Cutting Mechanism Rotor System Dynamic Characteristics of Cantilever Roadheader under Random Hard Rock Load

Abstract

Accurately mastering the power transmission characteristics of the cutting arm transmission shaft system is key to improving the reliability and working capacity of the cantilever roadheader. Based on the rigid–flexible coupling vibration characteristic modeling of the roadheader cutting arm, the vibration characteristics of different substructures in the transmission shaft system of the roadheader cutting arm were considered, the dynamic characteristic model was comprehensively constructed, and the numerical analysis was carried out with the parameters of the XTR260 tunnel hard rock roadheader to compare the vibration characteristics of the cutting head under different cutting conditions. The experiment was carried out by using an artificial concrete wall, and the measurement results verify the established dynamic model that lays the foundation for the dynamic design of a high-performance roadheader.

1. Introduction

The cantilever roadheader is an important piece of equipment in mine and tunnel construction. It provides the advantages of selective mining, strong mobility, less excavation, less disturbance to the ground, eliminating blasting vibration, reducing ventilation requirements, and lowering initial investment costs [1,2]. With increasing mechanization in the field of tunnel construction, the growing needs of hard rock crushing require roadheaders to become more heavy duty and high powered [3]. When the cutting head of the roadheader peels off the rock, it causes strong shock and vibration, which is then transmitted to the cantilever, turntable, fuselage, and traveling mechanism, posing a great threat to the reliable operation of the whole machine [4]. Vibration and impact specifically accelerate the wear and tear of connecting parts in the roadheader, reducing service life, causing failure of the control system [5], and affecting the smooth progress of roadheading work [6]. In addition, the vibration generated by the roadheader during operation carries a lot of energy, which can be regarded as a potential seismic source to detect mine shocks in advance [7]. Therefore, studying the vibration response of the cutting process holds practical significance in coming up with an optimal design of the roadheader’s structure, and the prevention and control of hazards.

Accurate modeling and analysis of vibration characteristics are not only an important factor in cutting performance optimization but also the basis for further automatic control of the roadheader. Yang et al. established an overall system model consisting of the electric motor model, hydraulic pump–motor model, torsional planetary group model and hybrid powertrain model. With the influence of the pick, the characteristics of its working model were simulated, and the dynamic response characteristics of the new hybrid cutting system and the traditional cutting system under impact load were compared [8]. Aiming to prevent instability of the roadheader during the process of cutting rock, Zhang et al. planned the cutting path through the gangue-containing section to reduce the vibration amplitude of the roadheader body [9]. Wang et al. conducted a modeling analysis of the vibration characteristics of a roadheader using the power reflux hydraulic transmission system to realize continuous variable-speed cutting, and concluded that the vibration of the motor and transmission system of the roadheader could be significantly attenuated [10]. To improve the vibration isolation effect of the roadheader electric control box, Li et al. deduced the overall vibration model considering the cutting effect, and obtained the stiffness-damping matching strategy of the vibration isolator on this basis [11]. Shen et al. proposed an error compensation method based on the vibration characteristics of the roadheader to reduce the attitude and position measurement errors of the roadheader caused by the complex cutting vibration [12].

During the boring process of the roadheader, various forms of complex angular vibration and linear vibration impact the roadheader body. To obtain its specific characteristics, it is necessary to carry out a dynamic analysis of the roadheader. Li et al. deduced the dynamic differential equation of the roadheader by using the Lagrange equation and obtained the mathematical model of the lateral and longitudinal vibration response of the roadheader [13]. By analyzing the crushing mechanism of the pick, Wei established a backlash-free vibration model between the pick and the tooth holder. Considering the randomness of the coal-rock force, the nonlinearity of the gap between the pick and the tooth holder, and the nonlinearity of the planetary gear transmission, he established the nonlinear dynamic torsional vibration model of the roadheader [14]. Zhao et al. employed Matlab to compile a simulation program for calculating the cutting head instantaneous load, simulated the impact load on the cutting head under the yaw condition, and imported it into the rigid–flexible coupled vibration model of the roadheader established in a collaborative simulation environment. The modal parameters of the system were obtained, and the easily excited mode shapes were pointed out through forced vibration analysis [15]. Li et al. established the roadheader kinematic differential equation based on the Lagrangian mechanics formula, used the virtual excitation method to obtain the vibration response of the roadheader for the random vibration generated by the cutting head, and obtained the displacement response curves of cutting head, cutting arm, and machine [16]. Huang et al. established a nonlinear dynamic model of the 13-degrees-of-freedom transverse torsional coupled cutting head-rotor-bearing system (CHRBS), which mainly considered the nonlinear coupling factors of spline clearance and bearing contact force, and carried out the dynamic analysis of CHRBS under time-varying loads [17].

