# Experimental and Theoretical Analysis for Isolation Performance of New Combined Isolation Devices under Blast Loading

Authors: Yang Liu; Zhen Liao; Yulong Xue; Zhizhong Li; Luzhong Shao; Degao Tang
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The strong shock and vibration effect caused by explosion may pose a serious threat to the surrounding environment and the safety of personnel and equipment. This also makes the problem of vibration isolation and absorption of the structures subjected to blast loading increasingly prominent. In this paper, three kinds of new combined isolation devices with high resistance are designed and manufactured, and the characteristic parameters such as natural vibration period, frequency, and damping ratio are obtained through drop hammer impact test. Based on the Duhamel integral principle, analytical solutions of dynamic response of the combined isolation devices under rectangular pulse blast loading are derived, and the calculation expressions of transmissibility and vibration isolation rate are proposed. Combined with the test results, the isolation performance of three kinds of combined isolation devices under blast loading is obtained by using the theoretical calculation formula, and the influencing factors of isolation performance are further analyzed parametrically. The research results provide a reference for the application of combined isolation devices in isolation and shock absorption of structures under blast loading.


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## Body

## 1. Introduction

In recent years, more and more attentions have been attracted to the field of vibration isolation and absorption of structures [1–3]. As early as 1976, Harris Cyril et al. [4] compiled the handbook of impact and vibration, which comprehensively and systematically discussed all aspects of impact and vibration problems, including the theoretical basis of impact and vibration, measurement means, shock absorption and isolation technology, and the impact of vibration on human body. Shang et al. [5] studied the variation law of horizontal stiffness of three-dimensional isolated piers by carrying out quasi-static tests, established the isolation and nonisolation models of frame structures, and conducted time-history analysis to study the isolation performance of three-dimensional isolated piers. The results show that the horizontal acceleration can be attenuated by more than 40% and the internal force of the beam can be reduced by more than 30% under different ground motions. Li and Xue [6] conducted a shaking table test and study on the three-dimensional tribo-disc spring combined isolation bearing suitable for long-span structures. By applying simple harmonic excitation load and ground motion excitation load in the horizontal direction, they analyzed the influence of load change and ground motion intensity on the isolation performance of the isolation bearing. The results show that the tribo-disc spring combined isolation has good hysteresis performance in both horizontal and vertical direction, and the vertical equivalent damping ratio is between 0.10 and 0.15. Zeng [7] conducted an experimental study on the vertical and horizontal stiffness characteristics of lead-core isolation combined supports and analyzed the influence of different combined supports on the isolation effect of the superstructure under the same ground motion. The results show that the lead in lead-core rubber bearings increases the shear stiffness of the bearings to some extent, but the effect on the bending stiffness can be neglected. Xiang et al. [8] adopted simulation software MATLAB Simulink to analyze the antiexplosion and isolation performance of the isolation system with MR damper. The results show that MR damper can effectively reduce displacement, acceleration, and vibration dose value (VDV) response of the structures subjected to blast loading, showing better isolation performance. Based on Runge-Kutta method, Yan et al. [9] calculated the dynamic response of double-layer isolation system and studied the isolation performance of the system under blast loading according to VDV isolation standard. Xia [10] analyzed the vibration response of large-scale storage tank under the earthquake action by using the finite element method. After using the combination of rubber bearing and sliding bearing, it can significantly improve the isolation efficiency for the structure. The results show that the combination of lead-core rubber isolation bearing and sliding bearing can bring better isolation effect to the structure after the earthquake, and the isolation efficiency increases with the increase of oil storage, with the maximum close to 50%. In order to explore the mechanism of continuous collapse of isolated structures during explosions in underground chambers, Du and Zeng [11] used LS-DYAN to establish a computational model for the isolated structures with reinforced concrete frames. The results show that the main reason for the continuous collapse of the structure is the failure of the transmission path caused by the initial damage of adjacent members after the failure of the target member, and the degree of initial damage is the key factor affecting the collapse scale of the structure.At present, many research results have been obtained on the isolation performance of isolation devices under general seismic load or earthquake load, but few researches on the isolation performance of isolation devices subjected to explosion and impact loads. The strong vibration effect of the structure or the surrounding environment caused by the explosion load put forward higher requirements for the isolation performance of the isolation device. For this reason, three new kinds of combined isolation devices with high resistance are developed in this paper, and the isolation performance of the combined isolation device is studied experimentally and theoretically, so as to provide a reference for the application of the new combined isolation device developed in this paper in the field of structural antiexplosion.The experimental research and experimental results of the combined isolation devices are described in Section2. The formula for calculating the isolation performance of combined isolation system under explosion load is theoretically derived, and the vibration isolation rate of three kinds of combined isolation devices is calculated in Section 3. The parametric analysis for isolation performance of the isolation device is developed in Section 4, while Section 5 contains the conclusions.


