Modelling of transverse vibration of conveyor belt in aspect of the trough angle

The transverse vibration of conveyor belts is a crucial aspect in their proper and safe design, as the correct determination of vibration frequencies is essential to avoid unsafe operating conditions and premature wear of components. This study presents the Moving Compressed Beam (MCB) transverse vibration model, that is based on the interpretation of a conveyor belt as a beam known from the literature. This approach allows you to include in the model the transverse flexural rigidity of a troughed belt-a parameter that is closely related to the belt geometry. While not commonly used in engineering practice, the MSB model was compared with other models during the laboratory tests based on changing of trough geometry. This study emphasizes the significance of considering the transverse flexural rigidity of conveyor belts during the design process, and the MCB model which offers a promising approach in aspect of vibration control.


List of Symbols
Bending moment θ 1y , θ 2y The angles between the horizontal axis and the direction of forceT g Standard acceleration due to gravity equilibrium position v Belt speed ϕ The angle between the vertical axis and the direction of the centrifugal force T 0 Equivalent belt tensile force EI Flexural rigidity r Radius of curvature l Support distance (spacing) θ Angle of inclination from equilibrium T 1y , T 2y Vertical components of the T force ξ Distance between center of gravity of lateral section of belt trough and the center of gravity of belt trough cross section y s Location of gravity center relative to the vertical axis y of the coordinate system

