Keywords

1 Introduction

1.1 A Subsection Sample

As society has progressed, rail transit has greatly developed as a more rapid mode of transportation. However, when a train passes through a building or a factory, the continuous vibration adversely affects the health of humans, the building stability, and the normal use and maintenance of precision instruments [1,2,3,4]. Many scholars have conducted various degrees of research on the effect and mechanism of the isolation of various isolation measures. Kumar et al. [5] found that the propagation mode of vibration waves in soil is based on two forms of waves, i.e., physical waves and surface waves, at the same time and that 70% of the energy is carried by surface waves. In an investigation presented by Murillo et al. [6], surface waves were the primary cause of vibration in railway operation; blocking the propagation path of surface waves in the soil can effectively reduce the impact of vibration on the surrounding environment. Wang et al. [7,8,9,10] used model tests and field tests to study the vibration characteristics of the crushed rock embankment during train operation. Hou et al. [11] used ANSYS to establish a model of an empty trench and analyzed the isolated vibration effect of empty trenches. The study found that the depth of the trenches had the greatest impact on the vibration isolation effect and that a high-frequency vibration has a better vibration isolation effect for open trenches. Zhang et al. [12] proposed a numerical method that can determine the vibration law of the soil behind open trenches in high-speed railways. The analysis of the open trench depth has a significant effect on the vibration isolation effect. Woods et al. [13] proposed using open trenches as barriers to isolate vibrations. The wavelength was studied as a parameter, and the relationship between the trench depth and the wavelength of the vibration wave on the mechanism was analyzed. Liu et al. [14, 15] used an indirect boundary integration equation method to simulate vibration isolation from several rows of elastic piles; the study found that the multiple rows of piles should be adopted for low-frequency waves, but more than three rows of piles will not significantly improve the vibration isolation effect for high-frequency waves.

However, in the above study on the isolation of vibrations, the vibration isolation effect of open trenches or row piles was regarded as a single-factor analysis. Most studies focus on a comparison of the vibration isolation effects and an analysis of the influencing factors and lack multivariate analysis of the isolation mechanisms for the vibration isolation effects.

A single-row vibration-isolated pile is used as a vibration isolation barrier in this test. The pile section size, pile length, pile spacing, and other factors are taken into account, and the test result adopts the reduction ratio of the amplitude (Ar) [13] to indicate the vibration isolation effect of the miniature concrete vibration-isolated piles.

2 Field Site and Instrumentation of the Test Layout

2.1 Field Site

Considering that the nature of sand is relatively stable in this test and the test variables are easy to control, the planar dimensions of the sand foundation form a rectangle with a size of 3 m × 2 m × 1.5 m. The planar particle size of sand is less than 5.0 mm, and the moisture content of the sand is controlled from 9 to 10%. With sand layer backfilling and then consolidation, the density of the sand is controlled from 9 to 10%.

2.2 Equipment Instrumentation

The main equipment is composed of a set of control systems named WS-Z30, which consists of a data acquisition controller, a power amplifier, an electric charge amplifier, a signal generator, an accelerometer amplifier, a signal amplifier, an electromagnetic exciter, a computer, and a few accelerometers (the sensitivity is 4 pc/ms−2, the frequency response is 0.2 ~ 8000 Hz, the measurement range is 50 m/s2, and the quality is 28.5 g). The field site and the equipment used in the experiment are shown in Figs. 1 and 2.

Fig. 1
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Miniature concrete vibration-isolated pile and accelerometer arrangement

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Fig. 2
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Details of the test equipment

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The pile is poured using a strength grade of C30; in this test, the pile section is designed as a square with section widths of 5, 10, and 15 cm corresponding to the length of each section size of 30, 40, and 50 cm, respectively. There are a total of four accelerometers, and the details of the test site are shown in Fig. 3.

