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Modeling of nonlinear interaction and its effects on the dynamics of a vibrator-ground system
Soil Dynamics and Earthquake Engineering 132 (2020) 106064
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Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn
Modeling of nonlinear interaction and its effects on the dynamics of a vibrator-ground system Xun Peng a, Zhiqiang Huang a, *, Lei Hao b a b
Electromechanical Engineering College, Southwest Petroleum University, Chengdu, Sichuan, 610500, PR China Bureau of Geophysical Prospecting, China National Petroleum Corporation, Zhuozhou, Hebei, 072750, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibrator–ground system Nonlinear interaction Coupled vibration Field test
The nonlinear interaction between a vibrator and the ground is one of the main sources of signal distortion in land exploration. So, it is crucial to describe this nonlinear interaction and study how it affects the outgoing wave radiated by the vibrator. For this purpose, an approach is developed to obtain the equivalent ground stiffness and damping using a fractal contact model with the half-space method. Both the impacts of surface topography and the dynamics of ground are accounted for innovatively in this approach. The results show that both the ground stiffness and damping increase with the total deformation load but decrease with the excitation frequency and roughness. In addition, a modified two-degree-of-freedom vibration model of the vibrator–ground system is proposed to calculate the output force of the vibrator for different rough surface topographies. A field test is used to verify the theoretical model, and it is concluded that compared with the theoretical excitation signal, the output force varies with the excitation frequency. The amplitude of the output force in the high-frequency phase decreases for a rougher contact surface, as does the resonance frequency of the system. Moreover, reducing the mass of the baseplate is an efficient way to improve the high-frequency output.
1. Introduction Seismic vibrator is the main source of seismic energy in land explo ration, and its wide application has resulted in great improvements in oil and gas exploration [1,2]. The idea of the vibroseis method is to radiate a theoretically prescribed frequency-modulated (sweep) signal—which provides the best identification of the returning seismic reflections—into the earth. Therefore, the accuracy and frequency bandwidth of the radiated signal are the main restrictions on the development of seismic vibrators [3,4]. Nevertheless, the true source signal (the signal that is realistically radiated into the earth) is never identical to the prescribed theoretical shape because of nonlinear and unpredictable distortions of the original idealized sweep that greatly reduce the imaging accuracy of complex layers and increase exploration costs. Meanwhile, the insuffi cient output of seismic vibrators in the high-frequency phase has nega tive impacts on the expansion to higher frequency. Previous studies have indicated that one of the main sources of these problems is the nonlinear interaction between a vibrator and the ground [5,6]. Various research efforts have been made to study the nonlinear interaction between a vibrator and the ground and assess how it affects the source signal. The main research methods are (i) experimental
analysis, (ii) finite-element simulation, and (iii) theoretical modeling. Regarding experiments, many field tests have been devoted to the har monic distortion of outgoing waves and have discussed the impacts of the nonlinear response of an elastic half space and some parts of the radiating system [3,7–10]. In addition, finite-element analysis has evolved as an efficient tool. Wei et al. [11,12] studied the vibration characteristics of a vibrator by developing a finite-element model of the vibrator–ground (VG) system. Jiang et al. [13] used finite-element analysis to optimize the hydraulic oil ducts and reduce the vibration noise. The key to theoretical modeling is to establish a reasonable dy namic description of the coupled VG system. Using what is known as the “weighted-sum” method, Castanet et al. [14–16] were the first to develop a practical linear model with which to calculate the force applied to the ground surface by a seismic vibrator. However, because of the nonlinearity in the complex VG system, many practical issues arising from field tests have remained unexplained by this model. The fact that the vibrator is placed on a rough surface is at odds with the assumed uniform pressure distribution beneath the baseplate in the weighted-sum method. To describe the nonlinear contact, Lebedev and colleagues [17,18] proposed a model in which the VG contact acts as a nonlinear spring. Noorlandt and Drijkoningen [19] and Huang et al. [20] extended this model to investigate the effects of ground surface
* Corresponding author. E-mail address: [email protected] (Z. Huang). https://doi.org/10.1016/j.soildyn.2020.106064 Received 20 September 2019; Received in revised form 16 December 2019; Accepted 21 January 2020 0267-7261/© 2020 Elsevier Ltd. All rights reserved.
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Notation A AðtÞ a’ ae ap a’c a’L a0 Cz c1 D E E* F Fc Fe Fp Fs FðtÞ fe fp f1 f2 fin fen f G G0 J0 K Kz k1 L
M mb m1 m2 n Peiωt Δpeiωt Qeiωt R Reiωt r’ r’c r’L T Tc t u w wd Y ys z1 z2 γ δ λ
amplitude of sweep excitation The cosine window function is A(t) truncated contact area real elastic microcontact area real plastic microcontact area critical truncated contact area largest truncated contact area dimensional frequency equivalent ground damping hydraulic damping coefficient fractal dimension elastic modulus of soil composite elastic modulus elastic contact force of an asperity total deformation load total elastic contact load total plastic contact load static load sweep excitation elastic contact load of an asperity plastic contact load of an asperity functions defined in Eq. (10) functions defined in Eq. (11) starting frequency ending frequency excitation frequency fractal roughness shear modulus zero-order Bessel function factor of yield strength equivalent ground stiffness hydraulic stiffness coefficient sample length
ν ρ σ τ ε ωθ ω
with differing roughness. Meanwhile, part of the ground is captured by the vibrator’s baseplate during the vibration process, thereby potentially changing the stiffness and damping of the vibration system. Some re searchers have used the elastic half-space method to establish the dy namic stiffness and damping of the VG system and have studied how the system parameters affect the dynamic response [21–23]. Of the three aforementioned methods, the most essential is theoret ical modeling, through which the vibrator ground force can be estimated in the vibration process. However, it is noticed that previous theoretical models have considered only single aspects of the nonlinear interaction and have failed to combine multiple factors, while in reality the nonlinear contact springs and the dynamics of the ground work together to affect the dynamic response of the VG system. To date, a few refer ences have presented a three-dimensional (3D) lumped-parameter vibrator system comprising nonlinear springs and captured ground mass in which it can consider both the impacts of contact and the dy namics of the ground, but they did not explain how to determine the mass and spring parameters [12]. The present study is aimed at describing the nonlinear interaction of a vibrator in contact with rough half-space ground by using a fractal contact model with the half-space method. The roughness and the dy namics of the ground are both accounted for, and how the excitation parameters and surface topography affect the ground stiffness and damping is discussed. A modified two-degree-of-freedom model with nonlinear stiffness and damping is developed to study the output of the VG system, and a field test is carried out to validate the theoretical
number of superposed ridges mass of baseplate pad mass of reaction mass mass of baseplate frequency index total dynamic load dynamic load on an asperity external load on the baseplate asperity radius reaction force radius of truncated microcontact area radius of critical truncated contact area radius of largest truncated contact area sweep length duration of the cosine window sweep time radial displacement vertical displacement vertical displacement of the center point Y ¼ Fc Fs yield strength displacement of reaction mass displacement of baseplate scaling parameter asperity interference Lam� e constant Poisson’s ratio of soil mass density of soil normal stress shear stress volumetric strain rotating component circular frequency
model. The present research results will help engineers to optimize the vibrator structure and system parameters to adapt to different surface topographies and improve exploration efficiency. The proposed approach could also be used for other vibration systems that correspond to a half-space model with a rough contact interface. 2. Model formulation and solution method 2.1. Problem description and solution approach Vibrator is the key component of seismic vibrators, which generates ground force through coupled vibration with ground. As shown in Fig. 1, the main components of the vibrator are the reaction mass and the baseplate (the latter comprising a top plate, supporting columns, piston, and baseplate pad). When the vibrator is operating, there are two loading phases, namely the static loading phase and the dynamic loading phase. In the static loading phase, the weight of vehicle (hold-on load) is applied to the vibrator baseplate to establish contact. In the dynamic loading phase, a time-varying excitation load is added to the afore mentioned static load and transferred into the ground through the baseplate. The output force is translated into a seismic wave at the contact between the baseplate and the ground, and the coupling ground is modeled as a half space with a rough surface. However, the actual VG contact area, which affects the surface displacement of the half-space model, changes with the excitation load. There is therefore an interac tion between the surface topography and the mechanical response of the 2
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Fig. 1. Schematic of vibrator–ground (VG) system.
half space. The solution approach is to decompose the problem into two subproblems: the contact problem and the half-space problem. The con tact problem is to determine the real contact area of the contact interface during the vibration process. Given the real contact area, the half-space problem is then to solve the surface displacement due to the vibration of ground generated by excitation load. Solving these two sub-problems gives the equivalent ground stiffness and damping.
