Mathematical model for optimising drive suspension of a conveyor with suspended belt and distributed drive

. This article is devoted to a topical issue related to the optimisation of the metal structure of conveyor drive hangers with suspended belt and distributed drive. The paper briefly reflects the current directions of research and development work carried out in the field of development of conveyors with suspended belt and distributed drive. The main focus is on the design of conveyor drive suspensions, performance conditions consisting in providing sufficient traction and grip, as well as the drawbacks of the existing designs of roller suspensions. An original technical solution of a roller suspension equipped with an individual drive in the form of a geared motor and a leading roller clamping device is presented, its main advantages are described. The authors have developed and presented a mathematical model which enables optimisation of metalwork of one of the most metal-intensive elements - load-bearing section of a drive hanger taking into account structural, strength and rigidity limitations. The realization of offered mathematical model is considered: by the example of the conveyor base design the calculations of the stress-deformed condition of the drive hanger metal structure are made. It is established that the base drive hanger has excessive strength, weight, and exceeded geometric parameters of the rods. It is concluded that it is necessary to perform multi-criteria optimisation of the metal structure of the drive hanger, as well as to study the influence of configuration parameters of the drive hanger on its mass-dimensional characteristics.


Introduction
Current trends in the development of conveyor transport involve the development of reliable, cost-effective, high-performance machines that provide uninterrupted transportation of goods along routes of varying complexity and length [1]. Sufficiently modern, developed, and tested in production conditions at the beginning of this century is a clip conveyor (CSC) [2].
Characterized by an original design combining design features of classical belt conveyors and rail transport, it has a number of inherent advantages [3] that have been revealed over a long period of operation. Nevertheless, the rigid closed rolling guides, whose length is difficult to change during conveyor operation, create certain difficulties in regulating the conveyor belt tension and in installing an intermediate drive on the track, which together impose restrictions on the length of the conveyor installation.
The solution to this problem has emerged with the development of a new conveyor design with a fundamentally revised kinematic arrangement. CSC with a distributed drive has no stationary concentrated drive and tensioning stations and is distinguished by the fact that a certain part of the hangers is equipped with built-in individual drive mechanisms which are powered by an electric bus duct and provide movement of the load-carrying belt [4]. The design of the CSC with distributed drive has been partially implemented in the form of a test bench [5]. Individual researchers are working on the design and development of possible technical solutions, as well as research and development work. Thus, to date, some variants of execution of conveyors, the drive of which operates under the distributed scheme, are known [4; 6, 7, 8]. The approximate calculation method [4], mathematical models of the structural group, suspensions of the discrete section [6,9] and the whole CSC with the distributed drive [8,10,11], as well as research results of some rational design parameters [12,13,14] are presented in the early scientific publications.
A special role in the design of CSCs with distributed drive is played by roller suspensions: driven and nondriven. On the one hand, they are the main supporting elements that keep the loaded belt suspended between symmetrical closed guideways and serve to move along the respective guideways along the track. On the other hand, they exercise a traction function by driving the mechanical system using lead hangers equipped with individual drive mechanisms. Consequently, the design and technical characteristics of the hangers determine the performance of the entire conveyor system.
The spatial configuration, design features, and geometrical parameters of hangers are determined by the shape and size of the cross section of the rolling guides. To date, a wide variety of CSC hanger designs are known [2,13,15], differing in the configuration of the supporting metal structure (MC), the number and type of rollers, the type of drive, and the design of the belt attachment unit. Nevertheless, a significant part of the presented hangers has the following disadvantages: they are not designed for use in the conveyor with a vertically closed configuration of the route; they are distinguished by complexity of the used profile of roller guides; they do not have sufficient resistance to distortions, do not provide qualitative coupling of a drive roller with the roller guide, do not provide for equipping the construction with an integrated individual drive.
Considering that the traction force of distributed drive CSC suspensions is realised by means of a friction transmission from the drive roller to the roller guide, it is important for slip-free travel of the drive roller to ensure its qualitative and reliable adhesion to the track surface. For this reason, the existing range of suspension designs is being supplemented with new technical solutions.
For example, the team of authors has developed a distributed drive CSC suspension design used in conjunction with rolling guides of rectangular cross-sectional shape (see Figure 1) [13]. Together, this geometric configuration contributes to the suspension's resistance to angular misalignment throughout the entire track. The vertical spatial orientation of the guide pulley eliminates a possible increase in resistance from sliding friction forces acting at the end pivot points in the case of its horizontal arrangement. The double-section arrangement of the leading hangers allows the main load of the belt and the transported load to be distributed between the elements of the bearing section (BS). The load of the weight of the clamping unit and the spring force of the drive roller clamping unit are taken up by the bearing supports, and the radial load on the gearmotor components is In addition, the location of the drive in a separate section equipped with a clamping device ensures quality grip of the drive roller with the rolling guide along the entire route, especially when driving on the idle branch when the hangers are inverted.
Studies of the influence of the design parameters of a hanger on the dynamic characteristics of the conveyor have shown that the weight of drive hangers is of significant importance, especially when their number on the track increases [12,14]. Therefore, quite relevant is the task of minimising the weight while ensuring an optimal combination of its geometric parameters, preserving the strength and rigidity of the structure. Let us analyse the stress-strain state MW BS of the CSC's distributed drive suspension, and form a mathematical model to carry out the subsequent optimisation.

