# Dynamic Force Identification Problem Based on a Novel Improved Tikhonov Regularization Method

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

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## Abstract

The main purpose of this paper is to identify the dynamic forces between the conical pick and the coal-seam. According to the theory of time domain method, the dynamic force identification problem of the system is established. The direct problem is described by Green kernel function method. The dynamic force is expressed by a series of functions superposed by impulses, and the dynamic response of the structure is expressed as a convolution integral form between the input dynamic force and the response of Green kernel function. Because of the ill-conditioned characteristics of the structure matrix and the influence of measurement noise in the process of dynamic force identification, it is difficult to deal with this problem by the usual numerical method. In present content, a novel improved Tikhonov regularization method is proposed to solve ill-posed problems. An engineering example shows that the proposed method is effective and can obtain stable approximate solutions to meet the engineering requirements.


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## Body

## 1. Introduction

Various mechanical and electrical products are often subjected to a variety of dynamic loads, including vibration, impact, noise, and thermal environment [1]. It is important for the safety and accurate design of structures to accurately determine these loads. In many cases, it is difficult to directly measure the external loads of large structures such as missiles, aircraft, and offshore platforms under the action of wind waves or alternating excitations. Dynamic load identification is based on the dynamic response of the measured system and the known dynamic characteristics of the system to solve the dynamic loads on the structure. In practical engineering, it is very difficult to directly measure dynamic loads. Therefore, load identification is an indirect method to obtain the required load [2–4].At present, load identification methods mainly include frequency domain method and time domain method [5–7]. Frequency domain method was put forward earlier. Its basic idea is to identify excitation spectrum by response spectrum. It is mainly realized by inversion of frequency response function between excitation and response. However, the ill-conditioned problem of coefficient matrix and singular value decomposition problem are often encountered in matrix inversion [8–10]. Based on the back analysis of the complex convolution relationship between load and response, the time domain method directly determines the time history of dynamic force, which has a good application prospect in engineering. For instance, [11] adopted the singular value decomposition (SVD) method to predict hourly load and peak load for the next selected time span. Reference [12] presented a new approach for solving specific classes of inverse source identification problems. Reference [13] used a modified inverse patch transfer function (IPTF) method to reconstruct the normal velocities of the target source in a noisy environment, and [14] used neural network in combination with genetic programming to implement the load identification of electric equipment. Reference [15] proposed the finite element and wavelet-based method for reconstructing the moving force. In [16], a new and effective algorithm for optimization problems, GPSA, is utilized in civil engineering. Reference [17] presented a single-point method for the identification of prevailing disturbing loads in power systems. But we may encounter some difficulties. It is very difficult to use the above methods for large-scale and complex load identification problems. Meanwhile, in practical engineering problems, it is necessary to identify the random dynamic force on pick and coal-seam structure. Unfortunately, they are complex inverse problems with inherent ill-posedness.Fortunately, there are many popular regularization methods to deal with the ill-posed problem in practical engineering applications [18–22]. It is well known that the Tikhonov regularization method [23], as a traditional technique, has been widely used in various mathematical and engineering problems to deal with ill-posedness [24], for instance, parameter reconstruction, force source reconstruction problem, and so on [25–29]. Reference [30] applied the Tikhonov regularization method to reconstruct the dynamic force. Reference [31] demonstrated that the Tikhonov regularization method can solve the inverse problem in a stable manner despite the presence of noisy data. Reference [32] identified the shock load on an electroelastic bimorph disk using the Tikhonov regularization method. However, the Tikhonov regularization method is not completely perfect; there are numerous unavoidable limitations and disadvantages, which is described as follows: (a) the approximate solution provided by Tikhonov regularization method is too smooth; (b) the approximate solution provided by Tikhonov regularization method may lack some details that the desired real solution might possess. Therefore, it is very necessary to discuss other regularization methods to improve the limitations and disadvantages of the regularization method mentioned above.Considering the limitations of the above methods in a certain case, we propose a novel improved Tikhonov regularization method to offer a stable solution for the ill-posed problems in practical engineering applications. Compared with the conventional methods, the uniqueness and improvement of the proposed method are obtained in this paper.First and foremost, the paper is different from the conventional methods, and the uniqueness and improvement of this paper can be described as follows: (a) a novel improved Tikhonov regularization method can make the error of the regularization solution reach the asymptotic optimal order; (b) a novel improved Tikhonov regularization method has better convergence, and the detailed information of the solution is clearly enhanced; (c) the method we propose can effectively overcome the smoothness of the solution of the ill-posed problem; (d) the proposed method not only eliminates the influence of large singular values but also ensures the filtering of small singular values; (e) the small perturbation of right-end data will not affect the accuracy of inversion results, which fully reflects the good stability of the improved method.In this study, our main purpose is to effectively identify the dynamic force between conical pick and coal-seam by using a novel improved Tikhonov regularization method. The rest of this study is structured as follows. In Section2, the inverse problem model is established. In Section 3, a novel improved Tikhonov regularization method is proposed. In Section 4, the stability of the proposed method is proved. In Section 5, an engineering application example is discussed. Some valuable conclusions are described in Section 6.


