# Vibration Reduction Design and Test of UAV Load Radar

Authors: Jia Fu; Chaohui Fan; Changshuai Yu; Guangming Liu; Haitao Luo
Journal: International Journal of Aerospace Engineering
 (2022)

---

## Abstract

It is challenging to design complex structures and calculate their dimensions only on the basis of theory. This issue was resolved efficiently by the FEM. Using the applicable test, the mechanical properties of the materials used in the damper are estimated, effectively guiding the design of the device and boosting its vibration-reducing performance. A large-load unmanned aerial vehicle (UAV) airborne radar is the subject of this research article. According to particular use case, vibration dampers for the radar-mounting platform are designed using passive vibration isolation technology to reduce aircraft vibration transmission to the radar. A conference paper has already presented prior research on the same topic. Earlier research confirmed the vibration-isolating efficacy of the T-type rubber isolator, which can greatly improve the operational conditions of the airborne radar. In contrast, earlier studies relied more on experimental validation and did not use numerical simulations. This research’s primary objective is the development of a T-type rubber isolator, whose vibration reduction performance is proven using FEM and experiments. Utilizing a suitable simulation analysis method to investigate the vibration reduction of an unmanned aerial vehicle’s (UAV) airborne component is a crucial technical task. By comparing multiple material models, a more precise model for characterizing rubber was developed.


---

## Body

## 1. Introduction

Due to advancements in communication capability and control algorithm development, the UAV business is increasing rapidly. UAVs are flexible because they may be remotely commanded and carry a range of mission equipment. On the one hand, the primary military applications of unmanned aerial vehicles are the detection of hostile activity, the delivery of alarms, and the relay of communications. On the other hand, the principal applications of UAVs in the civilian sector are plant cultivation, disaster assessment, forest fire control, and artificial precipitation [1–3]. Typically, UAVs are equipped with optical instruments, radar, inertial measurement units (IMUs), and other sensors for data collecting and flight control [4–7]. The precision of the data collected by these sensors affects the UAV’s ability to control its flight and complete its mission.The high-frequency vibrations generated by the operation of the UAV engine are a significant external component that affects the accuracy of the data. In the absence of sufficient damping in the system, high-frequency vibrations might cause sensor drift and reduce the signal’s precision [8]. Combining the aforementioned factors, vibration-dampening techniques for the on-board sensing apparatus are necessary for the UAV to operate effectively in a demanding operating environment. Vibration can cause data gathered from on-board equipment to be distorted [7, 9]. There are four primary types of stabilization solutions for UAVs: optical image stabilization, digital image stabilization, inertia-based stabilization, and mechanical vibration dampening [10]. La Rosa et al. [11] outlined the fundamental framework of the control system for optical image stabilization. Voice-coil actuators [12] and piezo actuators [13] are used in a wide range of applications. The objective of digital image stabilization is to remove the effects of global motion from an image sequence, leaving only smooth camera movement [14]. It can compensate for a wide variety of vibrations, excluding only those with a large amplitude. When reliable tracking of a target is necessary, inertia-based stabilization is most often employed. Typically, this stabilizing technology is in the form of a gimbal. Passive or active compensation methods exist [15]. Joints having a low coefficient of friction and a high internal axis of inertia can assist in passively maintaining the correct body position [16]. This stabilizing method is effective against a broad spectrum of vibrations, but not high-frequency vibrations. The first three stabilizing procedures account for the effect of vibration on the measuring device to compensate for the obtained data.The most prevalent way is isolating the collecting data device from drone vibrations. This practice is known as mechanical stabilization. It can be used alone or as a supplement to the three aforementioned techniques. According to Verma et al. [17], mechanical stabilization is the sole stabilization technique that can perform satisfactorily over the whole frequency range for all amplitudes. Passive isolation, semiactive isolation, and active isolation are the three types of mechanical stabilization procedures. Passive vibration reduction is commonly utilized, since it does not require the system to be supplied with external energy to regulate the vibration reduction device. Typically, active isolation calls for sensors, controllers, actuators, and power supply. On the one hand, active isolation systems, compared to passive isolation systems or other compensating stabilizing strategies, are more sophisticated and can result in major stability issues. On the other hand, they can greatly increase the payload’s weight. A semiactive isolation method can fix a portion of the problem, but the most effective answer is to employ a passive isolation system that functions well [18].Examples of well-known damping components include springs, dampers, rubber mounts, and viscoelastic polymers. Inspired by birds, Pete et al. [19] employed spring dampers to replicate a bird’s neck, therefore isolating the payload from the drone. Patrizi et al. [7] examined the Kyosho Zeal sheet and its effectiveness in UAV vibration isolation. Using digital low-pass filters and the Kyosho Zeal sheet, they discovered a 97.1% improvement in amplitude reduction. Small size, a broad damping frequency range, and a good damping effect at the moment of resonance are among the advantages of silicone rubber dampers, which are frequently employed in flight equipment [20–22]. Chen et al. created a rubber isolator, which was represented as a dual isolation system, to lessen the microvibration caused by the optical satellite’s flywheel. Using theoretical reasoning, they determined an acceptable cutoff frequency for the isolator and determined the RMS attenuation of the pixel offset under the dual isolation system [23]. Chen et al. discovered that the dual isolation system was effective in isolating vibrations; however, they did not specify how accurately the dual isolation system characterized the rubber vibration isolator. Xu et al. examined rubber dampers with various substrates and discovered that the mechanical properties of silicone rubber vibration isolators are rather stable and are not significantly impacted by the frequency and amplitude of the excitation [24]. The majority of damping system designs are now based on experience and refined through trial and error. This style of design requires a great deal of time and resources. In this study, the application of numerical analysis in the early design phase and the validation of the vibration isolator are the focal points. The constitutive model of the rubber material utilized in a numerical simulation has an immediate effect on the accuracy of the simulation.Rubber materials have complex mechanical properties. Rubber is a nearly incompressible substance. When loaded, it exhibits nonlinear elastic deformation, and when unloaded, the deformation returns to zero. As a result, its behavior is hyperelastic. Numerous hyperelastic models of rubber exist, including Arruda-Boyce [25], Neo-Hookean [26], Mooney-Rivlin [26–27], Yeoh [28], Gent [29], Blatz-Ko [30], and Ogden [31], among others. The Mooney-Rivlin model is widely employed and serves as the basis for a number of models based on strain tensor invariants. It can more correctly characterize the mechanical properties of rubber materials across a narrow range of strain (tensile strain within 100 percent and compressive strain within 30 percent). Before numerical simulations are performed on rubber materials, tensile tests are typically performed to get the rubber’s stress-strain curves. These curves are then fitted to the hyperelastic model’s parameters [32–33]. Rubber is viscoelastic because its stress-strain relationship exhibits hysteresis and strain rate dependence under dynamic loading. The conventional integer-order viscoelastic model has been used extensively to explore viscoelastic materials [34]. Aabqus’s viscoelastic constitutive model is the generalized Maxwell model, which accurately represents the dynamic mechanical properties of rubber across a broad frequency range. Zhou et al. employed DMA to determine the viscoelastic characteristics of silicone rubber and finite element analysis to create a model of the silicone rubber vibration isolator [35]. Zhang et al. utilized DMA to determine the parameters of the generalized Maxwell viscoelastic model and control variables in order to investigate the dynamic mechanical behavior of rubber at various frequencies, temperatures, and amplitudes [36]. In the past, numerical simulations of rubber either defined just hyperelasticity and ignored viscoelasticity or defined only viscoelasticity and replaced the nonlinear stress-strain relationship of rubber with linear elasticity.This research investigates a radar installed on a heavy payload UAV. Several typical rubber materials are summarized by analyzing the constitutive model of rubber utilized for vibration isolators. The parameters of the various constitutive models were determined by carrying out various experiments on the rubber and fitting the test results. Then, the T-type rubber vibration isolator is analyzed by the finite element method. Finally, the T-type rubber isolator is attached to the UAV-damping platform for analysis and testing inside. The performance of the damping platform is confirmed by the UAV’s actual flight test outside.