Overview of the literature: The concentrated mass method is often used to establish the vibration theoretical model of the roadheader, and the roadheader cutting arm is simplified into a mass block. Although some studies consider the flexible deformation of the cutting arm in the vibration, they still analyze it as a deformable continuum, ignoring the power transmission characteristics of the cutting arm transmission shaft system, so they cannot accurately obtain the vibration response of the system. This study takes the cantilever roadheader as the research object, fully considers the power transmission characteristics of the driving shaft system of the roadheader cutting arm with vibration, and studies the roadheader vibration characteristics under different cutting conditions through theoretical modeling, numerical simulation and experimental measurement.

2. Random Cutting Load of Cutting Head

2.1. Random Load of Pick

When the cantilever roadheader cuts complex rock strata, the cutting resistance Z_i, traction resistance Y_i, and lateral resistance X_i [18] (shown in Figure 1) of the i-th pick on the cutting head are respectively

$$\left.\begin{array}{l}{Z}_i =Z\left[n\right] =Z_{\max }\left[n\right]k_{CZ}{k}_{bZ}{k}_{YZ}{k}_{\phi Z}\\ Y_i =Y\left[n\right] =Y_{\max }\left[n\right]k_{bY}{k}_{YK}{k}_{\phi Y}\\ X_i =X\left[n\right] =Z\left[n\right]k_X\end{array}\right\}$$

(1)

where, Z_max[n] is the average peak value of cutting resistance of the standard pick, Y_max[n] is the average peak value of traction resistance, k_CZ is the influence coefficient of pick arrangement, k_bZ is the width coefficient of the cut section in the z-direction, k_bY is the width coefficient of the cut section in the y-direction, k_YZ is the influence coefficient of cutting angle, k_φZ is the front edge shape coefficient in the z-direction, k_φY is the front edge shape coefficient in the y-direction, k_YK is the shape coefficient of cutting edge, and k_X is the ratio of lateral resistance to cutting resistance [19,20,21].

The components in the rock stratum are complex, and the load fluctuates violently. The random load of the pick follows the χ² distribution random process with two-degree freedom and the correlation coefficient $R_{\tau }\left(\tau \right) ={\sigma }_z{e}^{-\alpha \left(\tau \right)}$. The process Z_cp is obtained by the sum of two independent stationary random processes ξ₁(t) and ξ₂(t) with mathematical expectation μ = 0, standard deviation D = 1, and standard coherence function ${\gamma }_0 =e^{-\alpha \left(\tau \right)/2}$. Namely,

$$Z_{cp}\left(t\right) ={\xi }_1\left(t\right)+{\xi }_2\left(t\right)$$

(2)

The center value of the random process at the time of n⋅Δt is

$$Z_{cp}\left[n\right] ={\xi }_1^2\left[n\right]+{\xi }_2^2\left[n\right]-2,n =1,2,3,4\cdots \cdots$$

(3)

where

$${\xi }_1\left[n\right] =\sqrt{1-{\rho }_1^2\left[n\right]}\cdot {\eta }_1\left[n\right]+{\rho }_1\left[n\right]\cdot {\xi }_1\left[n-1\right],n =2,3,4\cdots \cdots$$

(4)

$${\xi }_2\left[n\right] =\sqrt{1-{\rho }_1^2\left[n\right]}\cdot {\eta }_2\left[n\right]+{\rho }_1\left[n\right]\cdot {\xi }_2\left[n-1\right],n =2,3,4\cdots \cdots$$

(5)

${\eta }_1\left[n\right]$ and ${\eta }_2\left[n\right]$ are two mutually independent normal random number sequences, ${\xi }_1\left[1\right] ={\eta }_1\left[1\right]$, ${\xi }_2\left[1\right] ={\eta }_2\left[1\right]$, ${\rho }_1\left[n\right]$ is a sequence of uniformly distributed random numbers on the interval (0,1).

