# A Novel Method of Dynamic Force Identification and Its Application

Authors: Nengjian Wang; Qinhui Liu; Chunping Ren; Chunsheng Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, an efficient mixed spectral conjugate gradient (EMSCG, for short) method is presented for solving unconstrained optimization problems. In this work, we construct a novel formula performed by using a conjugate gradient parameter which takes into account the advantages of Fletcher–Reeves (FR), Polak–Ribiere–Polyak (PRP), and a variant Polak-Ribiere-Polyak (VPRP), prove its stability and convergence, and apply it to the dynamic force identification of practical engineering structure. The analysis results show that the present method has higher efficiency, stronger robust convergence quality, and fewer iterations. In addition, the proposed method can provide more efficient and numerically stable approximation of the actual force, compared with the FR method, PRP method, and VPRP method. Therefore, we can make a clear conclusion that the proposed method in this paper can provide an effective optimization solution. Meanwhile, there is reason to believe that the proposed method can offer a reference for future research.


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## Body

## 1. Introduction

It is of great significance to solve the engineering problem about the identification of the dynamic loads acting on the practical engineering structure with the improvement of engineering requirements and the progress of engineering technology [1–3]. Load identification is a kind of inverse problem, which is ill posed. At the same time, in many practical engineering problems, it is not easy to promptly obtain the dynamic loads in virtue of technical limitations [4, 5]. In general, we will tend to use indirect methods to restore the expected load on actual engineering structures.As well as known, the regularization method can be used to solve the ill-posed problem procedure which can be translated into a kind of unconstrained optimization problems [6–10]; then, some optimization algorithms are used to solve it, such as conjugate gradient (CG) method, memory gradient (MG) method, and supermemory gradient (SMG) method [11–13]. It is generally known that the CG method does not achieve global convergence for common functions under inexact line searches [14], whereas the MG method needs to deal with the trust region subproblem [15]. Although the SMG method is superior to CG and MG methods, in some cases, it is also not globally convergent [16]. However, the spectral conjugate gradient (SCG) method does a lot of important works among various methods for solving unconstrained optimization problems [17]. The SCG method combines the spectral gradient and the conjugate gradient. The choice of spectral parameters is crucially important for the SCG method [18]. In this paper, we propose an efficient mixed spectral conjugate gradient (EMSCG) method, a novel formula performed by using a conjugate gradient parameter which takes into account the advantages of Fletcher–Reeves (FR) method [19], Polak–Ribiere–Polyak (PRP) method [20], and variant Polak–Ribiere–Polyak (VPRP) method [21] is constructed. The purpose of this paper is to find an effective method to deal with unconstrained optimization problems. In this way, an unconstrained optimization problem can be represented as [22](1)minfxx∈ℜn,where the nonlinear function fx:ℜn⟶ℜ is a continuously differentiable function whose gradient is denoted by gx:ℜn⟶ℜ.Furthermore, a sequencexk can be obtained in an algorithm for solving (1) and has the following forms [23–25]:(2)xk+1=xk+αkdk,where dk is a search direction and αk is the step size which is achieved by the one-dimensional search method. Usually, the exact line search is described as follows [26]:(3)fxk+αkdk=minα≥0fxk+αdk.It is well known that there are different ways to determinedk. In the conjugate gradient (CG) method, dk can be expressed by [27](4)dk=−g0,k=0,−gk+βkdk−1,k≥1,where βk denotes a scalar parameter characterizing the conjugate gradient method. The best-known expressions of βk are Hestenes–Stiefel (HS) [28], Fletcher–Reeves (FR) [19], Polak–Ribiere–Polyak (PRP) [20], and Dai–Yuan (DY) [29] formulas, which are defined by(5)βkHS=gkTyk−1dk−1Tyk−1,βkFR=gk2gk−12,βkPRP=gkTyk−1gk−12,βkDY=gk2dk−1Tyk−1,respectively, where ⋅ denotes the Euclidean norm and yk=gk+1−gk. At the same time, (5) corresponds to four different conjugate gradient methods, and different methods have a lot of deformations. For example, in [20], a variant of PRP is given; βk is expressed as(6)βkVPRP=gk2−gk/gk−1gkTgk−1gk−12.The VPRP method inherits some excellent properties of the PRP method, such as good numerical performance. Liu and Li [30] proved that the VPRP method has global convergence and descending property under strong Wolfe line search.In recent years, some scholars developed a new method, spectral conjugate gradient (SCG) method, for solving (1). In [31], the spectral gradient method for large-scale unconstrained optimization was introduced, which combined a nonmonotone line search strategy that guarantees global convergence with the Barzilai and Borwein method. Utilizing spectral gradient and conjugate gradient ideas, Chen and Jiang [32] proposed a spectral conjugate gradient method. In these algorithms, the search direction has the following form:(7)dk=−gk,k=1,−θkgk+βkdk−1,k≥2.However, the search direction of the spectral conjugate gradient method proposed by Birgin and Martinez does not satisfy the descent property, and of course, it does not have global convergence. In order to obtain global convergence, Wang et al. [19, 33] constructed the FR-type spectral conjugate gradient method; spectral parameters θk and βk are expressed as(8)θk=dk−1Tyk−1gk−12,βk=βkFR.An important feature of this method is to satisfy the sufficient descent condition:(9)dkTgk≤−tgk2,where t>0 is a constant.The CD-type spectral conjugate gradient method is proposed in [34]. Reference [35] provided a HS-type spectral conjugate gradient method. A spectral conjugate gradient method for the PRP type is introduced in [36]. All the above methods meet the sufficient conditions of decline (3).Based on the above literature, an efficient mixed spectral conjugate gradient method with sufficient descent is proposed, which was abbreviated as the EMSCG method, and parametersθk and βk are expressed as(10)βkEMSCG=gk2−max0,gk/gk−1gkTgk−1,gkTgk−1gk−12,θk=c+βEMSCGgkTdk−1gk2,where c is a parameter and c>0.As we can see from the algorithm framework above, the EMSCG method is similar to other spectral conjugate gradient algorithms. However, we choose two different spectral parametersβkEMSCG and θk which is the main difference between the EMSCG method and the others.The present paper is organized as follows. In Section2, the properties of an efficient mixed spectral conjugate gradient method are introduced. The global convergence is proved in Section 3, while the analysis results compared with the FR method, PRP method, and VPRP method, which are given in Section 4. In the last section, we draw some conclusions about an efficient mixed spectral conjugate gradient method.