Materials and methods
At the initial stage of developing an optimal design methodology, MW BS The CSC's distributed drive suspension has been structured (see Figure 2).
Weight of a single rod MW BS of the drive hanger depends on the geometric dimensions of the cross-sectional profile. Possible rod profiles and the calculated relationships to find their weights are shown in Table 1.
A conveyor with a distributed drive hanger can have a track of any length, complexity and configuration containing straight, sloping and turning sections. By analysing the key positions of the drive hanger on the route, the possible design cases, and acting loads on the MW BS drive suspension (see Figure 3).
Such characteristics include those that are determined by other parameters of the steel structure or have already been determined in previous stages of calculation or specified in the design specification. Parameters that must be taken into account during the calculation and design of the MC and cannot be controlled also include the operating loads, modes and operating conditions determined by the design specification of the conveyor [14].
In addition to uncontrollable parameters, there are controllable parameters, which can be varied to find the optimum result. They form a vector of unknown dimensions {x}, to be defined in the optimisation process. A vector of controllable {x} and a vector of unmanageable {z} parameters fully determine the geometric characteristics of the optimum MW BS drive suspension.
Setting a conditional parametric optimisation problem MW BS CSC's distributed drive suspension consists of selecting varying parameters such that the weight MW BS of the drive suspension is minimised subject to design, strength and stiffness constraints. A mathematical model that includes a target function (2), with a system of constraints imposed on it, is constructed (3), (4), and (5).

T({x}, {z}) → min;
(2) where Ttarget function (mass MW BS drive suspension); em, fn, gpsystems of structural, strength and stiffness constraints respectively. During the optimal design procedure, the MW BS of the conveyor drive hanger is tested according to the first and second limit state groups.
In the target function MW BS The possibility of taking into account the cross-sectional shapes of rods from a wide range of standard profiles (see Table 1) has been implemented in the drive hanger. Using the coefficient ji, the availability of the following is taken into account (ji =1) or lack thereof (ji =0) cross-sectional profile i-th details in the MW BS drive suspension.
The uncontrollable parameter vector therefore has the form The problem of optimisation of a conveyor suspension metal structure with a suspended belt and a distributed drive taking into account varying (6) and uncontrollable (7) where r -cross sectional area (r = rrectangular tube; r = chchannel section; r = уcorner; r = ctcircular tube; r = ctsround section; r = rcsrectangular cross section); inode element MW [17]. Structural constraints em, imposed on the target function T ({x}, {z}), are the geometric relationships associated with the dimensions of the elements MW BS of the drive suspension. For an optimised suspension design, length limitations must be met: -vertical rods l1, which must be greater than or equal to the sum of the height Hrt drive roller clamping device, half the diameter Ddr drive roller, height НBSt rectangular tube roll guide, design clearance lз between the idle roller and the raceway, half the idle roller diameter dх: cross rods l2, which must be greater than or equal to the sum of the cross-sectional profile width ВBSt rectangular tube raceway, idle roller diameter dx, structural clearance lз between the idle roller and the raceway: vertical rods l3, which must be greater than or equal to the sum of the height Hrt drive roller clamping device, half the diameter Ddr of the drive roller and half the width of the idle roller bх: 6 E3S Web of Conferences 383, 01031 (2023) https://doi.org/10.1051/e3sconf/202338301031 TT21C-2023 -longitudinal rods l4, l5 and l6, which must be larger than the diameter Drol of the suspension drive pulley: where l1, l2, l3, l4, l5, l6 -rod lengths 1, 2, 3, 4, 5 and 6 respectively; Hrt -drive roller clamping devices; Ddrdrive roller diameter; d -idle roller diameter; b -idle roller width; BSt -height of the rectangular tube of the rolling bearing guide; BSt -rectangular tube width of the rolling bearing guide; l -design clearance between the idle roller and the rolling element guide. In addition to the developed design constraints related to the lengths of the rod elements, the cross-sectional dimensions of the rods must comply with those given in GOST for the profile used.
Strength limitations fn take into account conditions [12,13]. The rigidity constraints consist in maintaining the deflection values f of the rod elements 1-6 MW BS of the actuated suspension below the limit values fu [7,9].

