# Vibration Test and Shock Absorption of Coal Crusher Chambers in Thermal Power Plants (II): Numerical Analysis

Authors: Hanquan Yuan; Lihua Zhu; Haoyi Zhou; Dong Jiang; Baoquan Liu; Chunzhu Fan; Liujiu Tang
Journal: Shock and Vibration
 (2020)

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## Abstract

The coal crusher generates large vibrations when crushing coal blocks, which can affect the equipment itself, as well as the safety of the structure. In order to study the dynamic characteristics of coal crusher, a finite element model of the coal crusher chamber in the Shangluo power plant was built by using ABAQUS. Firstly, modal and harmonic response analyses were conducted, and the comparison shows that the numerical results are basically in accordance with the test results. Then, shock absorption research was performed using a parametric analysis that included the stiffness and position of the spring vibration isolator, the mass, and material of the vibration-isolation platform. Finally, the dynamic coefficient of a coal crusher was discussed. The results showed that, compared with the stiffness of the spring vibration isolator, the mass of the vibration-isolation platform had more influence on the vibration displacement of the coal crusher. To achieve better vibration isolation, the concrete platform is suggested, and the eccentricity of the spring vibration isolator should not exceed 5%. When static design method is adopted to calculate the bearing capacity of the supporting structure subjected to the dynamic load of the coal crusher, the dynamic coefficient of a coal crusher is suggested as 1.5.


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## Body

## 1. Introduction

Compared with other industrial plants, the vibration source in a coal crusher chamber is more complex, and the vibration mechanism of the dynamic machine in the coal crusher chamber is still not very clear. Because of the lack of a relevant theoretical basis, designers can only learn from the research results of other similar dynamic machines in both the design stage and the later reinforcement stage.Theoretically, vibration-isolation technology can effectively reduce the vibration of dynamic machines [1, 2]. It has been found that the vibration of the floor of a coal crusher chamber can be controlled by adopting this kind of technology, but the vibration displacements of coal crushers are very large. The current codes and specifications [3, 4] only limit the vibration displacements at the top surfaces of dynamic machine foundations, whereas those of the machine bodies are not specified. If the vibration displacement of a machine is too large, the machine may be subjected to rigid damage, which will reduce its reliability and service life.Finite element method is a very timely and effective meshing method. This method has been widely used for analysis. In this study, a finite element model of a coal crusher chamber is established based on the data obtained in coal crusher chamber tests. This article consists of four sections, the dimensions of coal crusher chamber introduced in Section 2. Then, modal and harmonic response analyses of coal crusher chamber are analyzed by the finite element method. Section 3further discusses the vibration reduction parameters of coal crusher. Finally, some conclusions are listed in Section 4.


## 2. Finite Element Analysis

### 2.1. Modeling

The coal crusher chamber in the Shangluo power plant was used as the prototype for the developed finite element model. A three-dimensional finite element model of a coal crusher system and coal crusher chamber was built using ABAQUS [5, 6].Beam elements were used for the frame, and shell elements were used for the floor slabs. Solid elements were used to simulate the coal crusher, hydraulic coupler, motor, and steel vibration-isolation platform. Springs and dashpots elements were used for the spring vibration isolators. Based on the actual structure, rigid connections were used for equipment-equipment connections and equipment-platform connections, whereas the platform and coal crusher chamber were connected by spring vibration isolators. In the coordinate system of the model,X represented the transverse direction of the coal crusher chamber, Y represented the longitudinal direction, and Z represented the vertical direction.The mesh size of the finite element model was 100–200 mm. The finite element model is shown in Figure1. Because of the simplification of the model, equivalent densities were adopted for the equipment and platform to keep their quality consistent with reality. The primary parameters of the equipment and platform are listed in Table 1. The primary parameters of the spring vibration isolators are listed in Table 2. The disturbing forces of the coal crusher system are listed in Table 3, and the action positions are shown in Figure 2.Figure 1
Finite element model. (a) Structure of coal crusher chamber. (b) Coal crusher system and vibration-isolation platform.
(a)(b)Table 1
Parameters of equipment and platform.
PartsSize (mm)NumberWeight (kg)Equivalent density (kg/m3)Coal crusher3585 × 2200 × 21001235001418Motor2000 × 1200 × 1300192002950Hydraulic coupler1500 × 500 × 500111663100Vibration-isolation platformSteel plate 17385 × 2400 × 30280005447Steel plate 2800 × 2400 × 302I-beam 1330 × 300 × 104I-beam 2370 × 300 × 104Table 2
Parameters of spring vibration isolator.
Vertical stiffness of the vibration isolator (kN/m)1150Horizontal stiffness of the vibration isolator (kN/m)897Vertical damping factor (kN·s/m)9.8Horizontal damping factor (kN·s/m)9.8Damping ratio of the structure0.05Table 3
Disturbing forces of coal crusher system.
EquipmentLocationDirectionX (kN)Y (kN)Z (kN)Coal crusherD194.59D294.59MotorD3212D4212Figure 2
Positions of the disturbing forces.


### 2.2. Modal Analysis

A modal analysis was performed using the Block Lanczos method [7]. The first three structural modes are shown in Figure 3, and the natural frequencies of the structure found in the numerical results and test results are listed in Table 4.Figure 3
First three modes of the structure. (a) First modal shape, (b) second modal shape, and (c) third modal shape.
(a)(b)(c)Table 4
Natural frequencies of structure from numerical results and test results.
OrderCalculation frequency (Hz)Test frequency (Hz)Vibration modal shapeError11.0200.987Y direction horizontal vibration3.24%21.0521.012X direction horizontal vibration3.80%31.211—Z direction rotational vibration—42.8872.763Y direction horizontal vibration4.30%52.8952.788X direction horizontal vibration3.70%63.626—Z direction rotational vibration—As can be observed in Table4, the numerical results are close to the test results, and the maximum error is not greater than 5%, which indicates that the model is reasonable and can be used for further calculations and analyses.


