Influence of Two Mass Variables on Inertia Cone Crusher Performance and Optimization of Dynamic Balance

: Inertia cone crushers are widely used in complex ore mineral processing. The two mass variables (fixed cone mass and moving cone mass) affect the dynamic performance of the inertia cone crusher. Particularly the operative crushing force of the moving cone and the amplitude of the fixed cone are affected, and thus the energy consumption of the crusher. In this paper, the process of crushing steel slag is taken as a specific research object, to analyze the influence of two mass variables on the inertia cone crusher performance. A real-time dynamic model based on the multibody dynamic (MBD) and the discrete element method (DEM) is established. Furthermore, the influence of the fixed cone mass and moving cone mass on the operative crushing force, amplitude and average power draw are explored by the design of simulation experiments. The predictive regression models of inertia cone crusher performance are obtained using response surface methodology (RSM). After increasing the fixed cone mass, the optimized amplitude, average power and moving cone mass are decreased by 37.1%, 33.1% and 10%, respectively, compared to without the adjustment. Finally, a more effective dynamic balancing mechanism of inertia cone crusher is achieved, which can utilize the kinetic energy of a balancer, and minimize the mass of the fixed and moving cone. The fixed cone mass and moving cone mass of a balancing crusher are decreased by 78.9% and 22.8%, respectively, compared to without the balancing mechanism.


Introduction
Inertia cone crushers are widely used in the secondary and tertiary crushing stages of complex ore processing, such as the comprehensive recovery of steel slag [1,2]. A mantle rotates and swings in the crushing chamber, which is due to an eccentric vibrator transferring the rotational motion to the main shaft. As it flows downward between the mantle and concave, the ore particle is crushed several times. The total crushing force for the inertia cone crusher is provided by the eccentric vibrator and mantle. As the concave is located above several rubber absorbers, the concave can move and roll in three-dimensional space. Therefore, the operative crushing force is less than the theoretical force, and the energy consumption increases [3]. The subgroup including the concave and subsidiary components is defined as the fixed cone, and the subgroup including the mantle and eccentric vibrator is defined as the moving cone. At the condition of keeping other parameters invariable, the fixed cone mass and moving cone mass have a great impact on the operative crushing force, amplitude of fixed cone and energy consumption, whereas the increase in moving cone mass can increase the theoretical crushing force and amplitude directly. The decrease of fixed cone mass can decrease the operating crushing force and increase energy consumption indirectly. Here, from a manufacturer's perspective, how to determine the two mass parameters is a key problem. Furthermore, at the guarantee of reasonable crushing force achievement rate and energy consumption, minimizing the mass of the fixed and moving cone is one of the main ways to reduce manufacturing cost.
Savov et al. [4] and Xia et al. [5] contributed to an initial mathematical modeling of the crushing force achievement rate. However, the models do not have the ability to take into account the effect of ore particles on the crusher. Additionally, no research regarding the mass of inertia cone crusher optimization has yet been published to our knowledge. Cleary et al. [6] and Andre et al. [7] studied the effect of feed properties (material strength, particle friction) and machine controls (CSS, speed) on cone crusher performances (particle distribution, throughput, power) based on the particle replacement model (PRM) in the software EDEM. Chen et al. [8] took the throughput and crushing force as the multiobjective optimization, and studied the effect of the parameters of the crushing chamber and speed on gyratory crusher performances based on the bonded particle model (BPM). These above studies can provide the main ways to optimize the variables (operation, chamber shape and feed properties). However, these simulation methods do not have the ability to take into account the effect of inertia parameters (fixed cone and moving cone mass) on operation performance (crushing force, amplitude and average power) for an inertia cone crusher. Cheng et al. [9] provided a powerful method whereby the coupling multi-body dynamics (MBD) [10,11] and discrete element method (DEM) [12,13] simulate the crushing behavior response for an inertia cone crusher. Currently no research using coupled MBD-DEM dynamic models for an inertia cone crusher has been published, except for our publication [9]. Barrios et al. [14] and Chung et al. [15], regarding coupled MBD-DEM models, provided useful attempts for high-pressure grinding rolls (HPGR). Furthermore, at the same industrial scale, the mass of the inertia cone crusher is much heavier than other cone crushers, such as hydraulic crushers and spring crushers. The reason is that the eccentric vibrator leads to the violent vibration of the crusher and the increase of energy consumption. Znamenskll et al. [16], regarding the dynamic balance of an inertia cone crusher, put forward a preliminary design. However, the dynamic balance design neither completely counteracts the excitation force nor utilizes the kinetic energy of the balancer. Ren et al. [17] utilized the kinetic energy of the balancer in the design. Nevertheless, the dynamic balancing mechanism is unstable and cannot completely counteract the excitation force, so it is difficult to widely use it in industry.
As such, this paper takes the process of crushing steel slag as the analysis object, and the crushing force achievement rate, amplitude of the fixed cone and average power of the drive shaft are explored by the MBD-DEM coupling method. Moreover, we propose an approach in which the use of response surface methodology (RSM) and analysis of variance (ANOVA) optimizes the fixed and moving cone mass to achieve the optimum operation performance. The results show that the operation performance is greatly improved by increasing the fixed cone mass, which increases the manufacturing cost. Finally, in order to reduce the manufacturing cost for manufacturers and the running cost for users, a more effective dynamic balancing mechanism of inertia cone crushers is achieved. Such a mechanism not only utilizes the kinetic energy of the balancer, but also minimizes the mass of the fixed cone and the moving cone.