where L is the sample length, D is the fractal dimension, G is the fractal roughness, M is the number of superposed ridges used to construct the fractal surfaces, n is a frequency index with nmax ¼ int½logðL =Ls Þ =log γ�, φm;n is a random phase in the range ½0; 2π�, and γðγ > 1Þ is a scaling parameter. Considering the surface flatness and the frequency distri bution density suggests that γ ¼ 1:5 is typical for most surfaces. The surface topography of ground is determined by the values of D and G. The D values determine the contribution of high and low fre quency components in the surface function Eq. (1). The physical sig nificance of D is the extent of space occupied by the rough surface, i.e., larger D values correspond to denser profiles (smoother topography). Physically, G is the height scaling parameter and higher G values correspond to rougher surface. The microcontact between the baseplate pad and ground is shown in Fig. 2(a), and it can be reduced to contact between an asperity of the ground surface and the opposing rigid plane, as shown in Fig. 2(b). With the change of the total deformation load, the number of the asperities which are in contact with the baseplate pad is various. Thus, the real total contact area between the baseplate pad and the ground is related to the total deformation load. In Appendix A, details on the derivation process of the relationship between the total deformation load and the largest truncated microcontact area, are given. The rela tionship is written as
2.2. Governing equations for contact problem In order to construct the rough surface of ground, the fractal geom etry is used to describe the surface topography of ground [24–26]. Fractal geometry can describe geometric features of various length scales and provide a means of characterizing asperities of a large range of sizes. The 3D ground-surface topography can be simulated deter ministically using the modified two-variable Weierstrass–Mandelbrot fractal function, which is written as [27]. zðx; yÞ ¼ L 8 > > < : cosφm;n > > :
� �ðD G L
� M X nmax ln γ X γðD M m¼1 n¼0
2Þ �
3Þn
2 n
2
2
62πγ x þ y cos6 4 L
�12
h ⋅cos tan
1
�y� x
39 > > 7= þ φm;n 7 ; 5 > M > ;
π mi
(1)
Fig. 2. Description of the contact model: (a) microcontact between the baseplate pad and ground; (b) microcontact between the pad and an asperity.
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� �pffiffiffiffiffiffiffi ! 8 25:5 D D 1 ln γ D 1 > � D 2 ’ 0:5D 0:5 ’2:5 D ’2:5 D > � a þ Kys a’0:5D G a a E > L L c L < 3 D 3π 2 0:5D 5 2D Fc ¼ > � � > > : � pffiffiffiffiffiffiffiffiffiffiffiffi�a’L �0:75 a’L ’0:25 2E G ln γ ln þ 3Kys a’0:75 a D ¼ 2:5 L c π ac
For the VG system, the total deformation load is determined by the static load Fs and the excitation load in the vibration processes. The static load Fs can generate the initial contact area between the baseplate pad and ground, and the added dynamic load brings about the change of the contact area. Thus, when the static load and amplitude of the exci tation load is given, the range of the total deformation load can be calculated and a’L at the contact interface is obtained from Eq. (2).
�
Previous research considered the initial loading phase to correspond to plastic deformation, after which elastic behavior becomes dominant [11,19,28]. The microcontact plastic deformation arises mainly as a static displacement offset that has little influence on the ground vibra tion, and the methods provided by the theory of elasticity are applicable. For axisymmetric problems as shown in Fig. 3, the wave equations are given by � � 8 2 > > ρ ∂ u ¼ λ þ 2G0 ∂ε þ 2G0 ∂ωθ > < ∂t2 ∂r ∂z ; (3) � � 2 > > ∂w ∂ε 1 ∂ rωθ Þ > : ρ 2 ¼ λ þ 2G0 2G0 ∂z r ∂r ∂t
pκ2 α σp ; G0 φðpÞ
(7)
function,κ2 ¼ ω2 ρ=G0 α ¼ p2 h2 , and h2 ¼ ω2 ρ=ðλ þ 2G0 Þ. Substitut ing Eq. (6) into Eq. (7) yields Z ∞ pακ2 p wðra ; 0Þ ¼ σ J0 ðprÞdp: (8) G0 φðpÞ 0 When considering the solution of Eq. (8), previous studies approxi mated the surface displacement with the displacement of the center point (r ¼ 0) to obtain a vertical displacement in the form [29]. wd ðtÞ ¼
ΔPeiωt ðf1 þ if2 Þ; G0 r
(9)
pffiffiffiffiffiffiffiffiffiffi where a0 ¼ ωr ρ=G0 is the dimensionless frequency and f1 , f2 are functions of a0 given by � f1 ¼ 0:238733 0:59683a20 þ 0:004163a40 ⋯ ; (10)
The microelement volumetric strain ε and rotating component ωθ are given by (4) (5)
⋯:
(11)
f2 � G0 rwd ðtÞ: f 21 þ f 22 ω
(12)
0:017757a30 þ 0:000808a50
f2 ¼ 0:148594a0
�
∂w : ∂r
(6)
where σ p is related to the boundary conditions (namely σp ¼ R∞ where J0 ðpra Þ is the zero-order Bessel 0 σ ðra Þra J0 ðpra Þdra ],
ρ is the soil density.
� 1 ∂u ωθ ¼ 2 ∂z
(2)
σz ðra ; 0; tÞ ¼ σðra Þeiωt ; τrz ðra ; 0; tÞ ¼ 0
wðp; 0Þ ¼
where u; w are the radial and vertical displacements, respectively, G0 is 2G0 ν E � the shear modulus [G0 ¼ 2ð1þ ν2 Þ], λ is the Lame constant [λ ¼ 1 2ν], and
∂u u ∂w þ þ ; ∂r r ∂z
D 6¼ 2:5
8 iωt > < ΔPe r < r a 2 where σðra Þ ¼ . πr > : 0 ra > r The Hankel transform of the vertical surface displacement can be derived as
2.3. Governing equations for half-space problem
ε¼
! 0:5 ’1:5 0:5D ac
Hence, we have
Assuming that the dynamic force acting on an asperity is ΔPeiωt , the boundary conditions are written as
ΔPeiωt ¼
f1 G0 rwd ðtÞ f 21 þ f 22
iω
For the entire contact interface between the baseplate pad and the ground, the total load can be obtained as Z Peiωt ¼ wd ðtÞ Z iωwd ðtÞ
r’L
r’c r’L r’c
f1 G0 r2πr’ nðr’ Þdr’ f 21 þ f 22 f2 � G0 r2πr’ nðr’ Þdr’ ; f 21 þ f 22 ω
(13)
where Peiωt is the total dynamic load, the radius of the critical truncated contact arear’c and the radius of the largest truncated contact area r’L are be determined by contact analysis, and nðr’ Þ is obtained from Eq. (A.8). 2.4. Formulas for ground stiffness and damping Assuming that the external force loaded on the baseplate pad is Qeiωt , as shown in Fig. 4, then the kinetic equation of the baseplate pad is € þ Reiωt ¼ Qeiωt ; mb wðtÞ
Fig. 3. Schematic of axisymmetric problem. 4
(14)
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Fig. 4. Force diagram for baseplate pad.
where mb is the mass of the baseplate pad and Reiωt ¼ action force. Substituting Eq. (13) into Eq. (14) yields
� � mb w€d t þ wd t
Z
r’L r’c
� � � f1 G0 r 2πr’ n r’ dr’ þ iωwd t f 21 þ f 22
Z
Fig. 6 shows how the ground damping varies with Fc and f for D ¼ 2.4 and G ¼ 6 � 10 7 m. As can be seen, the ground damping also increases with Fc and decreases with f. The projection curves in the Fc Cz plane show that the slope decreases when f is less than 60 Hz but increases
Peiωt is the re
r’L
r’c
� � f2 � G0 r 2πr’ n r’ dr’ ¼ Qeiωt : f 21 þ f 22 ω
otherwise. In Fig. 6(c), the f Cz curves decrease slowly and then drop significantly. With the increase of Fc , the reduction ratio of Cz increase.
Thus, the equivalent ground stiffness and damping can be respec tively written as pffiffi � Z r’ � L 2 f1 ’ðD 1Þ Kz ¼ (16) ðD 1ÞG0 rL r’ð1 DÞ dr’ ; 2 f 21 þ f 22 r’c Cz ¼
pffiffi 2 ðD 2
Z ’ðD 1Þ
1ÞG0 rL
r’L �
r’c
f 21
� f2 � r’ð1 2 þ f2 ω
DÞ
dr’ :
3.2. Effects of surface topography To determine how the surface topography affects the ground stiffness and damping, the fractal parameters are chosen as (i) D ¼ 2.3, 2.4, 2.5, and 2.6 for G ¼ 6 � 10 7 m and (ii) G ¼ 6 � 10 6, 6 � 10 7, 6 � 10 8, and 6 � 10 9 m for D ¼ 2.4. How the ground stiffness and damping vary for different values of D and G is shown in Figs. 7 and 8, respectively. It is shown that the rougher the surface topography (lower D and higher G), the smaller the ground stiffness and damping, and vice versa. Fig. 9 shows the ground stiffness projected in the Fc Kz plane and the ground damping projected in the f Kz plane for different values of D. The area of each figure in Fig. 9(a) indicates the variation laws of ground stiffness with f, and it can be seen that f affects the ground stiffness more as the ground surface becomes rougher. In Fig. 9(b), the reduction ratio of Cz increases initially with D until D ¼ 2.5 and de creases when D ¼ 2.6.