## 2. Establishment of Inverse Problem Model

The inverse problems encountered in engineering can be expressed by the following [33, 34]:(1)yt=∫0tkt-τxτdτ,where y(t) is the response which can be displacement, velocity, acceleration, strain, etc. k(t) is the corresponding Green’s function that is the kernel of impulse response. x(t) is the identified dynamic force.Using the rectangular formula, we can get the discrete form of (1). (2)∑i=1nktk-τixτiΔT=yti.where ΔT denotes the time interval and it can be rewritten as follows:(3)xi=xτi,(4)yi=yti,(5)kk-i=ktk-τi.So (6)X=x1,x2,x3,⋯xnT,(7)Y=y1,y2,y3,⋯ynT,(8)K=kk-iΔT.Hence, (2) can be simply written as(9)Yt=KtXt,or equivalently,(10)y1y2⋮ym=k1k2k1⋮⋮⋱kmkm-1…k1x1x2⋮xmΔt,Where yi,ki, and xi denote structural dynamic response, Green’s function matrix, and input force at time t=iΔt, respectively.Because the condition number of K is very large and the response signal Y measured in practice always contains noise, a very small disturbance can result in a huge solution deviation. So dynamic force identification is a typical ill-conditioned problem, and it is inappropriate to directly invert K in (3).The above characteristics of ill-conditioned problems do not mean that ill-conditioned problems are not solvable but that traditional linear algebraic methods cannot be directly applied to solve such problems.Therefore, in order to make the solution meaningful, the approximate solution is usually obtained by regularization technique.


## 3. The Regularization Method for Solving Inverse Problem

### 3.1. Preliminaries

As we all know, we have the following lemma for regularized filter functions.Lemma 1 (see [35]).
LetX and Y be Hilbert spaces, K:X→Y be a compact operator, μi,xi,yi be a singular system for the linear operator K:X→Y, and the function q be defined: 0,+∞×(0,K]→R; then we have the following properties:(11)qα,μ≤1,∀α0,+∞,∀μ∈0,K.∀α0,+∞, there is a constant cα>0, such that qα,μ≤cαμ and ∀μ∈0,K;(12)limα→0qα,μ=1,∀μ∈0,K.Operator Rα:Y→X is defined by the following formula:(13)Rαy=∑i=1∞qα,μiμiy,yixi.which is a regularization method, and Rα≤cα.
The functionqα,μ with the above properties is called a regularized filter function of K.For the famous Tikhonov regularization method, the Tikhonov regularization solution is the minimum value of Tikhonov functional; a singular system with compact operators can be expressed as(14)xαδ=Rαyδ=∑i=1∞qα,μiμiyδ,yixi.qα,μ is the Tikhonov regularized filter function, and qα,μ=μ2/α+μ2, α>0, 0<μ≤K. Next, we will get a new regularization method by giving a new regularization filter function.