Fig. 3
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Vibration-isolated pile and acceleration sensor arrangement

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The excitation frequency of the exciter ranges from 10 to 150 Hz, the sampling frequency was set to 5000 times per second in the test, and the sampling time was 5 s. The charge amplifier values must be consistent in the test. The wave velocity of the excitation wave in the sand was tested in advance, and then the effect of the wavelength on the vibration isolation effect of single-row pile was analyzed.

3 Determination of the Wave Velocity of a Soil Layer

3.1 Test Planning

This paper proposes a test method (spectral analysis of surface waves, SASW) [16, 17] to determine the wave velocity. The equipment layout that is involved is shown in Fig. 4.

Fig. 4
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Layout diagram of the test equipment

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Under the effect of the pulse load of the exciter, signal 1 and signal 2 were collected by the #1 accelerometer and the #2 accelerometer, respectively, and then a spectrum diagraph was obtained. Then, SASW is used to obtain the coherence function and the reciprocal power spectra of the two signals; if the coherence function value at a certain frequency exceeds the threshold value of 0.85 [16], these two signals have a significant correlation. Then, the reciprocal power spectra are used to determine the corresponding phase difference φ. Finally, the wave velocity definition is shown in Eq. (1).

$$V_{R} = 360 \cdot f \cdot x/\varphi$$
(1)

where VR denotes the wave velocity, f denotes the excitation frequency, x denotes the distance between the #1 accelerometer and the #2 accelerometer, and φ denotes the phase difference.

The corresponding wavelength is defined as Eq. (2).

$$L_{R} = 360 \cdot x/\varphi$$
(2)

In the test, the consideration of the frequency of the source and the pile spacing mostly depends on the depth of the soil being tested [17]. Therefore, when measuring the shallow wave velocity (VR), the vibration source is dominated by high frequencies, and the test signal is excited 4 times. The spacing (x) is in turn taken from Table 1. The forward test direction is defined as being from the #1 accelerometer to the #2 accelerometer. After the forward test is finished, the position of the accelerometer is not changed, the exciter is moved symmetrically to the other side of the centerline (MN), and then the above operation is repeated in the backward. The testing conditions are shown in Table 1.

Table 1 Testing conditions of the wave velocity test
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3.2 Data Processing

Taking testing condition 5 from Table 1 as an example, the exciter is arranged on the right side of the centerline (MN). The coherence function to analyze the measured signals, i.e., signal 1 and signal 2, and the coherent coefficient corresponding to the excitation frequency of 150 Hz from the derived frequency-domain files are screened. The coherence function is shown in Fig. 5.

Fig. 5
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Coherent function of testing condition 5

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The value of 0.956 marked in Fig. 5 represents the coherence coefficient value of the corresponding signal when the excitation wave propagates from the #1 accelerometer to the #2 accelerometer under testing condition 5. The requirement that the coherence coefficient value is greater than 0.85 is satisfied [16]; then, the phase difference φ was found using the method of the cross-power spectrum. The cross-power spectrum is shown in Fig. 6.

Fig. 6
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Reciprocal power spectrum of testing condition 5

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The value of 146.457 rad marked in Fig. 6 represents the phase difference (φ) value of the corresponding signal when the excitation wave propagates from the #1 accelerometer to the #2 accelerometer under testing condition 5.

Using the same method for the remaining conditions in Table 1, the arithmetic mean of the test results is taken in both directions as the measured wave velocities on the surface of the soil. The wave velocities are shown in Table 2.

Table 2 Results of the wave velocity test
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After averaging the wave velocities (VR) that were measured at each operating condition, the arithmetic mean was 109.989 m/s.

4 Size Optimization of a Single Factor

4.1 Single-Factor Data Processing

The reduction ratio of the amplitude (Ar) is adopted to indicate the vibration isolation effect of a single-row vibration-isolated pile. A smaller Ar indicates that the vibration isolation effect of single-row piles is better [13].

The Ar is defined as in Eq. (3).

$$Ar = a_{1} /a_{o}$$
(3)

where a1 denotes the vertical vibration acceleration of the ground after setting a single-row vibration-isolated pile and ao denotes the vertical vibration acceleration of the free field that is excited in the same position.