The prior art
Belt vibration models are generally classified into two types: static and dynamic 29 .In static models, the main assumption for solving the equations is that the system is in equilibrium and there are no vibrations 30 .Such models are used to determine the tensioning forces of the belt 31 , calculate deflection, and determine the frequency of natural vibrations 32 .The external load on the belt is taken into consideration, and boundary conditions are determined.In contrast, dynamic models consider the movement and vibration of the belt, and are used to study the belt's dynamic behavior under different operating conditions 33 .The belt is treated as a continuous elastic system subject to complex forces in different directions.Discrete elements into which the belt is divided 34 or a small section of the belt can be modeled as a continuous system 35 .Static models are simpler to use than dynamic models, but do not consider the dynamic behavior of the belt.Dynamic models, on the other hand, provide more detailed information about the motion of the belt, but are more complicated to use.It's worth noting that many of the currently used models are static models modified with motion parameters, making the concept of "static" in such systems vague.The string model is a commonly used computational model for predicting transverse vibrations in belts 36 .It is assumed in the calculations that the belt behaves like a perfectly flexible string, transmitting longitudinal forces but experiencing no bending stresses 37 .The wave equation relates the transverse displacement to the wave propagation speed, tension, and linear mass density of the string 38 .In most models, it is assumed that the vibrating string is homogeneous, and that the cross-sectional area and density are constant along the length of the string.By making this assumption, a single value for the linear mass density of the string can be used in the wave equation, thereby simplifying the analysis.However, if the string exhibits non-homogeneity, this assumption may require reconsideration, leading to necessary modifications in the analysis 39 .The string model is particularly developed for belt gears, and many of them consider nonlinear material parameters, damping and belt speed 6 .Some of them also take into account long-term changes in the speed of the belt 40 .The main disadvantage of string models is that they neglect the bending stiffness of the cross-section and the bending moments that occur in a loaded conveyor belt.Prior research 41,42 indicates that this parameter exerts a substantial influence on the transverse vibrations of the belts.Hence, these models do not always allow for a correct prediction.
A beam model is a more advanced approach for analyzing the transverse vibrations of a belt, as it takes into account the bending stiffness of the belt cross-section and the bending moments that occur in a loaded conveyor belt 43 .The model presume that the material will undergo bending when subjected to a load, which is why the term flexural rigidity is employed.In this model, the transverse displacement of the beam is described by a equation that takes into account the bending moment and shear force acting on the beam, as well as the transverse loading applied to the beam 44 .The flexural rigidity of the beam, which is a function of the cross-sectional area and material properties, determines how the beam resists bending deformation.Conveyor belts are subject to the influence of both transverse and longitudinal stiffness, which have a significant impact on their behavior 45,46 .In particular, transverse stiffness governs the belt's ability to be troughed 47,48 .The standard beam models used to analyze the transverse vibrations of a conveyor belt have a significant drawback, namely, they do not take into account the longitudinal stretching forces that are present in the belt during normal operation.These forces are crucial for ensuring the proper functioning of the conveyor under steady state conditions 49 , and their omission can lead to inaccurate results.Therefore, more advanced beam models that incorporate longitudinal stretching forces should be considered in order to obtain a more accurate representation of the belt's behavior.
Lodewijks proposed a solution 50 to the problem of accurately modeling conveyor belt vibrations by developing a specific model for this purpose.The model takes into account various parameters such as longitudinal tensile forces, belt speed and stress wave speed, transverse displacement of the belt or the influence of the load distributed on the belt on its deflection.The model is based on a set of partial differential equations that describe the motion of the belt, including both longitudinal and transverse vibrations.The equations are solved numerically using finite difference method (FEM).In this model, the belt is assumed to be a thin and flexible beam, and the transverse vibrations of the belt are modeled as standing waves.Lodewijsk presents a solution under the following assumptions: the cross-section rigidity of the conveyor belt is neglected, the transverse displacements are considerably smaller than the distance between the idlers, and the belt's length change due to transverse deformation is negligible compared to the belt's initial length.In essence, Lodewijks' model can be considered a hybrid of string and beam models, as it incorporates some of the assumptions of both models.
Harrison's conveyor belt vibrations model 51 is a complex mathematical model used to calculate the frequency of transverse vibrations in conveyor belts.The model is based on the mechanics of the plate and considers the stiffness of the belt and the boundary conditions.Some solutions of the model also include a belt stretching force.It involves developing a flat belt and a belt supported on two or three rollers to determine the vibrations.The model assumes that the belt is fixed on the bend line between the idlers and recommends the use of software to determine the frequency of vibrations 52 .Harrison's model shows greater accuracy than the string model, but it is complicated and requires many assumptions to be made for the calculations.
In terms of designing belt conveyors, the string model which includes Lodewijks beam elements, is used in computer-aided calculations and is favored by some design companies for conveyor route design.Empirical evidence suggests that under certain operational conditions, the inaccuracies observed in the predictions generated by the FEM model may be attributed to the absence of belt rigidity in the model.Harrison's vibration model, on the other hand, is known for its high degree of complexity and is typically used for more detailed calculations where recalculating auxiliary coefficients and selecting determinants defining the types of vibrations is necessary and justified.Overall, both models have their uses and are employed in different scenarios based on their strengths and limitations.Accordingly, efforts have been directed towards formulating a novel model that accounts for all the pertinent features of conveyor belts, while simultaneously possessing sufficient simplicity to be utilized in fundamental belt conveyor design.One of the fundamental assumptions underlying the development of a new model was the simplification of calculations and the elimination of subjective parameters.This was motivated by the fact that several existing models involve the incorporation of stiffness coefficients 53,54 of both the conveyor and the belt into their equations.

Flexural rigidity of belt
Flexural rigidity is a parameter that determines the line of bending of the conveyor belt on the idler supports.Proper alignment of the unloaded belt is a prerequisite for the correct operation of the conveyor.If the belt is not properly aligned, it may slip or become damaged, leading to downtime and increased maintenances.Usually, the transverse rigidity is determined analytically based on the geometry of the cross-sectional profile and the material parameters of the belt.It can be expressed as the product of the moment of inertia of the cross-section I and the modulus of elasticity E. The transverse rigidity of the belt depends on the way the belt is positioned on the idlers.The number and arrangement of idlers on the carrying and return sides of the belt loop are different, which is why the rigidity is also variable.The most general case of guiding the belt on idler supports is the symmetrical arrangement, three-roller support, used on the carrying side of the belt loop, which was considered in the article (Fig. 1).Additional parameters that affect changes in transverse rigidity include the trough angle, belt thickness and width, and idler length.In the calculations of the flexural rigidity of the cross-section of the belt, it is assumed that the neutral bending axis is a horizontal line passing through the center of gravity of the cross-sectional profile of the belt.
In the symmetrical system of the three-roller basin, the location of the center of gravity of the cross-section is determined by the coordinate: (1) The centre of gravity of the lateral section of the trough l 2 ts is shifted vertically from the neutral axis of bending by the size of: To determine the moment of inertia, the calculations must be divided into two parts.In the first stage, the moment of inertia of the section of the trough inclined relative to the neutral axis is determined: In the second stage, the moment of inertia of the central part can be determined using the following formula: In the final calculation, the moment of inertia of the cross-section of the troughed belt is equal to:

Equation of motion
In the analyzed model, the conveyor belt was treated as a homogeneous beam with a constant cross-section.Double-sided support of the belt was used.The conveyor belt as a beam is simply supported, from left side ( x = 0 ), to the right ( x = l ).The belt performs bending vibrations and the cross-section displacement is defined as y = y(x, t) .Figure 2 shows a section of the belt cross-section of length dx, loaded with acting forces, which allowed to define the equation of motion.
The equation of motion for the analyzed cross-section in the displacement direction is equal: The equation ( 5) makes sense for assuming a linear-elastic material and ignoring the circular motion of the section.In real conditions, conveyor belts are composites with a nonlinear Young's modulus, with additional material damping 55,56 .Due to the fact that the estimation of the frequency of transverse vibrations is usually carried out for a limited range of belt stresses, a segmental linearization of the longitudinal elasticity characteristic can be performed.Considering these assumptions, the mass of the vibrating element in Equation ( 5) can be defined as dm = ρSdx , included in the equation ( 5) acceleration of the dx element is a = ∂x , and the bending moment can be related to the radius of curvature resulting from the cyclic displacements of the beam M EI = 1 r , we we can define the condition that small deflections determine  www.nature.com/scientificreports/small angles between the horizontal axis and the direction of force along the beam axis Taking into account all the presented assumptions, we can write that the shear force (transverse) is: All presented relations can now be substituted into the original equation of motion (5), and the ordering of the variables and the degree of derivatives gives the following solution: The first part of the equation ( 7) is the increment of the shear force in the beam along the length of dx.The second part describes the force with which the tensioned belt resists the movement of the element.The third part is the elementary force of inertia.The standard solution of the equation of motion (7), which does not take into account the axial stress, leads to the following formula for the frequency of bending vibrations of the beam:

Prestressing concept
The solution presented in the previous subsection completely ignores the influence of longitudinal forces in the belt and its speed of movement.The method of taking into account the axial forces assumes a two-sided action of the compressive force 57 .The force acting along the axis of the deflected beam (curve) is equal to: Due to the fact that the transverse deflections of the belt are relatively small, it can be assumed that the angles formed between the horizontal plane are also very small and thus their influence is neglected.For such assumptions, the following conditions can be written: sin θ ∼ = θ and sin θ 1 = θ 1 = θ(x, t) + ∂θ(x,t) ∂x dx .The equation ( 9) therefore takes the following form: For this reason (10) the equation of motion (7) is also appropriate for a beam under double compression.The normal form of vibration as a function of displacement and time can be written as follows 58 : y = η(x)f (t) = η sin αt .By inserting into the equation of motion (7), after performing a few transformations it is possible to derive the equation: In the equation (11) EI , and ξ = x l .To solve this equation, we first assume a solution of the form η(x) = A sin(ωx) + B cos(ωx) + C sinh(ωx) + D cosh(ωx) , where A, B, C, and D are constants to be determined and ω is the natural frequency of the system.After solving the equation, the frequency of bending vibrations of an axially prestressed beam under double compression is obtained as: In the case of conveyor belts, the use of material density and cross-sectional area is not practical.A better solution is to replace the term ρS with an expression Bm.