Therefore, the random cutting resistance of the i-th pick at the time of n⋅Δt is

$$Z_i =Z\left[n\right] ={\sigma }_0\cdot Z_{cp}+{\overline{Z}}_i =\frac{{\sigma }_Z}{\sqrt{2n}}+{\overline{Z}}_i$$

(6)

where ${\overline{Z}}_i$ is the average cutting resistance of the i-th pick at the moment of n⋅Δt, σ₀ is the coefficient of the central value of the cutting resistance of the pick in the complex rock stratum, and σ_Z is the standard deviation of the cutting resistance load of the pick obeying the χ² distribution.

Due to the large correlation between cutting resistance and traction resistance, the random traction resistance of the i-th pick at time n⋅Δt can be determined by the correlation number R_ZY of Z_cp and Y_cp

$$Y_i =Y\left[n\right] =\frac{{\sigma }_Y}{{\sigma }_0}\left(R_{ZY}\cdot Z_{cp}\left[n\right]+\sqrt{1-R_{ZY}^2}\cdot {\eta }_3\left[n\right]\right)+{\overline{Y}}_i$$

(7)

where ${\overline{Y}}_i$ is the average traction resistance of the i-th pick at the time of n⋅Δt, σ_Y is the standard deviation of the random process $Y\left[n\right]$, V_Y is the variation coefficient, and ${\eta }_3\left[n\right]$ is the normal random sequence with mathematical expectation E_η3 = 0 and standard deviation D_η3 = 1.

The random lateral resistance X_i is expressed as

$$X_i =Z_i\left(\frac{C_1}{C_2+h}+C_3\right)\cdot \frac{h}{t}$$

(8)

where C₁, C₂ and C₃ are the influence coefficients of the chip diagram, h is the average chip thickness of the pick, and s is the average intercept spacing.

2.2. Random Load of Cutting Head

The working process of the roadheader is mainly divided into axial drilling, horizontal cutting, and vertical cutting. Under the three motion modes, the pick on the cutting head is subject to the random cutting resistance Z_i, random traction resistance Y_i and random lateral resistance X_i [22]. According to the force projection on each pick participating in the cutting along with the spatial coordinates (a, b, c) of the cutting head [23] (as shown in Figure 1), the instantaneous three-way random cutting load and a load torque of the cutting head can be derived thus

$$R_a ={\displaystyle \sum _{i =1}^{n_j}\left(Z_i\sin {\theta }_i+Y_i\cos {\theta }_i\right)}$$

(9)

$$R_b ={\displaystyle \sum _{i =1}^{n_j}\left(Y_i\sin {\theta }_i-Z_i\cos {\theta }_i\right)}$$

(10)

$$R_c ={\displaystyle \sum _{i =1}^{n_j}{X}_i}$$

(11)

$$M ={\displaystyle \sum _{i =1}^{n_j}{Z}_i{r}_{gi}}$$

(12)

where n_j is the total number of picks participating in cutting at the same time, θ_i is the circumferential angle of the i-th pick, and r_gi is the tooth tip rotation radius of the i-th pick.

3. Dynamics Model

3.1. Bearing Vibration Model

Two groups of bearings are arranged on the cutting main shaft. The outer ring of the rolling bearing is fixed on the cantilever shell, and does not rotate. The inner ring is fixed on the main shaft, and rotates with the main shaft at an angular speed ω. v_i is the linear velocity of the contact point between the bearing roller and the bearing inner ring, and v₀ is the linear velocity of the contact point between the bearing roller and the bearing outer ring.

$$v_i ={\omega }_{i}r$$

(13)

$$v_0 ={\omega }_{0}R$$

(14)

where r and R are the rolling radius of the bearing’s inner and outer rings, respectively. If the angular velocity of the bearing cage is equal to the angular velocity of the bearing roller revolution, then ω₀ = 0, ω_i = ω. So, the angular velocity of the cage

$${\omega }_d =\left(v_i+v_0\right)/\left(R+r\right) ={\omega }_{i}r/\left(R+r\right)$$

(15)

Then the rotation angle α_i of the i-th bearing roller is

$${\alpha }_i ={\omega }_{d}t+2\pi \left(i-1\right)/N_b \left(i =1,2,\dots ,N_b\right)$$

(16)

The vibration displacement of the rolling bearing center in two directions is x_v and y_v respectively; γ₀ represents the bearing clearance, and then the contact deformation between the i-th bearing roller and the inner and outer race raceway can be expressed as

$${\delta }_i =x_v\cos {\alpha }_i+y_v\sin {\alpha }_i-{\gamma }_0 \left(i =1,2,\dots ,N_b\right)$$

(17)

where N_b is the number of bearing rollers.