## 2. Experimental Study on Combined Isolation Devices

### 2.1. Test Model Design

Three kinds of combined isolation devices are developed, each of which is composed of four steel springs with the same specification and symmetrical arrangement and a damping body. According to the selected damping material, the isolation device is named as spring-cylinder isolation device, spring-rubber isolation device, and spring-aluminum foam isolation device successively, as shown in Table1. In order to obtain reliable and effective test results, 2 models were made for each kind of isolation device. The schematic diagrams and physical diagrams of the three kinds of combined isolation devices are shown in Figure 1.Table 1
Type and quantity of combined isolation devices.
NameAbbreviationDamping materialQuantitySpring-cylinder isolation deviceTQGGCylinder damping2Spring-rubber isolation deviceTXJGRubber damping2Spring-aluminum foam isolation deviceTPMLGAluminum foam damping2Figure 1
Schematic diagrams and physical diagrams of combined isolation devices. (a) Schematic diagram of TQGG. (b) Schematic diagram of TXJG. (c) Schematic diagram of TPMLG. (d) Physical diagram of TQGG. (e) Physical diagram of TXJG. (f) Physical diagram of TPMLG.
(a)(b)(c)(d)(e)(f)Table2 lists the major specifications parameters of single steel spring, from which we can find that the maximum bearing capacity of the selected steel springs can reach 45.22 kN while the allowable maximum deformation is 340 mm. These steel springs have the characteristics of high bearing capacity and large deformation. As the major energy dissipation unit in the isolation devices, the rubber and aluminum foam damping are made of chlorinated butyl rubber and aluminum foam, respectively, whereas the cylinder damping is filled with the air at normal temperature. The cylinder damping compresses or absorbs air through lateral holes during the deformation to consume the vibration energy.Table 2
Main parameters of steel spring.
MaterialWire diameterD1 (mm)Pitch diameterD2 (mm)HeightH (mm)Bearing capacityF (kN)Allowable deformationd (mm)Stiffnessk (N/mm)60Si2MnA3522085045.22340133


### 2.2. Experimental Scheme

The test was carried out on the hard and flat ground by the drop hammer impact loading method, as shown in Figure2. The square steel counterweight was fixed on the top of the isolation device, with a side length of 70 cm and a thickness of 5 cm. A weight of 4.0 t was suspended at a certain height directly above the isolation device. When the isolation device was in a static state, the shock load was applied to the isolation device by releasing the hammer suddenly and making it fall freely. Two acceleration sensors were arranged on the lower surface of the upper plate of the isolation device. The vibration acceleration signal of each test isolation device was obtained by dynamic signal test system. The two sensors were arranged at the midpoint of the two sides of the square top plate, with a distance of 5 cm from the respective edges of the plate. Each isolation device underwent several tests until reasonable and effective acceleration signals were acquired.Figure 2
The diagram of experimental test. (a) Schematic diagram. (b) Actual test figure.
(a)(b)


### 2.3. Test Results and Analysis

In order to eliminate the burr phenomenon in the acceleration signal, five-point cubic smoothing method [12, 13] was used to smooth the original acceleration vibration signal, and then the frequency domain integration method [14, 15] to integrate the smoothed acceleration signal to obtain the displacement time-history curve of the isolation devices.The time-history curve of free vibration for typical low-damping system is shown in Figure3. Since the real damping characteristics of structure are very complex and difficult to determine, the equivalent viscous damping ratio ξ with the same attenuation rate under free vibration is usually used to represent the damping of the actual structure. For a low-damping system, higher accuracy can be obtained by calculating damping ratios of the response peaks or troughs with few intervals. The formulas for calculating the parameters can be found in [16].Figure 3
Time-history curve of free vibration for a typical low-damping system.


#### 2.3.1. Measured Acceleration Time-History Curves

According to the test results of the drop hammer impact test, the typical measured time-history curves of acceleration were selected for each isolation device, as shown in Figures4–6.Figure 4
Measured acceleration time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 5
Measured acceleration time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 6
Measured acceleration time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures4–6 that for the same type of combined isolation device, the measured acceleration time-history curves have different vibration amplitude. This is because the drop hammer height of each test and the initial impact speed of drop hammer on the isolation device is different. The counterweight of the isolation device does reciprocating movement in the vertical direction together during the vibration process, so that the acceleration time-history curve vibrates up and down at the equilibrium position, and there are multiple peaks. With the damping force, the vibration amplitude of the acceleration curve rapidly decays to zero. The vibration form of acceleration curve of the isolation device with different damping bodies is quite different. The vibration form of acceleration curve of the spring-cylinder isolation device and the spring-rubber isolation device is similar, which has many peaks, and the subsequent peaks even exceed the initial peaks. However, the acceleration curve of spring-aluminum foam isolation device shows a single peak value.


#### 2.3.2. Displacement Time-History Curves

Based on the frequency domain integration method, the measured acceleration time-history curves of isolation device are integrated twice to obtain the corresponding displacement time-history curves, as shown in Figures7–9.Figure 7
Displacement time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 8
Displacement time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 9
Displacement time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures7–9 that the displacement time-history curve of the counterweight of the isolation device oscillates up and down at the equilibrium position, the existence of the damping body in the isolation device causes the energy of the isolation system to be consumed gradually, and the peak value of the vibration displacement decreases rapidly with the increase of time. Within 3 s, the isolation system basically recovered to its original static state.The 3∼4 peaks in the displacement time-history curves of the free vibration stage of each isolation device are selected for analysis. The specific coordinates of the calculation area selected for the calculation of characteristic parameters of each isolation device are shown in Figures7–9.


### 2.4. Characteristic Parameters of Combined Isolation Devices

According to the analysis in Section2.3.2, the dynamic characteristic parameters of each combined isolation device can be obtained, and the average value of the dynamic characteristic parameters of the same type of isolation device is acquired, as shown in Table 3.Table 3
Dynamic performance parameters of combined isolation devices.
AbbreviationNatural vibration periodT (s)Frequencyf (Hz)Damping ratioξ (%)TQGG0.5261.90112.6TXJG0.4202.38113.2TPMLG0.3223.10612.5It can be seen from Table3 that the frequency range of the three kinds of combined isolation devices is between 1.901 Hz and 3.106 Hz, and the damping ratio range is between 12.5% and 13.2%. The damping material not only has a good absorption performance, but also plays a supporting role in the isolation devices, so it changes the natural period or frequency of the isolation device.


