Stabilization effect of dynamic stabilizer with different operation parameters

Abstract

Dynamic stabilization operation can effectively improve the mechanical performance of ballast bed by using stabilizer. To improve the stabilization effect, a rigid-flexible coupling dynamics simulation model of the stabilizer-ballast bed system was established according to the actual structure of the stabilizer, and the effect of different operation parameters of stabilizer on the stabilization operation was analyzed. The simulation results showed that: the maximum value of lateral displacement response of the sleeper decreased with the increase in clamping wheel force normal angle, excitation force frequency and downforce; and increased with the increase in excitation force amplitude. The optimized operation parameters were obtained as follows: clamping wheel force normal angle 0°, excitation force frequency 18 Hz, excitation force amplitude 180 kN, downforce 60 kN. A vibration test bench simulating the stabilization operation process was built according to the working principle of the stabilizer, and the reliability of the simulation was verified by testing.

Keywords

Dynamic stabilizer multi-body dynamics simulation rigid-flexible coupling model stabilization parameters optimization vibration test bench

Introduction

Nowadays, railroad lines are divided into ballast bed railroad and ballastless bed railroad, where ballast bed railroad consists of rails, rail fasteners, sleepers, and loose gravel ballast particles.¹ For ballasted railroad, the train loads will change the geometry of the ballast bed, and the mechanical properties of the ballast bed will be reduced after newly built or renovation processes such as sieving and tamping,² while the reduction of lateral resistance and support of the ballast bed will affect the safety of train operation. The improvement of lateral and vertical resistance of ballast bed after stabilization is an indicator to evaluate the effectiveness of stabilization, and the lateral displacement of sleeper in stabilization is positively related to the improvement of resistance of ballast bed.³ Stabilization operation can make the lateral and vertical resistance of ballast bed improved. After a stabilization operation, the lateral resistance of the ballast bed can be increased to 187.54% after tamping, reaching 77.26% of the standard condition. And the vertical resistance of the ballast bed can be increased to 171.18% after tamping, reaching 95.31% of the standard state.⁴ Therefore, it is necessary to carry out stabilization operations to improve the mechanical properties of the ballast bed quickly.⁵

For the dynamic stabilization operation, it has been studied by many scholars. Han et al.⁶ and Han⁷ used different input parameters such as excitation frequency and downforce to analyze the dynamic response of the stabilizer-rail system from the perspective of lateral excitation force. Wang⁸ obtained the stiffness of the ballast bed through experimental research and discrete element simulation, optimized the excitation frequency during the stabilization operation of the dynamic stabilizer according to the stiffness of the ballast bed after tamping. Wang et al.⁹ obtained the horizontal excitation force of the exciter under the working state by using the numerical simulation method and based on the simulation of Adams. Based on the equivalent stiffness and discrete element method, Yan et al.¹⁰ established the functional model of sleeper support stiffness, damping and cumulative settlement of ballast bed during the stabilization operation. Shi et al.¹¹ established a ballast bed model by using the discrete element method and multi-body dynamics coupling approach, and analyzed the influence of stabilization operation on the mechanical properties of ballast bed. These studies focus on the dynamic response of the ballast bed system in stabilization operations using simulation, and to obtain the effect of different operation parameters on stabilization operation.

For the ballast bed system, Al Shaer et al.¹² conducted a three-sleeper scaling model test to simulate a high-speed train passing scenario with the help of hydraulic jacks to investigate the dynamic response characteristics of the ballast bed. Ishikawa et al.¹³ compared the settlement of ballast particles in ballast bed under dynamic wheel loading tests and single-point loading tests. Anderson and Fair¹⁴ conducted triaxial tests on single particle size and layered ballast samples, and obtained the settlement mode of ballast through static test and monotonic cyclic load test.

In summary, the existing studies on dynamic stabilizer are based on changing a single operation parameter to study the response of the ballast bed in stabilization operations, but few studies have been conducted on the effect of stabilization operations under the combined influence of multiple parameters. Many studies on stabilization operation carried out simulation analysis without experimental verification. To fill these gaps, a rigid-flexible coupled simulation model of the stabilizer-ballast bed system has been established based on multi-body dynamics. Then, the effect of stabilization operation has been respectively carried out with single-factor and multi-factor perspectives. Finally, a vibration test bench has been built based on the actual operation principle of the stabilizer and the simulation results have been verified. Based on the simulation and test results, the optimal stabilization operation parameters were determined.

