# Mathematical Modeling and Multi-Criteria Optimization of Design Parameters for the Gyratory Crusher

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## Abstract

**:**

## 1. Introduction

- to obtain and solve the differential equation of the curve of the rational profile of the working chamber jaw of a single-roll gyratory crusher;
- to obtain analytical expressions for determining the speed and capacity of the considered type of the crusher;
- to develop an analytical mathematical model for determining the kinematic components of the roll load of a single-roll gyratory crusher;
- to optimize the parameters of the crusher working chamber.

## 2. Materials and Methods

_{i}—number of levels of the i parameter;

_{i}—sequence number of the value of the i parameter.

_{1}, k

_{2}, …, k

_{nc}}, where n

_{c}was the number of partial criteria.

_{j}(k

_{j})—membership function of a particular value of the j criterion for the best fuzzy set.

_{j}.

_{jB}, k

_{jM}—the largest and the smallest of the calculated values of the j criterion, respectively; α

_{j}—nonlinearity indicator that is selected by a person making the decision for heuristic reasons.

_{j}= 1, the membership function is linear, at α

_{j}> 1 it is concave, at 0 < α

_{j}< 1 it becomes convex. Obviously, the more stringent the requirements are to the approach of the given criterion to the local minimum k

_{jM}, the larger the α

_{j}parameter should be.

## 3. Results and Discussion

_{max}(approximately 300 mm for mine crushers), and keeping the piece during crushing in the crusher chamber. It is necessary to devise a calculation scheme to obtain the equations of the curve of the rational form of the jaw. This diagram should show the forces acting on a piece of rock in the crushing chamber on the parts of the roll and jaw. Choosing a rational shape for the jaw profile allows the equilibrium of a piece of rock in the crushing chamber to be achieved. This ensures retention of the piece of rock in the chamber during its crushing. The diagram of the interaction of the piece of rock with the roll and the jaw is shown in Figure 2, where R

_{R}is the radius of the roll and e is the eccentricity (e << R

_{R}).

_{1}is the normal component of the jaw reaction; F

_{fr}is the friction force of the piece against the jaw. Equilibrium is possible if the forces to be adjusted form a force triangle, as shown in Figure 2. This triangle shows the following:

_{fr}—coefficient of the rock friction against the jaw, β—angle of contact, i.e., the angle between the tangents to the surfaces of the roll and the jaw at the points of contact (at the recommended value of f

_{fr}= 0.3 β = 16.4°).

_{e}to the desired profile, so, while ${\Psi}_{\epsilon}=\frac{\pi}{2}+\phi -\beta $, let us present the following differential equation:

_{b}= R

_{R}+ d

_{max}− e.

_{b}considering the design or to treat it as one of the optimized parameters. Then, the value of the angle φ

_{e}is determined by the expression:

_{min}—minimum size of the crushed product piece.

_{fall}determining the dimensions of the “fall-out body”:

_{c}is the path DA of the movement of the particle located at point D, a

_{c}is the acceleration of the particle moving along the DA profile under the influence of gravity and the pressure of the overlying rock pieces. The duration of the rotor withdrawal from the jaw is equal to the time of half of the shaft turn ${t}_{o}=\frac{30}{{n}_{\mathrm{sh}}}$, where n

_{sh}is the shaft speed in rpm. Hence, using t

_{fall}and t

_{o}, we can derive an expression to determine the required shaft speed:

_{c}, it is possible to take a

_{c}= 0.5 (a

_{inc}+ g), where a

_{inc}is the acceleration of a particle moving under the influence of gravity, taking into account friction over the surface DA, which, in the first approximation, can be considered as an inclined plane, and g is gravity acceleration:

_{loos}= 0.5 … 0.6—coefficient of material loosening at the exit from the crusher, V

_{fall}—volume of the “fall-out body”, m

^{3};

_{1}—area of Figure ABCD; L

_{roll}—roll length.

_{b}, it is possible to determine the capacity of the crusher.

_{b}and end φ

_{e}.

_{b}shown in the figure has a positive value, and a negative value applies to angle φ

_{e}.