## 2. Properties of an Efficient Mixed Spectral Conjugate (EMSCG) Gradient Method

The following is a detailed description of an efficient mixed spectral conjugate gradient (EMSCG) method.First of all, let us state a question, the constructeddk is a search direction. The following theorems and proofs are obtained.Theorem 1.
Suppose thatdk and gk are generated by using Algorithm 1, and the sufficient descent condition holds. Then,(11)gkTdk<0,for all k≥1.Algorithm 1:
Step 1. Choose x1 and ε, and then calculate g1. If g1≤ε, then terminate; else go to the next step.
Step 2. dk is computed based on (7).
Step 3. αk is computed based on (3).
Step 4. Set xk+1=xk+akdk.
Step 5. Set k=k+1, and go to the step 2.Proof.
By Algorithm1, we have(12)gkTdk=gkT−gkθk+βkEMSCGdk−1=−θkgk2+βkEMSCGgkTdk−1=−c+βkEMSCGgkTdk−1gk2gk2+βkEMSCGgkTdk−1=−cgk2<0.
Hence, (11) is proved.Theorem 2.
ParameterβkEMSCG satisfies(13)0≤βkEMSCG≤gk2gk−12,for all k≥1.Proof.
We know that this is the mixture of the three methods according to formula (10), and now it is explained in three cases.(i)
WhengkTgk−1≤0, we have(14)βkEMSCG=gk2gk−12=βkFR.(ii)
WhengkTgk−1>0 and gk/gk−1≤1, we have(15)βkEMSCG=gk2−gkTgk−1gk−12=βkPRP.(iii)
WhengkTgk−1>0 and gk/gk−1>1, we have(16)βkEMSCG=gk2−gk/gk−1gkTgk−1gk−12=βkVPRP.
The proof is effectively completed. In addition, from the selection of spectral coefficient of Algorithm1, we can select the different parameters c to optimize the numerical effect of Algorithm 1.