Inertia Cone Crusher Theory
The inertia cone crusher consists of a main frame, a concave, a mantle, rubber absorbers, a main shaft and an eccentric vibrator. The ore particles fall from the feed chute to the crushing chamber; then, they are squeezed by the mantle and other particles. Finally, the particles are discharged from the discharge zone. Figure 1 shows a vertical cross-section and a simplified MBD model for an inertia cone crusher, where α and θ are the mantle angle and the nutation angle, respectively. l0, l1 and l2 are the axis of the crusher, concave and mantle, respectively. B1 is the fixed cone, which is fixed to the main frame. B2 is the moving cone, which is fixed to the main shaft. O1 is a spherical joint; O2 is a spherical joint between B1 and the globe bearing (B4); O3 is a cylindrical joint between B2 and the eccentric vibrator (B3); O4 is a planar joint between B3 and B4; O5 is a ball-pin joint between B3 and the connecting shaft (B5); O6 is a universal joint between B5 and the drive shaft (B6); O7 is a revolute joint between B6 and the ground (B0).

Crusher Dynamic Model Using MBD
The generalized coordinate qi of the rigid body Bi is shown in Equation (1).
where ri is the matrix of independent position variables (xi, yi, zi), and Λi is the matrix of independent angle variables (ψi, θi, φi). According to [18], the kinetic formula for Bi without any joint equations is derived in Equation (2).
where q̈i is the generalized acceleration of qi, and Mi and Qi can be expressed, respectively: where mi and Ji are the mass and the inertia matrix for Bi, respectively. E and 0 denote the identity and null matrix, respectively. Fia and Tia denote the equivalent absorber forces and torques, Fif and Tif denote the joint friction forces and torques, Fig is the gravity of Bi, Fip and Tip denote the equivalent particle forces and torques, and Di is the coordinate matrix. The formula of MBD for the inertia cone crusher without any joints can be shown as: 1  2  3  4  5  6  T  T  T  T  T  T T  1  2  3  4  5  6 diag( , , , (4) where q̈ is the generalized accelerations of the multi-body system. The joint Oj (j = 1,2,…,7) equations and driving motion constraints for inertia cone crusher are expressed as:


According to Equation (5), the velocity and acceleration equation can be expressed as: where Φq and Φt is the Jacobian matrix for Φ(q,t). According to the Lagrange multiplier λi (i =1,2,…,26), the formula of MBD for inertia cone crusher is derived in terms of the Lagrange multiplier matrix λ and generalized coordinate matrix q, and can be shown as:

BPM Theory
The BPM consists of bonding a packed distribution of particles, forming a breakage cluster [19]. As shown in Figure 2, a parallel bonding beam is created between each particle in contact, so the forces (torques) on the bonding beam are calculated from Equation (8) and Equation (9). BPM has been used in simulating the crushing behavior of particles [13,20,21] where Fbn, Tbn, Fbt and Tbt denote the bond normal force, normal torque, bond tangentialdirected force and torque, respectively. kbn and kbt are the normal and tangential stiffness per unit area. A and J are the area of the parallel bond cross-section and polar moment of inertia, respectively.
where Rb, Vn, Vt, ωn, ωt, and δt are parallel bond radius, normal velocity, tangential velocity, normal angular velocity, tangential angular velocity and time step, respectively. The maximum normal and tangential stress are calculated according to Equation (10).
where σbc and τbc are critical normal and critical shear strength, respectively. If the maximum stress exceeds the critical strength, the bond beam will disappear. The particle interaction depends on the Hertz-Mindlin contact model [22].

BPM Calibration
The feed material is a steel slag which has a complex shape and size distribution in our industrial experiments of the inertia cone crusher. The feed particle size range is 50 mm to 70 mm, and a 3D model of slag is shown in Figure 3. When using BPM for simulating the crushing behavior of steel slag, the key is to make sure that the relevant DEM parameters of the particle are calibrated. Therefore, the Hertz-Mindlin contact model parameters were determined by the uniaxial compression deformation and repose angle test, as shown in Figure 4. Table 1 shows the contact parameters.  Figure 3. The schematic illustration of the slag particle using DEM: (a) BPM of the slag particle is formed by the EDEM software, and (b) a realistic slag shape is used creating the packing structure of a normal distribution.   The BPM-relevant parameters were determined by the Brazilian test and simulation experiment, as shown in Figure 5. The calibration method had been described in detail in our publication [13]. By comparing the tensile strength simulation with the experiment values (10.6 MPa), we directly provide the BPM parameters that are shown in Table 1.

The Solution of the Coupled MBD-DEM Model in Software
Combining with Sections 2.1-2.3, Figure 6 shows the simulation flowchart of the inertia cone crusher using the MBD-DEM coupling method. The MBD of the geometries is calculated by RecurDyn software, and the DEM of the particles is calculated using EDEM. As slag clusters flow downward between the mantle and the concave, the size becomes successively smaller in the software EDEM, as shown in Figure 7. The operative performances (the operative crushing force, amplitude and average power) for the inertia cone crusher are obtained by the software RecurDyn.

Influencing Factors and Performance Goals
Basing on Section 2 and the previous publications [3][4][5], the fixed cone mass (FM) m1 and the moving cone mass (MM) m2 are taken as the influencing factors. The operative crushing force Fo is less than the theoretical crushing force Ft under normal operating conditions. The theoretical crushing force of the inertia cone crusher is provided by the moving cone when the fixed cone is not moving. As such, the crushing force achievement rate ηf is taken as one of the performance goals, according to Equation (11).
when the inertia cone crusher works, the fixed cone will move horizontally and deflect around a rotation point. As such, the amplitude of the rotation point displacement and the deflection angle can give a good indication for the vibration characteristics of the crusher. The theoretical and experimental results indicate that the amplitude As and deflection angle γd have a significant positive correlation [22,23]. Ignoring the deflection angle γd, the amplitude As is taken as a performance goal. Besides this, the two mass variables have a great impact on the energy consumption of a crusher, and the average power draw of drive shaft Pa is taken as a performance goal. Based on the response surface methodology (RSM) [24], the influence of the two mass variables on the performance goals is modeled with the SPSS software. The corresponding predictive regression models can be expressed as Equation (12), and the simulated experiment scheme is listed in Table A1.
where fη, fA, and fP are the predictive models of crushing force achievement rate, amplitude and average power, respectively. Because the driving speed is determined by the productivity, the driving speed of the crusher (model: GYP1200) should not exceed 550 rpm. As such, the driving speed is set to 550 rpm in this paper.