- -
- radial roll load for each of the jaws is proportional to the central angle of the sector in which the crushed rock mass is compressed;
- -
- resulting radial force for each of the working jaws is applied in the middle of the sector in which compression occurs;
- -
- the force line of action passes normal to the roll surface when determining the crushing force arm;
- -
- the line of action of the force passes through the axis of rotation of the shaft when determining the inclination angle of the crushing force, so the error thus introduced is small, since the eccentricity e is significantly less than the radius of the roll R
_{R}.

_{F}= M

_{a}+ M

_{b}, where M

_{a}, M

_{b}are moments of forces a and b acting on the side of the jaws, is:

_{a}, F

_{b}are forces acting on the roll on the part of jaws a and b; ρ

_{a}, ρ

_{b}—arms of these forces acting relatively the shaft rotation axis:

_{a}, α

_{b}are inclination angles of force vectors F

_{a}and F

_{b}to the horizontal axis x (Figure 3).

_{a}and F

_{b}on the coordinate axis is determined by the expression:

_{a}, F

_{b}, α

_{a}and α

_{b}are determined as functions of the rotation angle of the shaft φ for the characteristic sections into which one shaft revolution is divided.

_{F}is the specific force acting on the roll relative to the single central angle:

_{max}—maximum force acting on the roll, which arises when the material is compressed over the entire surface of the jaw.

_{max}(Figure 4a), and by the value M

_{max}= F

_{max}e for the moments (Figure 4b).

_{2}—number of the harmonics taken into account (the analysis shows that for a typical mine crusher it is possible to take N

_{2}= 4). The values of the coefficients a

_{i}and b

_{i}in relation to a typical mine crusher, where φ

_{b}= 60° and φ

_{e}= −34°, are given in Table 2.

_{i}and b

_{i}depend on the mutual position and size of the jaws, i.e., on the angles φ

_{b}and φ

_{e}. The task of selecting values of these angles and shaft eccentricity, at which the minimum amplitudes of the kinematic load components and the maximum capacity of the crusher would be achieved, is of great interest.

_{m}and ν

_{F}, which require minimization (k

_{1}, k

_{2}and k

_{3}, respectively), are taken as the target functions. Therefore, the studied optimization problem is a multi-criteria problem.

_{b}and the eccentricity e (P

_{1}and P

_{2}, respectively) as objective variables in the considered optimization problem. The parametric restrictions are imposed on the values of the optimized parameters in terms of the possibility of their technical implementation (P

_{i min}≤ P

_{i}≤ P

_{i max}). The functional limitation on the speed of the interaction between the roll and a piece of rock under the friction spark condition plays an important role for the mine crushers considered:

_{add}= 0.3–0.4 m/s.

_{min}= 0.07 m, d

_{max}= 0.03 m, R

_{B}= 0.03 m, f

_{fr}= 0.3, φ

_{loos}= 0.55, L

_{roll}= 0.82 m, v

_{add}= 0.3 m/s. Table 3 shows the 10 best parameter combinations by value of the generalized criterion. The specified values are determined by the technical design assignment for the crusher and depend on the properties of the destroyed rock, on the specified overall dimensions of the machine and on the safety regulation requirements for working in the mine. For example, the friction coefficient f

_{fr}is a dimensionless scalar value equal to the relation of the force of the rock friction against the steel surface and the force of pressing the rock against the support surface. Numerous studies, conducted by other authors over the years, show that, in the case of most dry rocks, the coefficient of friction against steel f

_{fr}is 0.3 … 0.6. If the contacting pairs interact in a humid and dusty environment, the coefficient value may decrease significantly. During open-pit mining, when the rock is exposed to precipitation, the presence of moisture is difficult to exclude. Therefore, the specified lower value of f

_{fr}= 0.3 was accepted for further calculations. The remaining specified values of the crusher parameters are accepted for solid rocks having a hardness of up to 10 units, according to the Protodiakonov scale. The incoming rock size ranges from 100 to 300 mm. The geometric dimensions of the crusher elements were established on the basis of the design documentation provided for manufacturing the crusher. At the same time, the accepted values corresponded to the technical design assignment compiled for the crusher and the properties of the destroyed rock, as well as the requirements for safety regulations. According to the technical documentation concerning crushing rock in a range from 0 to 50 mm, productivity was up to 60 t/h, and the specific energy consumption was up to 0.15 MJ/t. When crushing rock in the range from 0 to 25 mm, productivity reduces to 30 t/h, and the specific energy consumption increases up to 0.30 MJ/t.