## 2. Airborne Radar Vibration Test

In accordance with the description of the radar’s performance in the specification, the amplitude of impact acceleration must not exceed 30 g with a period of 11 ms, and the root-mean-square of random vibration must not exceed 3grms in the frequency range of 5 Hz to 2000 Hz. Normal operation of the radar can continue under the approved conditions.An outside UAV flight test is conducted to examine the actual vibration acceleration response of the radar during flight. Place the acceleration sensor at the radar installation location in front of the UAV, as illustrated in Figure1. Then, we gathered the time domain and frequency domain acceleration signals of the measurement point and examined the vibration response’s frequency component and amplitude distribution. The LMS test system was used for signal acquisition in the test. An SCM209 data acquisition front end is shown in Figure 2(a). The B&K 4528-B acceleration sensor and the B&K 4524-B acceleration sensor selected for the test are shown in Figure 2(b).Figure 1
Flight vibration test.Figure 2
Signal acquisition equipment. (a) SCM209 data acquisition instrument. (b) Accelerometers.
(a)(b)Within 10 seconds of the UAV achieving flight stability, the vibration response curve is intercepted. Figure3 depicts the power spectral density (PSD) curve of acceleration in all directions.Figure 3
PSD response at radar installation location.During operation, the maximum acceleration amplitude of the UAV is close to 40 g, and the power spectral density response is 6.68 g in theX direction, 5.88 g in the Y direction, and 4.80 g in the Z direction. Analyzing the acquired vibration response reveals that the vibration response of the radar installation position during operation exceeded its bearing capacity. If the radar is mounted directly on the UAV, it will typically fail or be damaged. To preserve it, suitable vibration isolation measures must be implemented.In designing a vibration isolator, the frequency components of vibration response play a crucial role. As the UAV flies by rotating its propeller, its vibration manifests mostly as high-energy narrow-band random vibration or sinusoidal vibration superimposed by wide-band random vibration. Because the engine operates at full power during the flying phase, the influence of pneumatics on the UAV is minimal, and the broadband random vibration energy is low, while the sinusoidal vibration energy is considerable. As demonstrated in Figure4, the frequency components of vibration signals in the power spectral density curve are nearly identical in all three directions and are multiples of 100 Hz.Figure 4
Peak frequency corresponds of power spectral density.


## 3. Design of T-Type Rubber Vibration Absorber

### 3.1. Structure Design

We designed a T-type rubber vibration isolator that is symmetrically mounted on the radar mounting platform. The rubber vibration absorber comprises a retaining ring, a connecting sleeve, two gaskets, and two T-type rubbers. The structural schematic is shown in Figure5(a). Insert the connection sleeve into the mounting hole of the vibration isolation platform. The two T-shaped rubber sleeves are positioned outside of the connecting sleeve and are symmetrically oriented on both sides of the mounting hole. In addition, a gasket is positioned between the mounting plate and the T-shaped rubber on both sides. Maintain a specific distance between the retaining ring and connecting sleeve to ensure a 5% preload. In the horizontal direction, T-shaped rubber reduces vibrations more effectively. Moreover, the top and lower symmetrical structures enhance the overall damping performance and the effect of reducing unidirectional vibration. Adding a gasket between the mounting plate and the T-shaped rubber increases the bearing’s rigidity and prevents the rubber from collapsing. The processing and manufacturing of the isolator is mostly determined by the size of the T-rubber and the installed size. The dimensions are shown in Figure 5(b), and the parameter values are listed in Table 1. Table 2 displays the materials used for each component.Figure 5
T-type rubber isolator. (a) Schematic diagram. (b) Main dimensions.
(a)(b)Table 1
Main dimensions of T-type rubber isolator.
LabelD1D2L1L2L3Value (mm)208308.512.3Table 2
Material of each component.
ComponentGasketSleeveRetaining ringRubberMaterialQ235Q235Q235ZN-37The next test and simulation use the rubber isolator that was made with these parameters.


### 3.2. Dynamic Model

In the current research, the design of the vibration reduction device focuses primarily on designing the damping structure and elastic structure between the vibration source and precision device, as well as modifying their connection qualities to weaken their dynamic coupling relationship. In our research, the UAV’s engine is the cause of vibration, and the airborne radar is the device that requires vibration damping. The UAV bracket is the key structure to fixing the radar system and transmits the vibration to the radar. To simplify the vibration system, the vibration excitation in three directions might be equivalent to the vibration excitation in the horizontal and vertical directions in the spatial rectangular coordinate system.In this paper, a vibration isolation structure is designed on the front bracket of the UAV to reduce vibration transmitted to the radar. The vibration isolation structure can be simplified into a spring-damping system. According to the dynamic model structure, the spring-damping system is divided into four groups. The dynamic model of the front-mounted radar-mounting platform is shown in Figure6.Figure 6
Dynamic model of airborne radar-mounting platform.The vibration damping structure is simplified as a single-degree-of-freedom system, and the motion differential equation of the system is established:(1)mx¨+cẋ−ẏ+kx−y=0.Assuming that the input vibration signal of the UAV bracket is simple harmonic vibration, let the input vibration signal bey=Asinωt, and the complex number method is used to solve the differential equation. Both the input displacement vibration y and the steady-state displacement response x of the radar are expressed in complex form.
(2)x=aeiωt,(3)y=Beiωt−θ.Substitute into equation (1) and get
(4)k−mω2+icωBeiωt−θ=ak+icωeiωt,(5)η=BA=1+2ξλ21−λ22+2ξλ2,(6)ξ=c2mk,where η is the displacement transfer rate, λis the ratio of excitation frequency to the natural frequency, and ξ is the damping ratio. Take the frequency ratio λ as x-coordinate and the magnification factor η as y-coordinate to draw the amplitude-frequency response curves under different damping ratios, in which the displacement transfer rate η and frequency ratio λ are both dimensionless units, as shown in Figure 7.Figure 7
Amplitude-frequency response curve.It can be seen from Figure7 that when the frequency ratio λ is greater than2, the displacement transfer rate η for any size of the damping ratio ξ is less than 1. From the test results, we can know that the vibration response of UAV in actual work shows the peak response at 100 Hz and its multiple frequencies, and the PSD has a greater peak value at 100 Hz. Through the T-type rubber isolator, adjust the natural frequency of the radar structure and ensure that the transmission rate is less than one under the vibration response of the UAV. Moreover, the smaller the vibration transmission rate is, the better the vibration reduction performance of the structure will be. However, when the transmission rate is too low, the stiffness of the vibration reduction structure will become smaller, making the structure unable to bear too much load and prone to strength failure, which needs to be balanced.