### 3.2. A Novel Improved Tikhonov Regularization Method

According to the regularization theory, the regularization operatorRα can be constructed by regularization filter function qα,μ, which provides a theoretical basis for establishing a new regularization method.In this paper, a novel improved regularized filter function is constructed as follows:(15)qα,μ=1μσr≥αμσα+μσr1/rμσr<αwhere α>0, 0<μ≤K, and r>0,σ≥1.The method proposed in this paper can not only ensure that the large singular values are not corrected but also filter the small singular values without affecting the accuracy of the solutions.Theorem 2.
The functionqα,μ defined by formula (8) is a kind of regularized filter function.Proof.
① Whenμσr≥α, qα,μ=1≤1;
whenμσr<α, we all know (16)μσ=μσr1/r<α+μσr1/rTherefore, we get(17)qα,μ=μσα+μσr1/r<1.② When μσr≥α, μσr/α≥1⇒μ/α1/σr≥1, such that(18)qα,μ=1≤1α1/σr·μ;when μσr<α, qα,μ=μσ/α+μσr1/r.
Whenσ=1, we have(19)qα,μ=μα+μr1/r≤μα1/r.When σ>1, α+μσr≥α1/p·μσr/q.
Lettingp=σ,q=σ/σ-1, we have(20)α+μσr≥α1/σ·μrσ-1.Then (21)α+μσr1/r≥α1/σr·μσ-1,such that(22)μσα+μσr1/r≤μσα1/rσ·μσ-1=μα1/rσ.Hence, ∀σ≥1, we have qα,μ=μ/α1/σr.
And∀α>0; we can obtain(23)qα,μ=1α1/σr·μ=cαμ.③ When α→0, qα,μ=1.
It is known from Theorem2(24)qα,μ=1μσr≥αμσα+μσr1/rμσr<αwhich is a kind of regularized filter functions and its corresponding regularization operator is Rα:Y→X(25)Rαy=∑i=1∞qα,μiμiy,yixi.In order to express the advanced filtering operator of the proposed method more clearly, we give another representation of its filtering algorithm.
Substituting formula (13) into formula (14), we can get (26)Rαy=∑i=1k1μy,yixi+∑i=k+1∞1μi·μiβα+μiβi1/γy,yixiIntroduce the singular value decomposition (SVD) [36]:(27)A=U∑VT=∑i=1nuiσiviT,in which U=u1,u2,u3,⋯un+1 and V=v1,v2,v3,⋯vn+1 are orthogonal matrices and ∑=diag⁡σ1,σ2,σ3,⋯σn+1.(28)σ1≥σ2≥σ3≥⋯≥σn>σn+1.Substituting formula (27) into formula (26), we can get (29)Rαy=∑i=1k1σiy,uivi+∑i=k+1n+11σi·σiβα+σiβγ1/γy,uivi=∑i=1kuiTyσivi+∑i=k+1n+1σiβα+σiβγ1/γuiTyσiviThen, we have (30)Rαy=u1Tyσ1v1+u2Tyσ2v2+⋯ukTyσkvk+σk+1βα+σk+1βγ1/γuk+1Tyσk+1vk+1+⋯σn+1βα+σn+1βγ1/γun+1Tyσn+1vn+1=v1,v2⋯vn+11σ1⋱1σk⋱1σk+1·σk+1βα+σk+1βγ1/γ·u1Tu2Tu3T⋮un+1T·y=v1,v2⋯vn+1·1⋱1⋱1σk+1·σk+1βα+σk+1βγ1/γ·σ1-1σ2-1⋱σn+1-1·u1Tu2Tu3T⋮un+1T·y=V·Mβ,γα·∑-1Uy.in which (31)Mβ,γα=1⋱1⋱1σk+1·σk+1βα+σk+1βγ1/γHence, we can have(32)Rα=V·Mβ,γα·∑-1U.Comparing the Tikhonov regularization method with the improved Tikhonov regularization method, according to formula (31), we can see that, in part σiβγ≥α, change from σi2/α+σi2 to 1, 1≤i≤k, this eliminates the effect of large singular values.
Whenσiβγ<α change from σi2/α+σi2 to σiβ/α+σiβγ1/γ, k<i≤n+1; this not only guarantees the filtering of small singular values but also improves the approximation order of solutions.


## 3.1. Preliminaries

As we all know, we have the following lemma for regularized filter functions.Lemma 1 (see [35]).
LetX and Y be Hilbert spaces, K:X→Y be a compact operator, μi,xi,yi be a singular system for the linear operator K:X→Y, and the function q be defined: 0,+∞×(0,K]→R; then we have the following properties:(11)qα,μ≤1,∀α0,+∞,∀μ∈0,K.∀α0,+∞, there is a constant cα>0, such that qα,μ≤cαμ and ∀μ∈0,K;(12)limα→0qα,μ=1,∀μ∈0,K.Operator Rα:Y→X is defined by the following formula:(13)Rαy=∑i=1∞qα,μiμiy,yixi.which is a regularization method, and Rα≤cα.
The functionqα,μ with the above properties is called a regularized filter function of K.For the famous Tikhonov regularization method, the Tikhonov regularization solution is the minimum value of Tikhonov functional; a singular system with compact operators can be expressed as(14)xαδ=Rαyδ=∑i=1∞qα,μiμiyδ,yixi.qα,μ is the Tikhonov regularized filter function, and qα,μ=μ2/α+μ2, α>0, 0<μ≤K. Next, we will get a new regularization method by giving a new regularization filter function.