_{b}= 45° and e = 0.02 m. This implies that only criterion k

_{3}differs from its local optimum, and this difference is sufficiently insignificant (less than 1%). Compared to the worst case, that of 40 (not shown in Table 3) where φ

_{b}= 75° and e = 0.005 m, the productivity in the optimum alternative increases 2.05 times, and the variation coefficients of the torque and radial force on the roll reduce 1.67 and 1.20 times, respectively. In this case, performance depends mainly on the eccentricity, and the unevenness of kinematic components depends on the profile initial angle.

## 4. Conclusions

- The rational profile of the jaws of the working chamber of a single-roll gyratory crusher is realized by compiling and solving the differential equation of the profile curve. A rational profile is a segment of a logarithmic spiral that can be approximated, with practical accuracy, using a circular arc.
- Analytical expressions are provided to determine the rational speed of the roll and the capacity of the crusher, which are used in the mathematical model to optimize the parameters of the crusher.
- The developed mathematical model of kinematic load components established the fact that the kinematic load components of the crusher working member represent a periodic function of a shaft rotation angle. The Fourier transformation suggested that there were 1st-, 2nd-, 3rd- and 4th- order harmonics in the roll load. This should be taken into account in the dynamic analysis to prevent resonant phenomena during the crusher’s operation.
- The task of the multi-criteria optimization of the crusher parameters, according to the maximum productivity and minimum kinematic components of the radial load acting on the roll and the torque of resistance forces in the crusher drive, was set and solved. We recommend e = 0.01 … 0.02 m as a rational eccentricity value for mine crushers having a crushed rock size of d
_{min}= 0.07 m and a crusher feed rock size of d_{min}= 0.3 m. The optimal value of the central angle corresponding to the starting point of the profile for a typical mine crusher is 45°. It is possible to reduce the variation coefficients of kinematic components of the loads acting on the working member by 1.67 times for the torque and 1.2 times for the radial load, due to the optimal selection of the working chamber profile and angular coordinates of the fixed jaws. The performance depends mainly on the eccentricity, while the initial angle of the profile influences the amplitude of the kinematic components. - The findings were used in the design of an experimental sample of a single-roll gyratory crusher, the tests of which showed the effectiveness of the adopted technical solutions. The mathematical model used when selecting the parameters of the crusher working chamber were approximate, since it did not fully take into account the properties of the crushed material. Consequently, the obtained results are to be considered as the first approximation that will be further specified through the experience gained in designing and operating gyratory roller crushers.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Rylnikova, M.V.; Angelov, V.A.; Turkin, I.S. Features of technological and constructive solutions for the disposal of ore mining and processing waste in the worked-out space of mines. Min. J.
**2015**, 2, 59–66. [Google Scholar] - Kondrakhin, V.P.; Mizin, V.A.; Malorodov, V.G.; Olkhovsky, O.V. Mine roller crusher for hard rocks. Coal Ukr.
**1994**, 9, 15–16. [Google Scholar] - Holmberg, K.; Kivikytö-Reponen, P.; Härkisaari, P.; Valtonen, K.; Erdemir, A. Global energy consumption due to friction and wear in the mining industry. Tribol. Int.
**2017**, 115, 116–139. [Google Scholar] [CrossRef] - Marasanov, V.M. Optimization of crushing process in jaw crushers. Russ. Sci. Goals Object.
**2019**, 2, 35–41. [Google Scholar] [CrossRef] - Djokoto, S.S.; Karimi, H.R. Modeling and simulation of a High-Pressure Roller Crusher for silicon carbide production. In Proceedings of the International Conference on Electrical Power Quality and Utilisation (EPQU), Lisbon, Portugal, 17–19 October 2011; pp. 1–6. [Google Scholar] [CrossRef]
- Johansson, M.; Evertsson, M. A time dynamic model of a high-pressure grinding rolls crusher. Miner. Eng.