According to the nonlinear Hertz contact theory, F_i represents the contact pressure between the i-th bearing roller and the raceway in the case of rolling contact. The contact between the bearing roller and the raceway only produces normal positive pressure, and the nonlinear Hertz force is only valid when δ_i > 0.

$$F_i =k_c{\left({\delta }_i\right)}^{3/2}H\left({\delta }_i\right) =k_c{\left(x_v\cos {\alpha }_i+y_v\sin {\alpha }_i-{\gamma }_0\right)}^{3/2}\cdot H\left(x_v\cos {\alpha }_i+y_v\sin {\alpha }_i-{\gamma }_0\right)$$

(18)

where k_c is Hertz contact stiffness, H(x) is the Heaviside function. When the function variable is greater than 0, the function value is 1, otherwise, it is 0.

The nonlinear Hertz forces of bearing support force are

$$F ={\displaystyle \sum _{i =1}^{N_b}\left[k_c{\left(x\cos {\alpha }_i+y\sin {\alpha }_i-{\gamma }_0\right)}^{3/2}·H\left(x\cos {\alpha }_i+y\sin {\alpha }_i-{\gamma }_0\right)\right]}$$

(19)

$$\left\{{}_{F_{yj} = F\sin {\alpha }_i}^{F_{xj} = F\cos {\alpha }_i}\right. \left(j =1,2;i =1,2\dots ,N_b\right)$$

(20)

The support stiffness of the bearing in the x or y direction can be expressed as

$$\left\{{}_{k_y = d{F}_{yj}/dy}^{k_x = d{F}_{xj}/dx}\right. \left(j =1,2\right)$$

(21)

3.2. Dynamic Model of Cutting Head Rotor System

The cantilever transmission system of the roadheader provides stable rotating power for the cutting head to cut the rock stratum [24]. The simplified structure of the cutting head rotor system of the roadheader is shown in Figure 2. The cutting main shaft is supported by rolling bearings, and the cutting head is supported by a spline shaft and shell.

Considering the transverse and torsional vibration of the rotor system composed of cutting head, cutting spindle and bearing, the displacement vector of the system is

$$X ={\left[x_1,y_1,{\phi }_1,x_2,y_2,{\phi }_2,x_3,y_3,{\phi }_3\right]}^T$$

(22)

where x_i, y_i, φ_i (i = 1,2,…h) is the displacement in the x and y-axis and the rotation angle around the z-axis of mass point i. Applying Lagrange equation, according to the dynamic model in Figure 2, the dynamic differential equation of the cutting head rotor system is established as follows

$$m_1{\dot{x}}_1+k_{x1}{x}_1+k_b(x_1-x_2)+c_1{\dot{x}}_1+c_b({\dot{x}}_1-{\dot{x}}_2) =0$$

(23)

$$m_1{\ddot{y}}_1+k_{y1}{y}_1+k_b(y_1-y_2)+c_1{\ddot{y}}_1+c_b({\ddot{y}}_1-{\ddot{y}}_2) =0$$

(24)

$$I_1{\ddot{\phi }}_1+k_t({\phi }_1-{\phi }_2)+c_t({\dot{\phi }}_1-{\dot{\phi }}_2) =T_1$$

(25)

$$m_2{\ddot{x}}_2+k_{x2}{x}_2-k_b(x_1-x_2)+c_2{\ddot{x}}_2-c_b({\dot{x}}_1-{\dot{x}}_2)+k_h(x_2-x_h)+c_h({\dot{x}}_2-{\dot{x}}_h) =0$$

(26)

$$m_2{\ddot{y}}_2+k_{y2}{y}_2-k_b(y_1-y_2)+c_2{\dot{y}}_2-c_b({\dot{y}}_1-{\dot{y}}_2)+k_h(y_2-y_h)+c_h({\dot{y}}_2-{\dot{y}}_h) =0$$