## 3.2. A Novel Improved Tikhonov Regularization Method

According to the regularization theory, the regularization operatorRα can be constructed by regularization filter function qα,μ, which provides a theoretical basis for establishing a new regularization method.In this paper, a novel improved regularized filter function is constructed as follows:(15)qα,μ=1μσr≥αμσα+μσr1/rμσr<αwhere α>0, 0<μ≤K, and r>0,σ≥1.The method proposed in this paper can not only ensure that the large singular values are not corrected but also filter the small singular values without affecting the accuracy of the solutions.Theorem 2.
The functionqα,μ defined by formula (8) is a kind of regularized filter function.Proof.
① Whenμσr≥α, qα,μ=1≤1;
whenμσr<α, we all know (16)μσ=μσr1/r<α+μσr1/rTherefore, we get(17)qα,μ=μσα+μσr1/r<1.② When μσr≥α, μσr/α≥1⇒μ/α1/σr≥1, such that(18)qα,μ=1≤1α1/σr·μ;when μσr<α, qα,μ=μσ/α+μσr1/r.
Whenσ=1, we have(19)qα,μ=μα+μr1/r≤μα1/r.When σ>1, α+μσr≥α1/p·μσr/q.
Lettingp=σ,q=σ/σ-1, we have(20)α+μσr≥α1/σ·μrσ-1.Then (21)α+μσr1/r≥α1/σr·μσ-1,such that(22)μσα+μσr1/r≤μσα1/rσ·μσ-1=μα1/rσ.Hence, ∀σ≥1, we have qα,μ=μ/α1/σr.
And∀α>0; we can obtain(23)qα,μ=1α1/σr·μ=cαμ.③ When α→0, qα,μ=1.
It is known from Theorem2(24)qα,μ=1μσr≥αμσα+μσr1/rμσr<αwhich is a kind of regularized filter functions and its corresponding regularization operator is Rα:Y→X(25)Rαy=∑i=1∞qα,μiμiy,yixi.In order to express the advanced filtering operator of the proposed method more clearly, we give another representation of its filtering algorithm.
Substituting formula (13) into formula (14), we can get (26)Rαy=∑i=1k1μy,yixi+∑i=k+1∞1μi·μiβα+μiβi1/γy,yixiIntroduce the singular value decomposition (SVD) [36]:(27)A=U∑VT=∑i=1nuiσiviT,in which U=u1,u2,u3,⋯un+1 and V=v1,v2,v3,⋯vn+1 are orthogonal matrices and ∑=diag⁡σ1,σ2,σ3,⋯σn+1.(28)σ1≥σ2≥σ3≥⋯≥σn>σn+1.Substituting formula (27) into formula (26), we can get (29)Rαy=∑i=1k1σiy,uivi+∑i=k+1n+11σi·σiβα+σiβγ1/γy,uivi=∑i=1kuiTyσivi+∑i=k+1n+1σiβα+σiβγ1/γuiTyσiviThen, we have (30)Rαy=u1Tyσ1v1+u2Tyσ2v2+⋯ukTyσkvk+σk+1βα+σk+1βγ1/γuk+1Tyσk+1vk+1+⋯σn+1βα+σn+1βγ1/γun+1Tyσn+1vn+1=v1,v2⋯vn+11σ1⋱1σk⋱1σk+1·σk+1βα+σk+1βγ1/γ·u1Tu2Tu3T⋮un+1T·y=v1,v2⋯vn+1·1⋱1⋱1σk+1·σk+1βα+σk+1βγ1/γ·σ1-1σ2-1⋱σn+1-1·u1Tu2Tu3T⋮un+1T·y=V·Mβ,γα·∑-1Uy.in which (31)Mβ,γα=1⋱1⋱1σk+1·σk+1βα+σk+1βγ1/γHence, we can have(32)Rα=V·Mβ,γα·∑-1U.Comparing the Tikhonov regularization method with the improved Tikhonov regularization method, according to formula (31), we can see that, in part σiβγ≥α, change from σi2/α+σi2 to 1, 1≤i≤k, this eliminates the effect of large singular values.
Whenσiβγ<α change from σi2/α+σi2 to σiβ/α+σiβγ1/γ, k<i≤n+1; this not only guarantees the filtering of small singular values but also improves the approximation order of solutions.


