Determining the Power Consumption of the Automatic Device for Belt Perforation Based on the Dynamic Model

energies

Article Determining the Power Consumption of the Automatic Device for Belt Perforation Based on the Dynamic Model

Dominik Wojtkowiak * , Krzysztof Tala´ska,Dominik Wilczy ´nski , Jan Górecki and Krzysztof Wał˛esa

Institute of Machine Design, Poznan University of Technology, Piotrowo 3 Street, 61-138 Poznan, Poland; [email protected] (K.T.); [email protected] (D.W.); [email protected] (J.G.); [email protected] (K.W.) * Correspondence: [email protected]; Tel.: +48-61-224-4516

Abstract: The subject of the dynamic analysis presented in the article is the linear drive system with a timing belt utilized in the automatic device for polymer composite belt perforation. The analysis was carried out in two stages. In the first stage, the timing belt was modeled with all the relevant dynamic phenomena; subsequently, the tension force of the belt required for the correct operation of the belt transmission was determined. The necessary parameters for belt elasticity, vibration damping, and inertia are based exclusively on the catalog data provided by the manufacturer. During the second stage, equations of motion were derived for the designed drive system with a timing belt, and characteristics were identified to facilitate the optimal selection of electromechanical drives for the construction solution under analysis. The presented methodology allows for designing an effective solution that may be adapted for other constructions. The obtained results showed the influence of the kinematic parameters on the motor torque and proved the importance of reducing the mass of the components in machines that perform high-speed processes.

Keywords: dynamic analysis; timing belt; catalog mechanical properties; punching die; belt perforation; Lagrangian mechanics; linear drive

Citation: Wojtkowiak, D.; Tala´ska,K.; Wilczyński,D.; Górecki, J.; Wał˛esa,K. Determining the Power Consumption 1. Introduction of the Automatic Device for Belt In mechatronic design, the model which can focus on either functional or structural Perforation Based on the Dynamic Model. Energies 2021, 14, 317. form is a very important tool [1]. By juxtaposing these two models, we are able to examine https://doi.org/10.3390/en14020317 designs at an early stage [2]. The use of mathematical models to describe the dynamics in a given device not only improves the control process of the machine (e.g., choosing Received: 27 October 2020 the settings for control systems) but also makes it possible to verify its structural fea- Accepted: 30 December 2020 tures [3–5]. The main challenge is usually the determination of model parameters for the Published: 8 January 2021 basic mechanical components used in machine construction, such as guides, couplings, bearings, or belt transmissions [6,7]. Manufacturers provide a lot of different information Publisher’s Note: MDPI stays neu- in their product catalogs, but determining or finding the necessary parameters describing tral with regard to jurisdictional clai- elasticity, vibration damping or inertia for the machine components is challenging without ms in published maps and institutio- experimental studies. nal affiliations. Multiple examples of modeling the dynamics of mechanisms and drive systems with flat belts, V-belts, or timing belts can be found in the literature. Shirafuji et al. [8] proposed and analyzed a novel mechanism used for locking the flat belt and derived analytically the relationship between the tensile force and the frictional force. Balta et al. [9] Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. modeled the speed losses in V-ribbed belt drives and found the correlation between the This article is an open access article pulley size and the belt slip. Callegari et al. [10] tested the dynamic behavior of the distributed under the terms and con- timing belt transmissions in terms of acoustic radiation. Long et al. [11] analyzed the ditions of the Creative Commons At- dynamic performance of the timing belt and presented the influence of the tensioner design tribution (CC BY) license (https:// and belt pre-tension on the dynamic belt tension. Domek et al. [12] modeled the wear creativecommons.org/licenses/by/ of the timing belt pulley, while Hamilton et al. [13] presented the combination of the 4.0/). computer-based analytical simulation methods coupled with statistical tools to predict

