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Scientific Reports volume 14, Article number: 19631 (2024 ) Cite this article High Quality Cladding Steel Wear
The cross beam of a mining linear vibrating screen is prone to cracking under long-term cyclic load. In order to accurately predict the fatigue life of the cracked cross beam, a coupled analysis method of vibration and crack propagation is proposed. A 2D dynamic model of the vibrating screen is established based on the finite element method, which is verified by the vibration test platform. The cracked Euler beam element is used to model the cracked cross beam. The effects of crack depth, amplitude of excitation force, frequency of excitation force, and crack location on the crack-tip stress intensity factor of the cracked cross beam are investigated in detail. In addition, an iterative method is proposed on the basis of the Paris model. The residual service life of the beam under different frequencies of excitation force, amplitudes of excitation force, spring stiffness, crack positions, and degrees of stiffness imbalance are discussed. The results demonstrate that the fatigue life of the beam increases as the frequency of excitation force and spring stiffness increase. The increase of the amplitude of excitation force and spring permanent deformation reduces the fatigue life. The conclusion obtained provide some theoretical guidance for the design and routine maintenance of mining linear vibrating screens.
Vibrating screens are widely used in important industrial sectors such as coal and metallurgy1. The use of the vibrating screen is crucial for producing the refined coal concentrate required by customers2. Vibrating screens operating under high structural load and continuous vibration are prone to damage3. The damage to the bearing, loosening of the side plate, and permanent deformation of the support spring are the main malfunctions of the vibrating screen4. In addition, the failure of the cross beam of the vibrating screen can lead to unplanned shutdown5. The fatigue failure of the cross beam is not preceded by significant macroscopic plastic deformation6.
Fatigue fracture is one of the main failures of the cross beam, which has attracted the attention of many scholars. Li et al.7 used ANSYS to analyze the modal and harmonic response of the beam structure of the vibrating screen and obtained the dangerous area where the beam was prone to fracture under the action of fatigue. The improvement scheme of the beam was proposed, which was proved to improve the fatigue life and reliability of the vibrating screen. Wang et al.8 proposed a novel large vibrating screen with a duplex statically indeterminate mesh beam structure (VSDSIMBS) by improving the beam structure of the vibrating screen. The finite element method was applied to study the performance of VSDSIMBS and the traditional vibrating screen (TVS). Modal analysis results indicated that VSDSIMBS could avoid resonance and run more smoothly than TVS. Zhang et al.9 applied the critical plane method of multiaxial fatigue analysis to numerically analyze the fatigue life of a large linear vibrating screen and used electrodynamics to measure the dynamic strain of the fatigue hazardous nodes of a similar model screen. The results showed that the fatigue reliability of the local positions of the side plates and beams is relatively low. Li et al.10 established a simplified model and a prefabricated crack analysis model of the beam of the vibrating screen based on the extended finite element theory. The effects of the location, depth, and angle of the initial crack and spring stiffness on the crack extension rate were investigated using simulation analysis. Zhu et al.11 used ANSYS to conduct dynamic analysis on the beam of the vibrating screen. The remaining life of the cracked beam was estimated, and the curve between the crack length and the remaining life was calculated and drawn to obtain the crack propagation law using the linear elastic fracture mechanics method. Huang et al.12 used ANSYS and nCode fatigue analysis software, selected the material-corrected S–N curve, and conducted fatigue life analysis of the beam of the vibrating screen using fatigue cumulative damage theory and rain flow counting method. Fry et al.13 used the stroke angle, beam orientation and derived stress range in the middle of the beam to calculate the stress range in the corner of the beam. The life that would be expected for the new screen, with the original, profile blended and stress relieved weld detail could then be determined using the S–N curve for a BS 7608 Class E detail. Huo et al.14 used MSC.ADAMS to establish the rigid-flexible coupled dynamics model with the cross beams and side plates as flexible bodies and other parts of the vibrating screen as rigid bodies, respectively, which provided the modal displacement for the load spectrum of fatigue life analysis. Using the fatigue analysis software MSC.FATIGUE, the fatigue life analysis of the cross beams and side plates was carried out. Long et al.15 proposed a new stiffness matrix of a three-dimensional finite element for modelling a beam with a breathing crack of the vibrating screen and used the obtained stiffness matrix to derive a finite element model for the cracked beam subjected to a bidirectional base excitation. The simulation results indicated that the number of superharmonic components can be used as an indicator to detect the presence of the crack. Fu et al.16 applied low-frequency eddy current detection to detect the crack of the cross beam of the vibrating screen and carried out the corresponding theoretical analysis and experimental research. The optimal detection frequency was derived for the specific material and cladding thickness.