**2019**, 132, 27–38. [Google Scholar] [CrossRef] - Nikitin, A.; Epifantsev, Y.; Medvedeva, K.; Gerike, P. Power analysis of the process of brittle materials destruction in universal crushing machine with roll locker. Ferr. Metall.
**2019**, 62, 303–307. [Google Scholar] [CrossRef] - Lieberwirth, H.; Hillmann, P.; Hesse, M. Dynamics in double roll crushers. Miner. Eng.
**2017**, 103, 60–66. [Google Scholar] [CrossRef] - Bello, S.; Bajela, G.; Lamidi, S.; Oshinlaja, S. Design and Fabrication of Pneumatic Can Crushing Machine. Int. J. Adv. Sci. Res. Eng.
**2020**, 6, 154–161. [Google Scholar] [CrossRef] - Chakule, R.; Patil, S.; Talmale, P. Design and Development of Can Crushing Machine. Asian J. Eng. Appl. Technol.
**2020**, 9, 25–28. [Google Scholar] [CrossRef] - Bahre, K.; Krohm, R. Continuous Operation Crushing Machine. 2023. Available online: https://www.researchgate.net/publication/255043528_Continuous-operation_crushing_machine (accessed on 10 September 2022).
- Oke, K.; Alonge, A.; Olaiya, N. Performance optimization of jaw-type rock crushing machine through shaft eccentricity redesign. Afr. J. Sci. Technol. Innov. Dev.
**2019**, 12, 435–442. [Google Scholar] [CrossRef] - Kurbanov, X.; Yadgarov, S.; Turdiev, A.; Amanov, Z. Development of a Technological Image of a Stone-Crushing Machine. Int. J. Innov. Technol. Explor. Eng.
**2020**, 9, 313–315. [Google Scholar] [CrossRef] - Pacana, A.; Siwiec, D.; Bednarova, L.; Sofranko, M.; Vegsöova, O.; Cvoliga, M. Influence of Natural Aggregate Crushing Process on Crushing Strength Index. Sustainability
**2021**, 13, 8353. [Google Scholar] [CrossRef] - Rajan, B.; Singh, D. Understanding influence of crushers on shape characteristics of fine aggregates based on digital image and conventional techniques. Constr. Build. Mater.
**2017**, 150, 833–843. [Google Scholar] [CrossRef] - Bengtsson, M.; Hulthen, E.; Evertsson, C.M. Size and shape simulation in a tertiary crushing stage, a multi objective perspective. Miner. Eng.
**2015**, 77, 72–77. [Google Scholar] [CrossRef] - Ostroukh, A.; Surkova, N.; Varlamov, O.; Chernenky, V.; Baldin, A. Automated process control system of mobile crushing and screening plant. J. Appl. Eng. Sci.
**2018**, 16, 343–348. [Google Scholar] [CrossRef] - Koken, E.; Ozarslan, A. New testing methodology for the quantification of rock crushability: Compressive crushing value. Int. J. Miner. Metall. Mater.
**2018**, 25, 1227–1236. [Google Scholar] [CrossRef] - Cleary, P.W.; Sinnott, M.D.; Morrison, R.D.; Cummins, S.; Delaney, G.W. Analysis of cone crusher performance with changes in material properties and operating conditions using DEM. Miner. Eng.
**2017**, 100, 49–70. [Google Scholar] [CrossRef] - Chen, Z.; Wang, G.; Xue, D.; Cui, D. Simulation and optimization of crushing chamber of gyratory crusher based on the DEM and GA. Powder Technol.
**2021**, 384, 36–50. [Google Scholar] [CrossRef] - Chen, Z. Simulation and optimization of gyratory crusher performance based on the discrete element method. Powder Technol.
**2020**, 376, 93–103. [Google Scholar] [CrossRef] - Yamashita, A.S.; Thivierge, A.; Euzébio, T.A.M. A review of modeling and control strategies for cone crushers in the mineral processing and quarrying industries. Miner. Eng.
**2021**, 170, 107036. [Google Scholar] [CrossRef] - Wu, F. Chamber optimization for comprehensive improvement of cone crusher productivity and product quality. Math. Probl. Eng.
**2021**, 2021, 5556062. [Google Scholar] [CrossRef] - Khaled, A.A. A comparative study of regression model and the adaptive neuro-fuzzy conjecture systems for predicting energy consumption for jaw crusher. Appl. Sci.
**2019**, 9, 3916. [Google Scholar] - Błażej, D.; Król, R. Industry Scale Optimization: Hammer Crusher and DEM Simulations. Minerals
**2022**, 12, 244. [Google Scholar] - Atta, K.T.; Euzébio, T.; Ibarra, H.; Moreira, V.C.; Johansson, A. Extension, Validation, and Simulation of a Cone Crusher Model. IFAC-Pap.
**2019**, 52, 1–6. [Google Scholar] [CrossRef] - Duarte, R.A.; Yamashita, A.S.; da Silva, M.T.; Cota, L.P.; Euzébio, T.A.M. Calibration and Validation of a Cone Crusher Model with Industrial Data. Minerals
**2021**, 11, 1256. [Google Scholar] [CrossRef] - Efremenkov, E.A.; Martyushev, N.V.; Skeeba, V.Y.; Grechneva, M.V.; Olisov, A.V.; Ens, A.D. Research on the Possibility of Lowering the Manufacturing Accuracy of Cycloid Transmission Wheels with Intermediate Rolling Elements and a Free Cage. Appl. Sci.
**2022**, 12, 5. [Google Scholar] [CrossRef] - Sakawa, M. Fuzzy Sets and Interactive Multiobjective Optimization; Plenum Press: New York, NY, USA, 1993. [Google Scholar] [CrossRef]
- Odu, G. Review of Multi-criteria Optimization Methods—Theory and Applications. IOSR J. Eng.
**2013**, 3, 01–14. [Google Scholar] [CrossRef]