Results
Let us analyse the stress-strain state (SSS) MW BS drive hanger located on the track in design position 1 (see Figure 3) and with the specifications given in Table 2. In addition, based on the developed mathematical model using Siemens NX software, the following optimisation is carried out MW BS of the actuated suspension with the same layout and rod lengths.  Figure  4), which is converted into a finite element model (see Figure 5). For each rod element MW BS. The drive train is assigned a specific cross section depending on the selected profile type. The finite element model was then developed. The material (low alloy structural steel for welded structures 09G2S) and the mass of the rod elements has been determined MW BS of the drive hanger. The allowable stresses with an allowance for the safety factor are taken as 230 MPa. The locations and directions of the loads on the belt side are indicated, as well as the forces of the Fdr drive roller clamping device to the guide rail. The attachment points are set (see Figure 4) MW BS of the drive hanger at the conditional contact points of the track rollers with the raceway, limiting its movements: at points A and B linear along the OZ axis; at C and D linear along the axis ОХ.   Then the stress-strain states of the original variant are calculated MW BS The results obtained, as well as the profile sizes used in the initial and optimised version. The results obtained as well as the profile sizes used in the baseline and optimised variants MW BS of the actuated suspension, are summarised in Table 3.
In the original version MW BS In the drive hanger (see Figure 6, a), the greatest stresses occur in rods 2 and have a maximum of 142.96 MPa at the junction with vertical rods 1. Stresses in rods 1 and 3 are the same and decrease from 131.05 MPa to 11.92 MPa. Overall MW BS The drive suspension, with its original geometric dimensions, has a large safety margin which, due to the design of the CSC with distributed drive, can be considered unnecessary, since the excess weight of the suspensions results in higher driving resistances and increased dynamic loads on the conveyor belt during operation. Option MW BS of the drive hanger resulting from the optimisation procedure has a similar distribution of stresses in the rods (see Figure 6, b). The maximum stresses are 207.11 MPa. Thus, from the comparative analysis of the options, MW BS The weight of the drive train is 23% lower than that of the unoptimised suspension system as a result of optimising the cross-sectional areas of the frame rods MW drive suspension.

Conclusions
The developed mathematical model, including the target function and the systems of structural, strength, and stiffness constraints imposed on it, allows for an optimal design procedure MW BS of the CSC's distributed drive suspension.
As a result of the calculation of a typical variant MW BS of the CSC's distributed drive hanger, it was found that the hanger has excessive strength, mass and exceeded the geometric parameters of the rods. Optimisation in terms of cross-sectional dimensions of the profiles used alone has reduced the mass of the suspension by 23% compared to the unoptimised MW The need for a multicriteria optimisation of the suspension with the required safety margin. The need for multi-criteria optimisation is confirmed by calculation MW of the actuated suspension, taking into account structural, strength and stiffness constraints. Therefore, further research work will focus on the influence of geometric parameters (length and cross-