(17)
3. Influence factors of ground stiffness and damping 3.1. Effects of excitation parameters Eqs. (16) and (17) show that the ground stiffness and damping are related to the largest truncated contact area a’L and excitation frequency ω for the determined fractal parameters (D and G). The largest truncated contact area changes with the total deformation load Fc . Thus, the ground stiffness and damping are actually determined by Fc and ω for known surface topography. To investigate how these two parameters affect the ground stiffness and damping, the initial values are given as Fc ¼ ½225; 425� kN,Fs ¼ 325 kN, ω ¼ 2πf, f ¼ ½3; 120� Hz, E ¼ 2:2 � 7
3
(15)
4. Response analysis of coupled vibrator–ground system 4.1. Dynamic modeling and verification
7
10 Pa, ν ¼ 0:25, andρ ¼ 1800 kg=m . For D ¼ 2.4 and G ¼ 6 � 10 m, a’L ranges from 0.8321 m2 to 1.77255 m2 with the given Fc . Fig. 5 shows a 3D plot of how the ground stiffness varies with both Fc and ffor D ¼ 2.4 and G ¼ 6 � 10 7 m. Because the largest truncated contact area and also the real total contact area increases with the change of Fc , the ground stiffness increases with Fc (i.e., larger real total contact area). The trends of the curves of Kz versus Fc are consistent with the results in Ref. [30], and the decrease in the gradients of these curves indicates a nonlinear characteristic. Fig. 5 also shows that the ground stiffness decreases with the excitation frequency f. There is a 10.49% reduction in ground stiffness for Fc ¼ 425 kN compared to a 8.34% reduction in ground stiffness for Fc ¼ 225 kN, thus it can be inferred that the reduction increases with Fc .
The VG system can be expressed as a lumped-parameter model with two degrees of freedom, as shown in Fig. 10. The kinetic equations of the system are considered as � m1 z€1 þ c1 ðz_1 z_2 Þ þ k1 ðz1 z2 Þ ¼ FðtÞ ; (18) m2 z€2 þ c1 ðz_2 z_1 Þ þ Cz z_2 þ k1 ðz2 z1 Þ þ Kz z2 ¼ FðtÞ where m1 is the mass of the reaction mass, m2 is the mass of the base plate, k1 and c1 are the hydraulic stiffness and damping coefficients, respectively, FðtÞ is the sweep excitation generated by the hydraulic system, and z1 and z2 are the displacements due to the dynamic force. The sweep excitation is given by
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�� � fe FðtÞ ¼ AðtÞ ⋅ A⋅sin 2π fs þ
Soil Dynamics and Earthquake Engineering 132 (2020) 106064
2T
�� � � fs t t ; 0�t�T ;
(19)
Cz ¼ b1 þ b2 Y þ b3 f þ b4 Yf þ b5 f 2 ;
where the coefficients b1 ; b2 ; b3 ; b4 ; b5 are also determined by the surface topography. The kinetic equations of the VG system can be solved in MATLAB by using the fourth/fifth-order Runga–Kutta method. To validate the dy namic model of the VG system and calibrate the system parameters, a field experiment was developed in the Bureau of Geophysical Pro specting of China. The test vibrator was a 249-kN vibrator mounted on an EV56 vehicle, as shown in Fig. 11(a). The vibrator was lifted up during transportation [Fig. 11(a)] and placed on the ground for the vibrating process [Fig. 11(b)], and the acceleration of the reaction mass was measured using an accelerometer located on the top of the reaction mass. As given in Table 2, the system parameters used for the following simulation were obtained from the field experiment. A 3D laser scanner was used to measure the profile of the test ground surface and obtain the values of the fractal parameters. The fractal dimension for the ground profile is D ¼ 2.35. The acceleration of the reaction mass calculated using Eq. (18) is shown in Fig. 12, along with the results measured using the accelerometer. The comparison shows that the theoretical calcula tion results agree well with those of the field experiment.
where A is the amplitude of the sweep excitation, fin is the starting fre quency, fen is the ending frequency, t is the sweep time, T is the sweep length, and A(t) is the cosine window function, which is written as � � ��� 8 � t > > 0:5 1 þ cos π ⋅ þ1 > > Tc > < AðtÞ ¼ 1 ; (20) > � � � ��� > > > T t > : 0:5 1 cos π ⋅ Tc where Tc is the duration of the cosine window. For ease of calculation, Kz and Cz are obtained as functions of the deformation load and frequency by curve fitting. Note that the static load is ignored because it is not recorded by the geophones, although it is involved in determining the initial contact area. Assuming that Y ¼ Fc Fs , the equation for Kz ðY; fÞ is given by (21)
Kz ¼ a1 þ a2 Y þ a3 f þ a4 Yf þ a5 f 2 ;
where the coefficients a1 ; a2 ; a3 ; a4 ; a5 are determined by the sur face topography. Table 1 gives the values of a1 ; a2 ; a3 ; a4 ; a5 and the goodness of fit R2. The fitting results are satisfied with the accuracy requirements. Meanwhile, the equation for Cz ðY; fÞ is written as
Fig. 5. Ground stiffness for D ¼ 2.4 and G ¼ 6 � 10 curves in f Kz plane.
7
(22)
4.2. Analysis of the output force As a seismic-signal excitation source, the output force of the vibrator
m: (a) three-dimensional (3D) plot of ground stiffness; (b) projection curves in Fc
6
Kz plane; (c) projection
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Fig. 6. Ground damping for D ¼ 2.4 and G ¼ 6 � 10
7
m: (a) 3D plot of ground damping; (b) projection curves in Fc
baseplate is one of the most important performance indexes. The system parameters in Table 2 are used, and the excitation signal is plotted in Fig. 13. Some material damping is added to the system to avoid a loss of contact. The output forces for ground surfaces with D ¼ 2.3, 2.4, 2.5, and 2.6 for G ¼ 6 � 10 7 m are compared in Fig. 14 and similarly for ground surfaces with G ¼ 6 � 10 6, 6 � 10 7, 6 � 10 8, and 6 � 10 9 m for D ¼
Cz plane; (c) projection curves in f
Cz plane.
2.4 in Fig. 15. Fig. 13 shows that the amplitude of the output force for different surface topographies varies with time compared with the exciting force. The excitation frequency changes with the time due to the sweep exci tation, so the output amplitude actually varies with the excitation fre quency. It is seen that there is a decrease in the system resonance
Fig. 7. Ground (a) stiffness and (b) damping for different values of fractal dimension D. 7
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Fig. 8. Ground (a) stiffness and (b) damping for different values of fractal roughness G.
Fig. 9. Profile projections of ground (a) stiffness and (b) damping.
frequency for a rougher ground surface (lower D or higher G). And the decrease of the resonance frequency causes a loss of excitation force at higher frequencies for the vibration system with rougher contact in terfaces. Moreover, due to the nonlinearity of the stiffness, the ampli tude of the output force near the resonance region becomes asymmetric. 4.3. Harmonic distortion of the output force To further investigate the influence of the nonlinear interaction be tween the vibrator and ground on the signal distortion, the output force of the vibrator for different excitation frequencies is analyzed. The fractal parameters are chosen as D ¼ 2.4 and G ¼ 6 � 10 7 m, and the exciting frequencies are 5 Hz, 15 Hz, 30 Hz, 50 Hz, and 100 Hz. Fig. 16 shows the time-history output of the vibrator as well as the amplitudefrequency curves obtained by Fast Fourier Transform (FFT). It is seen that the distortion of the harmonic signals mainly appears near the resonance region as well as the asymmetry in the output curves. With the
Fig. 10. Lumped-parameter model of VG system. Table 1 Values of the coefficients of Kz for different values of D and G. D
G [m]
a1
2.3 2.4 2.5 2.6 2.4 2.4 2.4
6e-7 6e-7 6e-7 6e-7 6e-6 6e-8 6e-9
1.315e4 2.131e5 7.859e5 2.288e6 4.307e4 5.869e5 3.380e6
a2 48.13 221.1 1035 746.4 36.1 881.9 2551
8
a3
a4
0.02488 0.5914 2.286 6.962 0.1386 1.590 9.048
0.3188 0.3652 0.1024 0.5818 0.6267 1.204 6.233
a5 0.0001735 0.0009608 0.002953 0.003064 0.0004704 0.002430 0.005487
R2 0.9893 0.9987 0.9995 0.9998 0.9975 0.9992 0.9996
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Fig. 11. Field test of vibrator: (a) test seismic vibrator; (b) accelerometer located on reaction mass.
Fig. 12. Comparison between numerical and experimental results. Table 2 Values of experimental parameters. m1 [kg]
m2 [kg]
Fs [kN]
A [kN]
fs [Hz]
fe [Hz]
T [s]
Tc [s]
5905
1923
325
99.6
3
120
20
0.5
Fig. 14. Comparison of output force for different D values: (a) output force curves; (b) amplitude envelopes.