Stabilizer modeling

Structure and working principle of dynamic stabilizer

The dynamic stabilizer is the core working component of the dynamic stabilization vehicle. As shown in Figure 1(a), the gap between the traveling wheel and the rail is eliminated by the pressure of the horizontal cylinder in the stabilization operation. The traveling wheels transfer the lateral excitation force, vertical down force and guide the stabilizer on the track. Under the action of clamping cylinders and horizontal cylinders, one clamping wheel and two traveling wheels clamp the rail on one side of the stabilizer (the same on the other side) to prevent the stabilizer from derailment. The vertical cylinders exert downforce in the vertical direction to make the rails and sleepers sink in the ballast bed.

Figure 1.

Dynamic stabilizer: (a) model of the dynamic stabilizer (1 – Vertical cylinder; 2 – Stabilizer case; 3 – Clamping cylinder; 4 – Horizontal cylinder; 5 – Traveling wheel; 6 – Clamping wheel), (b) double shaft eccentric block exciter, and (c) vibration of double shaft eccentric block exciter (1 – Gear; 2 – Bearing; 3 – Follower shaft; 4 – Main shaft; 5 – Eccentric block).

Dynamic stabilizer produces lateral excitation force by rotation of a pair of eccentric wheels as shown in Figure 1(b).¹⁵ Under the combined action of lateral excitation force and downforce, the ballast bed produces lateral vibration which makes the rail-sleeper uniform settlement and ballast particles rearrangement. The stability of the ballast bed has been improved.

As shown in Figure 1(c), when the eccentric blocks on the two axes rotate to the farthest in the vertical direction, the centrifugal force generated by the eccentric block of the active axis and the eccentric block of the driven axis is in the opposite direction, the two centrifugal forces just cancel each other, and the overall external excitation force of the stabilizer is zero. When both eccentric blocks are not at the same time in the vertical direction to produce the lateral excitation force, and in the vertical direction of the excitation force cancelation, only the horizontal direction of the excitation force remains. When both eccentric blocks are in the horizontal direction at the same time, all centrifugal forces are horizontal forces with no vertical force, and the overall external lateral excitation force of the stabilizer is the largest.

Simulation model of stabilizer-ballast bed system and parameters selection

The ballast bed system can be regarded as an elastic damping structure support system,¹⁰ and the contact relationship between the rail-sleeper-ballast bed system can be represented by a spring system with equivalent stiffness-damping, as shown in Figure 2(a). The contact between the clamping wheel and the rail is pressed on the contact surface of the outer arc section of the rail as shown in Figure 2(b), and the change of θ in simulation model is achieved by modifying the 3D model of dynamic stabilizer, where N denotes the contact force between the rail and clamping wheel, and its direction is called the force normal direction; θ denotes the angle between the force normal direction and the horizontal plane, which is called the force normal angle.

Figure 2.

Dynamic stabilizer-ballast bed system: (a) model of the dynamic stabilizer-ballast bed system and (b) contact of clamping wheel and rail.

For the simulation model in Adams, nine Type III sleepers and two Type 60 steel rails were selected. The equivalent stiffness and damping of the spring dampers are used to substitute the fastener connection between the rails and the sleepers, the spring dampers connection positions are the same as reality. Similarly, the spring dampers with equivalent stiffness and damping are used to replace the connections between the sleepers and the ballast bed.

The clamping cylinders and horizontal cylinders of the dynamic stabilizer were replaced by the composite structure of sliding pair tension compression spring dampers. The working pressure of the horizontal cylinder is 6 MPa, and the working pressure of the clamping cylinder is in the range of 7.3–15 MPa to ensure good contact between the clamping wheel and the rail.¹⁶ Therefore, the working pressure of the clamping cylinder was selected as 10 MPa. Based on the actual size of the cylinder and the working pressure, the equivalent spring damper parameters of the cylinder were calculated. According to the studies,^8,16,17 the spring damper parameters between each component of the ballast bed system were selected, which are shown in Table 1.

Table 1.

Parameters of simulation system.