### 2.3. Harmonic Response Analysis

A harmonic analysis of a coal crusher under the service load was performed. The numerical results were then compared with the test results. The locations of the displacement measurement points on the coal crusher and platform are shown in Figure4. The vibration amplitude–frequency curves from 0 to 20 Hz were obtained through a harmonic response analysis, which included the working speed of the coal crusher and motor. The typical amplitude–frequency curves under various working conditions are shown in Figure 5.Figure 4
Locations of displacement measurement points.Figure 5
Typical displacement–frequency curves: (a) PT1X, (b) PT1Z, (c) SB1Z, and (d) FS1Z.
(a)(b)(c)(d)The vibration displacements of each point under each working condition, corresponding to the excitation frequency of 12.38 Hz, could be obtained from the amplitude–frequency curves. The square root of the sum of the squares (SRSS) method [8, 9] was used to process the data, which are listed in Table 5 (FS represents the displacement measurement points on the second floor slab, which correspond with those used in the vibration test).Table 5
Vibration displacements at points on coal crusher and platform.
LocationDirectionX (μm)Y (μm)Z (μm)SB160.229.7460.5SB260.281.542.32SB3124.3129.74126.78SB4124.3181.5119.39FS0.250.272.51PT142.3938.5361.08PT242.432.2341.15PT3117.0138.59127.18PT4117.0131.68119.1Table6 lists the numerical and test results for the maximum horizontal and vertical vibration displacements of the coal crusher body, platform, and second floor slab. The numerical results generally agree with the test results in terms of their orders of magnitude and variation tendencies, which verifies the accuracy of the finite element model and harmonic analysis.Table 6
Vibration displacements from numerical results and test results.
LocationDirectionNumerical results (μm)Test results (μm)Coal crusherX124.31149.09Y81.556.77Z126.78164.75Vibration-isolated platformX117.01154.09Y38.59130.93Z127.18138.34Second floorX0.252.44Y0.271.96Z2.512.75


### 2.4. Vibration-Isolation Efficiency

According to the Chinese Code [10], the single-degree-of-freedom model is used to design the vibration isolation of the coal crusher. In this model, the vibration-isolation efficiency was calculated by the frequency of the disturbing forces and the vibration-isolation system. The vibration displacements, velocities, and accelerations after vibration isolation are not involved in the calculation method of vibration-isolation efficiency. In the field test, the vibration-isolation efficiency was calculated by the dynamic response before and after vibration isolation. The spring vibration-isolation system was considered as a two-degree-of-freedom model in this calculation method. Both of these models are very different from a practical multi-degree-of-freedom model [11]. In order to study the differences between these three models, the numerical vibration-isolation efficiency of the spring vibration-isolation system was calculated and compared with the designed and measured vibration-isolation efficiencies.When there are multiple disturbing forces, the transmissibility (TR) of a spring vibration-isolation system [12] can be calculated using the SRSS method, where the vibration-isolation efficiency is found as follows:(1)η=1−TR=1−∑Fi2∑Pi2,where Fi is the reaction force of the spring vibration isolators under working condition i and Pi is the disturbing force of working condition i.The layout of the spring vibration isolators is shown in Figure6. The numerical vibration-isolation efficiencies based on (1) are listed in Table 7.Figure 6
Positions of spring vibration isolators.Table 7
Calculated vibration-isolation efficiencies.
Spring vibration isolatorsReaction force (kN)ConditionD1ConditionD2ConditionD3ConditionD410.1210.0420.0030.01320.1210.0390.0030.01430.1200.0360.0030.01540.0950.0320.0040.01750.1180.0290.0050.01860.1170.0250.0060.01970.0430.0530.0130.01780.0420.0490.0120.01690.0410.0460.0110.015100.0400.0430.0100.013110.0390.0390.0090.012120.0390.0360.0080.011130.0260.0550.0150.022140.0700.0570.0170.026150.0430.0600.0200.032160.0650.0630.0230.038170.0890.0380.0100.015180.0240.0400.0120.020190.0470.0430.0150.026200.0690.0460.0180.032Total of the reaction force1.370.870.220.39Disturbing force9922Vibration-isolation efficiency87.09%The results showed that the numerical vibration-isolation efficiency of the spring vibration-isolation system was 87.09%, which was slightly lower than the designed vibration-isolation efficiency of 90%, indicating that the single-degree-of-freedom model was effective in the design of the spring vibration-isolation system. The measured vibration-isolation efficiency was 73.59%, which was lower than the previous two values. This may have been related to the installation, aging, and other factors of the spring vibration isolators.