Fig. 13. Plot of excitation force.
increase of the output force, the asymmetry becomes more obvious and the amplitudes of higher harmonics also increase. It means that the nonlinear contact characteristics can cause a nonlinear distortion in the vibrator output and therefore affect the exploration accuracy. Familiar distortion laws also can be found in the output signal of other surface topographies.
Fig. 15. Comparison of output force for different G values: (a) output force curves; (b) amplitude envelopes.
(lower frequency) while the mass of the baseplate m2 affects the secondorder natural frequency (higher frequency) [20]. Figs. 17 and 18 show the influence of m1 on the vibrator output at about 6 Hz and the influ ence of m2 on the output at about 120 Hz for different fractal parame ters. When D ¼ 2.6 or G ¼ 6 � 10 7, the amplitude of the vibrator output at higher frequencies increases, the reason is that the resonance fre quencies of these systems are beyond 120 Hz. And the vibrator can maintain a steady output and barely have a loss of excitation force over most of the frequency sweep. Thus, increasing m1 and decreasing m2 are good ways to improve the output of the VG system at low-frequency phase and high-frequency phase, respectively. Using optimized
4.4. Output improvement measures As mentioned above, the output force of the vibrator in contact with a rougher surface is insufficient in the high-frequency phase, thereby reducing the propagation depth of the seismic wave and the intensity of the reflected signal. An efficient way to improve the output of vibrator is to increase the resonance frequency, which is related to the mass of the reaction mass and the baseplate. Previous analysis shows that the mass of the reaction mass m1 mainly affects first-order natural frequency 9
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Fig. 16. Output force of the vibrator for different excitation frequency: (a) time-history response; (b) amplitude-frequency curves.
Fig. 17. Variation laws of the amplitude of the output force with m1 for different fractal parameters: (a) different D values; (b) different G values.
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Fig. 18. Variation laws of the amplitude of the output force with m2 for different fractal parameters: (a) different D values; (b) different G values.
structures or lighter materials to decrease m2 helps to improve the vibrator performance for rougher surface topographies.
frequency of the output force decrease with a rougher ground sur face. The distortion of the output signal mainly appears near the resonance region as well as the asymmetry in the amplitude. In addition, decreasing the baseplate mass is an efficient way to improve the high-frequency output of the vibrator.
5. Conclusions In this paper, a dynamic model of the VG system was developed to account for two impact factors. The main conclusions are as follows.
Author statement
1) A theoretical approach to the problem, in which both the impacts of surface topography and the dynamics of ground are considered, was developed to calculate the ground stiffness and damping between the vibrator and the ground by using a fractal contact model with the half-space method. 2) How the ground stiffness and damping vary with the excitation pa rameters and surface topography was discussed. Increasing the total deformation load or decreasing the excitation frequency can improve the ground stiffness and damping. The impacts of surface topography were illustrated by choosing different values of D and G: both the ground stiffness and damping increase with decreasing surface roughness (higher D or lower G). 3) A two-degree-of-freedom dynamic model to simulate the dynamic response of the VG system was established with nonlinear ground stiffness and damping, and the simulation results were consistent with those from a field test. Both the amplitude and resonance
All authors warrant that this article is their original work, hasn’t received prior publication and isn’t under consideration for publication elsewhere. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National High Technology Research and Development Program of China (Grant No: 2012AA061201) and the National Natural Science Foundation of China (Grant No: 50474040).
Appendix. Fractal contact model of the baseplate and ground The asperity interference δ in Fig. 2(b) is pffiffiffiffiffiffiffi ð3 DÞ δ ¼ 2 ln γ GðD 2Þ ð2r’ Þ ; and the relationship between δ and the radius R is ðR ’ 2
(A.1) 2
δÞ2 þ ðr’ Þ ¼ R2 . Since δ is typically orders of magnitude smaller than R, the relationship can be
reduced to R ¼ ðr Þ =ð2δÞ. Substituting Eq. (A.1) into this simplified relationship gives
R¼
a’ð0:5D
π
2ð5 DÞ ð0:5D 0:5Þ
0:5Þ
pffiffiffiffiffiffiffi ðD ln γG
2Þ
(A.2)
;
where a’ ½a’ ¼ π r’2 � is the truncated microcontact area. The deformation of an asperity is given as pffiffiffiffiffiffiffi 2ð4 DÞ ln γ δ ¼ ð1:5 0:5DÞ GðD 2Þ a’ð1:5 0:5DÞ :
π
(A.3)
From the Hertz contact theory, the elastic contact force of a microcontact is given by F¼
4E� r3 ; 3R
(A.4)
11
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
whereE� ¼ E=ð1 ν2 Þis the composite elastic modulus, and E; ν is the elastic modulus and Poisson’s ration of soil, respectively. Hence, for the simple case of elastic and fully plastic asperity deformations, the contact load and contact area of an asperity are given by , pffiffiffiffiffiffiffi 25:5 D ln γ � ðD 2Þ ’ð2 0:5DÞ ’ 2; (A.5) E G a ; a ¼ a fe ¼ e 3π 2 0:5D and (A.6)
fc ¼ Kys a’ ; ap ¼ a’ ;
where the subscripts e and p represent elastic and plastic deformation, respectively. Here, ys is the yield strength of the soil and K is a factor. In addition, the critical truncated microcontact area a’c that distinguishes the elastic load from the plastic load is [31]. � ð11 2DÞ � � � �1=ðD 2Þ 2 E ð2D 4Þ a’c ¼ ln γ : (A.7) G 9π4 D Kys According to Mandelbrot, the distribution of the truncated microcontact area a’ can be written as [32,33]. nða’ Þ ¼ 0:5ðD
1Þa’0:5D L
0:5 ’ 0:5 0:5D
a
(A.8)
;
where a’L is the largest truncated microcontact area. While the contact interface apparently behaves elastically at larger displacements, the initial contact displacements are plastic in nature. Thus, the total contact load can be obtained by Z a’L Z a’c Fc ¼ F e þ Fp ¼ fe ða’ Þnða’ Þda’ þ fp ða’ Þnða’ Þda’ ; (A.9) a’c
0
which yields � �pffiffiffiffiffiffiffi ! 8 25:5 D D 1 ln γ D 1 > � D 2 ’ 0:5D 0:5 ’2:5 D ’2:5 D > � Kys a’0:5D E a þ G a a > L L c L < 3π 2 0:5D 5 2D 3 D Fc ¼ > � � > > pffiffiffiffiffiffiffiffiffiffiffiffi�a’ �0:75 a’L : ’0:25 ln þ 3Kys a’0:75 a D ¼ 2:5 2E� G ln γ L L c π ac
References
! 0:5 ’1:5 0:5D ac
D 6¼ 2:5 :
(A.10)
[17] Lebedev AV, Beresnev IA. Nonlinear distortion of signals radiated by vibroseis sources. Geophys 2004;69:968–77. [18] Lebedev AV, Beresnev IA, Vermeer PL. Model parameters of the nonlinear stiffness of the vibrator-ground contact determined by inversion of vibrator accelerometer data. Geophys 2006;71:25. [19] Noorlandt R, Drijkoningen G. On the mechanical vibrator-earth contact geometry and its dynamics. Geophys 2016;81:37–45. [20] Huang Z, Peng X, Li G, Hao L. Response of a two-degree-of-freedom vibration system with rough contact interfaces. Shock Vib 2019;2019:1–13. [21] Miller GF, Pursey H. The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid. Proc R Soc Lond 1954;223: 521–41. [22] Baeten GJM, Strijbos FPL. Wave field of a vibrator on a layered half-space: theory and practice. SEG Tech Progr Expand Abstr 1949;7:1359. [23] Liu J, Huang ZQ, Li G. Dynamic characteristics analysis of a seismic vibratorground coupling system. Shock Vib 2017;2017:1–12. [24] Armstrong AC. On the fractal dimensions of some transient soil properties. Eur J Soil Sci 2010;37:641–52. [25] Perfect E, Kay BD. Application of fractals in soil and tillage research: a review. Soil Res 1995;36:1–20. [26] Huang C, Bradford JM. Applications of a laser scanner to quantify soil microtopography. Soil Sci Soc Am J 1992;56:14–21. [27] Berry MV, Lewis ZV. On the Weierstrass–Mandelbrot function. Proc R Soc Lond A 1980;370:459–84. [28] Yan W, Komvopoulos K. Contact analysis of elastic-plastic fractal surfaces. J Appl Phys 1998;84:3617–24. [29] Sung T. Vibration in semi-infinite solids due to periodic surface loading. In: Symposium on dynamic testing of soils. Atlantic. ASTM International; 1954. p. 35–68. [30] Liu P, Zhao H, Huang K, Chen Q. Research on normal contact stiffness of rough surface considering friction based on fractal theory. Appl Surf Sci 2015;349:43–8. [31] Mandelbrot BB. How long is the coast of Britain? Science 1967;156:636–8. [32] Mandelbrot BB. The fractal geometry of nature. New York: W.H. Freeman; 1982. [33] Bycroft GN. Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum. Phil Trans Roy Soc Lond 1956;248:327–68.