## 3. Convergent Analysis of the Proposed Method

In order to study the convergence of the algorithm, some basic assumptions are given as follows:Assumption A.
(P1)
The objective function is bounded by the following level:(17)L=x∈ℜnfx≤fx0.(P2) The gradientg is Lipschitz continuous; that is, there exists a constant L>0 such that, for any x,y∈U, we obtain(18)gx−gy≤Lx−y.Lemma 1.
Suppose that the functionf has the properties (P1) and (P2). dk is a descent direction and αk is obtained by (3); then,(19)∑k=1∞gkTdk2dk2<+∞.
In [14, 37, 38], Dai and Yuan stated that (19) had been essentially proved by Zoutendijk and Wolfe.Lemma 2.
Suppose that assumption A holds, sequencesgk are generated by Algorithm 1; then,(20)limk⟶∞infgk=0.Proof.
Suppose that (19) is not established, there is a constant γ>0; for all k≥1, we have gk≥γ.
According to formula (7), we get(21)dk+θkgk=βkEMSCGdk−1.
Then,(22)dk+θkgkTdk+θkgk=βkEMSCG2dk−12.
Exploiting (22), we have(23)dk2+2θkdkTgk+θk2gk2=βkEMSCG2dk−12.
Then,(24)dk2=−2θkdkTgk−θk2gk2+βkEMSCG2dk−12.
Combining (24) and Theorem 1, we have(25)dk2=−2cθkgk2−θk2gk2+βkEMSCG2dk−12.
Dividing both sides bygkTdk2, we have(26)dk2gkTdk2=dk2c2gk4=2θkcgk2−θk2c2gk2+βkEMSCG2c2gk4dk−12.
Then, we have(27)dk2gkTdk2≤gk2gk−122dk−12c2gk4−θk−c2c2gk2=dk−12c2gk−14+1gk2−θk−c2c2gk2≤dk−12c2gk−14+1gk2≤∑i=1k1gk2+1c2g12≤kγ2+ll=1c2g12.
Hence, we can obtain(28)∑gkTdk2dk2≥γ2∑1k+γ2L⟶+∞.
From the above discussion, we can see that it is in contradiction with Lemma 1. Therefore, the conclusion is established.


## 4. An Example of Engineering Application

In this section, the present method is applied to an engineering example of the identification problem of dynamic force generated between conical pick and coal-seam structure.