Motion concept
Taking into account the variable stresses is a procedure that allows to distinguish the specific features of the string models, while maintaining the material properties of the belt.The Fig. 3 shows the load distribution for a short section of the belt length 59 .
The equation follows from the balance of forces in the vertical direction acting on the separated belt element: As in the case of the equation (10), an assumption of small, negligible deflections can be made: . In this case, equation ( 13) can take form: ∂Q(x) ∂x = mv 2 − mg .The transverse forces in the belt depend on the mass of the analyzed section and the speed.On this basis, it is possible to derive a parameter called the equivalent tensile force in the belt: T 0 = T − mv 2 .The equivalent belt tensile force is reduced in relation to the belt force by the normal inertia force (centrifugal force) acting on the transported material and the belt.By

Model sensitivity
The parameters that affect conveyor belt transverse vibration can be divided into three groups.The first group includes material parameters of the belt, such as flexural rigidity and mass.The second group comprises geometric or structural parameters, such as belt width and support spacing.Finally, the third group consists of technological and operational parameters, such as belt speed and tensioning force, which can affect the belt's behavior during normal operation.The sensitivity analysis results reveal the parameters that exert the most significant influence on the frequency of transverse vibrations of the belt, highlighting their crucial role in the initial stages of belt conveyor design By systematically varying these parameters and observing their impact on the belt's vibration behavior, sensitivity analysis can help optimize the design and operation of conveyor belt systems to reduce vibration and improve performance.Figure 4 shows the results of the model sensitivity analysis for three groups of input parameters.The range of parameters was selected based on the actual working conditions of the conveyors 60 .
The results indicate that the transverse flexural rigidity of the belt, one of the material parameters included in the model, has the greatest impact on the frequency outcome (Fig. 4a).In comparison to the effect of flexural rigidity, the impact of belt mass on the resulting frequency outcomes is insignificant.However, it is important to note that there is a considerable variation in the examined parameter ranges, which are derived from the conventional operating conditions of the conveyor.The mass of the belt is typically linked to its nominal strength, which in turn influences the elastic modulus.The flexural rigidity of a belt is dependent on the cross-sectional area's moment of inertia and the modulus of elasticity.The findings indicate that, among the examined parameters, the impact of the cross-sectional geometry has the most substantial effect on the vibration frequency (Fig. 4b).This effect is non-linear, highlighting a marked increase in the belt's vibration frequency at greater trough angles.
Moving on to construction parameters (Fig. 4c), the belt width was found to be the most decisive factor in changing the transverse vibration frequency.Furthermore, the idler spacing was identified as another design parameter that has a noticeable impact on the vibration frequency.Although the effect is smaller compared to belt width, it should not be overlooked, and designers must carefully consider this parameter when designing conveyor systems.Notably distinct alterations are noticeable for short support spacings.
Finally, among the operating parameters (Fig. 4d), the belt tensile force was identified as the main factor that affects the transverse vibration frequency.In contrast, the effect of speed on the vibration frequency was found to be minimal.Based on these findings, it can be confidently affirmed that selecting the appropriate belt tensioning force and managing belt tension during steady states are crucial for achieving optimal vibration characteristics of the conveyor.
The outcomes of the sensitivity analysis of the model provide clear evidence of the significant impact of the belt cross-sectional geometry on the transverse vibrations of the belt.Thus, when designing the conveyor capacity, the trough angle selection should also be considered during the stage of safeguarding the conveyor against resonance.The value of the trough angle ought to be incorporated into the calculations of the tensile resonance force.To demonstrate the effect of the trough angle on the vibration characteristics, Fig. 5 presents the model's solution for a brief time interval using standard trough angles, and assuming a sinusoidal course with a standard ( 14)

Materials and methods used in validation Equipment
The proposed model and theoretical results of the sensitivity analysis were verified through laboratory experimentation on a conveyor (Fig. 6a) with an approximate length of 8 meters.The conveyor consists of five idler sets, with each set having three flat-tubed steel rollers with a diameter of 133 mm and a length of 300 mm, that can be inclined in five different positions.The average distance between each set is 0.93 meters.The rig is capable of smoothly adjusting the belt speed up to 5.3 m/s.During the experimentation, a linear speed encoder with a wheel (Fig. 6b) was used to measure the belt's speed, while a hydraulic tension system applied a constant tensile force to the belt.The tensile force was measured by strain gauge force sensors located on the return pulley (Fig. 6c).The study utilized mobile measurement devices that traverse the entire conveyor route during operation, as depicted in Fig. 6d.These devices consist of a vibration sensor enclosed in a sealed casing that is mounted to the conveyor belt.A microcontroller performs the acquisition of measurement data, initial data processing, and data storage on an SD card.The system operates on independent battery power and features a free fall detection mechanism to monitor for unintended contact losses between the device and the belt.The device's weight is sufficiently low so as not to cause significant point changes in the weight of the belt.Sampling is conducted at a rate of 2,5 kHz.The experiments were conducted on a textile belt with EP250 specifications, with a length of approximately 17 meters.A Table 1 was prepared summarizing all the relevant parameters required for the computational model.