Crushing Force Achievement Rate Analysis
The influence of FM and MM on the crushing force achievement rate, and the prediction regression curves are shown in Figure 8. The theoretical force Ft is only determined by MM, and Figure 8a shows the influence of MM on the theoretical force Ft. Figure 8b shows the influence of FM on the crushing force achievement rate under three kinds of MM (3.5 t, 4.5 t, and 5.9 t), which indicates that the crushing force achievement rate ηf increases significantly with increasing FM. Figure 8c shows the influence of MM on ηf under three kinds of FM (15 t, 25 t, and 39 t), which indicates that ηf gradually decreases with increasing MM. At the 0.05 significance level, we can find that the influence of FM and MM on ηf is significant using the quadratic regression function.  (d) Figure 8d shows the influence of the interaction between FM and MM on ηf using the response surface methodology (RSM), and the corresponding results of ANOVA for predictive regression models are shown in Table 2. The prediction regression model of the crushing force achievement rate is expressed as:  The corresponding ANOVA for predictive regression coefficients is shown in Table  A2. At the 0.05 significance level, we can find that the linear term of MM has the weakest impact on ηf, and the p-value is more than 0.05. As such, the linear term of MM is ignored in the prediction regression model of crushing force achievement rate. As FM increases or MM decreases, ηf can increase. This is because the increase in FM or the decreases in MM will decrease the amplitude of the fixed cone, and increase the eccentric distance of the moving cone. However, when ηf is over 90%, the increase in ηf is very small with increasing FM, and for the different moving cone mass, the fixed cone mass required to reach 90% (ηf) is different.

Amplitude Analysis
The influence of FM and MM on the amplitude of the fixed cone, and the prediction regression curves, are shown in Figure 9. Figure 9a shows the influence of FM on the amplitude under three kinds of MM (3.5 t, 4.5 t, and 5.9 t), which indicates that the amplitude As decreases with increasing FM. Figure 9b shows the influence of MM on the amplitude under three kinds of FW (15 t, 25 t, and 39 t), which indicates that the amplitude As increases with increasing MM. Figure 9c shows the influence of the interaction between FM and MM on the crushing force achievement rate ηf using RSM, and Table 3 indicates that the prediction model of As has a good fitness using the quadratic regression function with the 0.01 significance level. The prediction regression model is shown in Equation (14).   It can be found that the quadratic term of MM has the weakest impact on amplitude, and the p-value is more than 0.01 from Table A2. So, the quadratic term of MM is ignored in the predictive regression model of amplitude. As FM increases, the amplitude As can decrease, and the fixed cone will be more difficult to move for a constant theoretical crushing force. Comparing Figures 8d and 9c, it can be found that when the ηf is over 90%, the decrease in amplitude As is very small with the increasing FM. However, with the change of MM, the amplitude As will change significantly for a constant FM.

Average Power Analysis
The influence of FM and MM on the average power draw, and the prediction regression curves, are shown in Figure 10. Figure 10a shows the influence of FM on the average power under three kinds of MM (3.5 t, 4.5 t, and 5.9 t), which indicates that the average power Pa decreases with increasing FM. Figure 10b shows the influence of MM on Pa under three kinds of FM (15 t, 25 t, and 39 t), which indicates that the average power Pa increases significantly with increasing MM. Figure 10c shows the influence of the interaction between FM and MM on Pa using RSM, and Table 3 indicates that the prediction model of average power Pa has a good fitness using the quadratic regression function with the 0.05 significance level. The prediction regression model is shown in Equation (15). Table A2 shows that the quadratic term of MM has the weakest impact on the average power Pa, and the p-value is more than 0.05.
Comparing Figures 9 and 10, we can see that as FM and MM increase, the variation in average power Pa is similar to that of the amplitude As. The reason for this is that as As increases, the energy consumption of the rubber absorbers can increase, and the kinetic energy of the steel slag particles also increases, resulting in the increase in friction heat energy.