## 3.1. Structure Design

We designed a T-type rubber vibration isolator that is symmetrically mounted on the radar mounting platform. The rubber vibration absorber comprises a retaining ring, a connecting sleeve, two gaskets, and two T-type rubbers. The structural schematic is shown in Figure5(a). Insert the connection sleeve into the mounting hole of the vibration isolation platform. The two T-shaped rubber sleeves are positioned outside of the connecting sleeve and are symmetrically oriented on both sides of the mounting hole. In addition, a gasket is positioned between the mounting plate and the T-shaped rubber on both sides. Maintain a specific distance between the retaining ring and connecting sleeve to ensure a 5% preload. In the horizontal direction, T-shaped rubber reduces vibrations more effectively. Moreover, the top and lower symmetrical structures enhance the overall damping performance and the effect of reducing unidirectional vibration. Adding a gasket between the mounting plate and the T-shaped rubber increases the bearing’s rigidity and prevents the rubber from collapsing. The processing and manufacturing of the isolator is mostly determined by the size of the T-rubber and the installed size. The dimensions are shown in Figure 5(b), and the parameter values are listed in Table 1. Table 2 displays the materials used for each component.Figure 5
T-type rubber isolator. (a) Schematic diagram. (b) Main dimensions.
(a)(b)Table 1
Main dimensions of T-type rubber isolator.
LabelD1D2L1L2L3Value (mm)208308.512.3Table 2
Material of each component.
ComponentGasketSleeveRetaining ringRubberMaterialQ235Q235Q235ZN-37The next test and simulation use the rubber isolator that was made with these parameters.


## 3.2. Dynamic Model

In the current research, the design of the vibration reduction device focuses primarily on designing the damping structure and elastic structure between the vibration source and precision device, as well as modifying their connection qualities to weaken their dynamic coupling relationship. In our research, the UAV’s engine is the cause of vibration, and the airborne radar is the device that requires vibration damping. The UAV bracket is the key structure to fixing the radar system and transmits the vibration to the radar. To simplify the vibration system, the vibration excitation in three directions might be equivalent to the vibration excitation in the horizontal and vertical directions in the spatial rectangular coordinate system.In this paper, a vibration isolation structure is designed on the front bracket of the UAV to reduce vibration transmitted to the radar. The vibration isolation structure can be simplified into a spring-damping system. According to the dynamic model structure, the spring-damping system is divided into four groups. The dynamic model of the front-mounted radar-mounting platform is shown in Figure6.Figure 6
Dynamic model of airborne radar-mounting platform.The vibration damping structure is simplified as a single-degree-of-freedom system, and the motion differential equation of the system is established:(1)mx¨+cẋ−ẏ+kx−y=0.Assuming that the input vibration signal of the UAV bracket is simple harmonic vibration, let the input vibration signal bey=Asinωt, and the complex number method is used to solve the differential equation. Both the input displacement vibration y and the steady-state displacement response x of the radar are expressed in complex form.
(2)x=aeiωt,(3)y=Beiωt−θ.Substitute into equation (1) and get
(4)k−mω2+icωBeiωt−θ=ak+icωeiωt,(5)η=BA=1+2ξλ21−λ22+2ξλ2,(6)ξ=c2mk,where η is the displacement transfer rate, λis the ratio of excitation frequency to the natural frequency, and ξ is the damping ratio. Take the frequency ratio λ as x-coordinate and the magnification factor η as y-coordinate to draw the amplitude-frequency response curves under different damping ratios, in which the displacement transfer rate η and frequency ratio λ are both dimensionless units, as shown in Figure 7.Figure 7
Amplitude-frequency response curve.It can be seen from Figure7 that when the frequency ratio λ is greater than2, the displacement transfer rate η for any size of the damping ratio ξ is less than 1. From the test results, we can know that the vibration response of UAV in actual work shows the peak response at 100 Hz and its multiple frequencies, and the PSD has a greater peak value at 100 Hz. Through the T-type rubber isolator, adjust the natural frequency of the radar structure and ensure that the transmission rate is less than one under the vibration response of the UAV. Moreover, the smaller the vibration transmission rate is, the better the vibration reduction performance of the structure will be. However, when the transmission rate is too low, the stiffness of the vibration reduction structure will become smaller, making the structure unable to bear too much load and prone to strength failure, which needs to be balanced.


## 4. Mechanical Property Testing of Rubber Materials

In the finite element analysis of rubber structures, the choice of material model and the accuracy of material parameters play a decisive role in the reliability of finite element analysis results. Rubber material has both hyperelasticity and viscoelasticity. Since rubber has nonlinear properties, it is important to test the rubber before FEA to obtain accurate stress-strain relationships and dynamic response characteristics.


### 4.1. Hyperelasticity Test

The primary purpose of the hyperelasticity test is to determine the stress-strain relationship of the rubber. To ensure the accuracy of finite element analysis, it is required to collect test data under various strain states.The uniaxial tensile test, biaxial tensile test, and plane tensile test were carried out using Tsinghua University’s testing equipment. The rubber used in the test is silicone rubber from the Institute of Aerospace Materials and Processes’ Z-37 series. The thickness of the rubber samples used for the three kind of tensile tests is depicted as 2 mm in the picture. The reflective strip is used to measure the strain of the rubber sample, and it is adhered to the location of the solid red line in Figure8. Each type of tensile test was conducted in three groups, with the mean value being the test result.Figure 8
Rubber specimens and dimensions. (a) Uniaxial tensile. (b) Equal biaxial tensile. (c) Plane tensile.
(a)(b)(c)Figure9 presents uniaxial tensile testing equipment, equal biaxial tensile testing equipment, and plane tensile testing equipment. The objective of the uniaxial tensile test is to determine the stress-strain curve of specimens of rubber under pure tensile stress. Utilizing sixteen in-plane actuators, the biaxial tensile test may evaluate the compression strain state of rubber material. During the test, it must be assured that all actuators apply the same force on the rubber sample and that the resultant force is 0. The objective of the plane tensile test is to evaluate the state of shear strain in rubber. In the plane tensile test, the dimensions of the specimens in the tensile direction are significantly lower than those in the breadth direction. In order to produce a strain state of simple shear deformation and measure the shear modulus of the rubber specimen, it is necessary to provide a complete constraint in the transverse direction of the specimens such that all shear occurs in the thickness direction. The specimens of rubber were examined at a strain rate of 1% per second. Six times each of 25%, 50%, 75%, and 100% strain tests were conducted in order to eliminate the Mullins effect.Figure 9
Hyperelasticity test equipment and results. (a) Uniaxial tensile test. (b) Uniaxial tensile curves. (c) Equal biaxial tensile test. (d) Equal biaxial tensile curves. (e) Plane tensile test. (f) Plane tensile curves.
(a)(b)(c)(d)(e)(f)The stress-strain curve obtained through the above material tests is the engineering stress-strain relationship. The relation between nominal stress and nominal strain should be obtained when defining the material’s stress-strain curve. It is assumed that the length before the tensile forceF is L0 and the cross-sectional area before the tensile action is A0. In the process of tensile action, it is assumed that the material volume remains unchanged, the length is L, and the cross-sectional area is A. We can get the nominal stress σn=F/A0, engineering stress σ=F/A, nominal strain εn=Δl/l0, and engineering strain ε=Δl/l.The relationship between engineering stress and strain and nominal stress and strain is as follows:(7)ε=Δll=∫l0ldll=lnll0=lnl0+Δll0=ln1+εn,(8)σ=FA0=FA0A0A=FA0ll0=FA0l0+Δll0=σn1+εn.According to equations (7) and (8), the curves of nominal stress and nominal strain of rubber in the uniaxial tensile test, equal biaxial tensile test, and plane tensile test are derived, as shown in Figure 9.Assuming that rubber is incompressible and isotropic, the above test data is fitted using the least-squares approach to the Mooney-Rivlin constitutive model. The expression for the constitutive model is as follows:(9)U=C10I¯1−3+C01I¯2−3+1D1Jel−12.The results of the constant parameters in the strain energy function of the constitutive model obtained by fitting are shown in Table3.Table 3
Mooney-Rivlin model constitutive parameters.
Material model coefficientC10C011/D1Value0.11970.01600.0