(27)

$$I_2{\ddot{\phi }}_2-k_t({\phi }_1-{\phi }_2)+k_{gc}({\phi }_2-{\phi }_h)-c_t({\dot{\phi }}_1-{\dot{\phi }}_2)+c_{gc}({\dot{\phi }}_2-{\dot{\phi }}_h) =0$$

(28)

$$m_h{\ddot{x}}_h-k_h(x_2-x_h)-c_h({\dot{x}}_2-{\dot{x}}_h) =R_b$$

(29)

$$m_h{\ddot{y}}_h-k_h(y_2-y_h)-c_h({\dot{y}}_2-{\dot{y}}_h) =R_a$$

(30)

$$I_h{\ddot{\phi }}_h-k_{gc}({\phi }_2-{\phi }_h)-c_{gc}({\dot{\phi }}_2-{\dot{\phi }}_h) =M$$

(31)

In order to accurately study the dynamic characteristics of the cutting mechanism and the rotor system of the cantilever roadheader on the basis of the original model with the torsional degrees of freedom of the cutting head and cutting arm, the new model also takes the transverse and longitudinal degrees of freedom into account. Therefore, the time-varying stiffness characteristic of the main shaft bearings in the cutting arm and the bending-torsion coupling characteristic of the main shaft can be reflected in the new model, so as to ensure the theoretical basis of the cutting mechanism rotor system dynamic analysis. The above equation presents a complex mathematical model [25,26]. It is a strong nonlinear system. After dimensionless, the Newmark- β Method is used for a numerical solution.

4. Results

This study takes the XTR06/260 roadheader as an example for calculation and analysis, and the parameter settings are shown in

Table 1. Parameters of the cutting mechanism rotor system of XTR06/260.

Parameters	Value
Spline teeth number of cutting spindle z	49
Spline modulus m (mm)	5
Spline meshing stiffness kgc (N·m·rad−1)	3.52 × 108
Spline meshing damping ratio ξgc	0.1
Cutting spindle weight m1, m2 (kg)	350, 469
Moment of cutting spindle inertia I1, I2 (kg·m2)	3.19, 4.06
Bending stiffness of cutting spindle kb (N·m−1)	6.14 × 108
Torsional stiffness of cutting spindle kt (N·m·rad−1)	2.39 × 107
Damping ratio of cutting spindle	0.03
Picks number of cutting head	38(Number of helices is 2)
Weight of cutting head mh (kg)	1307
Moment of cutting head inertia Ih (kg·m2)	93.53
Torsional stiffness of cutting head (N·m·rad−1)	1.24 × 108
Contact stiffness of rolling bearings kc1, kc2 (N·m−1)	6.52 × 108, 8.34 × 108
Rolling bearing clearances γ01, γ02 (m)	3 × 10−5, 2 × 10−5
Damping ratio of rolling bearing ξc	0.02
Number of bearing rollers Nb1, Nb2	24, 31

.

The rotation speed of the roadheader cutting head is inversely proportional to the output cutting force. To ensure that the cutting head has reasonable cutting force under different conditions, the dynamic response of the cutting head rotor system will be discussed. For the rock, the Protodyakonov coefficient f_p is 6 and 10.

Based on the commercial software MATLAB, the horizontal cutting rock stratum of the roadheader is set to simulate the random load of the cutting head under the following two typical working conditions: (1) f_p = 6, the cutting head rotate speed n = 55 r/min, the speed frequency f_r = 0.92 Hz, the spline meshing frequency f_m = 45.08 Hz, and the swing speed v_b = 1.5 m/min; (2) f_p = 10, the cutting head rotate speed n = 27 r/min, speed frequency f_r = 0.45 Hz, spline meshing frequency f_m = 22.05 Hz, and cutting head swing speed v_b = 1 m/min.

For the cutting head rotor system, each component has the typical characteristics of nonlinear vibration accompanied by instantaneous impact. The vibration acceleration of the cutting head is greater than that of the cutting spindle. According to the vibration transmission path from the cutting head to the cutting spindle, the cutting head is close to the excitation source and the vibration acceleration is large. In addition, the rolling bearing rotation at the rear of the cutting head rotor system introduces a large number of high-frequency components, which also make the vibration acceleration less than that at the front. Since most of the working time of the roadheader is in the cross-cutting condition, the vibration characteristics of the cutting head under the cross-cutting condition are mainly analyzed below. In the characteristic frequency analysis, frequency components with large power spectrum amplitude were mainly selected because these frequency components have the greatest impact on the vibration of the cutting mechanism, and the influence of the same frequency components on the cutting mechanism vibration in the x and y directions was considered.