Energies 2021, 14, 317. https://doi.org/10.3390/en14020317 https://www.mdpi.com/journal/energies Energies 2021, 14, 317 2 of 15

the timing belt life. Merghache and Ghernaout [14] focused on studying the heat transfer through the belt transmissions concerning different rotational speeds, engine torque, and setting tension. The computational methods were also used to model the belt passage by the driving drum [15] or to obtain the belt movement characteristics [16]. As can be observed in the above-mentioned research publications, the pre-tension of the belt affects the dynamic properties of the belt the most and should be precisely tested in order to gain the most effective exploitation parameters. Additionally, none of the papers presented above has been concerned with linear drive systems with timing belts. For that reason, before determining the dynamics model of the whole drive system, the focus should be put on modeling the belt itself. The dynamics of the drive systems have an influence on the power consumption of the designed machine. The rational design of the work drive system may greatly reduce the torque or the force, which has to be generated in order to properly perform the technological process [17]. If the machine is supplied with an electric motor, this value will have an impact on the power consumption of the machine. Electric motors are designed to work in the range between 50–100% of the rated load [18]. Below this value the power factor decreases sharply [19], resulting in inefficient power consumption of the electric device. This implies that performing a complex dynamic analysis of the designed structure may lead to improvements in the energetic efficiency of the device. Based on its results, it is possible to select a motor power very close to the real rated load. Of course, the method of controlling and using proper motor drivers or regulators will also have an effect on power consumption [20]. The dynamic model of the drive system can be also useful in selecting the parameters of the applied PI or PID regulators, which will increase the precision of positioning and prevent the occurrence of resonance. This article presents the methodology for modeling the dynamics of the designed device based on the catalog data and geometric parameters that are available at the design stage, using linear drive systems with timing belts employed in automatic devices for polymer composite belt perforation as an example. This approach makes it possible to design an effective device structure and analyze it at the design stage. In addition, the article provides the methodology of modeling the timing belt with a view to analyze aspects of drive system dynamics. All these aspects can be used to improve the efficiency and power consumption of the designed electric device.

2. Materials and Methods The object of research is the automatic device for composite polymer belt perforation (Figure1)[ 21]. It features two mirrored perforating heads and one separated working drive unit. The pressure plate connected with the working drive actuators presses on one or two heads at the same time, affecting the motion of the head-punch plate and cutting the hole in the belt using the punch and die. Both punching dies move along a shared linear die, whereas the working drive system has its own guiding system. Since both of the punching dies have signiﬁcant mass and the process works at rather high speed and acceleration, the inertia will have a crucial inﬂuence on selecting the proper drive [22]. Each of those subassemblies is driven by an individual, linear drive system with a timing belt spanning between the driven and idle wheel systems. Each of the belts features a bolt tensioner mechanism. Energies 2021, 14, x FOR PEER REVIEW 3 of 15 Energies 2021, 14, 317 3 of 15 Energies 2021, 14, x FOR PEER REVIEW 3 of 15