## 4. Stability Analysis of the Proposed Method

Theorem 3.
Suppose that the solutionx+of Kx=y satisfies x+=K∗Kνz∈RK∗Kν,z∈X, and z≤E; if choosing the regularization parameter αδ=cδ/Eσr/2v+1 (c is a positive constant), then there are the following error estimates:(33)xαδδ-x+=Oδ2v/2v+1.Proof.
As we all know, the error estimates of true solution and regular solution can be expressed as follows:(34)xαδ-x+≤Rα·δ+Rαy-x+.It is known from Theorem 2 that we have (35)Rα≤cα=1α1/σrBy using the singular system of operator K, then we have(36)Rαy-x+2=∑i=1∞qα,μi-12·x+,xi2=∑i=1∞qα,μi-12·K∗Kvz,xi2=∑i=1∞qα,μi-12·∑j=1∞μj2vz,xjxj,xi2=∑i=1∞qα,μi-12·μi2vz,xi2=∑i=1∞qα,μi-12·μi4v·z,xi2.when μσr<α and 0<qα,μ<1, such that qα,μi-1<1, hence, we can obtain (37)qα,μi-1·μi2v<μi2v=μiσr2v/σr<α2v/σr.When μσr≥α and qα,μ=1, then we have(38)qα,μi-1·μi2v=0·μi2v=0<α2v/σr.And (39)Rαy-x+2<α4v/σr∑i=1∞z,xi2=α4v/σr·z2≤α4v/σr·E2.So we can obtain(40)xαδ-x+≤1α1/σr+α2v/σr·E.If αδ=cδ/Eσr/2v+1, then(41)xαδδ-x+≤δ·cδEσr/2v+1-1/σr+cδEσr/2v+12v/σr·E=δ·c-1/σrδE-1/2v+1+c2v/σrδEσr/2v+1·E=c-1/σr+c2v/σr·E1/2v+1·δ2v/2v+1.Hence, we can get(42)xαδδ-x+=Oδ2v/2v+1.
From the above proofs, we can see that the method proposed in this paper can make the error of the regularization solution reach the asymptotically optimal order.


## 5. An Example of Engineering Application

In this section, the proposed method is used to an engineering example of the identification problem of dynamic force acting on conical pick and coal-seam structure.


### 5.1. The Establishment of Experimental System

The structure diagram of conical pick and coal-seam is made in Figure1. The parameters are as follows: the coal-seam was 2000 mm × 800 mm × 1750 mm. The density of conical pick and coal-seam is 7800 kg/m3 and 1607.98 kg/m3, respectively. The elastic modulus of conical pick and coal-seam is 600 MPa and 322.09 MPa, respectively. The Poisson ratio of conical pick and coal-seam is set as 0.25 and 0.25, respectively [37].Figure 1
The structure diagram of conical pick and coal-seam structure.Within our framework, the experimental system was shown in Figure2. The experimental system includes test-bed, motor, reducer, coupling, cutting mechanism, platform, coal-seam structure, the force-measured device, force sensor, the signal amplifier, and the DaspV10 intelligent data acquisition and signal processing system [38, 39].Figure 2
Experimental system.