## 2.1. Test Model Design

Three kinds of combined isolation devices are developed, each of which is composed of four steel springs with the same specification and symmetrical arrangement and a damping body. According to the selected damping material, the isolation device is named as spring-cylinder isolation device, spring-rubber isolation device, and spring-aluminum foam isolation device successively, as shown in Table1. In order to obtain reliable and effective test results, 2 models were made for each kind of isolation device. The schematic diagrams and physical diagrams of the three kinds of combined isolation devices are shown in Figure 1.Table 1
Type and quantity of combined isolation devices.
NameAbbreviationDamping materialQuantitySpring-cylinder isolation deviceTQGGCylinder damping2Spring-rubber isolation deviceTXJGRubber damping2Spring-aluminum foam isolation deviceTPMLGAluminum foam damping2Figure 1
Schematic diagrams and physical diagrams of combined isolation devices. (a) Schematic diagram of TQGG. (b) Schematic diagram of TXJG. (c) Schematic diagram of TPMLG. (d) Physical diagram of TQGG. (e) Physical diagram of TXJG. (f) Physical diagram of TPMLG.
(a)(b)(c)(d)(e)(f)Table2 lists the major specifications parameters of single steel spring, from which we can find that the maximum bearing capacity of the selected steel springs can reach 45.22 kN while the allowable maximum deformation is 340 mm. These steel springs have the characteristics of high bearing capacity and large deformation. As the major energy dissipation unit in the isolation devices, the rubber and aluminum foam damping are made of chlorinated butyl rubber and aluminum foam, respectively, whereas the cylinder damping is filled with the air at normal temperature. The cylinder damping compresses or absorbs air through lateral holes during the deformation to consume the vibration energy.Table 2
Main parameters of steel spring.
MaterialWire diameterD1 (mm)Pitch diameterD2 (mm)HeightH (mm)Bearing capacityF (kN)Allowable deformationd (mm)Stiffnessk (N/mm)60Si2MnA3522085045.22340133


## 2.2. Experimental Scheme

The test was carried out on the hard and flat ground by the drop hammer impact loading method, as shown in Figure2. The square steel counterweight was fixed on the top of the isolation device, with a side length of 70 cm and a thickness of 5 cm. A weight of 4.0 t was suspended at a certain height directly above the isolation device. When the isolation device was in a static state, the shock load was applied to the isolation device by releasing the hammer suddenly and making it fall freely. Two acceleration sensors were arranged on the lower surface of the upper plate of the isolation device. The vibration acceleration signal of each test isolation device was obtained by dynamic signal test system. The two sensors were arranged at the midpoint of the two sides of the square top plate, with a distance of 5 cm from the respective edges of the plate. Each isolation device underwent several tests until reasonable and effective acceleration signals were acquired.Figure 2
The diagram of experimental test. (a) Schematic diagram. (b) Actual test figure.
(a)(b)


## 2.3. Test Results and Analysis

In order to eliminate the burr phenomenon in the acceleration signal, five-point cubic smoothing method [12, 13] was used to smooth the original acceleration vibration signal, and then the frequency domain integration method [14, 15] to integrate the smoothed acceleration signal to obtain the displacement time-history curve of the isolation devices.The time-history curve of free vibration for typical low-damping system is shown in Figure3. Since the real damping characteristics of structure are very complex and difficult to determine, the equivalent viscous damping ratio ξ with the same attenuation rate under free vibration is usually used to represent the damping of the actual structure. For a low-damping system, higher accuracy can be obtained by calculating damping ratios of the response peaks or troughs with few intervals. The formulas for calculating the parameters can be found in [16].Figure 3
Time-history curve of free vibration for a typical low-damping system.


### 2.3.1. Measured Acceleration Time-History Curves

According to the test results of the drop hammer impact test, the typical measured time-history curves of acceleration were selected for each isolation device, as shown in Figures4–6.Figure 4
Measured acceleration time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 5
Measured acceleration time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 6
Measured acceleration time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures4–6 that for the same type of combined isolation device, the measured acceleration time-history curves have different vibration amplitude. This is because the drop hammer height of each test and the initial impact speed of drop hammer on the isolation device is different. The counterweight of the isolation device does reciprocating movement in the vertical direction together during the vibration process, so that the acceleration time-history curve vibrates up and down at the equilibrium position, and there are multiple peaks. With the damping force, the vibration amplitude of the acceleration curve rapidly decays to zero. The vibration form of acceleration curve of the isolation device with different damping bodies is quite different. The vibration form of acceleration curve of the spring-cylinder isolation device and the spring-rubber isolation device is similar, which has many peaks, and the subsequent peaks even exceed the initial peaks. However, the acceleration curve of spring-aluminum foam isolation device shows a single peak value.


### 2.3.2. Displacement Time-History Curves

Based on the frequency domain integration method, the measured acceleration time-history curves of isolation device are integrated twice to obtain the corresponding displacement time-history curves, as shown in Figures7–9.Figure 7
Displacement time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 8
Displacement time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 9
Displacement time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures7–9 that the displacement time-history curve of the counterweight of the isolation device oscillates up and down at the equilibrium position, the existence of the damping body in the isolation device causes the energy of the isolation system to be consumed gradually, and the peak value of the vibration displacement decreases rapidly with the increase of time. Within 3 s, the isolation system basically recovered to its original static state.The 3∼4 peaks in the displacement time-history curves of the free vibration stage of each isolation device are selected for analysis. The specific coordinates of the calculation area selected for the calculation of characteristic parameters of each isolation device are shown in Figures7–9.


## 2.3.1. Measured Acceleration Time-History Curves

According to the test results of the drop hammer impact test, the typical measured time-history curves of acceleration were selected for each isolation device, as shown in Figures4–6.Figure 4
Measured acceleration time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 5
Measured acceleration time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 6
Measured acceleration time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures4–6 that for the same type of combined isolation device, the measured acceleration time-history curves have different vibration amplitude. This is because the drop hammer height of each test and the initial impact speed of drop hammer on the isolation device is different. The counterweight of the isolation device does reciprocating movement in the vertical direction together during the vibration process, so that the acceleration time-history curve vibrates up and down at the equilibrium position, and there are multiple peaks. With the damping force, the vibration amplitude of the acceleration curve rapidly decays to zero. The vibration form of acceleration curve of the isolation device with different damping bodies is quite different. The vibration form of acceleration curve of the spring-cylinder isolation device and the spring-rubber isolation device is similar, which has many peaks, and the subsequent peaks even exceed the initial peaks. However, the acceleration curve of spring-aluminum foam isolation device shows a single peak value.