Figure 1. Schematic of the belt perforation device in a configuration using a punch and a die [23]: 1—punching die, 2— work drive unit (WDU), 3—punching die guide, 4—drive wheel assembly for punching die drive, 5—driven wheel as- Figure 1.1. SchematicSchematic ofof thethe belt belt perforation perforation device device in in a configurationa configuration using using a punch a punch and and a die a [die23]: [23]: 1—punching 1—punching die, 2—workdie, 2— semblywork drive punching unit (WDU), die drive 3—punching (PDD), 6—lower die guide, toothed 4—drive belt, 7— wheellower assemblybelt tensioner, for punching 8—punching die drive, die drive 5—driven mount, wheel9—WDU as- drive unit (WDU), 3—punching die guide, 4—drive wheel assembly for punching die drive, 5—driven wheel assembly guide,sembly 10—WDU punching diedrive drive wheel (PDD), assembly, 6—lower 11—WDU toothed drivenbelt, 7— whlowereel assembly, belt tensioner, 12—upper 8—punching toothed die belt, drive 13—upper mount, 9—WDU belt ten- punching die drive (PDD), 6—lower toothed belt, 7—lower belt tensioner, 8—punching die drive mount, 9—WDU guide, sioner,guide, 10—WDU14—frame, drive Belt—perforated wheel assembly, belt, PDDi—i-th11—WDU drivenpunching wh eeldie assembly,drive system 12—upper, WDUD—WDU toothed belt,drive 13—upper system. belt ten- 10—WDUsioner, 14—frame, drive wheel Belt—perforated assembly, 11—WDU belt, PDDi—i-th driven pu wheelnching assembly, die drive 12—upper system, WDUD—WDU toothed belt, 13—upper drive system. belt tensioner, 14—frame, Belt—perforated belt,To PDDi—i-th design an punching effective die automatic drive system, device WDUD—WDU for belt perforation, drive system. it is crucial to determine the peakTo design torque, an which effective will automatic be used todevice select for an belt electric perforation, motor and it is transmission.crucial to determine In the To design an effective automatic device for belt perforation, it is crucial to determine casethe peak of the torque, mechatronic which applications,will be used thisto select procedure an electric should motor be supported and transmission. by the dynamic In the the peak torque, which will be used to select an electric motor and transmission. In the modelcase of ofthe the mechatronic mechanical applications, structure [24]. this By pr ocedureanalyzing should the stiffness, be supported inertia, by and the dampingdynamic case of the mechatronic applications, this procedure should be supported by the dynamic ofmodel the machine, of the mechanical it is possible structure to perform [24]. theBy anparameteralyzing theoptimization stiffness, inertia,[25], which and resultsdamping in model of the mechanical structure [24]. By analyzing the stiffness, inertia, and damping designingof the machine, effective it is mechatronic possible to perform devices. the parameter optimization [25], which results in of the machine, it is possible to perform the parameter optimization [25], which results in designingThe analysis effective was mechatronic carried out devices. in two stages. In the first stage, the timing belt was mod- designing effective mechatronic devices. eled with all the relevant dynamic phenomena; subsequently, the tension force of the belt The analysis was carried out in two stages. In In the the first first stage, stage, the the timing timing belt belt was was mod- mod- requiredeled with for all thethe correctrelevant operation dynamic of phenomena; the belt transmission subsequently, was thedete tensionrmined. force The ofnecessary the belt parametersrequired for for the belt correct elasticity, operation vibration of the damping, belt transmission and inertia was are determined.dete basedrmined. exclusively The necessary on the catalogparameters data for provided belt elasticity, by the manufacturer. vibration damping, During and the inertia second are stage, based equations exclusively of motion on the werecatalog derived data provided for the designed by the manufacturer. drive system DuringDuwiringth a thetiming second belt stage,using equationsthe general of form motion of Lagrangianwere derived equations forfor thethe designeddesigned of the second drive drive system kind.system Then, with with athe timinga timing characteristics belt belt using using thewere the general identifiedgeneral form form to of La-fa- of cilitateLagrangiangrangian the equations optimal equations ofselection theof the second secondof electromecha kind. kind. Then, Then, thenical characteristicsthe drives characteristics for the were design were identified identifiedsolution to facilitate under to fa- analysis.cilitatethe optimal the optimal selection selection of electromechanical of electromecha drivesnical for drives the design for the solution design under solution analysis. under analysis. 3. Results Results 3.1.3. Results Modelling Modelling the the Timing Timing Belt Dynamic Behavior 3.1. ModellingInIn order order to the correctly Timing Belt model Dynamic the operation Behavior of the linear drive system designed on the basisIn of ordera toothed to correctly belt, one model should the begin operation with modelingof the linear the drive belt itself system (Figure designed2 2).). on the basis of a toothed belt, one should begin with modeling the belt itself (Figure 2).

Figure 2. ModelModel of of the the toothed toothed belt within the drive system. Figure 2. Model of the toothed belt within the drive system. Assuming that that the the tooth ii = 1 1 is acting as support, the tensile force F resultant from thethe drive driveAssuming torque torque that TM theof the tooth belt i wheel= 1 is acting with diameter as support, D is the applied tensile to force the tooth F resultant ii = z, where from zthez isis drivethe number torque of TM belt of teeth the belt on wheelthe section with with diameter length D isequal applied to the to distancethe tooth between i = z, where the beltz is thewheel number axles of a; belt hence, teeth the on toothed the section belt cancanwith bebe length modeledmodeled equal asas to aa systemthesystem distance comprisingcomprising between zz −the− 1 belt wheel axles a; hence, the toothed belt can be modeled as a system comprising z − 1