**Figure 1.**Working chamber of a single-roll gyratory crusher. Shaft 3 is located in housing, 1, equipped with reinforcing plates, 2. Working roll, 4, is installed on it eccentrically by means of bearings.

**Figure 4.**Kinematic components of the loads on the working roll of the gyratory crusher; (

**a**) graph of resulting force ${F}_{R}$ on the roll and its projections ${F}_{x}$ and ${F}_{y}$; (

**b**) graph of torque acting on the roll.

**Table 1.**Values of manufacturing clearances when changing the tolerance grade and combination of fits. Formulae for calculating the kinematic components of the roll loads.

No. | Boundary | Jaw a | Jaw b |
---|---|---|---|

1 | −(90° − φ_{b}); 90° − φ_{b} | F_{a} = 0M _{a} = 0 | F_{b} = F_{max}α _{b} = 0.5 (φ_{b} + φ_{K}) |

2 | 90° − φ_{b}; 90° + φ_{e} | F_{a} = P_{F}[φ − (90° − φ_{b})]α _{a} = 0.5[φ_{b} + (90° − φ)] | F_{b} = F_{max}α _{b} = 0.5 (φ_{b} + φ_{K}) |

3 | 90° + φ_{e}; 90° − φ_{e} | F_{a} = P_{F}[φ − (90° − φ_{b})]α _{a} = 0.5[φ_{b} + (90° − φ)] | F_{b} = P_{F} (φ_{b} − φ + 90°)α _{b} = 0.5 {φ_{b} + φ − 90°) |

4 | 90° − φ_{e}; 90° + φ_{b} | F_{a} = F_{max}α _{a} = 0.5 (φ_{e} + φ_{b}) | F_{b} = P_{F} (φ_{b} − φ + 90°)α _{b} = 0.5 {φ_{b} + φ − 90°) |

5 | 90° + φ_{b}; 270° − φ_{b} | F_{a} = F_{max}α _{a} = 0.5 (φ_{e} + φ_{b}) | F_{b} = 0M _{b} = 0 |

6 | 270° − φ_{b}; 270° + φ_{K} | F_{a} = P_{F}{φ_{b} − φ_{e} − [φ − (270° − φ_{b})]}α _{a} = 0.5[(270° − φ) + φ_{e}] | F_{b} = 0M _{b} = 0 |

7 | 270° + φ_{e}; 270° − φ_{e} | F_{a} = P_{F}{φ_{b} − φ_{e} − [φ − (270° − φ_{b})]}α _{a} = 0.5[(270° − φ) + φ_{e}] | F_{b} = P_{F}[φ − (270° + φ_{e})]α _{b} = 0.5[(270° − φ) + φ_{e}] |