## 2.1. Modeling

The coal crusher chamber in the Shangluo power plant was used as the prototype for the developed finite element model. A three-dimensional finite element model of a coal crusher system and coal crusher chamber was built using ABAQUS [5, 6].Beam elements were used for the frame, and shell elements were used for the floor slabs. Solid elements were used to simulate the coal crusher, hydraulic coupler, motor, and steel vibration-isolation platform. Springs and dashpots elements were used for the spring vibration isolators. Based on the actual structure, rigid connections were used for equipment-equipment connections and equipment-platform connections, whereas the platform and coal crusher chamber were connected by spring vibration isolators. In the coordinate system of the model,X represented the transverse direction of the coal crusher chamber, Y represented the longitudinal direction, and Z represented the vertical direction.The mesh size of the finite element model was 100–200 mm. The finite element model is shown in Figure1. Because of the simplification of the model, equivalent densities were adopted for the equipment and platform to keep their quality consistent with reality. The primary parameters of the equipment and platform are listed in Table 1. The primary parameters of the spring vibration isolators are listed in Table 2. The disturbing forces of the coal crusher system are listed in Table 3, and the action positions are shown in Figure 2.Figure 1
Finite element model. (a) Structure of coal crusher chamber. (b) Coal crusher system and vibration-isolation platform.
(a)(b)Table 1
Parameters of equipment and platform.
PartsSize (mm)NumberWeight (kg)Equivalent density (kg/m3)Coal crusher3585 × 2200 × 21001235001418Motor2000 × 1200 × 1300192002950Hydraulic coupler1500 × 500 × 500111663100Vibration-isolation platformSteel plate 17385 × 2400 × 30280005447Steel plate 2800 × 2400 × 302I-beam 1330 × 300 × 104I-beam 2370 × 300 × 104Table 2
Parameters of spring vibration isolator.
Vertical stiffness of the vibration isolator (kN/m)1150Horizontal stiffness of the vibration isolator (kN/m)897Vertical damping factor (kN·s/m)9.8Horizontal damping factor (kN·s/m)9.8Damping ratio of the structure0.05Table 3
Disturbing forces of coal crusher system.
EquipmentLocationDirectionX (kN)Y (kN)Z (kN)Coal crusherD194.59D294.59MotorD3212D4212Figure 2
Positions of the disturbing forces.


## 2.2. Modal Analysis

A modal analysis was performed using the Block Lanczos method [7]. The first three structural modes are shown in Figure 3, and the natural frequencies of the structure found in the numerical results and test results are listed in Table 4.Figure 3
First three modes of the structure. (a) First modal shape, (b) second modal shape, and (c) third modal shape.
(a)(b)(c)Table 4
Natural frequencies of structure from numerical results and test results.
OrderCalculation frequency (Hz)Test frequency (Hz)Vibration modal shapeError11.0200.987Y direction horizontal vibration3.24%21.0521.012X direction horizontal vibration3.80%31.211—Z direction rotational vibration—42.8872.763Y direction horizontal vibration4.30%52.8952.788X direction horizontal vibration3.70%63.626—Z direction rotational vibration—As can be observed in Table4, the numerical results are close to the test results, and the maximum error is not greater than 5%, which indicates that the model is reasonable and can be used for further calculations and analyses.


## 2.3. Harmonic Response Analysis

A harmonic analysis of a coal crusher under the service load was performed. The numerical results were then compared with the test results. The locations of the displacement measurement points on the coal crusher and platform are shown in Figure4. The vibration amplitude–frequency curves from 0 to 20 Hz were obtained through a harmonic response analysis, which included the working speed of the coal crusher and motor. The typical amplitude–frequency curves under various working conditions are shown in Figure 5.Figure 4
Locations of displacement measurement points.Figure 5
Typical displacement–frequency curves: (a) PT1X, (b) PT1Z, (c) SB1Z, and (d) FS1Z.
(a)(b)(c)(d)The vibration displacements of each point under each working condition, corresponding to the excitation frequency of 12.38 Hz, could be obtained from the amplitude–frequency curves. The square root of the sum of the squares (SRSS) method [8, 9] was used to process the data, which are listed in Table 5 (FS represents the displacement measurement points on the second floor slab, which correspond with those used in the vibration test).Table 5
Vibration displacements at points on coal crusher and platform.
LocationDirectionX (μm)Y (μm)Z (μm)SB160.229.7460.5SB260.281.542.32SB3124.3129.74126.78SB4124.3181.5119.39FS0.250.272.51PT142.3938.5361.08PT242.432.2341.15PT3117.0138.59127.18PT4117.0131.68119.1Table6 lists the numerical and test results for the maximum horizontal and vertical vibration displacements of the coal crusher body, platform, and second floor slab. The numerical results generally agree with the test results in terms of their orders of magnitude and variation tendencies, which verifies the accuracy of the finite element model and harmonic analysis.Table 6
Vibration displacements from numerical results and test results.
LocationDirectionNumerical results (μm)Test results (μm)Coal crusherX124.31149.09Y81.556.77Z126.78164.75Vibration-isolated platformX117.01154.09Y38.59130.93Z127.18138.34Second floorX0.252.44Y0.271.96Z2.512.75