### 4.1. Establishment of the Dynamic Force Model

An abridged general view of dynamic force between pick and coal-seam structure is plotted in Figure1. For a deterministic multi-degrees-of-freedom structures, we can use a unified dynamic equation to represent it, which can be expressed by the following form [39]:(29)My″t+Cy′t+Kyt=Fx,y,z,t,where M, C, and K denote the mass matrix, damping matrix, and stiffness matrix, respectively; Fx,y,z,t denotes the dynamic force column vector in different directions; yt, y′t, and y″t denote displacement response vector, velocity response vector, and acceleration response vector, respectively.Figure 1
Schematic of dynamic forces between pick and coal-seam. (a) Pick and coal-seam structure. (b) Dynamic forces.
(a)
(b)The displacement of the structure can be described as(30)yt=∑i=1nφiqit=ΦQ,where Φ denotes the matrix of mode and Φ=φ1,φ2,Λ,φn and Q denotes the displacement vector and time function in the generalized coordinates, Q=q1,q2,ΛqnT.Combined with equations (29) and (30), the dynamic equations represented by the physical coordinates are transformed into the modal coordinate system, and the n decoupled modal equations are described as(31)qi″+2ξiωiqi′+ωi2qi=FitMi,where ξi denotes the i-order modal damping ratio and Mi and Fit denote the generalized mass and the generalized force, respectively, and Mi=φiTMφi, Fit=φiTFt.x
0
′ and x0 denotes the initial velocity and initial displacement of the system, respectively. Hence, the qi in equation (30) is expressed as follows:(32)qit=q1it+q2it.q
1
i
t is obtained by the homogeneous equation of equation (29), which is described as follows:(33)q1it=e−ξiωitqi0tcosωidt+qi0′+ξiωiqi0tωidsinωidt.Here,qi0 and qi0′ are the corresponding values in modal coordinates; it is described as follows:(34)qi0=φiTMx0Mi.Here,q2i is expressed in the following form owing to it having nothing to do with the initial conditions of the system:(35)q2it=1Miωid∫0tFτe−ξiωit−τsinωidt−τdτi.The displacement of the system is expressed as follows:(36)y=y0+y∗.Here,y0=∑i=1nφiq1i and y∗=∑i=1nφiq2i.The displacement of the system can be represented as follows:(37)y∗=y−y0.The following expressions can be obtained by introducing equations (33) and (35) into equation (37):(38)y∗=∫0t∑i=1nφiφiTMiωidSiτe−ξiωit−τsinωidzt−τdτ.Usually,y∗t can be described as(39)y∗t=∫0tGt−τFτdτ.Hence,Gt can be described by comparing equations (38) and (39):(40)Gt=∑i=1nφiφiTMiωide−ξiωitsinωidt.Lety∗t=y1∗tΛym∗tT, Ft=F1tΛFmtT, and Gt=g1tΛgmtT; then, equation (39) was discretized, and it is transformed into a matrix form that can be expressed as follows:(41)y1∗y2∗y3∗Mym∗=g100Λ0g2g10Λ0g3g2g1Λ0MMMΛ0gmgm−1gm−2Λ0F1F2F3MFm.or simply noted as(42)Y=GF.Equation (42) can be expressed by the following form due to the responses containing noise:(43)Yc=GFid+e.Here,Fid is the identified dynamic force, e is the unknown noise, and e=l⋅stdyt⋅rand. l is a parameter level, std⋅ is the standard deviation, and rand is a random number which ranges from −1 to 1.Equation (43) is ill-conditioned, whose solution is usually astaticism. Then, it can be transformed into an optimization problem by using the Tikhonov regularization method [40]:(44)JFid=minGFid−Yc2+λFid.Equation (44) is considered as a kind of unconstrained optimization problem and a series of optimization algorithms, which can treat such unconstrained optimization problem. In our work, an efficient mixed spectral conjugate gradient (EMSCG) method was proposed to minimize equation (44).


### 4.2. Experimental Setup

In our work, a diagram of experimental setup is organized in Figure2. The size of the coal-seam structure specimen manufactured artificially was 2000 mm × 800 mm × 1750 mm; the pick is mounted on the rotary cutting arm, and it can be driven by the reducer and torque; its power driven by the motor is rated at 55 kW. The cutting bench achieves free forward and backward movement by the hydraulic pressure drive control system. System output responses can be measured by the corresponding sensors, then converted by a signal amplifier, and finally recorded by a V10Dasp data vibration signal acquisition system [41].Figure 2
A diagram of the experimental setup.The cutting force generated between pick and coal-seam structure is transmitted through the gear sleeve and measured by the force sensor at the back end. The measuring direction of the sensor and the cutting axis is defined as the axial loadFz. The measuring direction of the vertical the cutting axis is defined as the radial load Fy. The force diagram of pick is shown in Figure 3. Z denotes the cutting force, Y denotes the propulsion resistance, f denotes the friction force between the supporting structure and the pick sleeve, β is the tangential installation angle of the pick, O is the supporting point of the gear sleeve, l1 is the distance from the pick tip to the supporting point, and l2 is the distance from the sensor to the supporting point. It is clearly seen from Figure 3 that the balance equation of force and the equilibrium equation of moment can be obtained, which is shown in the following equation:(45)
Y
cos
β
+
Z
sin
β
−
f
=
F
z
,
Y
sin
β
−
Z
cos
β
⋅
l
1
+
F
y
l
2
=
0
,
l
1
f
=
l
1
+
l
2
f
n
F
y
.Figure 3
The force diagram of pick.By simplifying equation (45), the relationship between the cutting force Z and the axial load Fz, the radial load Fy and installation angle β are obtained, as shown in the following equation:(46)Z=Fzsinβ+Fyfnsinβ1+kl+klcosβ,where fn is the friction coefficient between the gear sleeve and the supporting structure and fn = 0.1 and kl is the size coefficient of pick and sensor and kl = 0.739.In this experimental, the installation angle of conical pick is 45°, the maximum cutting thickness of the pick is 20 mm, the hauling speed is 0.8 r/min, and the rotary arm speed is 41 m/min. The measured axial load and radial load of the pick are shown in Figures4(a) and 4(b), and the cutting force is converted by equation (15) according to the load curves shown in Figures 4(a) and 4(b), which is shown in Figure 4(c).Figure 4
Cutting load and cutting force. (a) Axial load. (b) Radial load. (c) Cutting force.
(a)
(b)
(c)Under the above experimental conditions, it can be seen that although the value of cutting force is different from the axial load, the change trend is similar and the radial load has little effect on the change rule; that is, the cutting forceZ is proportional to the axial load Fz. Therefore, the measured axial load can reflect the magnitude and variation of cutting force and can be approximately characterized by the measured axial load when analyzing the characteristics of cutting force.