8 | 270° − φ_{e}; 270° + φ_{b} | F_{a} = 0M _{a} = 0 | F_{b} = P_{F}[φ − (270° + φ_{e})]α _{b} = 0.5[(270° − φ) + φ_{e}] |

i | F_{x} | F_{y} | F_{∂} | M | ||||
---|---|---|---|---|---|---|---|---|

a_{i} | b_{i} | a_{i} | b_{i} | a_{i} | b_{i} | a_{i} | b_{i} | |

0 | 0 | 0 | 0 | −0.217 | 0 | 0.814 | 0 | 0.705 |

1 | 0 | −1.10 | −0.304 | 0 | 0.13 | 0 | 0.242 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 0.272 | 0 | 0.271 |

3 | 0 | 0.116 | 0.131 | 0 | −0.062 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 0 | 0 | −0.093 | 0 | 0 |

**Table 3.**Optimization of the parameters of the working chamber of a single-roll gyratory crusher (k

_{1}in m

^{3}/h).

No. | Objective Variables | Generalized Criterion | Criteria Membership Function | Quality Criteria | |||||
---|---|---|---|---|---|---|---|---|---|

P_{1} | P_{2} | ξ | ξ_{1} (k_{1})
| ξ_{2} (k_{2})
| ξ_{3} (k_{3}) | k_{1} | k_{2} | k_{3} | |

1 | 45 | 0.02 | 0.999 | 1.000 | 1.000 | 0.996 | 81.3 | 0.299 | 0.267 |

2 | 40 | 0.02 | 0.938 | 0.941 | 0.914 | 0.958 | 78.5 | 0.316 | 0.271 |

3 | 50 | 0.015 | 0.910 | 0.740 | 0.994 | 0.995 | 68.6 | 0.300 | 0.267 |

4 | 55 | 0.015 | 0.901 | 0.783 | 0.921 | 1.000 | 70.7 | 0.315 | 0.266 |

5 | 45 | 0.015 | 0.891 | 0.694 | 0.991 | 0.988 | 66.3 | 0.300 | 0.268 |

6 | 60 | 0.015 | 0.863 | 0.824 | 0.779 | 0.984 | 72.7 | 0.343 | 0.268 |

7 | 40 | 0.015 | 0.838 | 0.645 | 0.913 | 0.956 | 63.9 | 0.316 | 0.271 |

8 | 50 | 0.01 | 0.792 | 0.414 | 0.976 | 0.987 | 52.5 | 0.303 | 0.268 |

9 | 65 | 0.015 | 0.787 | 0.862 | 0.582 | 0.916 | 74.5 | 0.383 | 0.275 |

10 | 45 | 0.01 | 0.780 | 0.377 | 0.981 | 0.981 | 50.7 | 0.302 | 0.268 |

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## Share and Cite

**MDPI and ACS Style**

Kondrakhin, V.P.; Martyushev, N.V.; Klyuev, R.V.; Sorokova, S.N.; Efremenkov, E.A.; Valuev, D.V.; Mengxu, Q.
Mathematical Modeling and Multi-Criteria Optimization of Design Parameters for the Gyratory Crusher. *Mathematics* **2023**, *11*, 2345.
https://doi.org/10.3390/math11102345

**AMA Style**

Kondrakhin VP, Martyushev NV, Klyuev RV, Sorokova SN, Efremenkov EA, Valuev DV, Mengxu Q.
Mathematical Modeling and Multi-Criteria Optimization of Design Parameters for the Gyratory Crusher. *Mathematics*. 2023; 11(10):2345.
https://doi.org/10.3390/math11102345

**Chicago/Turabian Style**

Kondrakhin, Vitalii P., Nikita V. Martyushev, Roman V. Klyuev, Svetlana N. Sorokova, Egor A. Efremenkov, Denis V. Valuev, and Qi Mengxu.
2023. "Mathematical Modeling and Multi-Criteria Optimization of Design Parameters for the Gyratory Crusher" *Mathematics* 11, no. 10: 2345.
https://doi.org/10.3390/math11102345