## 2.4. Vibration-Isolation Efficiency

According to the Chinese Code [10], the single-degree-of-freedom model is used to design the vibration isolation of the coal crusher. In this model, the vibration-isolation efficiency was calculated by the frequency of the disturbing forces and the vibration-isolation system. The vibration displacements, velocities, and accelerations after vibration isolation are not involved in the calculation method of vibration-isolation efficiency. In the field test, the vibration-isolation efficiency was calculated by the dynamic response before and after vibration isolation. The spring vibration-isolation system was considered as a two-degree-of-freedom model in this calculation method. Both of these models are very different from a practical multi-degree-of-freedom model [11]. In order to study the differences between these three models, the numerical vibration-isolation efficiency of the spring vibration-isolation system was calculated and compared with the designed and measured vibration-isolation efficiencies.When there are multiple disturbing forces, the transmissibility (TR) of a spring vibration-isolation system [12] can be calculated using the SRSS method, where the vibration-isolation efficiency is found as follows:(1)η=1−TR=1−∑Fi2∑Pi2,where Fi is the reaction force of the spring vibration isolators under working condition i and Pi is the disturbing force of working condition i.The layout of the spring vibration isolators is shown in Figure6. The numerical vibration-isolation efficiencies based on (1) are listed in Table 7.Figure 6
Positions of spring vibration isolators.Table 7
Calculated vibration-isolation efficiencies.
Spring vibration isolatorsReaction force (kN)ConditionD1ConditionD2ConditionD3ConditionD410.1210.0420.0030.01320.1210.0390.0030.01430.1200.0360.0030.01540.0950.0320.0040.01750.1180.0290.0050.01860.1170.0250.0060.01970.0430.0530.0130.01780.0420.0490.0120.01690.0410.0460.0110.015100.0400.0430.0100.013110.0390.0390.0090.012120.0390.0360.0080.011130.0260.0550.0150.022140.0700.0570.0170.026150.0430.0600.0200.032160.0650.0630.0230.038170.0890.0380.0100.015180.0240.0400.0120.020190.0470.0430.0150.026200.0690.0460.0180.032Total of the reaction force1.370.870.220.39Disturbing force9922Vibration-isolation efficiency87.09%The results showed that the numerical vibration-isolation efficiency of the spring vibration-isolation system was 87.09%, which was slightly lower than the designed vibration-isolation efficiency of 90%, indicating that the single-degree-of-freedom model was effective in the design of the spring vibration-isolation system. The measured vibration-isolation efficiency was 73.59%, which was lower than the previous two values. This may have been related to the installation, aging, and other factors of the spring vibration isolators.


## 3. Parametric Analysis of the Vibration Reduction of Coal Crusher

### 3.1. Stiffness of Spring Vibration Isolators and Mass of Vibration-Isolation Platform

As reported in the previous paper in this series, the vibration displacement of the coal crusher body could be reduced by controlling the vibration displacement of the platform. In order to study the influences of the stiffness [13] of the spring vibration isolators and the mass of the vibration-isolation platform on its vibration, these two parameters were changed; finite element models at vibration-isolation efficiencies of 60%, 70%, 80%, and 90% were obtained; and a harmonic response analysis was performed.According to the Chinese code [10] for the vibration-isolation design of an auxiliary machine foundation in a thermal power plant, the design formulas for a spring vibration-isolation system are as follows:(2)Kz=ω2M1+1/η,(3)Cz=2ξzKzm,where ω is the circular frequency of the disturbing force; and Kz, M, Cz, and ξz are the vertical stiffness, mass, vertical damping coefficient, and vertical damping ratio of the spring vibration-isolation system, respectively.The parameters of the spring vibration-isolation system under each isolation efficiency could be obtained using formulas (2) and (3). In these calculations, ξz = 0.1; the horizontal stiffness of the system was 0.78 times the vertical stiffness; and the horizontal damping coefficient was the same as the vertical damping coefficient [14]. The parameters of the spring vibration-isolation system under each vibration-isolation efficiency are listed in Table 8.Table 8
Parameters of spring vibration-isolation system under each vibration-isolation efficiency.
Varying parameterParameters of the isolatorsVibration-isolation efficiency90%80%70%60%Stiffness of spring vibration isolatorsVertical stiffness (kN/m)1150210929203616Horizontal stiffness (kN/m)897164522782820Vertical damping factor (kN·s/m)9.813.315.617.4Horizontal damping factor (kN·s/m)9.813.315.617.4Mass of vibration-isolation platform (t)8Mass of vibration-isolated platformVertical stiffness (kN/m)2920Horizontal stiffness (kN/m)2278Vertical damping factor (kN·s/m)24.918.415.614.1Horizontal damping factor (kN·s/m)24.918.415.614.1Mass of vibration-isolation platform (t)72.424.180The discussion in this section is based on the values obtained at displacement measurement point PT4, as shown in Figure4. The vibration displacements at PT4, corresponding to each model, are shown in Figure 7.Figure 7
Vibration displacements of each point under each vibration-isolation efficiency. (a) PT4X and (b) PT4Z.
(a)(b)It can be observed that when the mass of the vibration-isolation platform remained unchanged, but the stiffness of the spring vibration isolators was decreased, the vibration-isolation efficiency of the spring vibration-isolation system increased from 60% to 90%, the horizontal vibration displacement of the platform decreased from 215.01μm to 117.01 μm, and the vertical vibration displacement decreased from 217.15 μm to 119.10 μm. When the stiffness of the spring vibration isolators remained unchanged, but the mass of the vibration-isolation platform was increased, the vibration-isolation efficiency of the spring vibration-isolation system increased from 60% to 90%, the horizontal vibration displacement of the platform decreased from 226.13 μm to 89.10 μm, and the vertical vibration displacement decreased from 270.17 μm to 92.65 μm. Therefore, decreasing the stiffness of the spring vibration isolators and increasing the mass of the vibration-isolation platform could improve the vibration-isolation efficiency of the spring vibration-isolation system. Thus, the vibration displacement of the vibration-isolation platform could be reduced.In order to compare the influences of these two parameters on the vibration displacement of the platform, the vibration displacement of the point at a 70% vibration-isolation efficiency was regarded as the reference value, and the ratio of the maximum vibration displacement of the point at each vibration-isolation efficiency to the reference value was calculated, with the results listed in Table9.Table 9
Ratio of vibration displacement of point to reference value at each vibration-isolation efficiency.
Varying parameterDirectionVibration-isolation efficiency90%80%70%60%Stiffness of spring vibration isolatorsX0.720.8311.32Z0.710.8311.29Mass of vibration-isolation platformX0.550.7411.39Z0.550.7111.6The results showed that, for the vibration-isolation platform, the variations in the horizontal vibration displacement and vertical vibration displacement were basically the same even if the stiffness of the spring vibration isolators was decreased or the mass of the vibration-isolation platform was increased. Regardless of the vibration displacement of the vibration-isolation platform in the horizontal or vertical direction, the influence of the mass of the vibration-isolation platform was obviously greater than that of the stiffness of the spring vibration isolators. Therefore, the method of increasing the mass of the vibration-isolation platform has a better vibration reduction effect.