### 4.3. Result Analysis

In order to study the influence of the proposed method on load identification results, and to compare with other methods, the results and methods are given. All codes were written in MATLAB 7.0 and run on a HP with 2.0 GB RAM and Windows 7 operating system. Stop the iteration if criterion is defined asgk≤10−5 is satisfied or run time is more than 500 seconds. And the step length αk is calculated by equation (3), the parameters are obtained as follows: c=0.1 and t=1.For performance analysis of dynamic force identification methods used, the performance measurement metrics for dynamic force are described as follows: restoration time, iterative steps, and root mean-square-error (RMSE), where RMSE is defined by(47)RMSE=1N∑i=1NZid−Zactual2,where Zactual is the actual force and Zid is the identified force.In order to effectively identify dynamic force generated between conical pick and coal-seam structure, first we provide the measured displacement response within 0.3 seconds which can be obtained by the finite element method, which is shown in Figure5.Figure 5
Measured displacement response.Since the measured displacement response can be given by the computed numerical solution, then we can use the present method in the paper to identify dynamic force based on the mathematical model of dynamic force identification [19, 20].The identified results of dynamic force are achieved by using the different methods in this paper, which are shown in Figures6(a)–6(d), respectively.Figure 6
Identified results under different methods. (a) FR method. (b) PRP method. (c) VPRP method. (d) EMSCG method.
(a)
(b)
(c)
(d)We can see from Figures6(a)–6(d) that the different methods can identify the dynamic force generated between conical pick and coal-seam structure from the measured displacement response by the finite element method. However, the performance of dynamic force identification results has a certain difference.We can draw a conclusion from Table1 that the present method makes it more successful to identify dynamic force generated between conical pick and coal-seam structure based on the measured displacement response.Table 1
Comparison of different identification methods.
Evaluation metrics
FR method
PRP method
VPRP method
EMSCG method
RMSE
0.4182
2.9087
0.9055
0.3665
Restoration time (s)
12.1442
40.3227
18.361
5.0116
Iterative steps
19
55
27
11