[1] Beresnev IA. Ground-force- or plate-displacement-based vibrator control? J Sound Vib 2012;331:1715–21. [2] Wei Z, Phillips TF, Hall MA. Fundamental discussions on seismic vibrators. Geophysics 2010;75:13–25. [3] Jeffryes BP. Far-field harmonic measurement for seismic vibrators. 66th Annual International Meeting, SEG, Expanded Abstracts, 60–63. [4] Lebedev A, Beresnev I. Radiation from flexural vibrations of the baseplate and their effect on the accuracy of travel time measurements. Geophys Prospect 2005;53: 543–55. [5] Beresnev I, Nikolaev A. Experimental investigations of nonlinear seismic effects. Phys Earth Planet In 2000;50(1):83–7. [6] Dimitriu PP. Preliminary results of vibrator-aided experiments in non-linear seismology conducted at Uetze, F. R. G. Phys Earth Planet In 1990;63:172–80. [7] Veen MVD, Brouwer J, Helbig K. Weighted sum method for calculating ground force: an evaluation by using a portable vibrator system. Geophys Prospect 2010; 47:251–67. [8] Wei Z. Design of a P-wave seismic vibrator with advanced performance. GeoArabia 2008;13:123–36. [9] Saragiotis C, Scholtz P, Bagaini C. On the accuracy of the ground force estimated in vibroseis acquisition. Geophys Prospect 2010;58:69–80. [10] Poletto F, Schlifer A, Zgauc F, et al. Borehole signals obtained using surface seismic and ground-force sensors. 81st Annual International Meeting, SEG, Expanded abstracts, 4298–4303. [11] Wei Z. Modeling and model analysis of seismic vibrator baseplate. Geophys Prospect 2010;58:19–31. [12] Wei Z. Pushing the vibrator ground-force envelope towards low frequencies. Geophys Prospect 2009;57:19–32. [13] Jiang JJ, Xiang Y, Jing Z, Zhang RC. Improving vibrator structure to eliminate vibration noise. Int J Adv Manuf Technol 2018;96:1741–7. [14] Castanet A, Lavergne M. Vibrator controlling system. U.S. Patent 1965:3208550. [15] Sallas J, Weber R. Comments on “The amplitude and phase response of a seismic vibrator” by Lerwill WE. Geophys Prospect 2010;30(6):935–8. [16] Sallas JJ. Seismic vibrator control and the downgoing P-wave. In: SEG Technical Program, Expanded Abstracts. 1; 1982. p. 732–40.
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Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: http://www.elsevier.com/locate/soildyn
Modeling of nonlinear interaction and its effects on the dynamics of a vibrator-ground system Xun Peng a, Zhiqiang Huang a, *, Lei Hao b a b
Electromechanical Engineering College, Southwest Petroleum University, Chengdu, Sichuan, 610500, PR China Bureau of Geophysical Prospecting, China National Petroleum Corporation, Zhuozhou, Hebei, 072750, PR China
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibrator–ground system Nonlinear interaction Coupled vibration Field test
The nonlinear interaction between a vibrator and the ground is one of the main sources of signal distortion in land exploration. So, it is crucial to describe this nonlinear interaction and study how it affects the outgoing wave radiated by the vibrator. For this purpose, an approach is developed to obtain the equivalent ground stiffness and damping using a fractal contact model with the half-space method. Both the impacts of surface topography and the dynamics of ground are accounted for innovatively in this approach. The results show that both the ground stiffness and damping increase with the total deformation load but decrease with the excitation frequency and roughness. In addition, a modified two-degree-of-freedom vibration model of the vibrator–ground system is proposed to calculate the output force of the vibrator for different rough surface topographies. A field test is used to verify the theoretical model, and it is concluded that compared with the theoretical excitation signal, the output force varies with the excitation frequency. The amplitude of the output force in the high-frequency phase decreases for a rougher contact surface, as does the resonance frequency of the system. Moreover, reducing the mass of the baseplate is an efficient way to improve the high-frequency output.
1. Introduction Seismic vibrator is the main source of seismic energy in land explo ration, and its wide application has resulted in great improvements in oil and gas exploration [1,2]. The idea of the vibroseis method is to radiate a theoretically prescribed frequency-modulated (sweep) signal—which provides the best identification of the returning seismic reflections—into the earth. Therefore, the accuracy and frequency bandwidth of the radiated signal are the main restrictions on the development of seismic vibrators [3,4]. Nevertheless, the true source signal (the signal that is realistically radiated into the earth) is never identical to the prescribed theoretical shape because of nonlinear and unpredictable distortions of the original idealized sweep that greatly reduce the imaging accuracy of complex layers and increase exploration costs. Meanwhile, the insuffi cient output of seismic vibrators in the high-frequency phase has nega tive impacts on the expansion to higher frequency. Previous studies have indicated that one of the main sources of these problems is the nonlinear interaction between a vibrator and the ground [5,6]. Various research efforts have been made to study the nonlinear interaction between a vibrator and the ground and assess how it affects the source signal. The main research methods are (i) experimental
analysis, (ii) finite-element simulation, and (iii) theoretical modeling. Regarding experiments, many field tests have been devoted to the har monic distortion of outgoing waves and have discussed the impacts of the nonlinear response of an elastic half space and some parts of the radiating system [3,7–10]. In addition, finite-element analysis has evolved as an efficient tool. Wei et al. [11,12] studied the vibration characteristics of a vibrator by developing a finite-element model of the vibrator–ground (VG) system. Jiang et al. [13] used finite-element analysis to optimize the hydraulic oil ducts and reduce the vibration noise. The key to theoretical modeling is to establish a reasonable dy namic description of the coupled VG system. Using what is known as the “weighted-sum” method, Castanet et al. [14–16] were the first to develop a practical linear model with which to calculate the force applied to the ground surface by a seismic vibrator. However, because of the nonlinearity in the complex VG system, many practical issues arising from field tests have remained unexplained by this model. The fact that the vibrator is placed on a rough surface is at odds with the assumed uniform pressure distribution beneath the baseplate in the weighted-sum method. To describe the nonlinear contact, Lebedev and colleagues [17,18] proposed a model in which the VG contact acts as a nonlinear spring. Noorlandt and Drijkoningen [19] and Huang et al. [20] extended this model to investigate the effects of ground surface
* Corresponding author. E-mail address: [email protected] (Z. Huang). https://doi.org/10.1016/j.soildyn.2020.106064 Received 20 September 2019; Received in revised form 16 December 2019; Accepted 21 January 2020 0267-7261/© 2020 Elsevier Ltd. All rights reserved.
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Notation A AðtÞ a’ ae ap a’c a’L a0 Cz c1 D E E* F Fc Fe Fp Fs FðtÞ fe fp f1 f2 fin fen f G G0 J0 K Kz k1 L
M mb m1 m2 n Peiωt Δpeiωt Qeiωt R Reiωt r’ r’c r’L T Tc t u w wd Y ys z1 z2 γ δ λ
amplitude of sweep excitation The cosine window function is A(t) truncated contact area real elastic microcontact area real plastic microcontact area critical truncated contact area largest truncated contact area dimensional frequency equivalent ground damping hydraulic damping coefficient fractal dimension elastic modulus of soil composite elastic modulus elastic contact force of an asperity total deformation load total elastic contact load total plastic contact load static load sweep excitation elastic contact load of an asperity plastic contact load of an asperity functions defined in Eq. (10) functions defined in Eq. (11) starting frequency ending frequency excitation frequency fractal roughness shear modulus zero-order Bessel function factor of yield strength equivalent ground stiffness hydraulic stiffness coefficient sample length
ν ρ σ τ ε ωθ ω
with differing roughness. Meanwhile, part of the ground is captured by the vibrator’s baseplate during the vibration process, thereby potentially changing the stiffness and damping of the vibration system. Some re searchers have used the elastic half-space method to establish the dy namic stiffness and damping of the VG system and have studied how the system parameters affect the dynamic response [21–23]. Of the three aforementioned methods, the most essential is theoret ical modeling, through which the vibrator ground force can be estimated in the vibration process. However, it is noticed that previous theoretical models have considered only single aspects of the nonlinear interaction and have failed to combine multiple factors, while in reality the nonlinear contact springs and the dynamics of the ground work together to affect the dynamic response of the VG system. To date, a few refer ences have presented a three-dimensional (3D) lumped-parameter vibrator system comprising nonlinear springs and captured ground mass in which it can consider both the impacts of contact and the dy namics of the ground, but they did not explain how to determine the mass and spring parameters [12]. The present study is aimed at describing the nonlinear interaction of a vibrator in contact with rough half-space ground by using a fractal contact model with the half-space method. The roughness and the dy namics of the ground are both accounted for, and how the excitation parameters and surface topography affect the ground stiffness and damping is discussed. A modified two-degree-of-freedom model with nonlinear stiffness and damping is developed to study the output of the VG system, and a field test is carried out to validate the theoretical
number of superposed ridges mass of baseplate pad mass of reaction mass mass of baseplate frequency index total dynamic load dynamic load on an asperity external load on the baseplate asperity radius reaction force radius of truncated microcontact area radius of critical truncated contact area radius of largest truncated contact area sweep length duration of the cosine window sweep time radial displacement vertical displacement vertical displacement of the center point Y ¼ Fc Fs yield strength displacement of reaction mass displacement of baseplate scaling parameter asperity interference Lam� e constant Poisson’s ratio of soil mass density of soil normal stress shear stress volumetric strain rotating component circular frequency
model. The present research results will help engineers to optimize the vibrator structure and system parameters to adapt to different surface topographies and improve exploration efficiency. The proposed approach could also be used for other vibration systems that correspond to a half-space model with a rough contact interface. 2. Model formulation and solution method 2.1. Problem description and solution approach Vibrator is the key component of seismic vibrators, which generates ground force through coupled vibration with ground. As shown in Fig. 1, the main components of the vibrator are the reaction mass and the baseplate (the latter comprising a top plate, supporting columns, piston, and baseplate pad). When the vibrator is operating, there are two loading phases, namely the static loading phase and the dynamic loading phase. In the static loading phase, the weight of vehicle (hold-on load) is applied to the vibrator baseplate to establish contact. In the dynamic loading phase, a time-varying excitation load is added to the afore mentioned static load and transferred into the ground through the baseplate. The output force is translated into a seismic wave at the contact between the baseplate and the ground, and the coupling ground is modeled as a half space with a rough surface. However, the actual VG contact area, which affects the surface displacement of the half-space model, changes with the excitation load. There is therefore an interac tion between the surface topography and the mechanical response of the 2
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
Fig. 1. Schematic of vibrator–ground (VG) system.