### 3.2. Positions of Spring Vibration Isolators

In order to study the influence of an eccentric arrangement of spring vibration isolators on the vibration of the coal crusher, their positions were changed, and the vibration displacements of the coal crusher were then compared with the original results.Eccentricity refers to the relative error between the center of the spring vibration isolator’s rigidity and the center of mass of the vibration-isolation platform. Its calculation formula is as follows:(4)ΔX=Xk−XmLX,where △X is the eccentricity in direction X; Xk is the center of rigidity of the spring vibration isolator in the X direction; Xm is the center of mass of the vibration-isolation platform in the X direction; and LX is the length of the vibration-isolation platform in the X direction.The position of spring vibration isolator No. 1 was used as the coordinate origin to establish a rectangular coordinate system. The corresponding coordinates of each spring vibration isolator are listed in Table10. The maximum vibration displacements in all directions on the machine were determined and compared with the calculated results for the original model, as listed in Table 11.Table 10
Positions of spring vibration isolators in model.
Spring vibration isolatorsOriginal modelThe model with an eccentricity of 5%The model with an eccentricity of 10%XYXYXY100000020.480000030.9600.4800.48041.4400.9600.96051.9201.4401.44062.401.9201.920703.58503.58503.58580.483.5850.483.58503.58590.963.5850.963.5850.483.585101.443.5851.443.5850.963.585111.923.5851.923.5851.443.585122.43.5852.43.5851.923.5851304.33504.33504.3351405.08505.08505.0851506.08506.08506.0851607.08507.08507.085172.44.3352.44.3352.44.335182.45.0852.45.0852.45.085192.46.0852.46.0852.46.085202.47.0852.47.0852.47.085Center of rigidity1.2—1.08—0.96—Center of mass1.2—1.2—1.2—Relative error0%—5%—10%—Table 11
Influence of eccentric positions of spring vibration isolators.
LocationType5%10%X (μm)Y (μm)Z (μm)X (μm)Y (μm)Z (μm)Coal crusherBefore eccentricity124.3181.5126.78124.3181.5126.78After eccentricity136.7493.72155.21188.9595.97190.88Change rate10.00%15.00%22.42%52.00%17.75%50.56%The results showed that when the eccentricity was 5%, the horizontal displacement of the machine increased by 15%, and the vertical displacement increased by 22.42%; when the eccentricity was 10%, the horizontal displacement of the machine increased by 52%, and the vertical displacement increased by 50.56%. The vibration displacement of the machine increased with the eccentricity but varied greatly. Considering machine operation safety, it is suggested that the eccentricity should not exceed 5%.


### 3.3. Comparison of Steel Platform and Concrete Platform

At present, steel platforms are widely used by equipment manufacturers in vibration-isolation design, while concrete platforms are adopted by design institutes. Compared with a concrete platform, a steel platform is lighter in mass and smaller in volume. To study the differences between steel and concrete platforms, a concrete platform finite element model was established. Then, the numerical results derived from the harmonic response analysis were compared with the numerical results for a steel platform reported in Section2.3.Based on the “Design Code of Vibration Isolation of Auxiliary Machine Foundation in Fossil Fuel Power Plant” (DL/T 5188-2004) [10], the following parameters were used for the concrete platform modeled in this study. The thickness was 1 m, the cross-sectional dimensions in the vertical direction were 7085 mm × 2400 mm, and the total mass was 47.31 t. The parameters of the spring vibration isolators at an vibration-isolation efficiency of 90% were calculated using formula (2) and are listed in Table 12. The finite element model of the concrete platform is shown in Figure 8. The maximum vibration displacements of the steel and concrete platforms are listed in Table 13.Table 12
Parameters of spring vibration isolators for concrete platform.
Vertical stiffness of the vibration isolator (kN/m)2200Horizontal stiffness of the vibration isolator (kN/m)1716Vertical damping factor (kN·s/m)50Horizontal damping factor (kN·s/m)50Figure 8
Concrete vibration-isolation platform.Table 13
Comparison of vibration displacements between the steel platform and concrete platform.
LocationMaterial of platformDirectionX (μm)Y (μm)Z (μm)Coal crusherSteel124.3181.5126.78Concrete100.853.6278.89Vibration-isolation platformSteel117.0138.59127.18Concrete72.5626.9880.19Based on the “Technical Code for the Design of Civil Structure of Fossil-Fired Power Plant” (DL 5022-2012) [4], the limit values of the vibration displacements of the vibration-isolation platform for a coal crusher are 0.15 mm in the vertical direction and 0.20 mm in the horizontal direction.The analysis results showed that as long as the steel and concrete platforms were designed in accordance with the code requirements, the vibration displacements of both could meet the code requirements. When the concrete platform was adopted, the displacements in all directions of the vibration-isolation platform were lower than those when a steel platform was adopted. Therefore, it is suggested that concrete platforms be adopted in the vibration-isolation design of coal crushers.