Parameters	Value
Stabilizer steel density (kg/m³)	7800
Ballast bed length (m)	5.4
Equivalent density of ballast bed (kg/m³)	2000
Type III sleeper spacing (m)	0.6
Density of Type III sleeper (kg/m³)	2500
Rail mass factor (kg/m)	60
Track gage (m)	1.435
Rail length (m)	5.4
Fastener lateral stiffness (N/mm)	2.94 × 10⁴
Fastener lateral damping (N·s/mm)	52
Fastener vertical stiffness (N/mm)	7.8 × 10⁴
Fastener vertical damping (N·s/mm)	50
Ballast bed lateral stiffness (N/mm)	1.4 × 10⁴
Ballast bed lateral damping (N·s/mm)	20
Ballast bed vertical stiffness (N/mm)	6.5 × 10⁴
Ballast bed vertical damping (N·s/mm)	29.4
Clamping cylinder working pressure (MPa)	10
Clamping cylinder equivalent stiffness factor (N/mm)	3780
Clamping cylinder equivalent damping factor (N·s/mm)	37.8
Horizontal cylinder working pressure (MPa)	6
Horizontal cylinder equivalent stiffness factor (N/mm)	3780
Horizontal cylinder equivalent damping factor (N·s/mm)	37.8
Contact stiffness of traveling wheel to rail (N/mm)	7.92 × 10⁴
Contact stiffness of clamping wheel to rail (N/mm)	5.56 × 10⁴
Contact stiffness of sleeper to rails (N/mm)	1 × 10⁵

The rail flexible body was generated directly using the Adams View Flex module based on the rigid 3D model of the rail, and the contact between other components in the system and the rails should be set as Flexible body – Rigid body contact. According to the study,¹⁷ the contact damping was 500 N·s/mm, the static friction coefficient was 0.3 and the dynamic friction coefficient was 0.25 in Adams.

The main operation parameters of the dynamic stabilizer are⁵: eccentricity (M_e) is 4 kg/m, operation traveling speed is 2.5 km/h, downforce is the range of 0–120 kN, excitation frequency is in the range of 0–42 Hz. The lateral simple harmonic force was applied in the mass center of stabilizer to simulate the lateral excitation force. Expression of excitation force is:

F = M_{e} ω^{2} \cos (ω t)

(1)

where F denotes excitation force, N; M_e denotes eccentricity, kg/m; ω denotes angular frequency of the exciter, Hz.

To simulate the process of stabilization operation, the stabilizer starts from the sleeper No. 4, gradually travels to the top of the sleeper No. 6 and then gradually moves toward the sleeper No. 8. This process takes 3 s under the condition that the stabilizer travels at a speed of 2.5 km/h. The simulation process duration was set to 3 s. The following discussions are focused on the sleeper No. 6 in the middle of the ballast bed in the simulation model.

The commonly used operation parameters for general operation were obtained¹⁷: clamping wheel force normal angle is 20°, downforce is 120 kN, excitation frequency is 34 Hz. The effect of a single parameter on stabilization operation can be obtained by changing one operation parameter with other parameters unchanged.

Influence of stabilizer parameters on stabilization operation

The vertical cylinders provide the downforce, the exciter produces the lateral excitation force, and the excitation force is transferred to the ballast bed by the clamping wheels and walking wheels. According to the operation principle of stabilizer, four operation parameters were selected to study their influences on the stabilization operation as follows: clamping wheel force normal angle, downforce, excitation force amplitude, and excitation force frequency.

Influence of clamping wheel force normal angle on stabilization operation

As shown in Figure 3, the maximum lateral displacement and the maximum lateral acceleration of the sleeper are decreasing trend with the increasing angle of the clamping wheel force normal angle. And there is an inflection point in the acceleration curve. Downward trend is not obvious before the inflection point, but the curve is rapidly declining after the inflection point. When the clamping wheel force normal angle is more than 50°, the maximum lateral acceleration of the rail sleeper decreases obviously. Therefore, the clamping wheel force normal angle should be maintained in the range of 0°–50° to ensure a better dynamic transmission effect in stabilization operation process.

Figure 3.

Trend of sleeper displacement and acceleration with different force normal angles: (a) displacement trend and (b) acceleration trend.