### 4.2. Viscoelasticity Test

Dynamic mechanical testing was employed in a DMA tester to investigate the frequency-dependent strain rate dependency of rubber materials. The maker of the analyzer is TA, and its model number is Q800, as indicated in Figure10. The frequency sweep experiment was performed on ZN-37 silicon rubber at a steady temperature. The experimental temperature was maintained at 25°C or room temperature. The sweep frequency range was 1–200 Hz. The step length of the frequency sweep was 1 Hz, and the amplitude was 20. Figure 11 depicts the frequency-dependent energy storage modulus, loss modulus, and damping factor of rubber material.Figure 10
Dynamic mechanical analyzer.Figure 11
Material dynamic viscoelastic test results. (a) Energy storage modulus and loss modulus. (b) Damping factor.
(a)(b)The results of the tests indicate that the energy storage modulus, loss modulus, and damping factor all vary with frequency, and the nonlinear properties are robust. For numerical simulation, the correlations between storage modulus, loss modulus, and frequency were directly incorporated into the Abaqus material model.


## 4.1. Hyperelasticity Test

The primary purpose of the hyperelasticity test is to determine the stress-strain relationship of the rubber. To ensure the accuracy of finite element analysis, it is required to collect test data under various strain states.The uniaxial tensile test, biaxial tensile test, and plane tensile test were carried out using Tsinghua University’s testing equipment. The rubber used in the test is silicone rubber from the Institute of Aerospace Materials and Processes’ Z-37 series. The thickness of the rubber samples used for the three kind of tensile tests is depicted as 2 mm in the picture. The reflective strip is used to measure the strain of the rubber sample, and it is adhered to the location of the solid red line in Figure8. Each type of tensile test was conducted in three groups, with the mean value being the test result.Figure 8
Rubber specimens and dimensions. (a) Uniaxial tensile. (b) Equal biaxial tensile. (c) Plane tensile.
(a)(b)(c)Figure9 presents uniaxial tensile testing equipment, equal biaxial tensile testing equipment, and plane tensile testing equipment. The objective of the uniaxial tensile test is to determine the stress-strain curve of specimens of rubber under pure tensile stress. Utilizing sixteen in-plane actuators, the biaxial tensile test may evaluate the compression strain state of rubber material. During the test, it must be assured that all actuators apply the same force on the rubber sample and that the resultant force is 0. The objective of the plane tensile test is to evaluate the state of shear strain in rubber. In the plane tensile test, the dimensions of the specimens in the tensile direction are significantly lower than those in the breadth direction. In order to produce a strain state of simple shear deformation and measure the shear modulus of the rubber specimen, it is necessary to provide a complete constraint in the transverse direction of the specimens such that all shear occurs in the thickness direction. The specimens of rubber were examined at a strain rate of 1% per second. Six times each of 25%, 50%, 75%, and 100% strain tests were conducted in order to eliminate the Mullins effect.Figure 9
Hyperelasticity test equipment and results. (a) Uniaxial tensile test. (b) Uniaxial tensile curves. (c) Equal biaxial tensile test. (d) Equal biaxial tensile curves. (e) Plane tensile test. (f) Plane tensile curves.
(a)(b)(c)(d)(e)(f)The stress-strain curve obtained through the above material tests is the engineering stress-strain relationship. The relation between nominal stress and nominal strain should be obtained when defining the material’s stress-strain curve. It is assumed that the length before the tensile forceF is L0 and the cross-sectional area before the tensile action is A0. In the process of tensile action, it is assumed that the material volume remains unchanged, the length is L, and the cross-sectional area is A. We can get the nominal stress σn=F/A0, engineering stress σ=F/A, nominal strain εn=Δl/l0, and engineering strain ε=Δl/l.The relationship between engineering stress and strain and nominal stress and strain is as follows:(7)ε=Δll=∫l0ldll=lnll0=lnl0+Δll0=ln1+εn,(8)σ=FA0=FA0A0A=FA0ll0=FA0l0+Δll0=σn1+εn.According to equations (7) and (8), the curves of nominal stress and nominal strain of rubber in the uniaxial tensile test, equal biaxial tensile test, and plane tensile test are derived, as shown in Figure 9.Assuming that rubber is incompressible and isotropic, the above test data is fitted using the least-squares approach to the Mooney-Rivlin constitutive model. The expression for the constitutive model is as follows:(9)U=C10I¯1−3+C01I¯2−3+1D1Jel−12.The results of the constant parameters in the strain energy function of the constitutive model obtained by fitting are shown in Table3.Table 3
Mooney-Rivlin model constitutive parameters.
Material model coefficientC10C011/D1Value0.11970.01600.0


## 4.2. Viscoelasticity Test

Dynamic mechanical testing was employed in a DMA tester to investigate the frequency-dependent strain rate dependency of rubber materials. The maker of the analyzer is TA, and its model number is Q800, as indicated in Figure10. The frequency sweep experiment was performed on ZN-37 silicon rubber at a steady temperature. The experimental temperature was maintained at 25°C or room temperature. The sweep frequency range was 1–200 Hz. The step length of the frequency sweep was 1 Hz, and the amplitude was 20. Figure 11 depicts the frequency-dependent energy storage modulus, loss modulus, and damping factor of rubber material.Figure 10
Dynamic mechanical analyzer.Figure 11
Material dynamic viscoelastic test results. (a) Energy storage modulus and loss modulus. (b) Damping factor.
(a)(b)The results of the tests indicate that the energy storage modulus, loss modulus, and damping factor all vary with frequency, and the nonlinear properties are robust. For numerical simulation, the correlations between storage modulus, loss modulus, and frequency were directly incorporated into the Abaqus material model.


## 5. Indoor Vibration Test and Finite Element Analysis

### 5.1. Indoor Vibration Test

As seen in Figure12, the vibration test of the block of equivalent mass was conducted on the shaking table to validate the vibration-damping effect of the T-rubber isolator. The T-rubber isolator was bolted to the disc fixture, and the mass block was bolted to the table of the shaking table via the disc fixture. Figure 13 shows the layout of measuring points on the vibration table and the state of the rubber shock absorber’s installation. Vibration tests were conducted in the horizontal (X and Y) and vertical (Z) directions, respectively, by altering the shaking table’s vibration direction. The frequency sweep range was 5–200 Hz, and the experimental vibration magnitudes were 0.2 g, 0.5 g, and 1 g, respectively.Figure 12
Vibration test in all directions. (a) Vibrate horizontally in theX and Y directions. (b) Vertical vibration in the Z direction.
(a)(b)Figure 13
Test preparation position and installation status.