### 3.4. Dynamic Coefficients of Coal Crushers

A dynamic coefficient [3] refers to a coefficient adopted in the static design of a structure or component under dynamic loads, and its value is the ratio of the maximum dynamic response of the structure or component to the corresponding static response. The values for the dynamic coefficients of coal crushers with vibration-isolation systems are not specified in the current specifications. Therefore, this study calculated the dynamic coefficients of the finite element models established in Section 3.1.The vibration displacements at the bottom of the spring vibration isolators (i.e., the support beams) were used to calculate the dynamic coefficients. Static vibration displacements were obtained by applying static forces equal to the disturbing forces after vibration isolation at the same position, where the finite element model was the same as that previously discussed. Table14 lists the calculation results for the dynamic coefficients of displacement measurement points L1–L12 (see Figure 6) under each vibration-isolation efficiency.Table 14
Dynamic coefficients of points under each vibration-isolation efficiency.
LocationDynamic displacement (μm)Static displacement (μm)Dynamic coefficients90%80%70%60%90%80%70%60%90%80%70%60%L12.374.676.889.011.783.575.357.141.331.311.291.26L21.092.153.144.091.082.153.234.311.011.000.970.95L32.474.887.209.481.973.945.927.891.251.241.221.20L41.132.223.274.301.212.423.634.840.930.920.900.89L51.152.273.364.451.312.613.925.230.880.870.860.85L61.152.273.384.531.362.724.085.440.850.830.830.83L71.122.233.334.501.362.734.095.450.820.820.810.83L81.072.133.204.351.322.633.955.260.810.810.810.83L92.394.747.099.492.144.276.418.541.121.111.111.11L102.494.947.369.832.214.416.618.811.131.121.121.12L112.555.037.489.932.194.386.578.761.161.151.141.13L122.545.007.419.802.114.226.348.451.201.181.171.16The results showed that the maximum values of the dynamic coefficients under the different vibration-isolation efficiencies were generally equal to 1.3. Theoretically, the dynamic coefficient is related to the frequency of the structure and machine; it has basically no relationship with the vibration-isolation efficiency. Based on Table10, it is recommended that a value of 1.5 be used for the dynamic coefficient of a coal crusher.The design value of the disturbing force generated by the machine should be used to calculate the dynamic internal force of the support structure. The calculation formula is as follows [15]:(5)Pe=Kd×P,where Pe is the design value of the disturbing force; P is the standard value of the disturbing force; and Kd is the dynamic overload coefficient, where the value for a rotary machine is four.Thus, the static design method could be adopted to calculate the bearing capacity of the support structure subjected to the dynamic load of the coal crusher, and the suggested value for the static load is the sum of the self-weight of the machine, platform, and six times the disturbing force.


## 3.1. Stiffness of Spring Vibration Isolators and Mass of Vibration-Isolation Platform

As reported in the previous paper in this series, the vibration displacement of the coal crusher body could be reduced by controlling the vibration displacement of the platform. In order to study the influences of the stiffness [13] of the spring vibration isolators and the mass of the vibration-isolation platform on its vibration, these two parameters were changed; finite element models at vibration-isolation efficiencies of 60%, 70%, 80%, and 90% were obtained; and a harmonic response analysis was performed.According to the Chinese code [10] for the vibration-isolation design of an auxiliary machine foundation in a thermal power plant, the design formulas for a spring vibration-isolation system are as follows:(2)Kz=ω2M1+1/η,(3)Cz=2ξzKzm,where ω is the circular frequency of the disturbing force; and Kz, M, Cz, and ξz are the vertical stiffness, mass, vertical damping coefficient, and vertical damping ratio of the spring vibration-isolation system, respectively.The parameters of the spring vibration-isolation system under each isolation efficiency could be obtained using formulas (2) and (3). In these calculations, ξz = 0.1; the horizontal stiffness of the system was 0.78 times the vertical stiffness; and the horizontal damping coefficient was the same as the vertical damping coefficient [14]. The parameters of the spring vibration-isolation system under each vibration-isolation efficiency are listed in Table 8.Table 8
Parameters of spring vibration-isolation system under each vibration-isolation efficiency.
Varying parameterParameters of the isolatorsVibration-isolation efficiency90%80%70%60%Stiffness of spring vibration isolatorsVertical stiffness (kN/m)1150210929203616Horizontal stiffness (kN/m)897164522782820Vertical damping factor (kN·s/m)9.813.315.617.4Horizontal damping factor (kN·s/m)9.813.315.617.4Mass of vibration-isolation platform (t)8Mass of vibration-isolated platformVertical stiffness (kN/m)2920Horizontal stiffness (kN/m)2278Vertical damping factor (kN·s/m)24.918.415.614.1Horizontal damping factor (kN·s/m)24.918.415.614.1Mass of vibration-isolation platform (t)72.424.180The discussion in this section is based on the values obtained at displacement measurement point PT4, as shown in Figure4. The vibration displacements at PT4, corresponding to each model, are shown in Figure 7.Figure 7
Vibration displacements of each point under each vibration-isolation efficiency. (a) PT4X and (b) PT4Z.
(a)(b)It can be observed that when the mass of the vibration-isolation platform remained unchanged, but the stiffness of the spring vibration isolators was decreased, the vibration-isolation efficiency of the spring vibration-isolation system increased from 60% to 90%, the horizontal vibration displacement of the platform decreased from 215.01μm to 117.01 μm, and the vertical vibration displacement decreased from 217.15 μm to 119.10 μm. When the stiffness of the spring vibration isolators remained unchanged, but the mass of the vibration-isolation platform was increased, the vibration-isolation efficiency of the spring vibration-isolation system increased from 60% to 90%, the horizontal vibration displacement of the platform decreased from 226.13 μm to 89.10 μm, and the vertical vibration displacement decreased from 270.17 μm to 92.65 μm. Therefore, decreasing the stiffness of the spring vibration isolators and increasing the mass of the vibration-isolation platform could improve the vibration-isolation efficiency of the spring vibration-isolation system. Thus, the vibration displacement of the vibration-isolation platform could be reduced.In order to compare the influences of these two parameters on the vibration displacement of the platform, the vibration displacement of the point at a 70% vibration-isolation efficiency was regarded as the reference value, and the ratio of the maximum vibration displacement of the point at each vibration-isolation efficiency to the reference value was calculated, with the results listed in Table9.Table 9
Ratio of vibration displacement of point to reference value at each vibration-isolation efficiency.
Varying parameterDirectionVibration-isolation efficiency90%80%70%60%Stiffness of spring vibration isolatorsX0.720.8311.32Z0.710.8311.29Mass of vibration-isolation platformX0.550.7411.39Z0.550.7111.6The results showed that, for the vibration-isolation platform, the variations in the horizontal vibration displacement and vertical vibration displacement were basically the same even if the stiffness of the spring vibration isolators was decreased or the mass of the vibration-isolation platform was increased. Regardless of the vibration displacement of the vibration-isolation platform in the horizontal or vertical direction, the influence of the mass of the vibration-isolation platform was obviously greater than that of the stiffness of the spring vibration isolators. Therefore, the method of increasing the mass of the vibration-isolation platform has a better vibration reduction effect.