The influences on the vibration isolation effect of the length of the pile (L), the size of the pile section (B × B), the pile spacing (s), the distance from the vibration source (r), and the wavelength (LR) are investigated. The following parameters are introduced to evaluate the effects of vibration isolation and are used as a measure of the vibration isolation effect of the vibration-isolated piles.

  1. (1)

    Depth parameter D

The ratio D between the length (L) of the pile and the wavelength (LR) represents the effect of different lengths of the piles on the vibration isolation effect of the vibration-isolated piles. The formula is shown in Eq. (4).

$$D = L/L_{R}$$
(4)
  1. (2)

    Width parameter W

The ratio W between the width (B) of the pile and the wavelength (LR) represents the influence of the different widths of the piles on the vibration isolation effect of the vibration-isolated piles. The formula is shown in Eq. (5).

$$W = B/L_{R}$$
(5)
  1. (3)

    Pile spacing parameter S

The ratio S between the pile spacing (s) and the wavelength (LR) indicates the influence of the density of the pile arrangement on the isolation effect of the vibration-isolated piles. The formula is shown in Eq. (6).

$$S = s/L_{R}$$
(6)
  1. (4)

    Vibration source distance parameter K

The ratio K between the distance from the vibration source (r) and the wavelength (LR) indicates the influence of the distance from the vibration-isolated pile to the vibration source on the vibration-isolated effect of the piles. The formula is shown in Eq. (7).

$$K = r/L_{R}$$
(7)

4.2 Influence of the Depth Parameter on the Vibration Isolation Effect of Piles

The length of the pile is the main factor that influences the vibration isolation effect [18]. First, the distance from the vibration source (r), the width of the pile section (B), and the pile spacing (s) were controlled. The effects of the pile length (L) on the vibration isolation of the pile were studied separately; the test arrangement is shown in Table 3.

Table 3 Test arrangement of the parameter D
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The excitation frequency of the exciter ranges from 10 to 150 Hz; 10 Hz is used as the step length for gradual excitations. After the discrete data are removed, the remaining data are fitted using MATLAB. Under certain conditions, the regression equations of the depth parameter (D) and the Ar are shown in Table 4. Significance tests on the regression equation were calculated to verify the correlations between the variables. The correlation coefficient R and the F value were derived using MATLAB. The correlation coefficient R and the F value are used to judge the statistical significance of the regression equation; if the correlation coefficient R is greater than the critical value Rα,f and closer to 1 and if the F value is greater than the critical value Fα, the regression equation is more obvious and the fitting is better. The Rα,f and Fα values of each testing condition are identified by using the tables of correlation coefficient critical values [19] and the F distribution table [19], as shown in Table 4.

Table 4 Correlation analysis for D and Ar
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Under testing conditions 1, 2, and 3, the correlation coefficient R is greater than the critical value of Rα,f, and the value of F is greater than the value of Fα from Table 4. The results show that the regression equation is significant in each testing condition; thus, the regression equation is meaningful and can be accepted.

The fitting graph of the regression equation for various testing conditions is shown in Fig. 7.

Fig. 7
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Fitting curve of D and Ar

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Figure 7 shows D rapidly increasing from approximately 0.18 to 0.8 and the value of Ar decreasing rapidly. When D increasing to approximately 0.8, the change in Ar tends to flatten gradually, and when D is in the approximate range of 1.042–1.125, the slope of the fitting curve is approximately Ar′ ≈ 0. When the length of the pile (L) increases, the change in Ar tends to be stable. When the length of the pile (L) continues to increase, the influence of the vibration isolation effect will be weakened. Under the conditions of controlling other influencing factors, D = 1.042 can be used as the limiting range of the length of the effective pile body.

4.3 Influence of the Pile Spacing Parameter on the Vibration Isolation Effect of Piles

The distance between the nearest edges of adjacent piles obviously influences the vibration isolation effect of the vibration-isolated pile. According to the optimization method of controlling other variables, the pile spacing parameter S is processed. The specific testing conditions are shown in Table 5.