Optimization Results
Combining with the above sections, it can be seen that when the crushing force achievement rate ηf is over 90%, the decrease in amplitude As and average power Pa are very small with increasing FM. Table 1 shows the FM is 20 t and the MM is 5.5 t for the industrial inertia cone crusher (model: GYP1200), and the operative crushing force Fo is 697.92 kN from the simulated experiment (Table A1). If the prediction values of Fo and ηf are 697.92 kN and 90%, the MM is set to 4.95 t (Figure 8a). According to Equation (13), the optimized fixed cone mass is 30.54 t.
According to Equations (14) and (15), the optimized amplitude As and average power Pa are 5.29 mm and 125.38 kW, respectively. Compared with the simulated experiment (Table A1), it can be seen that the optimized amplitude, average power and MM are decreased by 37.1%, 33.1% and 10.2%, respectively. Finally, we can see that the decrease in amplitude can effectively decrease the average power, and increase the crushing force achievement rate. However, the optimized FM is increased by 52.7%, so the optimized mass of the inertia cone crusher is about three times as much as the hydraulic cone crusher or spring crusher for the same industrial scale, which increases the manufacturing cost. In this paper, we design a more effective dynamic balancing mechanism for the inertia cone crusher, which decreases the amplitude and minimizes the mass of the inertia cone crusher.

Mechanics Principle of Dynamic Balancing
The total crushing force of the single exciter GYP-type inertia cone crusher is provided by the eccentric vibrator and the mantle, as shown in Equation (16): where Fcr is the total crushing force, Fice is the equivalent centrifugal force generated by the mantle, and Fucve is the equivalent exciting force generated by the eccentric vibrator.
The inertia cone crusher with the dynamic balancing mechanism is shown in Figure  11a. The dynamic balancing mechanism is mainly composed of a balancer and feedback mechanism. A balancer is added to the GYP-type inertia cone crusher, which can similarly rotate on the opposite side of the vibration exciter. In this way, the vibration of the crusher can be minimized to decrease the mass of the fixed and moving cone. Through a feedback mechanism to increase the crushing force, the kinetic energy of the balancer can be utilized efficiently. The planar layout of forces acting on the mantle cone is shown in Figure 11b. Fic, Fucv and Fbec are the centrifugal forces generated by the mantle, eccentric vibrator and balancer, respectively.
where mic, mucv and mbec are the mass of the mantle, eccentric vibrator and balancer, respectively; eic, eucv and ebec are the eccentric distance of the mantle, eccentric vibrator and balancer, respectively; ω is the drive shaft speed. The value of Fbec should conform to Equation (18).
where Fuic is the equivalent force of the centrifugal force generated by the moving cone, and eui is the equivalent eccentric distance of the moving cone. Based on the lever principle, the vector force Fbeci can be expressed as where Fbeci is the force acting on the moving cone by the feedback mechanism, R1 is the constraint reaction of spherical joint O1, and Gc is the gravity of the moving cone. Compared with Equations (16) and (19), it can be seen that the inertia cone crusher with the dynamic balancing mechanism adds the feedback force compared with the single exciter crusher. Therefore, the dynamic balancing mechanism can obviously decrease the amplitude of the fixed cone and increase the crushing force. In this way, the mass of the fixed and moving cone is minimized, and the manufacturing cost decreases significantly.

Experimental Devices
Corresponding experiments are validated to verify the dynamic balancing mechanism. However, it is an impossible task for us to manufacture an industrial-scale inertia cone crusher with the dynamic balancing mechanism. In this paper, a laboratory prototype with the same dynamic balancing mechanism is developed. The amplitude, power draw and product size distribution of the laboratory crusher can be collected by some experimental devices, and the results are compared with the crusher without the dynamic balancing mechanism. The crusher without the dynamic balance can provide the same theoretical crushing force as the dynamic balance prototype. The experimental devices and dynamic signal acquisition systems are shown in Figure 12. The feed material is a 7.5-10 mm white marble, and the amplitude, power draw and product size distribution for the two driving speed levels (450 and 650 rpm) are compared in the following sections.