## 3.2. Positions of Spring Vibration Isolators

In order to study the influence of an eccentric arrangement of spring vibration isolators on the vibration of the coal crusher, their positions were changed, and the vibration displacements of the coal crusher were then compared with the original results.Eccentricity refers to the relative error between the center of the spring vibration isolator’s rigidity and the center of mass of the vibration-isolation platform. Its calculation formula is as follows:(4)ΔX=Xk−XmLX,where △X is the eccentricity in direction X; Xk is the center of rigidity of the spring vibration isolator in the X direction; Xm is the center of mass of the vibration-isolation platform in the X direction; and LX is the length of the vibration-isolation platform in the X direction.The position of spring vibration isolator No. 1 was used as the coordinate origin to establish a rectangular coordinate system. The corresponding coordinates of each spring vibration isolator are listed in Table10. The maximum vibration displacements in all directions on the machine were determined and compared with the calculated results for the original model, as listed in Table 11.Table 10
Positions of spring vibration isolators in model.
Spring vibration isolatorsOriginal modelThe model with an eccentricity of 5%The model with an eccentricity of 10%XYXYXY100000020.480000030.9600.4800.48041.4400.9600.96051.9201.4401.44062.401.9201.920703.58503.58503.58580.483.5850.483.58503.58590.963.5850.963.5850.483.585101.443.5851.443.5850.963.585111.923.5851.923.5851.443.585122.43.5852.43.5851.923.5851304.33504.33504.3351405.08505.08505.0851506.08506.08506.0851607.08507.08507.085172.44.3352.44.3352.44.335182.45.0852.45.0852.45.085192.46.0852.46.0852.46.085202.47.0852.47.0852.47.085Center of rigidity1.2—1.08—0.96—Center of mass1.2—1.2—1.2—Relative error0%—5%—10%—Table 11
Influence of eccentric positions of spring vibration isolators.
LocationType5%10%X (μm)Y (μm)Z (μm)X (μm)Y (μm)Z (μm)Coal crusherBefore eccentricity124.3181.5126.78124.3181.5126.78After eccentricity136.7493.72155.21188.9595.97190.88Change rate10.00%15.00%22.42%52.00%17.75%50.56%The results showed that when the eccentricity was 5%, the horizontal displacement of the machine increased by 15%, and the vertical displacement increased by 22.42%; when the eccentricity was 10%, the horizontal displacement of the machine increased by 52%, and the vertical displacement increased by 50.56%. The vibration displacement of the machine increased with the eccentricity but varied greatly. Considering machine operation safety, it is suggested that the eccentricity should not exceed 5%.


## 3.3. Comparison of Steel Platform and Concrete Platform

At present, steel platforms are widely used by equipment manufacturers in vibration-isolation design, while concrete platforms are adopted by design institutes. Compared with a concrete platform, a steel platform is lighter in mass and smaller in volume. To study the differences between steel and concrete platforms, a concrete platform finite element model was established. Then, the numerical results derived from the harmonic response analysis were compared with the numerical results for a steel platform reported in Section2.3.Based on the “Design Code of Vibration Isolation of Auxiliary Machine Foundation in Fossil Fuel Power Plant” (DL/T 5188-2004) [10], the following parameters were used for the concrete platform modeled in this study. The thickness was 1 m, the cross-sectional dimensions in the vertical direction were 7085 mm × 2400 mm, and the total mass was 47.31 t. The parameters of the spring vibration isolators at an vibration-isolation efficiency of 90% were calculated using formula (2) and are listed in Table 12. The finite element model of the concrete platform is shown in Figure 8. The maximum vibration displacements of the steel and concrete platforms are listed in Table 13.Table 12
Parameters of spring vibration isolators for concrete platform.
Vertical stiffness of the vibration isolator (kN/m)2200Horizontal stiffness of the vibration isolator (kN/m)1716Vertical damping factor (kN·s/m)50Horizontal damping factor (kN·s/m)50Figure 8
Concrete vibration-isolation platform.Table 13
Comparison of vibration displacements between the steel platform and concrete platform.
LocationMaterial of platformDirectionX (μm)Y (μm)Z (μm)Coal crusherSteel124.3181.5126.78Concrete100.853.6278.89Vibration-isolation platformSteel117.0138.59127.18Concrete72.5626.9880.19Based on the “Technical Code for the Design of Civil Structure of Fossil-Fired Power Plant” (DL 5022-2012) [4], the limit values of the vibration displacements of the vibration-isolation platform for a coal crusher are 0.15 mm in the vertical direction and 0.20 mm in the horizontal direction.The analysis results showed that as long as the steel and concrete platforms were designed in accordance with the code requirements, the vibration displacements of both could meet the code requirements. When the concrete platform was adopted, the displacements in all directions of the vibration-isolation platform were lower than those when a steel platform was adopted. Therefore, it is suggested that concrete platforms be adopted in the vibration-isolation design of coal crushers.