half space. The solution approach is to decompose the problem into two subproblems: the contact problem and the half-space problem. The con tact problem is to determine the real contact area of the contact interface during the vibration process. Given the real contact area, the half-space problem is then to solve the surface displacement due to the vibration of ground generated by excitation load. Solving these two sub-problems gives the equivalent ground stiffness and damping.
where L is the sample length, D is the fractal dimension, G is the fractal roughness, M is the number of superposed ridges used to construct the fractal surfaces, n is a frequency index with nmax ¼ int½logðL =Ls Þ =log γ�, φm;n is a random phase in the range ½0; 2π�, and γðγ > 1Þ is a scaling parameter. Considering the surface flatness and the frequency distri bution density suggests that γ ¼ 1:5 is typical for most surfaces. The surface topography of ground is determined by the values of D and G. The D values determine the contribution of high and low fre quency components in the surface function Eq. (1). The physical sig nificance of D is the extent of space occupied by the rough surface, i.e., larger D values correspond to denser profiles (smoother topography). Physically, G is the height scaling parameter and higher G values correspond to rougher surface. The microcontact between the baseplate pad and ground is shown in Fig. 2(a), and it can be reduced to contact between an asperity of the ground surface and the opposing rigid plane, as shown in Fig. 2(b). With the change of the total deformation load, the number of the asperities which are in contact with the baseplate pad is various. Thus, the real total contact area between the baseplate pad and the ground is related to the total deformation load. In Appendix A, details on the derivation process of the relationship between the total deformation load and the largest truncated microcontact area, are given. The rela tionship is written as
2.2. Governing equations for contact problem In order to construct the rough surface of ground, the fractal geom etry is used to describe the surface topography of ground [24–26]. Fractal geometry can describe geometric features of various length scales and provide a means of characterizing asperities of a large range of sizes. The 3D ground-surface topography can be simulated deter ministically using the modified two-variable Weierstrass–Mandelbrot fractal function, which is written as [27]. zðx; yÞ ¼ L 8 > > < : cosφm;n > > :
� �ðD G L
� M X nmax ln γ X γðD M m¼1 n¼0
2Þ �
3Þn
2 n
2
2
62πγ x þ y cos6 4 L
�12
h ⋅cos tan
1
�y� x
39 > > 7= þ φm;n 7 ; 5 > M > ;
π mi
(1)
Fig. 2. Description of the contact model: (a) microcontact between the baseplate pad and ground; (b) microcontact between the pad and an asperity.
3
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� �pffiffiffiffiffiffiffi ! 8 25:5 D D 1 ln γ D 1 > � D 2 ’ 0:5D 0:5 ’2:5 D ’2:5 D > � a þ Kys a’0:5D G a a E > L L c L < 3 D 3π 2 0:5D 5 2D Fc ¼ > � � > > : � pffiffiffiffiffiffiffiffiffiffiffiffi�a’L �0:75 a’L ’0:25 2E G ln γ ln þ 3Kys a’0:75 a D ¼ 2:5 L c π ac
For the VG system, the total deformation load is determined by the static load Fs and the excitation load in the vibration processes. The static load Fs can generate the initial contact area between the baseplate pad and ground, and the added dynamic load brings about the change of the contact area. Thus, when the static load and amplitude of the exci tation load is given, the range of the total deformation load can be calculated and a’L at the contact interface is obtained from Eq. (2).
�
Previous research considered the initial loading phase to correspond to plastic deformation, after which elastic behavior becomes dominant [11,19,28]. The microcontact plastic deformation arises mainly as a static displacement offset that has little influence on the ground vibra tion, and the methods provided by the theory of elasticity are applicable. For axisymmetric problems as shown in Fig. 3, the wave equations are given by � � 8 2 > > ρ ∂ u ¼ λ þ 2G0 ∂ε þ 2G0 ∂ωθ > < ∂t2 ∂r ∂z ; (3) � � 2 > > ∂w ∂ε 1 ∂ rωθ Þ > : ρ 2 ¼ λ þ 2G0 2G0 ∂z r ∂r ∂t
pκ2 α σp ; G0 φðpÞ
(7)
function,κ2 ¼ ω2 ρ=G0 α ¼ p2 h2 , and h2 ¼ ω2 ρ=ðλ þ 2G0 Þ. Substitut ing Eq. (6) into Eq. (7) yields Z ∞ pακ2 p wðra ; 0Þ ¼ σ J0 ðprÞdp: (8) G0 φðpÞ 0 When considering the solution of Eq. (8), previous studies approxi mated the surface displacement with the displacement of the center point (r ¼ 0) to obtain a vertical displacement in the form [29]. wd ðtÞ ¼
ΔPeiωt ðf1 þ if2 Þ; G0 r
(9)
pffiffiffiffiffiffiffiffiffiffi where a0 ¼ ωr ρ=G0 is the dimensionless frequency and f1 , f2 are functions of a0 given by � f1 ¼ 0:238733 0:59683a20 þ 0:004163a40 ⋯ ; (10)
The microelement volumetric strain ε and rotating component ωθ are given by (4) (5)
⋯:
(11)
f2 � G0 rwd ðtÞ: f 21 þ f 22 ω
(12)
0:017757a30 þ 0:000808a50
f2 ¼ 0:148594a0
�
∂w : ∂r
(6)
where σ p is related to the boundary conditions (namely σp ¼ R∞ where J0 ðpra Þ is the zero-order Bessel 0 σ ðra Þra J0 ðpra Þdra ],
ρ is the soil density.
� 1 ∂u ωθ ¼ 2 ∂z
(2)
σz ðra ; 0; tÞ ¼ σðra Þeiωt ; τrz ðra ; 0; tÞ ¼ 0
wðp; 0Þ ¼
where u; w are the radial and vertical displacements, respectively, G0 is 2G0 ν E � the shear modulus [G0 ¼ 2ð1þ ν2 Þ], λ is the Lame constant [λ ¼ 1 2ν], and
∂u u ∂w þ þ ; ∂r r ∂z
D 6¼ 2:5
8 iωt > < ΔPe r < r a 2 where σðra Þ ¼ . πr > : 0 ra > r The Hankel transform of the vertical surface displacement can be derived as
2.3. Governing equations for half-space problem
ε¼
! 0:5 ’1:5 0:5D ac
Hence, we have
Assuming that the dynamic force acting on an asperity is ΔPeiωt , the boundary conditions are written as
ΔPeiωt ¼
f1 G0 rwd ðtÞ f 21 þ f 22
iω
For the entire contact interface between the baseplate pad and the ground, the total load can be obtained as Z Peiωt ¼ wd ðtÞ Z iωwd ðtÞ
r’L
r’c r’L r’c
f1 G0 r2πr’ nðr’ Þdr’ f 21 þ f 22 f2 � G0 r2πr’ nðr’ Þdr’ ; f 21 þ f 22 ω
(13)
where Peiωt is the total dynamic load, the radius of the critical truncated contact arear’c and the radius of the largest truncated contact area r’L are be determined by contact analysis, and nðr’ Þ is obtained from Eq. (A.8). 2.4. Formulas for ground stiffness and damping Assuming that the external force loaded on the baseplate pad is Qeiωt , as shown in Fig. 4, then the kinetic equation of the baseplate pad is € þ Reiωt ¼ Qeiωt ; mb wðtÞ
Fig. 3. Schematic of axisymmetric problem. 4
(14)
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
Fig. 4. Force diagram for baseplate pad.
where mb is the mass of the baseplate pad and Reiωt ¼ action force. Substituting Eq. (13) into Eq. (14) yields
� � mb w€d t þ wd t
Z
r’L r’c
� � � f1 G0 r 2πr’ n r’ dr’ þ iωwd t f 21 þ f 22
Z
Fig. 6 shows how the ground damping varies with Fc and f for D ¼ 2.4 and G ¼ 6 � 10 7 m. As can be seen, the ground damping also increases with Fc and decreases with f. The projection curves in the Fc Cz plane show that the slope decreases when f is less than 60 Hz but increases
Peiωt is the re
r’L
r’c
� � f2 � G0 r 2πr’ n r’ dr’ ¼ Qeiωt : f 21 þ f 22 ω
otherwise. In Fig. 6(c), the f Cz curves decrease slowly and then drop significantly. With the increase of Fc , the reduction ratio of Cz increase.