### 5.2. Finite Element Simulation Analysis

#### 5.2.1. Selection of Rubber Material Model

In the absence of appropriate testing equipment to evaluate the material’s nonlinear properties, the rubber analysis must be simplified, i.e., using the parameters in the material performance specification. The rubber material utilized in this test is the ZN-37 series of silicone rubber developed by the Institute of Aerospace Materials and Technology. Table4 displays the parameters of the silicone rubber’s material performance specification.Table 4
Silicone rubber material parameters.
Material no.Density (t/mm3)Elastic shear modulus (MPa)Loss factorZN-371E−910.2The rubber’s material parameters in Abaqus require elastic Young’s modulusE and shear modulus G, and the relationship between them can be expressed as follows:
(10)E=2G1+μ.For incompressible materials, Poisson ratioμ is 0.5.According to the viscoelastic test results, the damping property of rubber also exhibits nonlinearity. When the loss factor of rubber is known, it can be expressed by the relationship between the material loss factor and Rayleigh damping.(11)C=αM+βK.The parametersα and β are related to the free vibration frequencies ω1, ω2, and damping ratio ξ, which can be written as follows:
(12)αβ=2ξω1+ω2ω1ω21.The damping ratioξ at the resonance peak position is half of the material loss factor. The free vibration frequencies ω1 and ω2 should be selected within the required range. The parameters in Rayleigh damping are obtained: α=0.78431, β=0.00025.The description of material model 1 follows. In the numerical analysis, rubber is described using the linear elastic material model. In Abaqus, the mode superposition approach is utilized to solve steady-state dynamics and modal response in the frequency domain.The description of material model 2 follows. To improve the precision of rubber analysis, its nonlinear stress-strain characteristics must be considered. The nonlinear stress-strain characteristic of rubber is defined using the hyperelastic model determined by the tensile experiments mentioned in Section4. The loss factor is used to characterize the material’s damping property.The description of material model 3 follows. The tensile test measures the hyperelastic model, which represents the rubber’s nonlinear response to stress and strain. Under dynamic load, the dynamic analyzer analyzes viscoelasticity, which characterizes the nonlinear behavior of the complex modulus of the rubber.


#### 5.2.2. Finite Element Model

To ensure the dependability of the analysis and test, a basic equivalent mass block model is created at the early design phase, adopting the same mass and installation size as the radar system. In addition, the T-rubber isolator’s characteristics can be evaluated and tested using the finite element model. The model of the rubber isolation platform is depicted in Figure14.Figure 14
Finite element model with 0.8 mm element size.The material parameters of each component of the isolator used in the test are shown in Table5. Input the following material parameters into the finite element software for analysis calculation. Vibration response curves for the three analysis methods are obtained using finite element analysis. The dynamic implicit analysis is implemented in Abaqus with a time step of 0.0001 and an analysis duration of 5 s. The frequency range of the sine sweep is 0–200 Hz, with an amplitude of 0.2 g, 0.5 g, and 1.0 g, respectively.Table 5
Material parameters of each component of the mass block.
No.ComponentMaterialDensity (t/mm3)Elastic shear modulus (MPa)1Mass blockAL70502.8E−97.2E42GasketQ2357.8E−92.1E53SleeveQ2357.8E−92.1E54Retaining ringQ2357.8E−92.1E55RubberZN-371E−9/In addition, mesh convergence is explored under the frequency sweep condition. Using elements ranging in size from 0.8 mm to 1.5 mm, the location of the rubber isolator is discretized. Figure15 demonstrates that there is no statistically significant difference in the peak response of elements of different sizes. In terms of peak frequency, Figure 15 illustrates that convergence can be attained with an element size of 1 mm. It will take longer to calculate the results if you reduce the size of the pieces further. There will be little change in the outcomes.Figure 15
Results of different element sizes under sweep frequency condition with 1.0 g amplitude.


### 5.3. Results and Discussion

Figure16 shows a comparison between the numerical and experimental results of three material models in three orientations. Comparing the results of the three material models in the Z direction to those in the X and Y directions, it is evident that significant differences exist. The numerical model with material model 3 is able to more accurately characterize the rubber material and produce findings that are more comparable to the experiment. Consequently, the results in the Z direction were employed for subsequent data comparisons.Figure 16
Comparison curves of simulation test with different magnitudes. (a) Material model 1 andX direction. (b) Material model 1 and Y direction. (c) Material model 1 and Z direction. (d) Material model 2 and X direction. (e) Material model 2 and Y direction. (f) Material model 2 and Z direction. (g) Material model 3 and X direction. (h) Material model 3 and Y direction. (i) Material model 3 and Z direction.
(a)(b)(c)(d)(e)(f)(g)(h)(i)The experimental and simulation results reveal the vibrational features of the mass block model. The resonance phenomenon is formed close to the first peak frequency, and the vibration response has an amplifying impact. After the peak frequency, the intensity of the vibration decreases, indicating that the shock absorber is functioning.The material of model 1 rubber is characterized by linear elasticity and material loss factor. The analysis approach reveals that the peak frequency of the first order hardly varies with vibrations of 0.2 g, 0.5 g, and 1 g. At 100 Hz and maximal magnification, the effect of vibration reduction remains approximately linear. However, experimental observations indicate that the first-order peak frequency drops with increasing amplitude and frequency drift occurs at various magnitudes. The peak magnification likewise falls as the magnitude increases. The 100 Hz drop in vibration amplitude increases as the magnitude of the vibration increases.For model 2, the peak frequency drift phenomenon may be accurately quantified using the hyperelastic model and material-damping factor in the calculation for vibration magnitudes of 0.2 g, 0.5 g, and 1 g.For model 3, using the hyper-viscoelastic model to calculate, the frequency drift phenomena of the peak frequency can be well defined in the calculation under vibration magnitudes of 0.2 g, 0.5 g, and 1 g, demonstrating the material’s nonlinear nature. The vibration reduction impact at 100 Hz increases with increasing magnitude, which is likewise well defined and most compatible with actual test results. Clearly, model 3’s data error is smaller than model 2’s, as evidenced by the graph.The first-order frequency peak, peak amplification factor, and vibration reduction effect at 100 Hz are determined by examining the vibration response curves of the three material models in theZ direction generated by finite element analysis. Tables 6–8 depict, accordingly, the comparison of simulation and test results for the three models.Table 6
Z direction simulation and test comparison of model 1.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation70.0770.0670.003.893.873.927.927.967.90Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)16.7223.0035.7429.1422.758.8459.5771.8978.63Table 7
Z direction simulation and test comparison of model 2.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation63.4859.5753.223.913.933.9432.7443.8360.11Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)5.754.583.2028.7821.568.3767.1354.7762.59Table 8
Z direction simulation and test comparison of model 3.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation63.2360.3053.534.894.904.9727.3938.5746.15Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)5.335.863.8010.932.2015.5839.8236.1924.83From the aforementioned three tables, we can get Figure17, which depicts the greatest discrepancy between the outcomes of an experiment and a computer simulation, using the same evaluation index for each model.Figure 17
The maximum error under the corresponding evaluation index of three material models.Intuitively, Figure17 suggests that material model 3, i.e., the hyper-viscoelastic material model, is better appropriate for the characterization of rubber. The numerical simulation with material model 3 demonstrates good performance across all three assessment indices, which is the most congruent with the experimental findings. In terms of first-order peak frequency, the maximum inaccuracy of material model 1 is 35.74 percent, whereas those of material models 2 and 3 are 5.75 percent and 5.86 percent, respectively. In terms of maximum magnification, material model 2 is 28% whereas model 3 is only 15.58%. Finally, the 100 Hz inaccuracy in the amplitude reduction was detected. The greatest error of material models 1 and 2 was 78% and 67%, respectively, whereas material model 3 had a maximum error of only 39.82%. The frequency response curve also reveals that the three simulation findings of the material model are comparable to the experimental data. Using material model 3, a more precise finite element model can be created. Specifically, the hyperelastic model describes the nonlinear features of stress and strain of rubber material, whereas the viscoelastic model describes the nonlinear properties of the complex modulus of rubber under dynamic load. In addition, testing and finite element analysis demonstrate that the rubber isolator can dampen the vibration transmitted from the fuselage to the radar at 100 Hz and its multiple frequencies.