Amplitudes of Test Points
The displacements of two test point are sampled by displacement sensors in Figure  12a. The experimental data of two test points in the x direction are displayed for the drive speeds of 450 rpm and 650 rpm, as shown in Figure 13. In Table 3, the test data from the balancing crusher and without balancing mechanism are summarized. The results show that the amplitude and deflection angle of the balancing crusher are decreased by 80.6% and 64.2%, compared with the crusher without a balancing mechanism. Therefore, the good vibration reduction performance of the dynamic balancing mechanism is verified by experimental comparison.

Power draw and Product Size Distribution
The input power of the motor is sampled by an electrical parameter test instrument in Figure 12b. The experimental data are displayed for the drive speeds of 450 rpm and 650 rpm, as shown in Figure 14. Figure 14 shows that the average power of the balancing crusher is decreased by 20.9%, compared with the crusher without a balancing mechanism. Therefore, the dynamic balancing mechanism can effectively reduce the energy consumption and running cost of the inertia cone crusher. The product size distribution is collected by square sieves in Figure 12c. The experimental data are displayed for the drive speeds of 450 rpm and 650 rpm, as shown in Figure  15. The product size distribution displays a relatively good correspondence between the balancing mechanism and without the balance. It can be seen that the mechanism realizes the purpose of utilizing the inertia force and kinetic energy of the balancer.

Optimization Verification of Industrial-Scale Inertia Cone Crusher
To verify the optimization performances and two mass variables (FM and MM), the simulated experiments using MBD-DEM coupling are performed on the industrial-scale inertia cone crusher with the dynamic balancing mechanism. The fixed cone mass and the moving cone mass (including the dynamic balancing mechanism mass) are 6.44 t and 3.82 t, respectively. Compared with Section 3.4, the optimized crushing force achievement rate is over 95% using with the dynamic balancing mechanism. Furthermore, the optimized amplitude and average power are decreased by 33.1% and 10.2%, respectively, as shown in Figure 16. In Figure 16c shows the product size distributions for the case balancing mechanism and the case without are compared. It can be seen that the balance case is slightly finer than the without-balance case. Due to the amplitude, the average power and product size distribution have been improved, and the good crushing performance of the dynamic balancing mechanism is verified in the industrial-scale inertia cone crusher. Furthermore, the fixed cone mass and the moving cone mass are decreased by 78.9% and 22.8%, respectively. The dynamic balancing mechanism significantly reduces the manufacturing cost.

Conclusions
In the inertia cone crusher design, an inevitable problem concerns how to determine the two mass variables (fixed cone mass and moving cone mass), which can affect the crusher dynamic performances. Firstly, the crushing process of the inertia cone crusher is simulated using the MBD-DEM coupling. Predictive regression models, in which the two mass variables are taken as influencing factors, and the crushing force achievement rate, amplitude and average power are taken as the performance goals, are explored by the design of simulation experiments. In addition, it is found that when the achievement rate ηf is over 90%, the decrease in amplitude As and average power Pa, and the increase in ηf, are very small with increasing FM. Due to the optimized values of ηf and FM being 90% and 30.54 t, the optimized amplitude As, average power Pa, and MM are decreased by 37.1%, 33.1% and 10.2%, respectively, compared with the original crusher.
The optimized FM is increased by 52.7%, which increases the manufacturing cost. In this paper, a new and more cost-effective dynamic balancing mechanism of inertia cone crusher is achieved in order to reduce FM. The vibration reduction and inertia force utilization of the dynamic balancing mechanism are verified by the elementary prototype of the laboratory experiment. Moreover, the effect of FM and MM reduction is verified by the MBD-DEM simulation of the industrial inertia cone crusher. Compared with the without-balance case (Section 3.4), the amplitude, average power and product size distribution have been improved, and the FM and MM are decreased by 78.9% and 22.8%. As such, from a manufacturer's perspective, the manufacturing cost decreases significantly. In order to reduce the running costs for users, the future work will prioritize manually using the design of simulation experiments around the optimum drive shaft speed, the eccentric distance of eccentric vibrator and the discharge gap.

Conflicts of Interest:
The authors declare no conflict of interest.