## 2.3.2. Displacement Time-History Curves

Based on the frequency domain integration method, the measured acceleration time-history curves of isolation device are integrated twice to obtain the corresponding displacement time-history curves, as shown in Figures7–9.Figure 7
Displacement time-history curves of spring-cylinder isolation device. (a) TQGG1. (b) TQGG2.
(a)(b)Figure 8
Displacement time-history curves of spring-rubber isolation device. (a) TXJG1. (b) TXJG2.
(a)(b)Figure 9
Displacement time-history curves of spring-aluminum foam isolation device. (a) TPMLG1. (b) TPMLG2.
(a)(b)It can be seen from Figures7–9 that the displacement time-history curve of the counterweight of the isolation device oscillates up and down at the equilibrium position, the existence of the damping body in the isolation device causes the energy of the isolation system to be consumed gradually, and the peak value of the vibration displacement decreases rapidly with the increase of time. Within 3 s, the isolation system basically recovered to its original static state.The 3∼4 peaks in the displacement time-history curves of the free vibration stage of each isolation device are selected for analysis. The specific coordinates of the calculation area selected for the calculation of characteristic parameters of each isolation device are shown in Figures7–9.


## 2.4. Characteristic Parameters of Combined Isolation Devices

According to the analysis in Section2.3.2, the dynamic characteristic parameters of each combined isolation device can be obtained, and the average value of the dynamic characteristic parameters of the same type of isolation device is acquired, as shown in Table 3.Table 3
Dynamic performance parameters of combined isolation devices.
AbbreviationNatural vibration periodT (s)Frequencyf (Hz)Damping ratioξ (%)TQGG0.5261.90112.6TXJG0.4202.38113.2TPMLG0.3223.10612.5It can be seen from Table3 that the frequency range of the three kinds of combined isolation devices is between 1.901 Hz and 3.106 Hz, and the damping ratio range is between 12.5% and 13.2%. The damping material not only has a good absorption performance, but also plays a supporting role in the isolation devices, so it changes the natural period or frequency of the isolation device.


## 3. Calculation of Isolation Performance of Isolation System

### 3.1. Excitation Load of Isolation System

In this paper, the excitation loadFt of the isolation system is a short-duration explosion load with exponential attenuation. The following formula gives the specific attenuation form of the explosion load [17]:(1)Ft=1.5×1050.530.006t+0.00616.5+0.470.006t+0.0061.261−t0.029.When the duration of explosion loadF (t) is much less than the natural vibration period of the isolation system, the response of the isolation system x (t) mainly depends on the impulse of the explosion load, and the shape of the time-history curve of the explosion load has little influence on the system response [18]. In order to obtain the analytical solution of the response of the isolation system subjected to blast load, the rectangular impulse load Ft with equal impulse is constructed to replace the real blast load Ft. If the peak value of rectangular pulse load is the same as the peak value of explosion load Fmax, then the duration time of rectangular pulse load is as follows:(2)t1=1Fmax∫0t+Ftdt,where Fmax and t+ are the peak value and positive overpressure duration time of explosion load, respectively.The equivalent rectangular pulseft is expressed as(3)ft=Fmax,0≤t≤t1,0,t>t1.


### 3.2. Analytical Solution of Isolation Rate of Isolation System

#### 3.2.1. Analytical Solution of Motion Equation of Isolated System

The shock and vibration isolation system in this paper belongs to a single-stage active isolation system. It is assumed that the mass of the isolation system ism, the blast load received by the isolation system is F (t), the stiffness of the isolation system is k, the damping coefficient is c, and the response displacement of the isolation system along the vertical direction is x, as shown in Figure 10. Then the motion differential equation of the isolation system is as follows:(4)mx¨+cx˙+kx=Ft.Figure 10
Mechanical model of combined isolation system.By introducing the damping ratio (ξ=c/2mωn), the motion equation can be rewritten as(5)x¨+2ξωnx˙+ωn2x=Ftm,where ωn=k/m is the undamped angular frequency of isolation system.The Duhamel integral formula of the response of the isolation system with viscous damping under the general explosion loadFt is as follows [16]:(6)xt=1mωD∫0tFτsinωDt−τexp−ξωnt−τdτ,where ωD=ωn1−ξ2 and ωD is the damped angular frequency of the isolation system.(1)
When0≤t≤t1, the equivalent rectangular pulse load ft=Fmax=const. The isolation system is in the forced vibration stage:(7)xt=FmaxmωD∫0tsinωDt−τexp−ξωnt−τdτ,After second times of partial integration, the expression of response displacementxt of isolation system is obtained:(8)xt=Fmaxk1−exp−ξωntcosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(9)x˙t=Fmaxωnk1−ξ2sinωDt⋅exp−ξωnt.(2)
Whent>t1, the equivalent rectangular pulse load ft=0. At this time, the isolation system is in the stage of free vibration. The displacement xt1 and velocity x˙t1 at the time t1 of forced vibration stage are taken as the initial conditions of damped free vibration. The displacement response is calculated as follows:(10)xt=xt1cosωDt−t1+x˙t1+xt1ξωnωDsinωDt−t1exp−ξωnt−t1,where(11)xt1=Fmaxk1−exp−ξωnt1cosωDt1+ξ1−ξ2sinωDt1,(12)x˙t1=Fmaxωnk1−ξ2sinωDt1⋅exp−ξωnt1.The displacementxt1 and velocity x˙t1 at time t1 are substituted into equation (10). The following results are obtained:(13)xt=FmaxkA−Be−ξωnt1e−ξωnt−t1,where(14)A=cosωDt−t1+ξ1−ξ2sinωDt−t1,B=cosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(15)x˙t=−Fmaxωk1−ξ2sinωDt−t1−sinωDt⋅e−ξωnt1e−ξωnt−t1.Equations (8) and (9) and equations (13)–(15) are the analytical solutions of response displacement xt and response velocity x˙t of the differential motion equation of the isolated system during 0≤t≤t1 and t>t1, respectively.