Influence of vertical cylinders downforce on stabilization operation

The excitation force is generated by the eccentric wheel pair at the center of the stabilizer, the excitation force provides the dynamic stabilizer a tendency to twist laterally relative to the ballast bed system. It may even cause the stabilizer to roll over and derail with small downforce. Therefore, the downforce should not be too small even if the downforce of vertical cylinders is in the range of 0–120 kN.¹⁸

The traveling wheel-rail vertical contact force was obtained by simulation as shown in Figure 4. It is required that the traveling wheel should be in close contact with rail in stabilization operation, and the contact force curve should be completely on one side of the x-axis to ensure close contact in Figure 4. When the downforce was lower than 40 kN, the contact force curve between the traveling wheel and the rail intersected with the x-axis, which meant the traveling wheel and the rail were no longer in contact and there was a gap at a certain time. When the downforce was higher than 50 kN, the contact force curve between the traveling wheel and the rail did not intersect with the x-axis, and the number of intersection points between the contact force curve and the x-axis gradually decreased with the increase in downforce. This indicated that the downforce of 40 kN was a critical value. When the downforce was lower than 40 kN, stabilization operation would cause great impact and friction loss on the wheel-rail contact point, which should be avoided. To study the effect of the downforce on the stabilization operation, downforce in the range of 40–120 kN was selected.

Figure 4.

Vertical contact force between traveling wheel and rail: (a) vertical contact force between traveling wheel and rail at downforce 30 kN, (b) vertical contact force between traveling wheel and rail at downforce 40 kN, and (c) vertical contact force between traveling wheel and rail at downforce 50 kN.

As shown in Figure 5, the overall trend of the response of the sleeper increased and then decreased. The maximum lateral displacement and maximum acceleration both appeared in the downforce of 60 kN, while the maximum lateral acceleration was maintained in the higher range at 60–80 kN. Thus, the downforce should not be as large as possible in stabilization operation. But when the vertical downforce was small, the vibration center of the sleeper deviated from the starting point as shown in Figure 6. Such vibration would cause damage to the ballast bed and make the rail straightness decrease. Sum up the analysis: to obtain the best excitation effect, the downforce should be maintained within the range of 60–80 kN; to protect the ballast bed, the downforce should be increased to more than 80 kN. Therefore, the downforce should be adjusted according to the actual operational requirements. It is recommended to maintain the downforce in the range of 60–100 kN to make a good balance between excitation effect and protection of the ballast bed.

Figure 5.

Trend of sleeper displacement and acceleration with different downforce: (a) displacement trend and (b) acceleration trend.

Figure 6.

Displacement and acceleration of sleeper with different downforce: (a) sleeper displacement at downforce 60 kN, (b) sleeper acceleration at downforce 60 kN, (c) sleeper displacement at downforce 100 kN, and (d) sleeper acceleration at downforce 100 kN.

Influences of the excitation force amplitude and frequency on stabilization operation

It can be seen from equation (1), the excitation force frequency and excitation force amplitude have a mutual coupling relationship. However, in the dynamic simulation model, coefficients can be added to the excitation force expression to fix one of the parameters and study the effect of the other parameter on the operation alone.

The excitation force amplitude was chosen to be in the range of 36–180 kN, with a uniform interval of 18 kN. And the dynamic simulation was carried out for the sleeper in the middle of the model ballast bed.

As shown in Figure 7(a) and (b), with the increase in the excitation force amplitude, the lateral displacement and acceleration of the sleeper are monotonically increased. And the lateral response amplitude of the sleeper was positively correlated with the excitation force amplitude.

Figure 7.

Trend of sleeper displacement and acceleration with different amplitude or frequency of excitation force: (a) displacement trend under different excitation force amplitude, (b) acceleration trend under different excitation force amplitude, (c) displacement trend under different excitation force frequency, and (d) acceleration trend under different excitation force frequency.

In the simulation model, the excitation frequency was selected as 18–40 Hz interval, with 2 Hz interval uniformly taken. As shown in Figure 7(c) and (d), it could be seen that with the dynamic stabilizer excitation frequency increases, the lateral displacement of the sleeper monotonically decreased, the lateral acceleration first decreased and then increased.

As shown in Figure 8, the lateral displacement curve of the sleeper could be plotted against the excitation force curve. The red solid line was the displacement curve of the sleeper, and the blue dashed line was the excitation force curve. Comparing Figure 8(b) with Figure 8(d), the phase difference between the center-of-mass displacement curve and the excitation force curve of the sleeper gradually increased with the increase in excitation frequency. The application of the excitation force to the sleeper displacement has a time lag effect due to the excitation force in the dynamic stabilizer-ballast bed system for the step-by-step transfer, so the excitation force curve and the sleeper displacement curve are not synchronized. When the excitation force frequency increased, the phase difference between the two curves tend to inverse phase and the sleeper has not yet moved to the maximum position that was pulled back by the reverse of the excitation force. In a way, the increase in excitation frequency hindered the lateral movement of sleeper. Hence, it could be reasonably presumed that when the excitation force frequency continued increasing, the displacement of the sleeper should gradually increase until the phase difference of excitation force curve and sleeper displacement curve again gradually tend to zero.