To sum up, the previous studies on the cross beam of the vibrating screen focused on the crack fault diagnosis and estimating the fatigue life of the cross beam using the S–N curve. However, predicting the residual service life of the cracked cross beam is crucial, and can be used to develop reasonable maintenance plans to prevent sudden breakage of the cross beam. The fatigue life predictive model for the cracked cross beam is not built using the finite element method. Therefore, the main innovation of this paper is to propose a 2D dynamic model of the mining linear vibrating screen and a predictive model for crack propagation of the cross beam under cyclic load based on the Paris model to accurately estimate the residual service life of the cracked cross beam.
The mining linear vibrating screen is designed to vibrate at approximately 45° to the horizontal so that material moves forward along the screening deck17. The two vertically positioned side plates are joined by the cross beams 18. The whole structure is supported by springs to reduce vibration and shock19. The connections and joints between components are made using bolts, welds or swage lock bolts20. This paper mainly focuses on the crack propagation characteristics of the cross beam. Therefore, it is necessary to establish a simplified vibrating screen model to improve computational efficiency. The motion of the vibrating screen is simplified to a plane motion (z-o-w plane). Except for the springs and the cross beam, the other parts of the vibrating screen are simplified as mass blocks attached to both ends of the beam, which are connected to the cross beam by fixed connections. The cross beam is subjected to distributed load F(z, t) from the saddle seats, and the mass blocks are subjected to excitation forces as shown in Fig. 1.
Diagram of simplified cross beam model.
The 2D cracked beam element model is used to establish the dynamic equation of the vibrating screen, the length l, the crack depth a and the distance from the crack to the left end node of the beam lc. The crack beam element consists of two nodes, each with two degrees of freedom, including lateral displacement v and rotation angle θ. The corresponding nodal force components are shear force T and bending moment M, respectively, as shown in Fig. 2. Assuming that the direction of crack propagation is perpendicular to the crack front.
The strain energy of intact beam element can be written as21
where E is the Young's modulus, I is the moment of inertia.
The additional strain energy caused by the crack is expressed as22
Where KIM, KIT, KIIT, and are the crack-tip stress intensity factors of mode I and mode II cracks generated by bending moment M and shear force T, respectively.
The compliance coefficient matrix of the crack-free beam element can be written as23
in which \(c_{ij}^{0} = \frac{{\partial^{2} U_{0} }}{{\partial T_{i} \partial T_{j} }},\left( {i,j = 1,2} \right)\) , where T1 = T, T2 = M.
The additional compliance coefficient matrix caused by the crack is24
in which \(c_{ij}^{1} = \frac{{\partial^{2} U_{T} }}{{\partial T_{i} \partial T_{j} }},\left( {i,j = 1,2} \right)\) .
According to the principle of virtual work, the stiffness matrix of the cracked beam element can be represented as25
where T is the transformation matrix.
The stiffness matrix \({\varvec{k}}_{\textbf{c}}\) of the cracked beam element is closely related to the compliance matrix \({\varvec{C}}^{1}\) caused by the crack. When \({\varvec{C}}^{1}\) is 0, the stiffness matrix \({\varvec{k}}_{\textbf{c}}\) of the cracked beam element is equal to the stiffness matrix \({\varvec{k}}_{\textbf{e}}\) of the intact beam element, as shown below
Under the action of the external forces, the crack of the cross beam is not always in the open state, but in a periodic open and closed state. It is assumed that the crack can only be in a fully open or fully closed state during the dynamic analysis to simplify the complexity of the calculation. When the crack is completely closed, the stiffness matrix of the cracked beam element is the same as that of the intact beam element. The basis for judging the open and closed state of the crack is
The effect of the crack on the mass and damping of the beam element is neglected. The mass matrix of the cracked and intact beam element can be expressed as26
where \(\rho\) is the density, A is the cross-sectional area, S is the shape function matrix expressed as follows
The mass matrix and stiffness matrix of the two end elements are obtained by modifying the beam element mass matrix and stiffness matrix based on the mass blocks and supporting springs. The dynamic equation of the vibrating screen is formed by assembling the element matrices and solved using Newmark-β method27.