Methods
The study involved capturing vibration time series data using a device that was placed on a conveyor belt under different trough angles.By adjusting the trough angle, the stiffness of the belt cross-section was modified.Data was collected for five different trough angles: 0 • , 27  , with the flat belt run designated as angle 0 • .Although the flat belt run is not commonly used for the upper belt during bulk material transportation (except in specific transition sections), it serves as an interesting reference point for model validation since it leads to a collapse of the applied geometrical relations.The measurement device traversed the entire route from the return pulley to the drive pulley, capturing the vibration signal of the entire conveyor.However, only short time segments located between the idler sets were selected for processing.The exact distances between the supports in this location were known.Short time periods were subjected to processing (filtration and spectral analysis) in order to extract the characteristic frequencies of the components.Filtering is necessary to separate interference in the extremely low frequency range.To accomplish this, a high-pass filter was utilized.The final course of the transverse vibration frequency as a function of the trough angle was determined and compared with the model proposed in the article.Full statistics of measured frequencies and amplitudes of vibrations for all series and measurement points are presented.The average forecast error of the proposed model and the popular models used so far were determined.The discussion section formulates conclusions based on the results obtained.The tests were carried out for a constant belt tension force of 2.0 kN and speed 1.3 m/s .Under the specific operating conditions of the conveyor, an estimated measurement error of ± 0.1 m/s was determined for belt speed, and ± 0.1 kN for belt force.

Results of model validation
Figure 7 shows an exemplary full time signal (Fig. 7a) and spectrum (Fig. 7b) of the recorded vibration signal from the entire conveyor.The detected peaks are attributed to direct interaction between the measuring device and the idler, as the device passes through the idler.The recorded frequencies of various phenomena along the conveyor path, including belt vibrations in different directions, interaction between the belt and idler, and transmission of vibrations from other systems, fall within the frequency range of 125 Hz, with intensification up to 50 Hz.The observed peaks in frequency resulting from excitation generated by the rotating rollers are within the range of 10 to 20 Hz.
The following Fig. 8 depicts a sample recorded signal for the segment of the conveyor belt located between the idler.

Discussion
By filtering out short waveforms and removing disturbances recorded directly from the belt, it becomes possible to identify trends in the amplitudes of transverse vibrations between the supports.The frequencies of these vibrations for each trough angle are determined and presented in a box plot in Fig. 9a, along with full statistics.In the Fig. 9b, the statistics of the recorded average vibration amplitudes are presented in a similar way as in the case of the transverse vibration frequency.The recorded vibration amplitudes pertain to the entire structure of the conveyor and not just to small sections of the belt between the idlers.
The measured values of the frequency of the transverse vibrations of the belt between the idler supports, for these specific settings of the conveyor, were in the approximate ranges from 10 Hz to 12 Hz.With the increase in the trough angle (and thus the exponen-tial increase in the rigidity of the belt cross-section), the measured vibration frequencies slightly increased.The amplitude of the belt vibrations decreased with the increase of the trough angle.This is due to the increase in the stiffness of the belt cross-section and the lower tendency of the belt to lateral and horizontal movements.In the study of the tests carried out for the unloaded belt, it can be concluded that large trough angles better stabilize the belt.
The measured transverse vibration frequencies were averaged and compared on a common plot (Fig. 10) with the forecasts of the MCB model, the standard string model (SS model) 61 , the string model used so far for the design of conveyors (SC model) 59 , the standard model of the axially unloaded beam (SB model) 57 , the string model with beam elements (FEM model) 50 and the vibrating plate model (PM model) 51 .The results are shown as a function of the trough angle (Fig. 10a) and as a function of rigidity (Fig. 10b).
The study found that the MCB model has a mean absolute error of 0.79 Hz in predicting the vibration frequency of conveyors, with respect to the expected values based on validation data.The study revealed that the     SS, FEM, and PM models exhibited the highest accuracy among the other models evaluated, as they consistently predicted a constant vibration frequency around 10 Hz.The level of proximity to the validation measurements was sufficiently close that the suitability of these models in conveyor design cannot be dismissed.The study found that for the flat belt, the FEM model exhibited superior accuracy compared to the MCB model.However, it is not possible to conclusively state whether the FEM model provides better prediction accuracy for small trough angles, due to the absence of additional validation points and the unique characteristics of the flat belt operation.Within the range of typical trough angles used in conveyor design, the other models evaluated in the study lacked sensitivity to changes in the flexural rigidity of the belt cross-section.On the other hand, the MCB model exhibited a high degree of accuracy, closely tracking the validation data and providing the most precise predictions among the design tools tested.The SS and SB models were found to consistently predict vibration frequencies approximately 1.5 to 2 times lower than the actual validation results 62 .