Figure 3 is the time-domain curve and spectrum of cutting head vibration acceleration in x and y directions when f_p = 6. Under the excitation of rock breaking dynamic force, the time-domain curve of cutting head vibration acceleration shows irregular yet strong oscillation. The vibration frequency is mainly concentrated in low frequency, and the frequency is complex. The main characteristic frequencies in the x-direction are 1.95 Hz, 5.86 Hz, 9.12 Hz and 50.81 Hz respectively, while the main characteristic frequencies in the y-direction are 1.95 Hz, 3.91 Hz, 9.12 Hz and 25.4 Hz respectively, and their vibration frequencies are close to the frequency doubling of the rotation frequency of the main shaft. This frequency distribution feature is related to the distribution of the pick. There are 38 picks on the cutting head of the analysis model, most of which are arranged on multiple helixes. The top of the cutting head is more densely arranged than the bottom pick, which may produce a vibration frequency that is multiple times the conversion frequency.

Figure 4 is the time-domain curve and spectrum of cutting head vibration acceleration in x and y directions when f_p = 10. Due to the hard rock and the strong impact characteristics of rock breaking force, the maximum vibration acceleration exceeds 100 m/s² at a few time points. The vibration frequency is still concentrated in the low frequency, the main frequencies in the x-direction are 0.97 Hz, 1.93 Hz, 2.9 Hz, 9.35 Hz and 22.53 Hz respectively, the main frequencies in the y-direction are 0.97 Hz, 1.93 Hz, 2.58 Hz, 11.92 Hz and 24.17 Hz respectively, and their vibration frequencies are close to the frequency doubling of the spindle rotation frequency.

Figure 5 and Figure 6 are the vibration phase diagram and Poincaré section of the cutting head under two kinds of hardness respectively. It can be seen from the figures that the phase diagram trajectories of the cutting head rotor system are composed of a large number of approximate ellipses, and the phase diagram trajectories intersect, but the phase diagram trajectory curves are not smooth due to severe vibration, discontinuous speed, and displacement. There are many scattered points in the Poincaré section, which indicates that there is chaotic behavior in the vibration of the cutting head rotor system.

5. Experiments

5.1. Experimental Setup

The experimental object XTR260 roadheader has a mass of 90 t and cutting power of 200~260 kW. The loading is realized by cutting the artificial concrete wall with protodyakonov coefficient f_p = 6. The speed of the cutting head is set to 55 r/min, the swing speed of the cutting head is 1.5 m/s, and the size of the artificial concrete wall is 6 m × 2.8 m × 1.2 m. The experimental site is shown in Figure 7.

5.2. Test Devices

The vibration signal of the roadheader during operation is collected through sensors, which include DONGHUA 1A313E three-way sensors and LZDL1-Y vibration sensors. The test system consists of sensors, Siemens LMS SCADAS data acquisition system, and the LMS test lab software. The sensors’ arrangement is shown in Figure 8. Since it is difficult to install the sensors stably on the cutting head, the sensors for measuring the cutting head are installed on the cantilever shell close to the cutting head without rotation.

5.3. Experimental Results

As the cutting mechanism of the roadheader adopts a two-stage planetary gear reducer, many gears mesh with each other. The time-varying meshing stiffness, tooth side clearance, meshing impact, and design and manufacturing error of the gears cause internal excitation and vibration of the cutting head. Under the no-load condition, the vibration fluctuation is relatively small, and the vibration of each component is stable, which is a quasi-periodic form. Due to the long cantilever of the cutting mechanism, the three-dimensional vibration amplitude of the cutting head is the largest, as shown in

Table 2. The peak vibration acceleration data in the time domain measured in the experiment.