### 5.2. The Principle of Force Measurement

The cutting resistance of the pick when cutting the coal-seam is transmitted through the gear sleeve, and its size is measured by the force sensor at the back end. The force direction of the sensor is defined as the axial load Fz in accordance with the axis of the pick, and the force direction measured is defined as the radial load Fy perpendicular to the axis of the pick.The force state of the pick in the test force measuring device is shown in Figure3. Z is the cutting resistance, Y is the propulsion resistance, f is the friction resistance between the support structure and the pick sleeve, beta is the tangential installation angle of the pick, O is the support point of the pick sleeve, l1 is the distance between the tip of the tooth and the support point, and l2 is the distance between the sensor and the support point. According to Figure 3, the force balance and moment balance equations of the pick are obtained. (43)Ycos⁡β+Zsin⁡β-f=FzYsin⁡β-Zcos⁡β·l1+Fyl2=0l1f=l1+l2fnFySetting kl=l2/l1, so we have (44)Z=Fzsin⁡β+Fyfnsin⁡β1+kl+klcos⁡β.fn and kl are called friction coefficient and structural dimension coefficient, respectively.Figure 3
The force diagram of pick.Experimental conditions: the installation angle of the pick is45∘, the cutting impedance of coal - seam is set to 180 kN/m, the speed of the cutting arm is set to 40.8 r/min, and the traction speed is set to 0.8 m/min. The measured axial and radial loads of the pick are shown in Figures 4(a) and 4(b). According to the load curves of Figures 4(a) and 4(b), the cutting force is converted from (19) at the sampling discrete points, as shown in Figure 4(c), and the measured displacement response is shown in Figure 5.Figure 4
Cutting load and cutting force.
(a)
Axial load (b)
Radial load (c)
Cutting forceFigure 5
The measured displacement response.Under the above experimental conditions, it can be seen that the cutting force obtained by conversion is similar to that of axial load, and the radial load has little effect on its variation. That is to say, the cutting force Z is proportional to the axial load Fz. Therefore, the measured axial load can reflect the size and variation of cutting resistance and can be approximately characterized by the measured axial load when analyzing the characteristics of cutting resistance.


### 5.3. Result Analysis

Adding a random distributed perturbation to displacement response data, we obtain(45)yδ=y+εrandn⁡sizey,where ε indicates the noise level of the measurement displacement response and the randn⁡• generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1, and standard deviation σ=1. randn⁡sizey returns an array of random entries that is of the same size as y [40]. The noise level δ can be measured in the sense of root mean square error (RMSE) according to (46)δ=yδ-yl2=1N+1∑i=0Ny-yδ21/2.This process is known as Morozov’s discrepancy principle [41, 42].Relative error (RE)(47)RE=x^-xx.In order to consider the effect of noise level, we setε=0.01,0.05,0.1,0.20. All algorithmic programs are set in MATLAB 7.0 and Windows 7 operating system. The identified results of dynamic force are achieved by using different regularization methods in this paper, which are shown in Figures 6, 7, 8, and 9, respectively.Figure 6
Identified resultsε=0.01.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Ref. [33] method (d)
Present methodFigure 7
Identified resultsε=0.05.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Ref. [33] method (d)
Present methodFigure 8
Identified resultsε=0.10.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization (c)
Reference [33] method (d)
Present methodFigure 9
Identified resultsε=0.20.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Reference [33] method (d)
Present methodFigures6(a), 6(b), 6(c), and 6(d) illustrate that the identified force is obtained by adopting various different regularization methods under ε=0.01, respectively. Figures 7(a), 7(b), 7(c), and 7(d) show the identified force by adopting various different regularization methods under ε=0.05, respectively. The identified force can be seen from Figures 8(a), 8(b), 8(c), and 8(d), which gives that the identified force under ε=0.10, and Figures 9(a), 8(b), 8(c), and 9(d) offer the identified force by adopting various different methods under ε=0.20, respectively.It can be seen from Figures6, 7, 8, and 9 that the various different methods can identify the dynamic force. On the contrary, the effect of dynamic force identification is different.To explore the quality of dynamic force identification by adopting various different methods, we give the value of RE and CC and iterative steps, which are shown in Tables1, 2, 3, and 4. It is easy to see from Table 1 that RE and iterative steps of Figure 6(d) are smaller than that of Figures 6(a), 6(b), and 6(c), On the contrary, CC value presents opposite result. Table 2 shows that RE and iterative steps of Figures 7(a), 7(b), and 7(c) are larger than that of Figure 7(d), and the CC value presents opposite result. Table 3 shows that RE and iterative steps of Figure 8(d) are smaller than that of Figures 8(a), 8(b), and 8(c), but the CC value is the opposite. Table 4 offers that RE and iterative steps of Figure 9(d) are smaller than that of Figures 9(a), 9(b), and 9(c); nevertheless, the CC value presents opposite result.Table 1
Analysis of identification results (ε=0.01).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.2719 0.2011 0.1897 0.1832 CC 0.8939 0.9037 0.9145 0.9276 Iterative steps 22 15 10 8Table 2
Analysis of identification results (ε=0.05).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.3265 0.2327 0.1994 0.1871 CC 0.7940 0.8138 0.9033 0.9104 Iterative steps 30 22 12 9Table 3
Analysis of identification results (ε=0.10).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.3994 0.3011 0.2144 0.1957 CC 0.7291 0.8084 0.8996 0.9001 Iterative steps 37 28 14 10Table 4
Analysis of identification results (ε=0.20).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.5673 0.4033 0.2938 0.2011 CC 0.4497 0.5988 0.8177 0.8846 Iterative steps 50 33 20 13Hence, compared with the other methods, we find from Tables1–4 that the proposed method has the smallest RE, the largest CC, and the least iterative steps, so the optimal value is ε=0.01.In addition, with the increase ofε, although RE and iterative steps increase, they do not affect the identification accuracy; that is to say, the small perturbation of the right-hand data will not affect the accuracy of the inversion results, which fully reflects that the present method has a good stability.Through the analysis, we draw the following conclusions that the present method is successful to identify dynamic force acting on conical pick and coal-seam.