#### 3.2.2. Calculation of Isolation Rate of Isolation System

For a single-stage active isolation system, its input is the external excitation loadFt, and its output is the disturbance force nt transmitted by the isolation system to the foundation. From the mechanical model of single-stage active isolation system in Figure 10, it can be seen that(16)nt=kx+cx˙.According to the definition of the transmission rate of the isolation system, the transmission rate of any isolation system is equal to the maximum value of the ratio of the system’s output and input [19]. For the combined isolation system proposed in this paper, the transmission rate of the system is the ratio of the disturbance dynamic amplitude of the given foundation to the initial excitation load amplitude Fmax. Therefore, the time-history curves of the transmission rate Tqt can be defined as(17)Tqt=ntFmax,where Fmax is the peak value of explosion load.By substituting the analytical solutions of the response displacementxt and the response speed x˙t into equation (16), it can be concluded that the disturbance force of the foundation is as follows:(18)nt=Fmax1−Ce−ξωnt,0≤t≤t1,FmaxD−Ce−ξωnt1e−ξωnt−t1,t>t1,where(19)C=cosωDt−ξ1−ξ2sinωDt,D=cosωDt−t1−ξ1−ξ2sinωDt−t1.Combining formulas (17) and (18), we can obtain(20)Tqt=1−Ce−ξωnt,0≤t≤t1,D−Ce−ξωnt1e−ξωnt−t1,t>t1,In the process of vibration, the time-history curvesηt of isolation rate have the following relationship with Tqt:(21)ηt=1−Tqt.The isolation rateη and the transmission rate Tq of the isolation system are obtained by taking the maximum values of ηt and Tqt, respectively.


### 3.3. Analysis of Isolation Rate of Combined Isolation Device

In order to further compare and analyze the isolation effect of the isolation devices, the basic parameters of the combined isolation devices obtained from the tests have been used to calculate the transmissibility and vibration isolation rate using the isolation rate formula derived in this paper. The main performance parameters of the three kinds of isolation devices are shown in Table4. It is worth mentioning that the stiffness k of the combined isolation device in Table 4 is calculated based on the natural vibration period obtained from the dynamic test and the mass (4191 kg) during the test, which is the dynamic stiffness of the isolation device. The mass of the combined isolation device under normal working condition is only 2,700 kg. Based on this mass, the angular frequency ωn of the isolation device is recalculated. After calculation, the transmissibility and isolation rate curves of three kinds of combined isolation devices under the same rectangular pulse load are shown in Figure 11.Table 4
Primary performance parameters of combined isolation devices.
AbbreviationMassm (kg)Stiffnessk (N·mm−1)Angular frequencyωn (rad·s−1)Damping ratioξ (%)Duration timet1 (ms)TQGG270061715.11712.62.44TXJG270094218.67913.22.44TPMLG2700159624.31312.52.44Table 5
The transmissibility and isolation rate of the combined isolation devices.
AbbreviationTransmissibilityTq (%)Vibration isolation rateη (%)TQGG3.196.9TXJG3.996.1TPMLG7.392.7Figure 11
Transmissibility and vibration isolation rate time-history curves of combined devices. (a) Transmissibility curve of spring-cylinder isolation device. (b) Vibration isolation rate curve of spring-cylinder isolation device. (c) Transmissibility curve of spring-rubber isolation device. (d) Vibration isolation rate curve of spring-rubber isolation device. (e) Transmissibility curve of spring-aluminum foam isolation device. (f) Vibration isolation rate curve of spring-aluminum foam isolation device.
(a)(b)(c)(d)(e)(f)As can be seen from Figure11, the transmissibility and isolation rate curves of the isolation devices under the same explosion load are similar in shape, but the peak value and attenuation speed of each curve are different. The spring-cylinder isolation device is better than spring-rubber isolation device and spring-aluminum foam isolation device. The higher the damping ratio of the isolation device, the less the frequency of subsequent wave peaks of the transmission rate and isolation rate and the faster the attenuation, indicating that the more obvious the energy dissipation effect is, the better the isolation and absorption performance is. The transmissibility and isolation rate of the three kinds of combined isolation devices are shown in Table 5.In the strong shock and vibration environment caused by explosion, a good isolation device not only requires a high isolation efficiency, but also requires a good performance of the reuse of the isolation device, especially the damping material. Due to the inevitable plastic deformation in the response process, aluminum foam is not suitable as an ideal damping material. It can be seen from Table5 that compared with aluminum foam damping body, rubber damping body and cylinder damping body have higher reliability in working state.


## 3.1. Excitation Load of Isolation System

In this paper, the excitation loadFt of the isolation system is a short-duration explosion load with exponential attenuation. The following formula gives the specific attenuation form of the explosion load [17]:(1)Ft=1.5×1050.530.006t+0.00616.5+0.470.006t+0.0061.261−t0.029.When the duration of explosion loadF (t) is much less than the natural vibration period of the isolation system, the response of the isolation system x (t) mainly depends on the impulse of the explosion load, and the shape of the time-history curve of the explosion load has little influence on the system response [18]. In order to obtain the analytical solution of the response of the isolation system subjected to blast load, the rectangular impulse load Ft with equal impulse is constructed to replace the real blast load Ft. If the peak value of rectangular pulse load is the same as the peak value of explosion load Fmax, then the duration time of rectangular pulse load is as follows:(2)t1=1Fmax∫0t+Ftdt,where Fmax and t+ are the peak value and positive overpressure duration time of explosion load, respectively.The equivalent rectangular pulseft is expressed as(3)ft=Fmax,0≤t≤t1,0,t>t1.