The vibration test platform is built to verify the correctness of the proposed dynamic model of the mining linear vibrating screen in this paper. The test system consists of a mining linear vibrating screen, piezoelectric accelerometer, NI DAQ card, constant current adapter, and a computer as shown in Fig. 3. The size of the vibrating screen is 4.8 m × 3.6 m (length × width). The length of the cross beam near the feeding end of the vibrating screen is 3.6 m, assuming l', and the Outer diameter and wall thickness are 0.356 m and 0.01 m, respectively. The motor speed of the vibrating screen is 900 r/min. Four accelerometers are evenly placed at l'/8, l'/4, (3 l')/8, and l'/2 from the left endpoint of the cross beam, respectively.
The acceleration signals collected by the piezoelectric accelerometers set at l'/4 and l'/2 are denoised, and are integrated into displacement signals using frequency-domain integration as shown in Fig. 4.
The parameters of the vibration test platform are substituted into the proposed rigid flexible coupling dynamic model of the vibrating screen. The displacement of the cross beam at l'/4 and l'/2 is obtained as shown in Fig. 5.
Displacement obtained through dynamic model. (a) l'/4, (b) l'/2.
It can be seen from Table 1 that the error between the displacement at l'/4 obtained through test and dynamic model is less than 5%, and also at l'/2. The vibration frequencies at l'/4 and l'/2 obtained through test and dynamic model are all 15 Hz. Therefore, the proposed rigid flexible coupling dynamic model of the mining linear vibrating screen is effective.
The fatigue life of the cracked cross beam of the mining vibrating screen is greatly affected by fatigue crack propagation. The crack-tip stress intensity factor determines the rate of crack propagation, and the peak-to-peak value within one cycle can be shown as28
where \(Y\left( s \right)\) is the crack correction factor, \(\Delta \sigma\) is the peak-to-peak value of dynamic stress within one cycle.
The Paris equation can be used to effectively describe the fatigue crack propagation rate of a straight crack when in a medium-stress state with low average stress. Therefore, the Paris equation is used to simulate fatigue crack propagation in this paper. Equation (11) is substituted into the Paris equation to obtain the vibration fatigue crack propagation rate model29.
where N is the number of fatigue cycles, C and m are the constants related to the material of the cross beam30.
The crack propagation calculation is taken for each vibration cycle to simplify the analysis. The crack propagation increment during the i-th cycle is
The forced vibration of a cross beam of the mining linear vibrating screen under external load causes fatigue crack propagation. The fatigue crack propagation also changes the stiffness of the cracked cross beam to change the vibration characteristics. Therefore, there is a coupling effect between fatigue crack propagation and vibration characteristics of the cross beam. An iterative updating method is used to predict the crack growth of the cross beam to improve the prediction accuracy. The specific steps for predicting crack propagation are summarized as shown in Fig. 6.
Schematic diagram of predicting crack propagation.
The failure conditions of the cross beam of the mining linear vibrating screen based on engineering experience are as follows.
(1) The crack extends to the neutral axis of the cross beam, ac = De/2.
(2) The maximum crack-tip stress intensity factor \(K_{{{\text{Imax}}}}\) reaches and exceeds the material fracture toughness of the cross beam \(K_{{{\text{IC}}}}\) .
(3) The maximum dynamic stress at the crack of the cross beam \(\sigma_{{\textbf{max}}}\) exceeds the strength limit of the material \(\sigma_{\textbf{b}}\) .
In this section, the effects of crack depth, crack position, amplitude of excitation force, frequency of excitation force and spring stiffness on the crack-tip stress intensity factor and fatigue life of the cracked cross beam are analyzed. The parameters of the mining linear vibrating screen are given in Table 2. Let the length of the beam be L.
The crack-tip stress intensity factor is closely related to the crack depth. Four crack depths (i.e., 2 mm, 4 mm, 6 mm, 8 mm) are selected to analyze how the crack depth affects that. There is only a single through-transverse crack in the middle of the cross beam. The amplitude and frequency of excitation force are 250 kN and 16 Hz, respectively. The spring stiffness is 180 kN/m.
As shown in Fig. 7, the crack-tip stress intensity factor shows positive and negative alternating rule of change in the working process of the vibrating screen, indicating that the crack is in a periodic open and closed state under the action of external load. The amplitude of the crack-tip stress intensity factor increases sequentially with the increase of the crack depth. The main reason is that the stiffness of the beam decreases as the crack depth increases, leading to an increase in the extreme value of the dynamic response and the stress near the crack tip.
Effect of crack depth on the crack-tip stress intensity factor. (a) Crack-tip stress intensity factor, (b) Amplitude of crack-tip stress intensity factor.
Assuming that the crack is located in the middle of the beam, with a depth of 4 mm. Four different amplitudes of excitation force (i.e., 150 kN, 200 kN, 250 kN, 300 kN) are selected for analysis, where the frequency of excitation force and spring stiffness are 16 Hz and 1800 kN/m, respectively.