Figure 8.

Displacement and acceleration of sleeper with different amplitude or frequency of excitation force: (a) 18 Hz comparison, (b) Figure (a) partially enlarged, (c) 40 Hz comparison, and (d) Figure (c) partially enlarged.

In reality, the variation of excitation frequency changes the excitation force amplitude. For this case, the trend of sleeper displacement with excitation force in range 18–40 Hz was obtained by simulation as shown in Figure 9. The lateral displacement of the sleeper increased with the increase in the excitation force. When the excitation frequency increased from 18 to 30 Hz, the lateral displacement of the sleeper increased faster. When the excitation force increased from 30 to 36 Hz, the lateral displacement of the sleeper increased slower. When the excitation force was higher than 36 Hz, the lateral displacement of the sleeper increased faster again, but it was still not as fast as the beginning.

Figure 9.

Trend of sleeper displacement with different excitation force.

According to the general mechanical operation requirements, it should try to avoid mechanical working under full load to extend the service life of machinery. Therefore, the recommended excitation frequency was in the range of 30–36 Hz.

Seeking optimal operation parameters by uniform test design method

The uniform test design method can be used to design the experiments when the variables and levels required a large quantity of values.¹⁹ For the research content of this paper, due to a large number of operation parameters and levels of the dynamic stabilizer, the number of tests using the orthogonal test design method is excessive. Therefore, the uniform test design method was chosen to design the experiment. Four factors had a greater impact on the effect of stabilization operation, and nine levels of each factor were selected for the test. As shown in Table 2, x₁ denotes clamping wheel force normal angle, °; x₂ denotes downforce, kN; x₃ denotes excitation force frequency, Hz; x₄ denotes excitation force amplitude, kN; y denotes lateral displacement of sleeper, mm.

Table 2.

Test result.

Serial number	x ₁ (°)	x ₂ (kN)	x ₃ (Hz)	x ₄ (kN)	y (Lateral displacement of sleeper) (mm)
1	0	110	26	90	1.4659
2	10	100	30	162	1.8656
3	20	90	34	72	0.7891
4	30	80	38	144	1.4381
5	40	70	24	54	1.2415
6	50	60	28	126	1.6462
7	60	50	32	36	0.4825
8	70	40	36	108	1.3780
9	80	120	40	180	1.6132

The test results of the uniform test design are generally analyzed by the intuitive analysis method similar to the orthogonal test design instead of ANOVA,¹⁹ because the uniform test design has more levels of each factor and fewer trials, and the uniform design table is not orthogonal. In addition to the intuitive analysis method, the test results can also be analyzed by linear regression. The two analysis methods are as follows:

1) Intuitive analysis method: Directly finding the value of the maximum lateral displacement of the sleeper and selecting the corresponding factors group in the test. In Table 2, the maximum value of the lateral displacement of the sleeper was 1.8656 mm, which corresponded to the factors group No. 2. The most optimal operation parameters in the test were determined.

2) Linear regression method: In Table 2, the number of factors x₁–x₄ corresponded to the clamping wheel force normal angle, downforce, excitation force frequency, and excitation force amplitude, one test result y denoted the lateral displacement of sleeper. The regression equation of this test could be derived using the multiple linear regression method and the regression function in MATLAB software with the significance level α = 0.05. The regression equation is as follow:

\hat{y} = 1.8908 - 0.0001 x_{1} - 0.0021 x_{2} - 0.0480 x_{3} + 0.0105 x_{4}

(2)

Using the F-test for this regression equation, the coefficient of determination r² = 0.944 which was very close to 1 and the regression equation is significant. Its p-value is 0.0089 of the F-test, the p-value is small and the fitted model is valid.

As can be seen from equation (2), the Index “y” decreased with factors x₁, x₂, x₃ increases and increased with factor x₄ increase. Compared with the results in section III A–C, the maximum value of the lateral displacement response of the sleeper decreased with the increase in the normal angle of the clamping wheel force, the excitation force frequency, and downforce. And increased with the increase in the excitation force amplitude. This trend was consistent with equation (2). The correctness of the fitted multiple linear regression equation was verified.