## 5.1. Indoor Vibration Test

As seen in Figure12, the vibration test of the block of equivalent mass was conducted on the shaking table to validate the vibration-damping effect of the T-rubber isolator. The T-rubber isolator was bolted to the disc fixture, and the mass block was bolted to the table of the shaking table via the disc fixture. Figure 13 shows the layout of measuring points on the vibration table and the state of the rubber shock absorber’s installation. Vibration tests were conducted in the horizontal (X and Y) and vertical (Z) directions, respectively, by altering the shaking table’s vibration direction. The frequency sweep range was 5–200 Hz, and the experimental vibration magnitudes were 0.2 g, 0.5 g, and 1 g, respectively.Figure 12
Vibration test in all directions. (a) Vibrate horizontally in theX and Y directions. (b) Vertical vibration in the Z direction.
(a)(b)Figure 13
Test preparation position and installation status.


## 5.2. Finite Element Simulation Analysis

### 5.2.1. Selection of Rubber Material Model

In the absence of appropriate testing equipment to evaluate the material’s nonlinear properties, the rubber analysis must be simplified, i.e., using the parameters in the material performance specification. The rubber material utilized in this test is the ZN-37 series of silicone rubber developed by the Institute of Aerospace Materials and Technology. Table4 displays the parameters of the silicone rubber’s material performance specification.Table 4
Silicone rubber material parameters.
Material no.Density (t/mm3)Elastic shear modulus (MPa)Loss factorZN-371E−910.2The rubber’s material parameters in Abaqus require elastic Young’s modulusE and shear modulus G, and the relationship between them can be expressed as follows:
(10)E=2G1+μ.For incompressible materials, Poisson ratioμ is 0.5.According to the viscoelastic test results, the damping property of rubber also exhibits nonlinearity. When the loss factor of rubber is known, it can be expressed by the relationship between the material loss factor and Rayleigh damping.(11)C=αM+βK.The parametersα and β are related to the free vibration frequencies ω1, ω2, and damping ratio ξ, which can be written as follows:
(12)αβ=2ξω1+ω2ω1ω21.The damping ratioξ at the resonance peak position is half of the material loss factor. The free vibration frequencies ω1 and ω2 should be selected within the required range. The parameters in Rayleigh damping are obtained: α=0.78431, β=0.00025.The description of material model 1 follows. In the numerical analysis, rubber is described using the linear elastic material model. In Abaqus, the mode superposition approach is utilized to solve steady-state dynamics and modal response in the frequency domain.The description of material model 2 follows. To improve the precision of rubber analysis, its nonlinear stress-strain characteristics must be considered. The nonlinear stress-strain characteristic of rubber is defined using the hyperelastic model determined by the tensile experiments mentioned in Section4. The loss factor is used to characterize the material’s damping property.The description of material model 3 follows. The tensile test measures the hyperelastic model, which represents the rubber’s nonlinear response to stress and strain. Under dynamic load, the dynamic analyzer analyzes viscoelasticity, which characterizes the nonlinear behavior of the complex modulus of the rubber.


### 5.2.2. Finite Element Model

To ensure the dependability of the analysis and test, a basic equivalent mass block model is created at the early design phase, adopting the same mass and installation size as the radar system. In addition, the T-rubber isolator’s characteristics can be evaluated and tested using the finite element model. The model of the rubber isolation platform is depicted in Figure14.Figure 14
Finite element model with 0.8 mm element size.The material parameters of each component of the isolator used in the test are shown in Table5. Input the following material parameters into the finite element software for analysis calculation. Vibration response curves for the three analysis methods are obtained using finite element analysis. The dynamic implicit analysis is implemented in Abaqus with a time step of 0.0001 and an analysis duration of 5 s. The frequency range of the sine sweep is 0–200 Hz, with an amplitude of 0.2 g, 0.5 g, and 1.0 g, respectively.Table 5
Material parameters of each component of the mass block.
No.ComponentMaterialDensity (t/mm3)Elastic shear modulus (MPa)1Mass blockAL70502.8E−97.2E42GasketQ2357.8E−92.1E53SleeveQ2357.8E−92.1E54Retaining ringQ2357.8E−92.1E55RubberZN-371E−9/In addition, mesh convergence is explored under the frequency sweep condition. Using elements ranging in size from 0.8 mm to 1.5 mm, the location of the rubber isolator is discretized. Figure15 demonstrates that there is no statistically significant difference in the peak response of elements of different sizes. In terms of peak frequency, Figure 15 illustrates that convergence can be attained with an element size of 1 mm. It will take longer to calculate the results if you reduce the size of the pieces further. There will be little change in the outcomes.Figure 15
Results of different element sizes under sweep frequency condition with 1.0 g amplitude.


## 5.2.1. Selection of Rubber Material Model

In the absence of appropriate testing equipment to evaluate the material’s nonlinear properties, the rubber analysis must be simplified, i.e., using the parameters in the material performance specification. The rubber material utilized in this test is the ZN-37 series of silicone rubber developed by the Institute of Aerospace Materials and Technology. Table4 displays the parameters of the silicone rubber’s material performance specification.Table 4
Silicone rubber material parameters.
Material no.Density (t/mm3)Elastic shear modulus (MPa)Loss factorZN-371E−910.2The rubber’s material parameters in Abaqus require elastic Young’s modulusE and shear modulus G, and the relationship between them can be expressed as follows:
(10)E=2G1+μ.For incompressible materials, Poisson ratioμ is 0.5.According to the viscoelastic test results, the damping property of rubber also exhibits nonlinearity. When the loss factor of rubber is known, it can be expressed by the relationship between the material loss factor and Rayleigh damping.(11)C=αM+βK.The parametersα and β are related to the free vibration frequencies ω1, ω2, and damping ratio ξ, which can be written as follows:
(12)αβ=2ξω1+ω2ω1ω21.The damping ratioξ at the resonance peak position is half of the material loss factor. The free vibration frequencies ω1 and ω2 should be selected within the required range. The parameters in Rayleigh damping are obtained: α=0.78431, β=0.00025.The description of material model 1 follows. In the numerical analysis, rubber is described using the linear elastic material model. In Abaqus, the mode superposition approach is utilized to solve steady-state dynamics and modal response in the frequency domain.The description of material model 2 follows. To improve the precision of rubber analysis, its nonlinear stress-strain characteristics must be considered. The nonlinear stress-strain characteristic of rubber is defined using the hyperelastic model determined by the tensile experiments mentioned in Section4. The loss factor is used to characterize the material’s damping property.The description of material model 3 follows. The tensile test measures the hyperelastic model, which represents the rubber’s nonlinear response to stress and strain. Under dynamic load, the dynamic analyzer analyzes viscoelasticity, which characterizes the nonlinear behavior of the complex modulus of the rubber.