Measuring Point	Direction	No-Load Condition			Horizontal Cutting Condition
		Maximum Value (m·s−2)	Variance (m2·s−4)	Valid Value (m·s−2)	Maximum Value (m·s−2)	Variance (m2·s−4)	Valid Value (m·s−2)
cutting head	x	11.7	24.3	4.93	32.9	44.5	6.67
	y	11.9	25.0	5.00	36.6	46.8	6.84
	z	10.4	23.4	4.83	21.1	28.8	5.37
cutting arm	y	3.3	0.3	0.51	13.5	7.5	2.74

.

It can be seen from Table 2 that the effective value of three-dimensional vibration of the cutting head under no-load condition is close. Under the cross-cutting condition, the vibration in y direction is the largest, followed by x direction, and the vibration in z direction is the smallest. According to the cutting theory of the roadheader, the three-dimensional force on the cutting head has great correlation. It can be seen from

Table 3. Acceleration power spectrum eigenfrequencies of the cutting head in the horizontal cutting experiment.

No.	Measuring Point	Eigenfrequency Point
(a)	x-direction of the cutting head	(0.8, −36.8), (1.9, −38.4), (3.4, −32.5), (8, −36.4), (9, −28), (50, −3.5), (250, −26)
(b)	y-direction of the cutting head	(0.8, −31.7), (1.9, −34.1), (3.4, −37.4), (8, −23.2), (9, −18), (50, −2.9), (250, −30)
(c)	z-direction of the cutting head	(0.8, −42.3), (3.1, −37.4), (3.4, −32.5), (8, −16.7), (9, −10), (50, −3.7), (250, −27)
(d)	y-direction of the cutting arm	(1.3, −36.1), (1.9, −34.5), (3.4, −44.7), (7, −27.9), (9, −27.4), (50, −32), (230, −43.5)

that the vibration frequency composition of the cutting head in three directions are basically the same, which is in line with the actual situation of the cutting mechanism.

Under the cutting condition, with the excitation of the rock breaking dynamic random load, the time-domain curve of cutting head vibration acceleration shows irregular strong oscillation, as shown in Figure 9. The vibration frequency is mainly concentrated in low frequency, and the frequency is complex, as shown in Figure 10. Its characteristic frequency points are shown in Table 3. The main characteristic frequencies in x and y directions are the same, which are 1.9 Hz, 3.4 Hz, 9 Hz and 50 Hz respectively. Their vibration frequencies are close to the frequency doubling and mixing of the main shaft frequency. Among them, 50 Hz is the mixing of f_m + 5 f_r, with a large amplitude, and its vibration of 5 times the frequency of 250 Hz is also prominent. This is because the sensors are installed near the spline of the cutting head, and cutting the main shaft results in a large mixing vibration amplitude of spline meshing frequency and frequency conversion. This frequency component is also prominent in the no-load test. The measuring point of the cutting arm is far away from the cutting head of the excitation source, and not at the spline location between the cutting head and the main shaft. In addition, the randomness of load and human operation makes the power spectrum amplitude at 50 Hz suddenly decrease as compared with the three-dimensional power spectrum amplitude of the cutting head.

6. Conclusions

It can be seen from the measured and simulated curves that the dynamic acceleration curve obtained by the multi-degree-of-freedom dynamic equation of the cutting head rotor system under the condition of random load force is in line with the change law of the measured curve, and fluctuates around a close mean value. The frequency characteristics of the simulated curve and the measured curve are also relatively close, which verifies that the established dynamic model conforms to the actual situation. The simulated random load can replace the measured load.

In this study, the dynamic mechanical response of the cutting head rotor system under different kinds of rock hardness has been analyzed. The cutting head rotor system displays chaotic motion under random load, with complex frequency components and obvious frequency amplitude fluctuations. The characteristic frequency is mainly the frequency conversion and frequency doubling of the spindle. In the design and development of a cantilever tunnel roadheader for hard rock, the vibration characteristics and rigid–flexible coupling vibration of different substructures of the system should be considered when modeling the dynamic theory of the cutting mechanism, to improve the reliability and safety of the roadheader during its service. The dynamic analysis method of the cutting head rotor system provides a theoretical basis for vibration reduction and dynamic design of the roadheader.

The dynamic model refinement also provides a new theoretical basis for the dynamic design and accurate measurement [27,28] of other cutting construction machinery, such as coalcutters. This also amounts to important basic work in the cooperative operation of heavy engineering equipment [29] and innovation of intelligent construction technology [5].