Table 5 Test arrangement of the parameter S
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The excitation method of the parameter S is the same as the parameter D in the test arrangement. Additionally, the discrete data are removed, and then the remaining data are fitted. Under certain conditions, the regression equations of the parameter S and the Ar are shown in Table 6. Statistical significance of the test is shown in Table 6.

Table 6 Correlation analysis of S and Ar
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Table 6 shows that the correlation coefficient R is greater than the critical value of Rα,f and that the F values are greater than the value of Fα. These results show that the regression equation is significant in each testing condition; thus, the regression equation is meaningful and can be accepted.

The fitting graphs of the regression equations for the various testing conditions are shown in Fig. 8.

Fig. 8
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Fitting curve of S and Ar

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Figure 8 shows that when S is increasing from approximately 0.013 to 0.0804, the value of Ar changes slowly and sustains a lower level, demonstrating that a single-row pile can obtain a better vibration isolation effect. After S increases to approximately 0.0804, the change in Ar tends to grow rapidly, and then the vibration isolation effect decreases gradually. Under the conditions of controlling other influencing factors, the pile spacing (s) can be maintained at a lower level (S = 0.013), while the better vibration isolation effect can be obtained.

4.4 Influence of the Width Parameter on the Vibration Isolation Effect of Piles

The influence of the width parameter (W) on the vibration isolation of vibration-isolated piles is studied; the test arrangement is shown in Table 7.

Table 7 Test arrangement of the parameter W
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The excitation method of the parameter W is the same as the parameter D in the test arrangement. Additionally, the discrete data are removed, and then the remaining data are fitted. Under certain conditions, the regression equations of the parameter W and the Ar are shown in Table 8. The statistical significance of the test is shown in Table 8.

Table 8 Correlation analysis of W and Ar
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Table 8 shows that the correlation coefficient R is greater than the critical value of Rα,f and that the F values are greater than the value of Fα. The results show that the regression equation is significant in each testing condition; thus, the regression equation is meaningful and can be accepted.

The fitting graphs of the regression equations for the various testing conditions are shown in Fig. 9.

Fig. 9
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Fitting curve of W and Ar

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Figure 9 shows that when W is increasing from approximately 0.004 to 0.058, the value of Ar increases rapidly. Additionally, the vibration isolation effect is weakened rapidly. W increases to approximately 0.058, and with an increase in W, the change in Ar tends to gradually shallow. In other words, with an increase in the width of a square pile, the change in the vibration isolation effect is not significant. This is similar to the conclusion of Hou En-ping’s [18] study that a single-row vibration-isolated pile section size change has no significant influence on the vibration isolation effect. When the vibration-isolated pile is practically engineered, choosing a reasonable section size of vibration-isolated pile is of guiding significance.

4.5 Influence of the Vibration Source Distance Parameter on the Vibration Isolation Effect of Piles

When a vibration wave is transmitted over a distance and the energy of the wave is consumed continuously, it is advantageous to improve the vibration isolation effect by arranging the positions of the vibration-isolated piles rationally. The specific testing conditions are shown in Table 9.

Table 9 Test arrangement of the parameter K
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The excitation method of the parameter K is the same as the parameter D in the test arrangement. Additionally, the discrete data are removed, and then the remaining data are fitted. Under certain conditions, the regression equations of K and Ar are shown in Table 10. The statistical significance of the test is shown in Table 10.

Table 10 Correlation analysis of K and Ar
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Table 10 shows that the correlation coefficient R is greater than the critical value of Rα,f and that the F values are greater than the value of Fα. The results show that the regression equation is significant in each testing condition; thus, the regression equation is meaningful and can be accepted. The fitting graphs of the regression equations for the various testing conditions are shown in Fig. 10.