According to equation (2), the factors x₁, x₂, x₃ were set as small as possible and the factor x₄ was set as large as possible to obtain the maximum lateral displacement y = 2.4488 in the test range. The optimal combination of operation parameters could be obtained by comparing the results of the visual analysis method and the linear regression method: clamping wheel force normal angle 0°, downforce 60 kN, excitation force frequency 18 Hz, excitation force 180 kN.

It is important to note that in reality, the excitation force is not decoupled from the frequency due to the fixed size of the exciter eccentric block. Therefore, the exciter should be designed with a larger eccentric block mass and eccentricity to obtain a larger excitation force at a lower frequency if a better excitation effect is required.

Experimental verification

The trend of the influence of different operation parameters on the stabilization operation was obtained by simulation analysis of the dynamic stabilizer-ballast bed system. To verify the correctness of the trends, a full-size dynamic stabilizer-ballast bed vibration test bench was built and tests were conducted.

Test bench construction

The vibration test bench of the dynamic stabilizer-ballast bed is shown in Figure 10.

Figure 10.

Vibration test bench (prototype): (a) parts of vibration test bench (1 – Ballast box; 2 – Standard installation of type 60 rails; 3 – Type III sleepers; 4 – Downforce adjustment mechanism; 5 – Dynamic stabilizer), (b) dynamic stabilizer (1 – Substrate; 2 – Motor base; 3 – Adjustable speed 22 kW three-phase asynchronous AC motor; 4 – Universal coupling; 5 – Vibration exciter; 6 – Vibration exciter mounting base; 7 – Vibration exciter drive gear; 8 – Adjustable clamping mechanism; 9 – Traveling wheel), (c) double shaft eccentric block exciter of the test bench, and (d) vibration test bench.

The eccentricity of a single steel eccentric block in the exciter is 25.35 mm and the mass is 51.794 kg. The relationship of excitation force amplitude and excitation force frequency obeys equation (1). The downforce spring can be adjusted during the test to get the required downforce.

Measurement and acquisition equipment

The DH187E acceleration sensor and the DH5923 dynamic signal analysis system of the Donghua-Test were used for the acquisition equipment. In the test, the acquisition frequency of the accelerometer was set to 256 Hz, which could meet the acquisition requirement below 100 Hz of excitation frequency according to Shannon sample theorem.²⁰

Analysis of test results

Three parameters including downforce, clamping force normal angle and excitation force frequency were selected, which influenced the stabilization operation. And three levels were selected for each parameter for an orthogonal test to verify the effect of each operation parameter on the lateral acceleration response of the sleeper. The results of the test are shown in Table 3.

Table 3.

Results of Box-Behnken design experiments.

Serial number	X ₁ (Downforce) (kN)	X ₂ (Force normal angle) (°)	X ₃ (Stimulating force frequency) (Hz)	Y (Lateral acceleration of sleeper) (m/s²)
1	44	80	10	4.1
2	44	80	14	10.8
3	44	40	12	12.0
4	48	40	14	8.2
5	48	80	12	8.0
6	44	40	12	11.0
7	44	0	14	10.3
8	44	0	10	4.4
9	44	40	12	11.8
10	44	40	12	12.7
11	40	40	14	16.9
12	40	40	10	5.5
13	48	40	10	4.4
14	40	0	12	11.0
15	40	80	12	8.8
16	48	0	12	5.2
17	44	40	12	10.6

The test was designed by using the Box-Behnken Design (BBD) method and the response surface methodology (RSM). The response Y in Table 3 was fitted by using Design Expert 12 software to obtain the regression equation:

\begin{matrix} Y = 11.62 - 2.05 X_{1} - 0.1 X_{2} + 3.48 X_{3} + 1.25 X_{1} X_{2} \\ - 1.9 X_{1} X_{3} + 0.2 X_{2} X_{3} - 1.01 X_{1}^{2} - 2.36 X_{2}^{2} - 1.86 X_{3}^{2} \end{matrix}

(3)

ANOVA was performed on equation (3) and the results as shown in Table 4.

Table 4.

Analysis of variance for response surface test results.