## 5.2.2. Finite Element Model

To ensure the dependability of the analysis and test, a basic equivalent mass block model is created at the early design phase, adopting the same mass and installation size as the radar system. In addition, the T-rubber isolator’s characteristics can be evaluated and tested using the finite element model. The model of the rubber isolation platform is depicted in Figure14.Figure 14
Finite element model with 0.8 mm element size.The material parameters of each component of the isolator used in the test are shown in Table5. Input the following material parameters into the finite element software for analysis calculation. Vibration response curves for the three analysis methods are obtained using finite element analysis. The dynamic implicit analysis is implemented in Abaqus with a time step of 0.0001 and an analysis duration of 5 s. The frequency range of the sine sweep is 0–200 Hz, with an amplitude of 0.2 g, 0.5 g, and 1.0 g, respectively.Table 5
Material parameters of each component of the mass block.
No.ComponentMaterialDensity (t/mm3)Elastic shear modulus (MPa)1Mass blockAL70502.8E−97.2E42GasketQ2357.8E−92.1E53SleeveQ2357.8E−92.1E54Retaining ringQ2357.8E−92.1E55RubberZN-371E−9/In addition, mesh convergence is explored under the frequency sweep condition. Using elements ranging in size from 0.8 mm to 1.5 mm, the location of the rubber isolator is discretized. Figure15 demonstrates that there is no statistically significant difference in the peak response of elements of different sizes. In terms of peak frequency, Figure 15 illustrates that convergence can be attained with an element size of 1 mm. It will take longer to calculate the results if you reduce the size of the pieces further. There will be little change in the outcomes.Figure 15
Results of different element sizes under sweep frequency condition with 1.0 g amplitude.


## 5.3. Results and Discussion

Figure16 shows a comparison between the numerical and experimental results of three material models in three orientations. Comparing the results of the three material models in the Z direction to those in the X and Y directions, it is evident that significant differences exist. The numerical model with material model 3 is able to more accurately characterize the rubber material and produce findings that are more comparable to the experiment. Consequently, the results in the Z direction were employed for subsequent data comparisons.Figure 16
Comparison curves of simulation test with different magnitudes. (a) Material model 1 andX direction. (b) Material model 1 and Y direction. (c) Material model 1 and Z direction. (d) Material model 2 and X direction. (e) Material model 2 and Y direction. (f) Material model 2 and Z direction. (g) Material model 3 and X direction. (h) Material model 3 and Y direction. (i) Material model 3 and Z direction.
(a)(b)(c)(d)(e)(f)(g)(h)(i)The experimental and simulation results reveal the vibrational features of the mass block model. The resonance phenomenon is formed close to the first peak frequency, and the vibration response has an amplifying impact. After the peak frequency, the intensity of the vibration decreases, indicating that the shock absorber is functioning.The material of model 1 rubber is characterized by linear elasticity and material loss factor. The analysis approach reveals that the peak frequency of the first order hardly varies with vibrations of 0.2 g, 0.5 g, and 1 g. At 100 Hz and maximal magnification, the effect of vibration reduction remains approximately linear. However, experimental observations indicate that the first-order peak frequency drops with increasing amplitude and frequency drift occurs at various magnitudes. The peak magnification likewise falls as the magnitude increases. The 100 Hz drop in vibration amplitude increases as the magnitude of the vibration increases.For model 2, the peak frequency drift phenomenon may be accurately quantified using the hyperelastic model and material-damping factor in the calculation for vibration magnitudes of 0.2 g, 0.5 g, and 1 g.For model 3, using the hyper-viscoelastic model to calculate, the frequency drift phenomena of the peak frequency can be well defined in the calculation under vibration magnitudes of 0.2 g, 0.5 g, and 1 g, demonstrating the material’s nonlinear nature. The vibration reduction impact at 100 Hz increases with increasing magnitude, which is likewise well defined and most compatible with actual test results. Clearly, model 3’s data error is smaller than model 2’s, as evidenced by the graph.The first-order frequency peak, peak amplification factor, and vibration reduction effect at 100 Hz are determined by examining the vibration response curves of the three material models in theZ direction generated by finite element analysis. Tables 6–8 depict, accordingly, the comparison of simulation and test results for the three models.Table 6
Z direction simulation and test comparison of model 1.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation70.0770.0670.003.893.873.927.927.967.90Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)16.7223.0035.7429.1422.758.8459.5771.8978.63Table 7
Z direction simulation and test comparison of model 2.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation63.4859.5753.223.913.933.9432.7443.8360.11Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)5.754.583.2028.7821.568.3767.1354.7762.59Table 8
Z direction simulation and test comparison of model 3.
First order peak frequency (Hz)Peak magnification100 Hz amplitude reduction effect0.2 g0.5 g1.0 g0.2 g0.5 g1.0 g0.2 g0.5 g1.0 gSimulation63.2360.3053.534.894.904.9727.3938.5746.15Test60.0356.9651.575.495.014.3019.5928.3236.97Error (%)5.335.863.8010.932.2015.5839.8236.1924.83From the aforementioned three tables, we can get Figure17, which depicts the greatest discrepancy between the outcomes of an experiment and a computer simulation, using the same evaluation index for each model.Figure 17
The maximum error under the corresponding evaluation index of three material models.Intuitively, Figure17 suggests that material model 3, i.e., the hyper-viscoelastic material model, is better appropriate for the characterization of rubber. The numerical simulation with material model 3 demonstrates good performance across all three assessment indices, which is the most congruent with the experimental findings. In terms of first-order peak frequency, the maximum inaccuracy of material model 1 is 35.74 percent, whereas those of material models 2 and 3 are 5.75 percent and 5.86 percent, respectively. In terms of maximum magnification, material model 2 is 28% whereas model 3 is only 15.58%. Finally, the 100 Hz inaccuracy in the amplitude reduction was detected. The greatest error of material models 1 and 2 was 78% and 67%, respectively, whereas material model 3 had a maximum error of only 39.82%. The frequency response curve also reveals that the three simulation findings of the material model are comparable to the experimental data. Using material model 3, a more precise finite element model can be created. Specifically, the hyperelastic model describes the nonlinear features of stress and strain of rubber material, whereas the viscoelastic model describes the nonlinear properties of the complex modulus of rubber under dynamic load. In addition, testing and finite element analysis demonstrate that the rubber isolator can dampen the vibration transmitted from the fuselage to the radar at 100 Hz and its multiple frequencies.