Fig. 10
figure 10

Fitting curve of K and Ar

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Figure 10 shows that when K rapidly increases from approximately 0.031 to 0.291, the value of Ar shows a decreasing phenomenon, and when K increases in the approximate range from 0.291 to 0.357, regions of minima occur, and the vibration isolation effect is relatively excellent. When K increases to approximately 0.357, the change in Ar tends to increase gradually. Under the conditions of controlling other influencing factors, K values between 0.291 and 0.357 can be used as a reference index for determining the reasonable positions of vibration-isolated pile.

5 Size Optimization of Two Factors

5.1 Two-Factor Data Processing

In single-factor analysis, the influences of the pile length (L), the pile cross-sectional size (B × B), the pile spacing (s), and the vibration source distance (r) on the vibration isolation effect of the single-row pile are analyzed individually. The influence of the pile length (L) on the vibration isolation effect is the most obvious.

Therefore, the parameters of the two-factor analysis are introduced into the length of the pile using the single-factor analysis results of the reasonable pile length (L) range and are then used to define the reasonable range of the section width (B), the pile spacing (s), and the vibration source distance (r).

The setting parameters are as follows:

  1. (1)

    Slenderness ratio δ

The ratio δ between the section width (B) and the length of the pile (L) represents the degree of elongation of the pile. The formula is shown in Eq. (8).

$$\delta = B/L$$
(8)
  1. (2)

    Relative spacing of piles β

The ratio β between the pile spacing (s) and the length of the pile (L) represents the single-row piles’ layout density. The formula is shown in Eq. (9).

$$\beta = s/L$$
(9)
  1. (3)

    Relative vibration source distance γ

The ratio γ between the distance from the vibration source (r) and the length of the pile (L) represents the position of a single-row pile relative to the vibration source. The formula is shown in Eq. (10).

$$\gamma = r/L$$
(10)

5.2 Influence of the Slenderness Ratio δ on the Vibration Isolation Effect of the Piles

The vibration isolation effect of the pile width and the length of the pile were studied by preparing a test group and a control group, respectively. The test results of the experimental group were validated by the test results of the control groups. The specific testing conditions are shown in Table 11.

Table 11 Test arrangement of the parameter δ
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In the test and control groups, the section width of the pile was 5, 10, or 15 cm corresponding to a length of each section size of 30, 40, and 50 cm, respectively; the ratio δ has nine kinds of values. Vibration frequencies of 10, 50, and 100 Hz were selected, and then the vibration frequencies were tested. The vibration isolation effect of each testing condition was compared. The results of the test are shown in Figs. 11 and 12.

Fig. 11
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Changing relation between Ar and δ in the experimental group

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Fig. 12
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Changing relation between Ar and δ in the control group

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Figure 11 shows that the corresponding graphs of the two groups show a similar trend. When δ is between 0.1 and 0.2, corresponding value of Ar changes gently. When δ is greater than 0.2, the value of Ar increases rapidly with the δ. For each high, medium, and low frequency, the rule is consistent insomuch that the value of Ar increases with δ, showing a smooth change in a certain interval and then an increasing phenomenon.

Figure 12 shows that the change laws of the corresponding figures in the test and control groups are consistent with those of the two groups.

Using both Figs. 11 and 12 and further analyzing the test results, it is found that when the size of the pile has a smaller δ for a vibration-isolated piles, it can obtain a better vibration isolation effect; when the size of the vibration-isolated pile is less than δ, the vibration isolation effect of a vibration-isolated pile is weakened. The vibration isolation effect of the vibration-isolated pile is maintained at a good level when δ is kept between 0.1 and 0.2.

5.3 Influence of the Relative Spacing of the Piles β on the Vibration Isolation Effect of the Piles

The relative spacing of the pile (β) reflects the influence of the density of the vibration-isolated piles on the vibration isolation effect. To study the relationship between β and Ar, the test group and control group were prepared; the test arrangement is shown in Table 12.

Table 12 Test arrangement of the parameter β
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The specifications of the vibration-isolated pile and the excitation frequencies used in the test are the same as the arrangement of the δ test, and the test results are shown in Figs. 13 and 14.