## 3.4. Dynamic Coefficients of Coal Crushers

A dynamic coefficient [3] refers to a coefficient adopted in the static design of a structure or component under dynamic loads, and its value is the ratio of the maximum dynamic response of the structure or component to the corresponding static response. The values for the dynamic coefficients of coal crushers with vibration-isolation systems are not specified in the current specifications. Therefore, this study calculated the dynamic coefficients of the finite element models established in Section 3.1.The vibration displacements at the bottom of the spring vibration isolators (i.e., the support beams) were used to calculate the dynamic coefficients. Static vibration displacements were obtained by applying static forces equal to the disturbing forces after vibration isolation at the same position, where the finite element model was the same as that previously discussed. Table14 lists the calculation results for the dynamic coefficients of displacement measurement points L1–L12 (see Figure 6) under each vibration-isolation efficiency.Table 14
Dynamic coefficients of points under each vibration-isolation efficiency.
LocationDynamic displacement (μm)Static displacement (μm)Dynamic coefficients90%80%70%60%90%80%70%60%90%80%70%60%L12.374.676.889.011.783.575.357.141.331.311.291.26L21.092.153.144.091.082.153.234.311.011.000.970.95L32.474.887.209.481.973.945.927.891.251.241.221.20L41.132.223.274.301.212.423.634.840.930.920.900.89L51.152.273.364.451.312.613.925.230.880.870.860.85L61.152.273.384.531.362.724.085.440.850.830.830.83L71.122.233.334.501.362.734.095.450.820.820.810.83L81.072.133.204.351.322.633.955.260.810.810.810.83L92.394.747.099.492.144.276.418.541.121.111.111.11L102.494.947.369.832.214.416.618.811.131.121.121.12L112.555.037.489.932.194.386.578.761.161.151.141.13L122.545.007.419.802.114.226.348.451.201.181.171.16The results showed that the maximum values of the dynamic coefficients under the different vibration-isolation efficiencies were generally equal to 1.3. Theoretically, the dynamic coefficient is related to the frequency of the structure and machine; it has basically no relationship with the vibration-isolation efficiency. Based on Table10, it is recommended that a value of 1.5 be used for the dynamic coefficient of a coal crusher.The design value of the disturbing force generated by the machine should be used to calculate the dynamic internal force of the support structure. The calculation formula is as follows [15]:(5)Pe=Kd×P,where Pe is the design value of the disturbing force; P is the standard value of the disturbing force; and Kd is the dynamic overload coefficient, where the value for a rotary machine is four.Thus, the static design method could be adopted to calculate the bearing capacity of the support structure subjected to the dynamic load of the coal crusher, and the suggested value for the static load is the sum of the self-weight of the machine, platform, and six times the disturbing force.


## 4. Conclusion

This study investigated methods for reducing the vibration of a coal crusher in a thermal power plant using finite element analyses. The major conclusions are summarized as follows:(1)
The finite element analysis results were shown to be basically consistent with the measured results, which verified that it was feasible to use the finite element method to analyze the vibration performance of a coal crusher.(2)
The dynamic characteristics and vibration displacements under the disturbing forces of the coal crusher chamber could be accurately estimated using modal and harmonic analyses. Thus, this method provided a basis for dynamically optimizing the design of the coal crusher chamber and vibration-isolated foundation of a coal crusher.(3)
The vibration-isolation efficiency of the multi-degree-of-freedom model was slightly less than the designed value, which indicated that a vibration-isolation design based on a single-degree-of-freedom model could meet the accuracy requirements. The measured vibration-isolation efficiency was less than the previous two values, which may have been related to the installation, aging, and other factors of the spring vibration isolators.(4)
The results involving decreasing the stiffness of the spring vibration isolator and increasing the mass of the vibration-isolation platform could reduce the vibration displacement of the platform of the coal crusher, but the influence of the mass of the vibration-isolation platform was obviously greater than that of the stiffness of the spring vibration isolators. Therefore, it is suggested that the latter be adopted in the displacement optimization of a coal crusher.(5)
Both steel and concrete platforms could meet the design requirements, but, compared to the steel platform, the vibration displacements of the concrete platforms were smaller, and they had a better vibration-isolation effect. Considering the machine operation safety, it is suggested that the eccentricity of the spring vibration isolator should not exceed 5%.(6)
The static design method could be adopted to calculate the bearing capacity of the support structure subjected to the dynamic load of the coal crusher, and the dynamic coefficient of a coal crusher is suggested as 1.5.



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*Source: 8885546-2020-09-15.xml*