## 4.1. Establishment of the Dynamic Force Model

An abridged general view of dynamic force between pick and coal-seam structure is plotted in Figure1. For a deterministic multi-degrees-of-freedom structures, we can use a unified dynamic equation to represent it, which can be expressed by the following form [39]:(29)My″t+Cy′t+Kyt=Fx,y,z,t,where M, C, and K denote the mass matrix, damping matrix, and stiffness matrix, respectively; Fx,y,z,t denotes the dynamic force column vector in different directions; yt, y′t, and y″t denote displacement response vector, velocity response vector, and acceleration response vector, respectively.Figure 1
Schematic of dynamic forces between pick and coal-seam. (a) Pick and coal-seam structure. (b) Dynamic forces.
(a)
(b)The displacement of the structure can be described as(30)yt=∑i=1nφiqit=ΦQ,where Φ denotes the matrix of mode and Φ=φ1,φ2,Λ,φn and Q denotes the displacement vector and time function in the generalized coordinates, Q=q1,q2,ΛqnT.Combined with equations (29) and (30), the dynamic equations represented by the physical coordinates are transformed into the modal coordinate system, and the n decoupled modal equations are described as(31)qi″+2ξiωiqi′+ωi2qi=FitMi,where ξi denotes the i-order modal damping ratio and Mi and Fit denote the generalized mass and the generalized force, respectively, and Mi=φiTMφi, Fit=φiTFt.x
0
′ and x0 denotes the initial velocity and initial displacement of the system, respectively. Hence, the qi in equation (30) is expressed as follows:(32)qit=q1it+q2it.q
1
i
t is obtained by the homogeneous equation of equation (29), which is described as follows:(33)q1it=e−ξiωitqi0tcosωidt+qi0′+ξiωiqi0tωidsinωidt.Here,qi0 and qi0′ are the corresponding values in modal coordinates; it is described as follows:(34)qi0=φiTMx0Mi.Here,q2i is expressed in the following form owing to it having nothing to do with the initial conditions of the system:(35)q2it=1Miωid∫0tFτe−ξiωit−τsinωidt−τdτi.The displacement of the system is expressed as follows:(36)y=y0+y∗.Here,y0=∑i=1nφiq1i and y∗=∑i=1nφiq2i.The displacement of the system can be represented as follows:(37)y∗=y−y0.The following expressions can be obtained by introducing equations (33) and (35) into equation (37):(38)y∗=∫0t∑i=1nφiφiTMiωidSiτe−ξiωit−τsinωidzt−τdτ.Usually,y∗t can be described as(39)y∗t=∫0tGt−τFτdτ.Hence,Gt can be described by comparing equations (38) and (39):(40)Gt=∑i=1nφiφiTMiωide−ξiωitsinωidt.Lety∗t=y1∗tΛym∗tT, Ft=F1tΛFmtT, and Gt=g1tΛgmtT; then, equation (39) was discretized, and it is transformed into a matrix form that can be expressed as follows:(41)y1∗y2∗y3∗Mym∗=g100Λ0g2g10Λ0g3g2g1Λ0MMMΛ0gmgm−1gm−2Λ0F1F2F3MFm.or simply noted as(42)Y=GF.Equation (42) can be expressed by the following form due to the responses containing noise:(43)Yc=GFid+e.Here,Fid is the identified dynamic force, e is the unknown noise, and e=l⋅stdyt⋅rand. l is a parameter level, std⋅ is the standard deviation, and rand is a random number which ranges from −1 to 1.Equation (43) is ill-conditioned, whose solution is usually astaticism. Then, it can be transformed into an optimization problem by using the Tikhonov regularization method [40]:(44)JFid=minGFid−Yc2+λFid.Equation (44) is considered as a kind of unconstrained optimization problem and a series of optimization algorithms, which can treat such unconstrained optimization problem. In our work, an efficient mixed spectral conjugate gradient (EMSCG) method was proposed to minimize equation (44).