Conclusion
In conclusion, the vibration analyses of the conveyor belt have revealed that existing calculation models often neglect the real influence of belt stiffness on the vibration frequency.The theoretical analyses suggest that the conveyor belt behaves like a beam with bending stiffness.The geometry of the belt, particularly the presence of a trough, increases the inertia of the cross-section, resulting in higher bending stiffness compared to a flat belt configuration.The stiffness of the belt cross-section is directly correlated with the angle of the trough.Laboratory tests have provided evidence that even for a belt of low strength, there is an observable increase in the stiffness of the cross-section bending as the trough angle increases.The model was found to be directly applicable to the conveyor system under discussion.Furthermore, changing the rigidity of the cross-section, achieved by modifying the trough angle, affected the frequency and amplitude of transverse vibrations of the unloaded belt, and the proposed model provided the smallest error while maintaining the correct changes according to the rigidity of the cross-section.
Due to the model's small number of determinable parameters and the lack of dimensionless coefficients regulating the stiffness of the conveyor-belt system, it can be effectively used in the engineering design stage to prevent resonance and is a more precise alternative to commonly used solutions.The research conducted using the developed model on a real object in the rock aggregates mine has yielded qualitative results, which have been presented in a publication.

I
Moment of inertia of the belt cross-section E Young's modulus of elasticity I s Moment of inertia of the inclined part of the cross-section of the belt h t Belt thickness l ts Length of the inclined side roller Angle of inclination of the belt trough B Belt width I m Moment of inertia of the central part of the cross-section of the belt l m Length of the center roller m

Figure 1 .
Figure 1.The cross-section of the belt at the symmetrical three-roller trough.

Figure 2 .
Figure 2. The section of the belt cross-section of length dx.

Figure 3 .
Figure 3.The load on the separated belt element with the length dx.

Figure 4 .
Figure 4. Sensitivity analysis of the MCB model: (a), (b) for changing material parameters, (c) for changing construction parameters, (d) for changing operating parameters.

Figure 6 .
Figure 6.Equipment used to MCB model validation: (a) test rig, (b) encoder for belt speed measurement, (c) load cell for measuring the tensile force of the belt, (d) measuring device on a moving conveyor bel.

Figure 7 .
Figure 7.The recorded signal of transverse vibrations from the entire conveyor: (a) signal over time with the identified peaks corresponding to the idler supports and the analysis areas, (b) frequency spectrum of a signal with the most significant frequencies.

Figure 8 .
Figure 8.The recorded transverse vibration signal for a particular section that is located between the supports of the idler: (a) time signal, (b) signal spectrum.

Figure 10 .
Figure 10.Measured vibration frequencies and comparison with other models: (a) as a function of the trough angle, (b) as a function of rigidity.

Table 1 .
Properties of the EP250 belt.