## 6. Vibration Performance Test of Airborne Radar with Rubber Isolator

### 6.1. Flight Test Preparation

As seen in Figure18(a), the T-type rubber isolator is mounted on an actual UAV prototype for flight testing. The radar and T-type rubber isolators are mounted on the UAV’s front. During flight, the acceleration sensor receives radar and fuselage vibration signals. Figure 18(b) illustrates the arrangement of measurement point 1 on the UAV fuselage and measuring point 6 on the radar.Figure 18
Outfield flight test. (a) UAV prototype. (b) Installation of platform and the location of the measuring points.
(a)(b)In Figure18(b), measuring point 6 corresponds to the radar installation location, where the vibration absorber is operational, and measuring point 1 corresponds to the UAV fuselage, where the vibration is generated.


### 6.2. Flight Test Result

The outside flight test of the UAV is equivalent to the actual working environment. Figure19 displays the time-domain response and frequency-domain response of the three directions acquired after flight stabilization.Figure 19
Field test result. (a)X direction time-domain response. (b) X direction frequency-domain response. (c) Y direction time-domain response. (d) Y direction frequency-domain response. (e) Z direction time-domain response. (f) Z direction frequency-domain response.
(a)(b)(c)(d)(e)(f)From the vibration response in the frequency domain of different directions, it can be shown that the vibration isolation platform at low frequency exhibits a resonance phenomenon that generates an amplification effect. After 100 Hz, however, the effect of reducing vibrations becomes increasingly apparent, and the vibration response at point 6 is clearly decreasing compared to the response at point 1, indicating that the vibration absorber is functioning properly.Table9 displays the vibration response results obtained during the airborne phase of the UAV’s actual flight test. At measuring point 1, where the vibration is generated, the values for the greatest acceleration response in the time domain are 44.20 g in the X direction, 35.39 g in the Y direction, and 31.28 g in the Z direction. The root-mean-square acceleration values in the X direction are 6.76 g, in the Ydirection, they are 5.53 g, and in the Zdirection, they are 5.03 g. The aforementioned vibration responses all exceed the radar environment’s criteria. At measuring point 6, which is affected by the vibration isolator, the vibration response results are as follows: the maximum acceleration response results in the time domain are 2.14 g in the X direction, 3.45 g in the Y direction, and 2.55 g in the Z direction, and the root-mean-square acceleration values are 0.51 in the X direction, 0.74 g in the Y direction, and 0.54 g in the Z direction. 92.46 percent, 86.62 percent, and 89.24 percent, respectively, are the vibration reduction efficiencies of the three directions. Following the application of the vibration absorber, a significant reduction in vibration is realized.Table 9
Data comparison of vibration reduction performance.
DirectionMeasuring point A1Measuring point A6Vibration reduction efficiencyTime-domain response (g)RMS acceleration (g)Time-domain response (g)RMS acceleration (g)X43.756.843.240.5192.55%Y33.884.885.450.9580.54%Z29.454.963.310.6087.91%


## 6.1. Flight Test Preparation

As seen in Figure18(a), the T-type rubber isolator is mounted on an actual UAV prototype for flight testing. The radar and T-type rubber isolators are mounted on the UAV’s front. During flight, the acceleration sensor receives radar and fuselage vibration signals. Figure 18(b) illustrates the arrangement of measurement point 1 on the UAV fuselage and measuring point 6 on the radar.Figure 18
Outfield flight test. (a) UAV prototype. (b) Installation of platform and the location of the measuring points.
(a)(b)In Figure18(b), measuring point 6 corresponds to the radar installation location, where the vibration absorber is operational, and measuring point 1 corresponds to the UAV fuselage, where the vibration is generated.


## 6.2. Flight Test Result

The outside flight test of the UAV is equivalent to the actual working environment. Figure19 displays the time-domain response and frequency-domain response of the three directions acquired after flight stabilization.Figure 19
Field test result. (a)X direction time-domain response. (b) X direction frequency-domain response. (c) Y direction time-domain response. (d) Y direction frequency-domain response. (e) Z direction time-domain response. (f) Z direction frequency-domain response.
(a)(b)(c)(d)(e)(f)From the vibration response in the frequency domain of different directions, it can be shown that the vibration isolation platform at low frequency exhibits a resonance phenomenon that generates an amplification effect. After 100 Hz, however, the effect of reducing vibrations becomes increasingly apparent, and the vibration response at point 6 is clearly decreasing compared to the response at point 1, indicating that the vibration absorber is functioning properly.Table9 displays the vibration response results obtained during the airborne phase of the UAV’s actual flight test. At measuring point 1, where the vibration is generated, the values for the greatest acceleration response in the time domain are 44.20 g in the X direction, 35.39 g in the Y direction, and 31.28 g in the Z direction. The root-mean-square acceleration values in the X direction are 6.76 g, in the Ydirection, they are 5.53 g, and in the Zdirection, they are 5.03 g. The aforementioned vibration responses all exceed the radar environment’s criteria. At measuring point 6, which is affected by the vibration isolator, the vibration response results are as follows: the maximum acceleration response results in the time domain are 2.14 g in the X direction, 3.45 g in the Y direction, and 2.55 g in the Z direction, and the root-mean-square acceleration values are 0.51 in the X direction, 0.74 g in the Y direction, and 0.54 g in the Z direction. 92.46 percent, 86.62 percent, and 89.24 percent, respectively, are the vibration reduction efficiencies of the three directions. Following the application of the vibration absorber, a significant reduction in vibration is realized.Table 9
Data comparison of vibration reduction performance.
DirectionMeasuring point A1Measuring point A6Vibration reduction efficiencyTime-domain response (g)RMS acceleration (g)Time-domain response (g)RMS acceleration (g)X43.756.843.240.5192.55%Y33.884.885.450.9580.54%Z29.454.963.310.6087.91%


## 7. Conclusions

In this paper, the vibration acceleration signal of a UAV during flight was collected, from which it was determined that the frequency of the UAV flight vibration signal is predominantly distributed at 100 Hz and its multiple frequencies. Under the frequency sweep condition of 200 Hz, numerical simulation and indoor experiment on the corresponding UAV vibration isolation platform are conducted. Then, it was determined that the proposed T-type rubber isolator reduces the vibration generated by the UAV to the radar effectively. In addition, we obtain a more accurate model of the rubber material by comparing the FEM findings with the experimental data. The nonlinear characteristics of stress and strain of rubber are characterized by the Mooney-Rivlin hyperelastic model, and the nonlinear characteristics of the complex modulus of rubber under dynamic load are characterized by the generalized Maxwell viscoelastic model when constructing an accurate finite element model. Finally, a real outdoor flight test was carried out. After the vibration absorber is applied, the vibration reduction phenomenon can be clearly seen through the frequency-domain PSD curves. Due to the complexity of the vibration environment, the vibration response amplitude at the first peak of the PSD curve increases with increasing vibration frequency, resulting in an amplifying effect at low-order frequency. The vibration absorber is effective at isolating vibrations at high frequencies. The frequency-domain PSD curves reveal that, despite the higher power spectral density peaks in the high-frequency region, the vibration response in the frequency range has a greater attenuation than the peak PSD without the T-type rubber isolator. During the flight test, it is evident from the time-domain curves that the vibration absorber operates effectively in the whole time domain.



---

*Source: 4186303-2022-12-02.xml*