Fig. 13
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Changing relation between Ar and β for the experimental group

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Fig. 14
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Changing relation between Ar and β for the control group

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Figure 13 shows that the corresponding figures of the two group testing conditions show a similar trend in the two group testing conditions for the test group. For each of the high, medium, and low excitation frequencies in the figure, the value of Ar increases rapidly with an increase in β.

Figure 14 shows that the change laws of the corresponding figures in the test and control groups are consistent with those of the two groups.

Using both Figs. 13 and 14, it is found that when β is small, the single-row pile has a better isolation effect when adopting a long pile and a dense arrangement; when β is large, the isolation effect of the vibration-isolated pile will weaken when a sparse arrangement of short piles is used.

5.4 Influence of the Relative Vibration Source Distance γ on the Vibration Isolation Effect of the Piles

The relative vibration source (γ) reflects the influence of the arranged positions of vibration-isolated piles on the vibration isolation effect. The test group and control group were prepared to study the relationship between γ and Ar; the test arrangement is shown in Table 13.

Table 13 Test arrangement of the parameter γ
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The specifications of the vibration-isolated pile and the excitation frequencies used in the test are the same as those in the arrangement of the δ test; the test results are shown in Figs. 15 and 16.

Fig. 15
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Changing relation between Ar and γ for the experimental group

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Fig. 16
figure 16

Changing relation between Ar and γ for the control group

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Figure 15 shows that the corresponding figures of two group testing conditions show similar trends in the two group testing conditions for the test group. For each of the high, medium, and low excitation frequencies in the figure, γ rapidly grows from approximately 0.7 to 0.9, and the value of Ar decreases gradually with an increase in γ. When γ increases from approximately 0.9 to 1.2, regional minima are presented, and when γ increases to approximately 1.2, with an increase in γ, the change in Ar tends to increase gradually.

Figure 16 shows that the change laws of the corresponding figures in the test and control groups are consistent with those of the two groups.

Using both Figs. 15 and 16, it is found that if the position of the vibration-isolated pile is not closer to the vibration sources, a better vibration isolation effect can be obtained by a vibration-isolated pile. However, when the layout of the vibration-isolated pile satisfies the requirement that the γ reaches the range of 0.9–1.2, it can obtain a better vibration isolation effect.

6 Conclusions

To study each of the factors that affect the vibration isolation effect of a concrete vibration-isolated pile, single-factor analysis and two-factor analysis methods are applied on model tests while considering the length of the pile and the pile cross section; the following conclusions are obtained:

  1. (1)

    When D increases to approximately 1.042–1.125, the change in Ar tends to become stable with an increase in the length of the pile (L). When the length of the pile (L) continues to increase, the enhancement of the vibration isolation effect will be nonsignificant. When the depth parameter (D = 1.042) can be used as the limit range of the length of the effective pile body, a better vibration isolation effect can be obtained.

  2. (2)

    When the width of a vibration-isolated pile increases, the change in the vibration isolation effect is not significant; when the vibration-isolated pile is furnished and the pile spacing (s) is bigger, the vibration isolation effect will weaken.

  3. (3)

    When the position of a vibration-isolated pile is not closer to the vibration source, a better vibration isolation effect can be obtained by a vibration-isolated pile. When K is between 0.291 and 0.357, it can be used as a reference index for determining reasonable positions of vibration-isolated piles.

  4. (4)

    The vibration-isolated pile length (L) has the most obvious influence on the vibration isolation effect when increasing the length of the vibration-isolated pile, and Ar can reach levels below 0.1.

  5. (5)

    When the slenderness ratio (δ) is between 0.1 and 0.2, a vibration-isolated pile can obtain a better vibration isolation effect, and the minimum value of Ar can reach 0.2. If the relative pile spacing (β) is larger, the vibration isolation effect of a vibration-isolated pile is worse.

  6. (6)

    If the location of a vibration-isolated pile is not close to the source of vibration, the vibration-isolated pile can obtain a better vibration isolation effect, and if the relative vibration source distance (γ) is between 0.9 and 1.2, the value of Ar can reach 0.3 ~ 0.4.