Figure 8 demonstrates that the greater the amplitude of excitation force, the greater the crack-tip stress intensity factor. The main reason is that the increase in the excitation force results in an increase in the strain of the cross beam, resulting in a more significant displacement gradient and dynamic stress at the crack tip. The effect of excitation force on the stress intensity factor is noticeable. For every 50 kN increase in amplitude of excitation force, the amplitude of stress intensity factor increases by 22.48%, 18.08% and 15.31%, respectively.
Effect of amplitude of excitation force on the crack-tip stress intensity factor. (a) Crack-tip stress intensity factor, (b) Amplitude of crack-tip stress intensity factor.
The crack is set at 0, L/6, L/3 and L/2 from the left endpoint of the cross beam, respectively, to analyze the effect of crack location on the crack-tip stress intensity factor of the cross beam. Assuming that the crack depth is 4 mm, the amplitude of excitation force is 250 kN, the frequency of excitation force is 16 Hz, and the spring stiffness is 1800 kN/m, the simulated results under different crack locations are obtained as shown in Fig. 9.
Effect of crack location on the crack-tip stress intensity factor. (a) Crack-tip stress intensity factor, (b) Amplitude of crack-tip stress intensity factor.
From the simulated results shown in Fig. 9, it can be deduced that the closer the crack is to the side plate, the greater the stress intensity factor is. The main reason is that the position near the side plate on the beam is subjected to greater bending moment and shear force, resulting in greater stress. The crack is more likely to occur near the side plate. Therefore, reinforcement materials need to be used to strengthen the structure of this area.
As shown in Fig. 10, the crack-tip stress intensity factor decreases when the frequency of excitation force increases. The main reason is that the higher frequency reduces the bending displacement and dynamic stress of the cross beam when the amplitude of the excitation force is constant, thereby reducing the stress intensity factor. High-frequency vibration makes it easier for the particles to pass through the screen mesh. The service life of the cross beam and the screening efficiency of the vibrating screen can be increased by increasing the motor speed appropriately while ensuring that the vibration amplitude meets the working requirements.
Effect of frequency of excitation force on the crack-tip stress intensity factor. (a) Crack-tip stress intensity factor, (b) Amplitude of crack-tip stress intensity factor.
When the crack is located at the left endpoint of the beam and the spring stiffness is 1800 kN/m, the fatigue life of cracks with different depths under different frequencies of excitation force and amplitudes of excitation force is shown in Fig. 11.
Fatigue life under different amplitudes and frequencies of excitation force. (a) 150kN, (b) 200kN, (c) 250kN, (d) 300kN.
It can be seen from Fig. 11 that as the depth of the crack increases, the fatigue life of the beam decreases, and the difference in life becomes smaller, which is consistent with the law of fatigue crack propagation. The main reason is that when the depth is within the range of 1 mm to 4 mm, the crack propagation rate is slow and is in the first propagation stage. When the crack depth is greater than 4 mm, the crack steadily grows, and there is a roughly linear relationship between fatigue life and crack depth. The fatigue life of the cross beam with visible cracks is extremely low, and the maximum value is about 6 days. The beam exhibits a more pronounced damping effect at high frequency. The damping absorbs the vibration energy of the system and reduces the vibration amplitude. Therefore, the fatigue life of the beam increases as the frequency of excitation force increases. On the contrary, the increase of the excitation force will shorten that.
The amplitude of excitation force and spring stiffness are assumed to be 250 kN and 1800 kN/m to analyze the effects of frequency of excitation force and crack location on the fatigue life of the cross beam as shown in Fig. 12.
Fatigue life under different frequencies of excitation force and crack locations. (a) 12 Hz, (b) 14 Hz, (c) 16 Hz, (d) 18 Hz.
From the simulated results shown in Fig. 12, the fatigue life of the cracked cross beam is affected significantly by the crack location. The life is lowest when the crack appears at the joint between the beam and the side plate. Taking the crack with the depth of 1 mm in Fig. 12 (a) as an example, the life of the beam with a crack at the connection between the beam and the side plate is reduced by about 55% compared to that of the beam with a crack in the middle. This is because the boundary effect restricts the deformation of the beam and increases its stiffness, resulting in significant displacement gradient and stress at the endpoint of the beam. It can be deduced that the shortest crack fatigue life is only 0.6 days. The main reason is that the strength and bearing capacity of the cross beam significantly decrease as the crack depth equals to the wall thickness, and the failure mode changes from bending failure to brittle fracture mode.
