundamental components, and then the digital filter completely removes the fundamental components, thereby basically eliminating vibration interference noise.
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