## 5.1. The Establishment of Experimental System

The structure diagram of conical pick and coal-seam is made in Figure1. The parameters are as follows: the coal-seam was 2000 mm × 800 mm × 1750 mm. The density of conical pick and coal-seam is 7800 kg/m3 and 1607.98 kg/m3, respectively. The elastic modulus of conical pick and coal-seam is 600 MPa and 322.09 MPa, respectively. The Poisson ratio of conical pick and coal-seam is set as 0.25 and 0.25, respectively [37].Figure 1
The structure diagram of conical pick and coal-seam structure.Within our framework, the experimental system was shown in Figure2. The experimental system includes test-bed, motor, reducer, coupling, cutting mechanism, platform, coal-seam structure, the force-measured device, force sensor, the signal amplifier, and the DaspV10 intelligent data acquisition and signal processing system [38, 39].Figure 2
Experimental system.


## 5.2. The Principle of Force Measurement

The cutting resistance of the pick when cutting the coal-seam is transmitted through the gear sleeve, and its size is measured by the force sensor at the back end. The force direction of the sensor is defined as the axial load Fz in accordance with the axis of the pick, and the force direction measured is defined as the radial load Fy perpendicular to the axis of the pick.The force state of the pick in the test force measuring device is shown in Figure3. Z is the cutting resistance, Y is the propulsion resistance, f is the friction resistance between the support structure and the pick sleeve, beta is the tangential installation angle of the pick, O is the support point of the pick sleeve, l1 is the distance between the tip of the tooth and the support point, and l2 is the distance between the sensor and the support point. According to Figure 3, the force balance and moment balance equations of the pick are obtained. (43)Ycos⁡β+Zsin⁡β-f=FzYsin⁡β-Zcos⁡β·l1+Fyl2=0l1f=l1+l2fnFySetting kl=l2/l1, so we have (44)Z=Fzsin⁡β+Fyfnsin⁡β1+kl+klcos⁡β.fn and kl are called friction coefficient and structural dimension coefficient, respectively.Figure 3
The force diagram of pick.Experimental conditions: the installation angle of the pick is45∘, the cutting impedance of coal - seam is set to 180 kN/m, the speed of the cutting arm is set to 40.8 r/min, and the traction speed is set to 0.8 m/min. The measured axial and radial loads of the pick are shown in Figures 4(a) and 4(b). According to the load curves of Figures 4(a) and 4(b), the cutting force is converted from (19) at the sampling discrete points, as shown in Figure 4(c), and the measured displacement response is shown in Figure 5.Figure 4
Cutting load and cutting force.
(a)
Axial load (b)
Radial load (c)
Cutting forceFigure 5
The measured displacement response.Under the above experimental conditions, it can be seen that the cutting force obtained by conversion is similar to that of axial load, and the radial load has little effect on its variation. That is to say, the cutting force Z is proportional to the axial load Fz. Therefore, the measured axial load can reflect the size and variation of cutting resistance and can be approximately characterized by the measured axial load when analyzing the characteristics of cutting resistance.