The amplitude and frequency of the excitation force are assumed to be 250 kN and 16 Hz, respectively. The fatigue life of the cracked cross beam is shown in Fig. 13.
Fatigue life under different crack locations and spring stiffness. (a) 0, (b) L/6, (c) L/3, (d) L/2.
As the spring stiffness increases, the vibration response of the beam decreases while more damping effects are introduced, resulting in less stress concentration at the crack tip and an increase of the fatigue life of the cracked cross beam. However, the overall trend remains unchanged.
When the crack is located at the connection between the beam and the side plate, the fatigue life of the cross beam with different depths under different spring stiffness and reduction rates of stiffness is shown in Fig. 14. The amplitude and frequency of the excitation force are the same as in Sect. 5.2.3.
Fatigue life under different spring stiffness and reduction rates of stiffness. (a) 2200 kN/m, (b) 1800 kN/m, (c) 1400 kN/m, (d) 1000 kN/m.
When the crack depth is 1 mm, the contribution of different spring stiffness and degree of stiffness imbalance to the reduction of fatigue life compared with the fatigue life under intact springs with the stiffness of 2200 kN/m is shown in Fig. 15.
Contribution of different spring stiffness and degrees of stiffness imbalance to the reduction of fatigue life.
Figure 14 demonstrates that the fatigue life of the cross beam continues to decrease as the spring stiffness imbalance intensifies. This is because the difference in spring stiffness leads to dynamic imbalance of the vibrating screen, which causes additional load and vibration of the beam, and increases the stress at the crack tip. As shown in Fig. 15, the stiffness imbalance has an increasing effect on the fatigue life with the increase of the spring stiffness, reducing life by up to 52.40%. Therefore, in the field production process, it is also necessary to promptly investigate the permanent deformation of the springs in addition to the monitoring and maintenance of the cross beam.
The difference in fatigue life is used to measure the influence of the above factors on the fatigue life of the cracked crossbeam of the vibrating screen. It can be seen from Fig. 16 that the amplitude of the excitation force has the greatest effect on the fatigue life of the cracked beam of the vibrating screen, followed by the crack location.
Effect of different factors on fatigue life.
An iterative method is proposed to predict the fatigue crack propagation of the cross beam of the mining linear vibrating screen under long-term dynamic load in this paper. The main conclusions are as follows:
(1) A 2-D dynamic model of the mining linear vibrating screen is established based on the finite element method and is verified by building the vibration test platform. The difference between the peak-to-peak values of theoretical and measured displacement is less than 5%. The theoretical and measured vibration frequencies are all 15 Hz. The accuracy of the proposed model can meet engineering requirements.
(2) A dynamic model of the vibrating screen with a cracked cross beam is established using the cracked beam element. It is found that the crack-tip stress intensity factor is proportional to the crack depth and amplitude of excitation force but inversely proportional to the frequency of excitation force. The closer to the side plate, the faster the crack propagation rate.
(3) A predictive model for crack propagation of the cross beam under cyclic load is proposed based on the Paris model. The fatigue life of the cracked beam increases as the frequency of excitation force and spring stiffness increase. The increase of the amplitude of excitation force and spring permanent deformation reduces the fatigue life. The amplitude of excitation force has the most significant impact on the life, and the variation is \({4}{\text{.68}} \times {10}^{{6}}\) cycles.
(4) The cross beam breaks down quickly once the visible crack appears, with a maximum life of less than 6 days. When the depth is within the range of 1 mm to 4 mm, the crack is in the first stage, and the propagation rate is slow. There is a linear relationship between crack depth beyond 4 mm and before 10 mm and fatigue life.
The data used to support the findings of this study are available from the corresponding author upon request.
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This work was supported by the National Natural Science Foundation of China (allotment grant number 51774193), and the Shandong Provincial Natural Science Foundation, China (allotment grant number ZR2017MEE025).
College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qingdao, 266590, China
Fangping Yan & Hao Lu
Swinburne College, Shandong University of Science and Technology, Jinan, Shandong Province, 250031, China
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F.P.Y. conceived the study; L.J.X. and H.L. were the principal investigators; L.J.X. and H.L. analyzed the data; F.P.Y. wrote the manuscript; H.L. revised the manuscript.
The authors declare no competing interests.
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Yan, F., Lu, H. & Xiao, L. Fatigue life prediction of cracked cross beam of mining linear vibrating screen under cyclic load. Sci Rep 14, 19631 (2024). https://doi.org/10.1038/s41598-024-70671-5
DOI: https://doi.org/10.1038/s41598-024-70671-5
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