## 3.2. Analytical Solution of Isolation Rate of Isolation System

### 3.2.1. Analytical Solution of Motion Equation of Isolated System

The shock and vibration isolation system in this paper belongs to a single-stage active isolation system. It is assumed that the mass of the isolation system ism, the blast load received by the isolation system is F (t), the stiffness of the isolation system is k, the damping coefficient is c, and the response displacement of the isolation system along the vertical direction is x, as shown in Figure 10. Then the motion differential equation of the isolation system is as follows:(4)mx¨+cx˙+kx=Ft.Figure 10
Mechanical model of combined isolation system.By introducing the damping ratio (ξ=c/2mωn), the motion equation can be rewritten as(5)x¨+2ξωnx˙+ωn2x=Ftm,where ωn=k/m is the undamped angular frequency of isolation system.The Duhamel integral formula of the response of the isolation system with viscous damping under the general explosion loadFt is as follows [16]:(6)xt=1mωD∫0tFτsinωDt−τexp−ξωnt−τdτ,where ωD=ωn1−ξ2 and ωD is the damped angular frequency of the isolation system.(1)
When0≤t≤t1, the equivalent rectangular pulse load ft=Fmax=const. The isolation system is in the forced vibration stage:(7)xt=FmaxmωD∫0tsinωDt−τexp−ξωnt−τdτ,After second times of partial integration, the expression of response displacementxt of isolation system is obtained:(8)xt=Fmaxk1−exp−ξωntcosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(9)x˙t=Fmaxωnk1−ξ2sinωDt⋅exp−ξωnt.(2)
Whent>t1, the equivalent rectangular pulse load ft=0. At this time, the isolation system is in the stage of free vibration. The displacement xt1 and velocity x˙t1 at the time t1 of forced vibration stage are taken as the initial conditions of damped free vibration. The displacement response is calculated as follows:(10)xt=xt1cosωDt−t1+x˙t1+xt1ξωnωDsinωDt−t1exp−ξωnt−t1,where(11)xt1=Fmaxk1−exp−ξωnt1cosωDt1+ξ1−ξ2sinωDt1,(12)x˙t1=Fmaxωnk1−ξ2sinωDt1⋅exp−ξωnt1.The displacementxt1 and velocity x˙t1 at time t1 are substituted into equation (10). The following results are obtained:(13)xt=FmaxkA−Be−ξωnt1e−ξωnt−t1,where(14)A=cosωDt−t1+ξ1−ξ2sinωDt−t1,B=cosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(15)x˙t=−Fmaxωk1−ξ2sinωDt−t1−sinωDt⋅e−ξωnt1e−ξωnt−t1.Equations (8) and (9) and equations (13)–(15) are the analytical solutions of response displacement xt and response velocity x˙t of the differential motion equation of the isolated system during 0≤t≤t1 and t>t1, respectively.


### 3.2.2. Calculation of Isolation Rate of Isolation System

For a single-stage active isolation system, its input is the external excitation loadFt, and its output is the disturbance force nt transmitted by the isolation system to the foundation. From the mechanical model of single-stage active isolation system in Figure 10, it can be seen that(16)nt=kx+cx˙.According to the definition of the transmission rate of the isolation system, the transmission rate of any isolation system is equal to the maximum value of the ratio of the system’s output and input [19]. For the combined isolation system proposed in this paper, the transmission rate of the system is the ratio of the disturbance dynamic amplitude of the given foundation to the initial excitation load amplitude Fmax. Therefore, the time-history curves of the transmission rate Tqt can be defined as(17)Tqt=ntFmax,where Fmax is the peak value of explosion load.By substituting the analytical solutions of the response displacementxt and the response speed x˙t into equation (16), it can be concluded that the disturbance force of the foundation is as follows:(18)nt=Fmax1−Ce−ξωnt,0≤t≤t1,FmaxD−Ce−ξωnt1e−ξωnt−t1,t>t1,where(19)C=cosωDt−ξ1−ξ2sinωDt,D=cosωDt−t1−ξ1−ξ2sinωDt−t1.Combining formulas (17) and (18), we can obtain(20)Tqt=1−Ce−ξωnt,0≤t≤t1,D−Ce−ξωnt1e−ξωnt−t1,t>t1,In the process of vibration, the time-history curvesηt of isolation rate have the following relationship with Tqt:(21)ηt=1−Tqt.The isolation rateη and the transmission rate Tq of the isolation system are obtained by taking the maximum values of ηt and Tqt, respectively.