## 4.2. Experimental Setup

In our work, a diagram of experimental setup is organized in Figure2. The size of the coal-seam structure specimen manufactured artificially was 2000 mm × 800 mm × 1750 mm; the pick is mounted on the rotary cutting arm, and it can be driven by the reducer and torque; its power driven by the motor is rated at 55 kW. The cutting bench achieves free forward and backward movement by the hydraulic pressure drive control system. System output responses can be measured by the corresponding sensors, then converted by a signal amplifier, and finally recorded by a V10Dasp data vibration signal acquisition system [41].Figure 2
A diagram of the experimental setup.The cutting force generated between pick and coal-seam structure is transmitted through the gear sleeve and measured by the force sensor at the back end. The measuring direction of the sensor and the cutting axis is defined as the axial loadFz. The measuring direction of the vertical the cutting axis is defined as the radial load Fy. The force diagram of pick is shown in Figure 3. Z denotes the cutting force, Y denotes the propulsion resistance, f denotes the friction force between the supporting structure and the pick sleeve, β is the tangential installation angle of the pick, O is the supporting point of the gear sleeve, l1 is the distance from the pick tip to the supporting point, and l2 is the distance from the sensor to the supporting point. It is clearly seen from Figure 3 that the balance equation of force and the equilibrium equation of moment can be obtained, which is shown in the following equation:(45)
Y
cos
β
+
Z
sin
β
−
f
=
F
z
,
Y
sin
β
−
Z
cos
β
⋅
l
1
+
F
y
l
2
=
0
,
l
1
f
=
l
1
+
l
2
f
n
F
y
.Figure 3
The force diagram of pick.By simplifying equation (45), the relationship between the cutting force Z and the axial load Fz, the radial load Fy and installation angle β are obtained, as shown in the following equation:(46)Z=Fzsinβ+Fyfnsinβ1+kl+klcosβ,where fn is the friction coefficient between the gear sleeve and the supporting structure and fn = 0.1 and kl is the size coefficient of pick and sensor and kl = 0.739.In this experimental, the installation angle of conical pick is 45°, the maximum cutting thickness of the pick is 20 mm, the hauling speed is 0.8 r/min, and the rotary arm speed is 41 m/min. The measured axial load and radial load of the pick are shown in Figures4(a) and 4(b), and the cutting force is converted by equation (15) according to the load curves shown in Figures 4(a) and 4(b), which is shown in Figure 4(c).Figure 4
Cutting load and cutting force. (a) Axial load. (b) Radial load. (c) Cutting force.
(a)
(b)
(c)Under the above experimental conditions, it can be seen that although the value of cutting force is different from the axial load, the change trend is similar and the radial load has little effect on the change rule; that is, the cutting forceZ is proportional to the axial load Fz. Therefore, the measured axial load can reflect the magnitude and variation of cutting force and can be approximately characterized by the measured axial load when analyzing the characteristics of cutting force.


## 4.3. Result Analysis

In order to study the influence of the proposed method on load identification results, and to compare with other methods, the results and methods are given. All codes were written in MATLAB 7.0 and run on a HP with 2.0 GB RAM and Windows 7 operating system. Stop the iteration if criterion is defined asgk≤10−5 is satisfied or run time is more than 500 seconds. And the step length αk is calculated by equation (3), the parameters are obtained as follows: c=0.1 and t=1.For performance analysis of dynamic force identification methods used, the performance measurement metrics for dynamic force are described as follows: restoration time, iterative steps, and root mean-square-error (RMSE), where RMSE is defined by(47)RMSE=1N∑i=1NZid−Zactual2,where Zactual is the actual force and Zid is the identified force.In order to effectively identify dynamic force generated between conical pick and coal-seam structure, first we provide the measured displacement response within 0.3 seconds which can be obtained by the finite element method, which is shown in Figure5.Figure 5
Measured displacement response.Since the measured displacement response can be given by the computed numerical solution, then we can use the present method in the paper to identify dynamic force based on the mathematical model of dynamic force identification [19, 20].The identified results of dynamic force are achieved by using the different methods in this paper, which are shown in Figures6(a)–6(d), respectively.Figure 6
Identified results under different methods. (a) FR method. (b) PRP method. (c) VPRP method. (d) EMSCG method.
(a)
(b)
(c)
(d)We can see from Figures6(a)–6(d) that the different methods can identify the dynamic force generated between conical pick and coal-seam structure from the measured displacement response by the finite element method. However, the performance of dynamic force identification results has a certain difference.We can draw a conclusion from Table1 that the present method makes it more successful to identify dynamic force generated between conical pick and coal-seam structure based on the measured displacement response.Table 1
Comparison of different identification methods.
Evaluation metrics
FR method
PRP method
VPRP method
EMSCG method
RMSE
0.4182
2.9087
0.9055
0.3665
Restoration time (s)
12.1442
40.3227
18.361
5.0116
Iterative steps
19
55
27
11


## 5. Conclusions

In this paper, an efficient mixed spectral conjugate gradient (EMSCG) method is proposed for the dynamic force identification. Moreover, the convergence of the proposed method based on the convergence analysis in Section3 is strictly proved. An engineering example is investigated in Section 4, in which the performance of the proposed method is also compared with that of the existing methods. The analysis results show that the present method has high efficiency and very robust convergence performance and reduces the number of iterations. In addition, the proposed method can provide more efficient and numerically stable approximation of the actual force, compared with the FR method, PRP method, and VPRP method. The results validate the stability and the effectiveness of the present method, and there is reason to believe that the present method can offer a reference for future research.



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*Source: 1534560-2019-12-14.xml*