Thus, the equivalent ground stiffness and damping can be respec tively written as pffiffi � Z r’ � L 2 f1 ’ðD 1Þ Kz ¼ (16) ðD 1ÞG0 rL r’ð1 DÞ dr’ ; 2 f 21 þ f 22 r’c Cz ¼
pffiffi 2 ðD 2
Z ’ðD 1Þ
1ÞG0 rL
r’L �
r’c
f 21
� f2 � r’ð1 2 þ f2 ω
DÞ
dr’ :
3.2. Effects of surface topography To determine how the surface topography affects the ground stiffness and damping, the fractal parameters are chosen as (i) D ¼ 2.3, 2.4, 2.5, and 2.6 for G ¼ 6 � 10 7 m and (ii) G ¼ 6 � 10 6, 6 � 10 7, 6 � 10 8, and 6 � 10 9 m for D ¼ 2.4. How the ground stiffness and damping vary for different values of D and G is shown in Figs. 7 and 8, respectively. It is shown that the rougher the surface topography (lower D and higher G), the smaller the ground stiffness and damping, and vice versa. Fig. 9 shows the ground stiffness projected in the Fc Kz plane and the ground damping projected in the f Kz plane for different values of D. The area of each figure in Fig. 9(a) indicates the variation laws of ground stiffness with f, and it can be seen that f affects the ground stiffness more as the ground surface becomes rougher. In Fig. 9(b), the reduction ratio of Cz increases initially with D until D ¼ 2.5 and de creases when D ¼ 2.6.
(17)
3. Influence factors of ground stiffness and damping 3.1. Effects of excitation parameters Eqs. (16) and (17) show that the ground stiffness and damping are related to the largest truncated contact area a’L and excitation frequency ω for the determined fractal parameters (D and G). The largest truncated contact area changes with the total deformation load Fc . Thus, the ground stiffness and damping are actually determined by Fc and ω for known surface topography. To investigate how these two parameters affect the ground stiffness and damping, the initial values are given as Fc ¼ ½225; 425� kN,Fs ¼ 325 kN, ω ¼ 2πf, f ¼ ½3; 120� Hz, E ¼ 2:2 � 7
3
(15)
4. Response analysis of coupled vibrator–ground system 4.1. Dynamic modeling and verification
7
10 Pa, ν ¼ 0:25, andρ ¼ 1800 kg=m . For D ¼ 2.4 and G ¼ 6 � 10 m, a’L ranges from 0.8321 m2 to 1.77255 m2 with the given Fc . Fig. 5 shows a 3D plot of how the ground stiffness varies with both Fc and ffor D ¼ 2.4 and G ¼ 6 � 10 7 m. Because the largest truncated contact area and also the real total contact area increases with the change of Fc , the ground stiffness increases with Fc (i.e., larger real total contact area). The trends of the curves of Kz versus Fc are consistent with the results in Ref. [30], and the decrease in the gradients of these curves indicates a nonlinear characteristic. Fig. 5 also shows that the ground stiffness decreases with the excitation frequency f. There is a 10.49% reduction in ground stiffness for Fc ¼ 425 kN compared to a 8.34% reduction in ground stiffness for Fc ¼ 225 kN, thus it can be inferred that the reduction increases with Fc .
The VG system can be expressed as a lumped-parameter model with two degrees of freedom, as shown in Fig. 10. The kinetic equations of the system are considered as � m1 z€1 þ c1 ðz_1 z_2 Þ þ k1 ðz1 z2 Þ ¼ FðtÞ ; (18) m2 z€2 þ c1 ðz_2 z_1 Þ þ Cz z_2 þ k1 ðz2 z1 Þ þ Kz z2 ¼ FðtÞ where m1 is the mass of the reaction mass, m2 is the mass of the base plate, k1 and c1 are the hydraulic stiffness and damping coefficients, respectively, FðtÞ is the sweep excitation generated by the hydraulic system, and z1 and z2 are the displacements due to the dynamic force. The sweep excitation is given by
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X. Peng et al.
�� � fe FðtÞ ¼ AðtÞ ⋅ A⋅sin 2π fs þ
Soil Dynamics and Earthquake Engineering 132 (2020) 106064
2T
�� � � fs t t ; 0�t�T ;
(19)
Cz ¼ b1 þ b2 Y þ b3 f þ b4 Yf þ b5 f 2 ;
where the coefficients b1 ; b2 ; b3 ; b4 ; b5 are also determined by the surface topography. The kinetic equations of the VG system can be solved in MATLAB by using the fourth/fifth-order Runga–Kutta method. To validate the dy namic model of the VG system and calibrate the system parameters, a field experiment was developed in the Bureau of Geophysical Pro specting of China. The test vibrator was a 249-kN vibrator mounted on an EV56 vehicle, as shown in Fig. 11(a). The vibrator was lifted up during transportation [Fig. 11(a)] and placed on the ground for the vibrating process [Fig. 11(b)], and the acceleration of the reaction mass was measured using an accelerometer located on the top of the reaction mass. As given in Table 2, the system parameters used for the following simulation were obtained from the field experiment. A 3D laser scanner was used to measure the profile of the test ground surface and obtain the values of the fractal parameters. The fractal dimension for the ground profile is D ¼ 2.35. The acceleration of the reaction mass calculated using Eq. (18) is shown in Fig. 12, along with the results measured using the accelerometer. The comparison shows that the theoretical calcula tion results agree well with those of the field experiment.
where A is the amplitude of the sweep excitation, fin is the starting fre quency, fen is the ending frequency, t is the sweep time, T is the sweep length, and A(t) is the cosine window function, which is written as � � ��� 8 � t > > 0:5 1 þ cos π ⋅ þ1 > > Tc > < AðtÞ ¼ 1 ; (20) > � � � ��� > > > T t > : 0:5 1 cos π ⋅ Tc where Tc is the duration of the cosine window. For ease of calculation, Kz and Cz are obtained as functions of the deformation load and frequency by curve fitting. Note that the static load is ignored because it is not recorded by the geophones, although it is involved in determining the initial contact area. Assuming that Y ¼ Fc Fs , the equation for Kz ðY; fÞ is given by (21)
Kz ¼ a1 þ a2 Y þ a3 f þ a4 Yf þ a5 f 2 ;
where the coefficients a1 ; a2 ; a3 ; a4 ; a5 are determined by the sur face topography. Table 1 gives the values of a1 ; a2 ; a3 ; a4 ; a5 and the goodness of fit R2. The fitting results are satisfied with the accuracy requirements. Meanwhile, the equation for Cz ðY; fÞ is written as
Fig. 5. Ground stiffness for D ¼ 2.4 and G ¼ 6 � 10 curves in f Kz plane.
7
(22)
4.2. Analysis of the output force As a seismic-signal excitation source, the output force of the vibrator
m: (a) three-dimensional (3D) plot of ground stiffness; (b) projection curves in Fc
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Kz plane; (c) projection
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
Fig. 6. Ground damping for D ¼ 2.4 and G ¼ 6 � 10
7
m: (a) 3D plot of ground damping; (b) projection curves in Fc
baseplate is one of the most important performance indexes. The system parameters in Table 2 are used, and the excitation signal is plotted in Fig. 13. Some material damping is added to the system to avoid a loss of contact. The output forces for ground surfaces with D ¼ 2.3, 2.4, 2.5, and 2.6 for G ¼ 6 � 10 7 m are compared in Fig. 14 and similarly for ground surfaces with G ¼ 6 � 10 6, 6 � 10 7, 6 � 10 8, and 6 � 10 9 m for D ¼
Cz plane; (c) projection curves in f
Cz plane.
2.4 in Fig. 15. Fig. 13 shows that the amplitude of the output force for different surface topographies varies with time compared with the exciting force. The excitation frequency changes with the time due to the sweep exci tation, so the output amplitude actually varies with the excitation fre quency. It is seen that there is a decrease in the system resonance
Fig. 7. Ground (a) stiffness and (b) damping for different values of fractal dimension D. 7
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Fig. 8. Ground (a) stiffness and (b) damping for different values of fractal roughness G.