## 3.2.1. Analytical Solution of Motion Equation of Isolated System

The shock and vibration isolation system in this paper belongs to a single-stage active isolation system. It is assumed that the mass of the isolation system ism, the blast load received by the isolation system is F (t), the stiffness of the isolation system is k, the damping coefficient is c, and the response displacement of the isolation system along the vertical direction is x, as shown in Figure 10. Then the motion differential equation of the isolation system is as follows:(4)mx¨+cx˙+kx=Ft.Figure 10
Mechanical model of combined isolation system.By introducing the damping ratio (ξ=c/2mωn), the motion equation can be rewritten as(5)x¨+2ξωnx˙+ωn2x=Ftm,where ωn=k/m is the undamped angular frequency of isolation system.The Duhamel integral formula of the response of the isolation system with viscous damping under the general explosion loadFt is as follows [16]:(6)xt=1mωD∫0tFτsinωDt−τexp−ξωnt−τdτ,where ωD=ωn1−ξ2 and ωD is the damped angular frequency of the isolation system.(1)
When0≤t≤t1, the equivalent rectangular pulse load ft=Fmax=const. The isolation system is in the forced vibration stage:(7)xt=FmaxmωD∫0tsinωDt−τexp−ξωnt−τdτ,After second times of partial integration, the expression of response displacementxt of isolation system is obtained:(8)xt=Fmaxk1−exp−ξωntcosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(9)x˙t=Fmaxωnk1−ξ2sinωDt⋅exp−ξωnt.(2)
Whent>t1, the equivalent rectangular pulse load ft=0. At this time, the isolation system is in the stage of free vibration. The displacement xt1 and velocity x˙t1 at the time t1 of forced vibration stage are taken as the initial conditions of damped free vibration. The displacement response is calculated as follows:(10)xt=xt1cosωDt−t1+x˙t1+xt1ξωnωDsinωDt−t1exp−ξωnt−t1,where(11)xt1=Fmaxk1−exp−ξωnt1cosωDt1+ξ1−ξ2sinωDt1,(12)x˙t1=Fmaxωnk1−ξ2sinωDt1⋅exp−ξωnt1.The displacementxt1 and velocity x˙t1 at time t1 are substituted into equation (10). The following results are obtained:(13)xt=FmaxkA−Be−ξωnt1e−ξωnt−t1,where(14)A=cosωDt−t1+ξ1−ξ2sinωDt−t1,B=cosωDt+ξ1−ξ2sinωDt.Take the derivative of time to get the expression of response velocity of the isolation system as follows:(15)x˙t=−Fmaxωk1−ξ2sinωDt−t1−sinωDt⋅e−ξωnt1e−ξωnt−t1.Equations (8) and (9) and equations (13)–(15) are the analytical solutions of response displacement xt and response velocity x˙t of the differential motion equation of the isolated system during 0≤t≤t1 and t>t1, respectively.


## 3.2.2. Calculation of Isolation Rate of Isolation System

For a single-stage active isolation system, its input is the external excitation loadFt, and its output is the disturbance force nt transmitted by the isolation system to the foundation. From the mechanical model of single-stage active isolation system in Figure 10, it can be seen that(16)nt=kx+cx˙.According to the definition of the transmission rate of the isolation system, the transmission rate of any isolation system is equal to the maximum value of the ratio of the system’s output and input [19]. For the combined isolation system proposed in this paper, the transmission rate of the system is the ratio of the disturbance dynamic amplitude of the given foundation to the initial excitation load amplitude Fmax. Therefore, the time-history curves of the transmission rate Tqt can be defined as(17)Tqt=ntFmax,where Fmax is the peak value of explosion load.By substituting the analytical solutions of the response displacementxt and the response speed x˙t into equation (16), it can be concluded that the disturbance force of the foundation is as follows:(18)nt=Fmax1−Ce−ξωnt,0≤t≤t1,FmaxD−Ce−ξωnt1e−ξωnt−t1,t>t1,where(19)C=cosωDt−ξ1−ξ2sinωDt,D=cosωDt−t1−ξ1−ξ2sinωDt−t1.Combining formulas (17) and (18), we can obtain(20)Tqt=1−Ce−ξωnt,0≤t≤t1,D−Ce−ξωnt1e−ξωnt−t1,t>t1,In the process of vibration, the time-history curvesηt of isolation rate have the following relationship with Tqt:(21)ηt=1−Tqt.The isolation rateη and the transmission rate Tq of the isolation system are obtained by taking the maximum values of ηt and Tqt, respectively.


## 3.3. Analysis of Isolation Rate of Combined Isolation Device

In order to further compare and analyze the isolation effect of the isolation devices, the basic parameters of the combined isolation devices obtained from the tests have been used to calculate the transmissibility and vibration isolation rate using the isolation rate formula derived in this paper. The main performance parameters of the three kinds of isolation devices are shown in Table4. It is worth mentioning that the stiffness k of the combined isolation device in Table 4 is calculated based on the natural vibration period obtained from the dynamic test and the mass (4191 kg) during the test, which is the dynamic stiffness of the isolation device. The mass of the combined isolation device under normal working condition is only 2,700 kg. Based on this mass, the angular frequency ωn of the isolation device is recalculated. After calculation, the transmissibility and isolation rate curves of three kinds of combined isolation devices under the same rectangular pulse load are shown in Figure 11.Table 4
Primary performance parameters of combined isolation devices.
AbbreviationMassm (kg)Stiffnessk (N·mm−1)Angular frequencyωn (rad·s−1)Damping ratioξ (%)Duration timet1 (ms)TQGG270061715.11712.62.44TXJG270094218.67913.22.44TPMLG2700159624.31312.52.44Table 5
The transmissibility and isolation rate of the combined isolation devices.
AbbreviationTransmissibilityTq (%)Vibration isolation rateη (%)TQGG3.196.9TXJG3.996.1TPMLG7.392.7Figure 11
Transmissibility and vibration isolation rate time-history curves of combined devices. (a) Transmissibility curve of spring-cylinder isolation device. (b) Vibration isolation rate curve of spring-cylinder isolation device. (c) Transmissibility curve of spring-rubber isolation device. (d) Vibration isolation rate curve of spring-rubber isolation device. (e) Transmissibility curve of spring-aluminum foam isolation device. (f) Vibration isolation rate curve of spring-aluminum foam isolation device.
(a)(b)(c)(d)(e)(f)As can be seen from Figure11, the transmissibility and isolation rate curves of the isolation devices under the same explosion load are similar in shape, but the peak value and attenuation speed of each curve are different. The spring-cylinder isolation device is better than spring-rubber isolation device and spring-aluminum foam isolation device. The higher the damping ratio of the isolation device, the less the frequency of subsequent wave peaks of the transmission rate and isolation rate and the faster the attenuation, indicating that the more obvious the energy dissipation effect is, the better the isolation and absorption performance is. The transmissibility and isolation rate of the three kinds of combined isolation devices are shown in Table 5.In the strong shock and vibration environment caused by explosion, a good isolation device not only requires a high isolation efficiency, but also requires a good performance of the reuse of the isolation device, especially the damping material. Due to the inevitable plastic deformation in the response process, aluminum foam is not suitable as an ideal damping material. It can be seen from Table5 that compared with aluminum foam damping body, rubber damping body and cylinder damping body have higher reliability in working state.