## 5.3. Result Analysis

Adding a random distributed perturbation to displacement response data, we obtain(45)yδ=y+εrandn⁡sizey,where ε indicates the noise level of the measurement displacement response and the randn⁡• generates arrays of random numbers whose elements are normally distributed with mean 0, variance σ2=1, and standard deviation σ=1. randn⁡sizey returns an array of random entries that is of the same size as y [40]. The noise level δ can be measured in the sense of root mean square error (RMSE) according to (46)δ=yδ-yl2=1N+1∑i=0Ny-yδ21/2.This process is known as Morozov’s discrepancy principle [41, 42].Relative error (RE)(47)RE=x^-xx.In order to consider the effect of noise level, we setε=0.01,0.05,0.1,0.20. All algorithmic programs are set in MATLAB 7.0 and Windows 7 operating system. The identified results of dynamic force are achieved by using different regularization methods in this paper, which are shown in Figures 6, 7, 8, and 9, respectively.Figure 6
Identified resultsε=0.01.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Ref. [33] method (d)
Present methodFigure 7
Identified resultsε=0.05.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Ref. [33] method (d)
Present methodFigure 8
Identified resultsε=0.10.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization (c)
Reference [33] method (d)
Present methodFigure 9
Identified resultsε=0.20.
(a)
Tikhonov regularization method (b)
Iterative Tikhonov regularization method (c)
Reference [33] method (d)
Present methodFigures6(a), 6(b), 6(c), and 6(d) illustrate that the identified force is obtained by adopting various different regularization methods under ε=0.01, respectively. Figures 7(a), 7(b), 7(c), and 7(d) show the identified force by adopting various different regularization methods under ε=0.05, respectively. The identified force can be seen from Figures 8(a), 8(b), 8(c), and 8(d), which gives that the identified force under ε=0.10, and Figures 9(a), 8(b), 8(c), and 9(d) offer the identified force by adopting various different methods under ε=0.20, respectively.It can be seen from Figures6, 7, 8, and 9 that the various different methods can identify the dynamic force. On the contrary, the effect of dynamic force identification is different.To explore the quality of dynamic force identification by adopting various different methods, we give the value of RE and CC and iterative steps, which are shown in Tables1, 2, 3, and 4. It is easy to see from Table 1 that RE and iterative steps of Figure 6(d) are smaller than that of Figures 6(a), 6(b), and 6(c), On the contrary, CC value presents opposite result. Table 2 shows that RE and iterative steps of Figures 7(a), 7(b), and 7(c) are larger than that of Figure 7(d), and the CC value presents opposite result. Table 3 shows that RE and iterative steps of Figure 8(d) are smaller than that of Figures 8(a), 8(b), and 8(c), but the CC value is the opposite. Table 4 offers that RE and iterative steps of Figure 9(d) are smaller than that of Figures 9(a), 9(b), and 9(c); nevertheless, the CC value presents opposite result.Table 1
Analysis of identification results (ε=0.01).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.2719 0.2011 0.1897 0.1832 CC 0.8939 0.9037 0.9145 0.9276 Iterative steps 22 15 10 8Table 2
Analysis of identification results (ε=0.05).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.3265 0.2327 0.1994 0.1871 CC 0.7940 0.8138 0.9033 0.9104 Iterative steps 30 22 12 9Table 3
Analysis of identification results (ε=0.10).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.3994 0.3011 0.2144 0.1957 CC 0.7291 0.8084 0.8996 0.9001 Iterative steps 37 28 14 10Table 4
Analysis of identification results (ε=0.20).
Evaluation metrics Tikhonov regularization Iterative Tikhonov regularization Ref. [33] method Present method RE 0.5673 0.4033 0.2938 0.2011 CC 0.4497 0.5988 0.8177 0.8846 Iterative steps 50 33 20 13Hence, compared with the other methods, we find from Tables1–4 that the proposed method has the smallest RE, the largest CC, and the least iterative steps, so the optimal value is ε=0.01.In addition, with the increase ofε, although RE and iterative steps increase, they do not affect the identification accuracy; that is to say, the small perturbation of the right-hand data will not affect the accuracy of the inversion results, which fully reflects that the present method has a good stability.Through the analysis, we draw the following conclusions that the present method is successful to identify dynamic force acting on conical pick and coal-seam.


## 6. Conclusions

In this paper, a novel improved Tikhonov regularization method is proposed to solve the ill-posed problem. Under the current framework, a new regularized filter function is constructed by using the singular system of compact operator, and we prove the stability of the proposed method effectively.An engineering example indicates that the proposed method has the smallest RE, the largest CC, and the least iterative steps and can obtain an efficient approximation of the actual force compared with the Tikhonov regularization method, iterative Tikhonov regularization method, and Reference [33] method. Meanwhile, the small perturbation of the right-hand data will not affect the accuracy of the inversion results, which fully reflects the present method has a good stability.Consequently, we can give a new conclusion that the proposed method can offer a unified stable solution for solving the ill-posed problem. Furthermore, we are confident that the proposed method will provide a reference for the next research work.



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*Source: 6095184-2019-04-02.xml*