Fig. 9. Profile projections of ground (a) stiffness and (b) damping.
frequency for a rougher ground surface (lower D or higher G). And the decrease of the resonance frequency causes a loss of excitation force at higher frequencies for the vibration system with rougher contact in terfaces. Moreover, due to the nonlinearity of the stiffness, the ampli tude of the output force near the resonance region becomes asymmetric. 4.3. Harmonic distortion of the output force To further investigate the influence of the nonlinear interaction be tween the vibrator and ground on the signal distortion, the output force of the vibrator for different excitation frequencies is analyzed. The fractal parameters are chosen as D ¼ 2.4 and G ¼ 6 � 10 7 m, and the exciting frequencies are 5 Hz, 15 Hz, 30 Hz, 50 Hz, and 100 Hz. Fig. 16 shows the time-history output of the vibrator as well as the amplitudefrequency curves obtained by Fast Fourier Transform (FFT). It is seen that the distortion of the harmonic signals mainly appears near the resonance region as well as the asymmetry in the output curves. With the
Fig. 10. Lumped-parameter model of VG system. Table 1 Values of the coefficients of Kz for different values of D and G. D
G [m]
a1
2.3 2.4 2.5 2.6 2.4 2.4 2.4
6e-7 6e-7 6e-7 6e-7 6e-6 6e-8 6e-9
1.315e4 2.131e5 7.859e5 2.288e6 4.307e4 5.869e5 3.380e6
a2 48.13 221.1 1035 746.4 36.1 881.9 2551
8
a3
a4
0.02488 0.5914 2.286 6.962 0.1386 1.590 9.048
0.3188 0.3652 0.1024 0.5818 0.6267 1.204 6.233
a5 0.0001735 0.0009608 0.002953 0.003064 0.0004704 0.002430 0.005487
R2 0.9893 0.9987 0.9995 0.9998 0.9975 0.9992 0.9996
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Fig. 11. Field test of vibrator: (a) test seismic vibrator; (b) accelerometer located on reaction mass.
Fig. 12. Comparison between numerical and experimental results. Table 2 Values of experimental parameters. m1 [kg]
m2 [kg]
Fs [kN]
A [kN]
fs [Hz]
fe [Hz]
T [s]
Tc [s]
5905
1923
325
99.6
3
120
20
0.5
Fig. 14. Comparison of output force for different D values: (a) output force curves; (b) amplitude envelopes.
Fig. 13. Plot of excitation force.
increase of the output force, the asymmetry becomes more obvious and the amplitudes of higher harmonics also increase. It means that the nonlinear contact characteristics can cause a nonlinear distortion in the vibrator output and therefore affect the exploration accuracy. Familiar distortion laws also can be found in the output signal of other surface topographies.
Fig. 15. Comparison of output force for different G values: (a) output force curves; (b) amplitude envelopes.
(lower frequency) while the mass of the baseplate m2 affects the secondorder natural frequency (higher frequency) [20]. Figs. 17 and 18 show the influence of m1 on the vibrator output at about 6 Hz and the influ ence of m2 on the output at about 120 Hz for different fractal parame ters. When D ¼ 2.6 or G ¼ 6 � 10 7, the amplitude of the vibrator output at higher frequencies increases, the reason is that the resonance fre quencies of these systems are beyond 120 Hz. And the vibrator can maintain a steady output and barely have a loss of excitation force over most of the frequency sweep. Thus, increasing m1 and decreasing m2 are good ways to improve the output of the VG system at low-frequency phase and high-frequency phase, respectively. Using optimized
4.4. Output improvement measures As mentioned above, the output force of the vibrator in contact with a rougher surface is insufficient in the high-frequency phase, thereby reducing the propagation depth of the seismic wave and the intensity of the reflected signal. An efficient way to improve the output of vibrator is to increase the resonance frequency, which is related to the mass of the reaction mass and the baseplate. Previous analysis shows that the mass of the reaction mass m1 mainly affects first-order natural frequency 9
Soil Dynamics and Earthquake Engineering 132 (2020) 106064
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Fig. 16. Output force of the vibrator for different excitation frequency: (a) time-history response; (b) amplitude-frequency curves.
Fig. 17. Variation laws of the amplitude of the output force with m1 for different fractal parameters: (a) different D values; (b) different G values.
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
Fig. 18. Variation laws of the amplitude of the output force with m2 for different fractal parameters: (a) different D values; (b) different G values.
structures or lighter materials to decrease m2 helps to improve the vibrator performance for rougher surface topographies.
frequency of the output force decrease with a rougher ground sur face. The distortion of the output signal mainly appears near the resonance region as well as the asymmetry in the amplitude. In addition, decreasing the baseplate mass is an efficient way to improve the high-frequency output of the vibrator.
5. Conclusions In this paper, a dynamic model of the VG system was developed to account for two impact factors. The main conclusions are as follows.
Author statement
1) A theoretical approach to the problem, in which both the impacts of surface topography and the dynamics of ground are considered, was developed to calculate the ground stiffness and damping between the vibrator and the ground by using a fractal contact model with the half-space method. 2) How the ground stiffness and damping vary with the excitation pa rameters and surface topography was discussed. Increasing the total deformation load or decreasing the excitation frequency can improve the ground stiffness and damping. The impacts of surface topography were illustrated by choosing different values of D and G: both the ground stiffness and damping increase with decreasing surface roughness (higher D or lower G). 3) A two-degree-of-freedom dynamic model to simulate the dynamic response of the VG system was established with nonlinear ground stiffness and damping, and the simulation results were consistent with those from a field test. Both the amplitude and resonance
All authors warrant that this article is their original work, hasn’t received prior publication and isn’t under consideration for publication elsewhere. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National High Technology Research and Development Program of China (Grant No: 2012AA061201) and the National Natural Science Foundation of China (Grant No: 50474040).
Appendix. Fractal contact model of the baseplate and ground The asperity interference δ in Fig. 2(b) is pffiffiffiffiffiffiffi ð3 DÞ δ ¼ 2 ln γ GðD 2Þ ð2r’ Þ ; and the relationship between δ and the radius R is ðR ’ 2
(A.1) 2
δÞ2 þ ðr’ Þ ¼ R2 . Since δ is typically orders of magnitude smaller than R, the relationship can be
reduced to R ¼ ðr Þ =ð2δÞ. Substituting Eq. (A.1) into this simplified relationship gives
R¼
a’ð0:5D
π
2ð5 DÞ ð0:5D 0:5Þ
0:5Þ
pffiffiffiffiffiffiffi ðD ln γG
2Þ
(A.2)
;
where a’ ½a’ ¼ π r’2 � is the truncated microcontact area. The deformation of an asperity is given as pffiffiffiffiffiffiffi 2ð4 DÞ ln γ δ ¼ ð1:5 0:5DÞ GðD 2Þ a’ð1:5 0:5DÞ :
π
(A.3)
From the Hertz contact theory, the elastic contact force of a microcontact is given by F¼
4E� r3 ; 3R
(A.4)
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Soil Dynamics and Earthquake Engineering 132 (2020) 106064
whereE� ¼ E=ð1 ν2 Þis the composite elastic modulus, and E; ν is the elastic modulus and Poisson’s ration of soil, respectively. Hence, for the simple case of elastic and fully plastic asperity deformations, the contact load and contact area of an asperity are given by , pffiffiffiffiffiffiffi 25:5 D ln γ � ðD 2Þ ’ð2 0:5DÞ ’ 2; (A.5) E G a ; a ¼ a fe ¼ e 3π 2 0:5D and (A.6)
fc ¼ Kys a’ ; ap ¼ a’ ;
where the subscripts e and p represent elastic and plastic deformation, respectively. Here, ys is the yield strength of the soil and K is a factor. In addition, the critical truncated microcontact area a’c that distinguishes the elastic load from the plastic load is [31]. � ð11 2DÞ � � � �1=ðD 2Þ 2 E ð2D 4Þ a’c ¼ ln γ : (A.7) G 9π4 D Kys According to Mandelbrot, the distribution of the truncated microcontact area a’ can be written as [32,33]. nða’ Þ ¼ 0:5ðD
1Þa’0:5D L
0:5 ’ 0:5 0:5D
a
(A.8)
;
where a’L is the largest truncated microcontact area. While the contact interface apparently behaves elastically at larger displacements, the initial contact displacements are plastic in nature. Thus, the total contact load can be obtained by Z a’L Z a’c Fc ¼ F e þ Fp ¼ fe ða’ Þnða’ Þda’ þ fp ða’ Þnða’ Þda’ ; (A.9) a’c
0
which yields � �pffiffiffiffiffiffiffi ! 8 25:5 D D 1 ln γ D 1 > � D 2 ’ 0:5D 0:5 ’2:5 D ’2:5 D > � Kys a’0:5D E a þ G a a > L L c L < 3π 2 0:5D 5 2D 3 D Fc ¼ > � � > > pffiffiffiffiffiffiffiffiffiffiffiffi�a’ �0:75 a’L : ’0:25 ln þ 3Kys a’0:75 a D ¼ 2:5 2E� G ln γ L L c π ac
References
! 0:5 ’1:5 0:5D ac
D 6¼ 2:5 :
(A.10)
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