## 4. Parametric Analysis of Isolation Performance

For the isolation system subjected to explosion loads, when its bearing quality is certain, the isolation effect of the isolation device is affected by a variety of factors [20, 21]. In order to further analyze the influencing factors of the isolation performance, the isolation rate of the isolation device with different stiffness, damping ratio, and angular frequency is calculated.


### 4.1. Effect of Stiffness

In order to study the influence of the stiffness of the isolation device on the isolation rate under the rectangular pulse load, the variation law of the isolation rate with different stiffness is obtained. In this paper, the calculation formula of the isolation rate of the isolation system derived from Section3.2 is used to obtain the isolation rate curves changing with the stiffness under three different damping ratios, as shown in Figure 12.Figure 12
Vibration isolation rate curve of isolation device with changing stiffness.From Figure12, it can be seen that the isolation rate decreases with the increase of stiffness. With the increase of damping ratio, the decrease trend of isolation rate decreases with the increase of stiffness. When the stiffness is small, the isolation rate of the isolation device with three damping ratios is basically the same.


### 4.2. Effect of Damping Ratio

In order to analyze the influence of damping ratio on isolation rate of isolation device, the change curves of isolation rate with damping ratio under three different stiffness conditions are calculated, as shown in Figure13.Figure 13
Vibration isolation rate curve of isolation device with damping ratio.It can be seen from Figure13 that the isolation rate of the isolation device increases first and then decreases with the increase of the damping ratio. Compared with the curvature of the three curves, the greater the stiffness is, the greater the influence of the damping ratio on the isolation rate is. There is an optimal damping ratio in the isolation device, which is independent of stiffness. The optimal damping ratio is 25.8%, and isolation device has the highest isolation rate and the best isolation effect under this condition.


### 4.3. Effect of Angular Frequency

In order to analyze the influence of the angular frequency on the isolation rate of the isolation device, the variation curve of the isolation rate with angular frequency under three different damping ratios is obtained, as shown in Figure14.Figure 14
Vibration isolation rate curve of isolation device with angular frequency.As can be seen from Figure14, the isolation rate of the isolation device decreases approximately linearly with the increase of natural vibration angular frequency. The slope of isolation rate is related to the damping ratio. The lower the damping ratio, the slower the decreasing trend of isolation with the increase of stiffness.


## 4.1. Effect of Stiffness

In order to study the influence of the stiffness of the isolation device on the isolation rate under the rectangular pulse load, the variation law of the isolation rate with different stiffness is obtained. In this paper, the calculation formula of the isolation rate of the isolation system derived from Section3.2 is used to obtain the isolation rate curves changing with the stiffness under three different damping ratios, as shown in Figure 12.Figure 12
Vibration isolation rate curve of isolation device with changing stiffness.From Figure12, it can be seen that the isolation rate decreases with the increase of stiffness. With the increase of damping ratio, the decrease trend of isolation rate decreases with the increase of stiffness. When the stiffness is small, the isolation rate of the isolation device with three damping ratios is basically the same.


## 4.2. Effect of Damping Ratio

In order to analyze the influence of damping ratio on isolation rate of isolation device, the change curves of isolation rate with damping ratio under three different stiffness conditions are calculated, as shown in Figure13.Figure 13
Vibration isolation rate curve of isolation device with damping ratio.It can be seen from Figure13 that the isolation rate of the isolation device increases first and then decreases with the increase of the damping ratio. Compared with the curvature of the three curves, the greater the stiffness is, the greater the influence of the damping ratio on the isolation rate is. There is an optimal damping ratio in the isolation device, which is independent of stiffness. The optimal damping ratio is 25.8%, and isolation device has the highest isolation rate and the best isolation effect under this condition.


## 4.3. Effect of Angular Frequency

In order to analyze the influence of the angular frequency on the isolation rate of the isolation device, the variation curve of the isolation rate with angular frequency under three different damping ratios is obtained, as shown in Figure14.Figure 14
Vibration isolation rate curve of isolation device with angular frequency.As can be seen from Figure14, the isolation rate of the isolation device decreases approximately linearly with the increase of natural vibration angular frequency. The slope of isolation rate is related to the damping ratio. The lower the damping ratio, the slower the decreasing trend of isolation with the increase of stiffness.


## 5. Conclusions

In this paper, three new kinds of combined isolation devices are designed and manufactured, and experimental and theoretical studies are carried out. Based on the Duhamel integral principle, the analytical expression of isolation rate of the isolation device is obtained, and the isolation performance of the combined isolation device and its influencing factors are analyzed. The main conclusions are as follows.(1)
The damping ratio of spring-cylinder isolation device, spring-rubber isolation device, and spring-aluminum foam isolation device developed in this paper ranges from 0.125 to 0.132, and the vibration isolation rate is above 92%, which has good antiexplosion and impact isolation performance.(2)
The three kinds of damping materials not only have absorption effect, but also play a supporting role in the isolation system. The damping effect of the cylinder damping body on the isolation device is the smallest, but the damping ratio is the largest. The supporting effect of the aluminum foam damping body is the largest, and its damping ratio is closer to the rubber damping body.(3)
The isolation rate of the isolation device decreases with the increase of stiffness and angular frequency. With the increase of the damping ratio, the isolation ratio increases first and then decreases, and there is an optimal damping ratio, which is independent of the stiffness of the isolation device.



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