Source	Sum of squares	df	Mean square	F-value	p-Value	Significance
Model	197.79	9	21.98	31.31	<0.0001	**
X ₁	33.62	1	33.62	47.90	0.0002	**
X ₂	0.08	1	0.08	0.1140	0.7456
X ₃	96.60	1	96.60	137.64	<0.0001	**
X ₁X₂	6.25	1	6.25	8.90	0.0204	*
X ₁X₃	14.44	1	14.44	20.57	0.0027	**
X ₂X₃	0.16	1	0.16	0.2280	0.6476
X ₁ ²	4.30	1	4.30	6.12	0.0426	*
X ₂ ²	23.45	1	23.45	33.41	0.0007	**
X ₃ ²	14.57	1	14.57	20.75	0.0026	**
Residual	4.91	7	0.7019
Lack of Fit	2.15	3	0.7150	1.03	0.4677
Pure Error	2.77	4	0.6920
Cor Total	202.70	16

Indicates a significant effect on the results (p < 0.05).

Indicates a highly significant effect on the results (p < 0.01).

As can be seen from Table 4, the developed model p < 0.0001, which was significant; the misfit term p > 0.05, which was not significant, indicating that the model had high reliability. The influence order of three individual factors on the sleeper lateral acceleration is X₃ > X₁ > X₂, which meant the excitation force frequency > downforce > force normal angle. The primary terms X₁ and X₃ had highly significant effects on the results. The interaction term X₁X₃ had a highly significant effect on the results. And X₁X₂ had a significant effect on the results. The secondary terms X₂² and X₃² had a highly significant effect on the results. The effect of X₁² on the results was significant.

The response surface of the three parameters on the sleeper lateral acceleration is shown in Figure 11.

Figure 11.

Response surface of interaction of various factors on lateral acceleration of sleeper.

As can be seen from Figure 11, when the excitation frequency was fixed, with the increases of downforce and force normal angle, the lateral acceleration of the sleeper was decreasing trend. When the force normal angle was fixed, with downforce decreased and excitation frequency increased, the lateral acceleration of the sleeper was increasing trend, and the increasing trend was obvious. When downforce was fixed, with the force normal angle decrease and the excitation frequency increase, the lateral acceleration of the sleeper was increasing trend. In comparison with the simulation results in section III, the simulation of operation parameters effects on the sleeper lateral acceleration were consistent with the test results. The test results verified the correctness of the simulation results.

Conclusion

A simulation model of the rigid-flexible coupling dynamics of the dynamic stabilizer-ballast bed system was established. The stabilization operation process was studied and simulated by changing operation parameters. The conclusions are as follows:

1) With the clamping wheel force normal angle increasing, the lateral maximum displacement and lateral maximum acceleration of the sleeper were decreasing trend. The clamping wheel force normal angle should be maintained in the range of 0°–50° to ensure better stabilization operation effect.

2) With the gradual increase in downforce, sleeper lateral displacement and acceleration trends first increased and then decreased. The maximum lateral displacement and maximum acceleration were in downforce 60 kN, and the maximum lateral acceleration was maintained in the high scope when downforce was in the range of 60–80 kN. The center of the sleeper deviated from the vibration starting point when the downforce was small, and the track straightness was reduced. To obtain the best excitation effect, the downforce should be maintained within the range of 60–80 kN; to protect the track bed, the downforce should be increased to more than 80kN. Therefore, it was recommended to select 60–100 kN as the operation range for downforce.

3) The effect of excitation force on stabilization operations is: with the increase in the excitation force amplitude, the lateral displacement and acceleration of the sleeper were monotonically increasing; with the increase in the excitation frequency, the lateral displacement of the sleeper monotonically decreased, and the lateral acceleration first decreased and then increased.

4) For the rigid-flexible coupled dynamics simulation model, a multifactorial analysis was performed using uniform test design method. The optimized operation parameters were obtained by comparing the results of the intuitive analysis method and the multiple linear regression method: clamping wheel force normal angle 0°, downforce 60 kN, excitation force frequency 18 Hz, and excitation force amplitude 180 kN. According to the obtained regression equation, it was concluded that the maximum value of sleeper lateral displacement response decreased with the increases of clamping wheel force normal angle, excitation force frequency and downforce; and increased with excitation force amplitude increase. The four operation parameters on the sleeper lateral displacement in the order of significance were the excitation force frequency > excitation force amplitude > downforce > clamping wheel force normal angle.

5) A vibration test bench was built to verify the trend of the effect of each parameter on stabilization operation. The three operation parameters on the sleeper lateral displacement in the order of significance were excitation force frequency > downforce > clamping wheel force normal angle.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China [grant number 51765023].

ORCID iDs

Jiongli Wang

Hongyi Huang

Zemin Zhao

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