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systems, including body-spring-damper (m-k-b) with parameters as provided on Figure2, connected in series [26,27]. For a system comprising sets of units connected in series, the displacement of the entire system is equal to the sum of displacement of every constituent, whereas each component is affected by an identical force equal to the global load. Within the m-k-b system, the constituents are connected in parallel, i.e., the displacement of each component is the same, whereas the force is distributed on all the components proportionally to displacement (for the spring), velocity (for the damper), and acceleration (for the body). As the structure of the belt is uniform along its entire length, the coefficients of elasticity kTB’, coefficients of damping bTB’ and single unit weights mTB’ are identical for each m-k-b system. If the first link is immobilized, and each system is displaced equally by the value x’ the 0 displacement of the solid body i is equal to xi = (i − 1)·x . Assuming the system number including body ni = i − 1, it can be described with the following set of equations:  .. . . 0 0 0 0 0 0  F = mTB x + bTB x + kTB x  .. . .  = 0 0 + 0 0 + 0 0 F mTB 2x bTB x kTB x (1) ...  .. . .  0 0 0 0 0 0  F = mTB (z − 1)x + bTB x + kTB x

After summing the equations, we arrive at:

z(z − 1) .. . (z − 1)F = m 0 x0 + b 0(z − 1)x0 + k 0(z − 1)x0 (2) TB 2 TB TB After dividing both sides of the equation by (z − 1) and introducing the dependency (3), the following characteristic is obtained (4).

...... x = (z − 1)x0; x = (z − 1)x0; x = (z − 1)x0 (3)

0 0 z .. b . k F = m 0 x + TB x + TB x (4) TB 2(z − 1) (z − 1) (z − 1) Applying the replacement model as a single system of body-spring-damper with parameters mTB, kTB, and bTB, we arrive at the following dynamic equation: .. . F = mTBx + bTBx + kTBx (5)

Considering that the weight of the entire belt on the examined section can be calculated with Equation (6), we receive the following substitute parameters for the model as provided in Equations (7)–(9). 0 mTBr = (z − 1)mTB (6) mTBrz mTB = (7) 2(z − 1)2 k 0 k = TB (8) TB (z − 1) b 0 b = TB (9) TB (z − 1)

3.2. Dynamic Analysis of the Timing Belt Linear Drive Systems The above model of the toothed belt is used in the dynamic models of linear drive systems for perforating heads as well as the working drive unit. All three systems are implemented in an analogous manner to Figures3 and4. In these systems, the moving component is rigidly connected via toothed plates to the toothed belt which spans between the two toothed belt wheels. The analyzed component travels along a proﬁled linear Energies 2021, 14, 317 5 of 15 EnergiesEnergies 2021 2021, 14, 14, x, xFOR FOR PEER PEER REVIEW REVIEW 5 5of of 15 15

guide on carriages with rolling components. The driven wheel uses a brushless direct currentcurrentcurrent drive drive (BLDC), (BLDC), its its its shaft shaft shaft is is is connected connected to to to the the wheel wheel shaft shaft via via the the dog dog clutch clutch with with a a polyurethanepolyurethane insert. insert. ToTo ensure ensure ensure the the the correct correct correct operation operation operation of of the ofthe thepresented presented presented linear linear linear drive drive drive system, system, system, correct correct correct belt belt ten- ten- belt sioningsioningtensioning is is required. required. is required. With With With a a too-low too-low a too-low initial initial initial belt belt belt tension, tension, tension, the the the teeth teeth teeth of of ofthe the the idle idle idle belt belt belt section section collidecollidecollide or or or overlay overlay with with the the teeth teeth of of the the wheel wheel and and increased increased surface surface wear wear occurs occurs due due to to frictionfrictionfriction as as as teeth teeth are are engaged. engaged. Furthermore, Furthermore, Furthermore, forced forced forced breakage breakage breakage of of of the the belt belt may may occur occur due due to to excessiveexcessive elongation elongationelongation caused causedcaused by by completecomplete complete overlapover overlaplap of of of teeth. teeth. teeth. On On On the the the other other other hand, hand, hand, excessive excessive excessive belt beltbelttension tension tension causes causes causes a high a a high high load load load on shaft on on shaft shaft bearings, bearings, bearings, reduces reduces reduces transmittable transmittable transmittable power, power, power, and increasesand and inin- creasescreaseswearof wear wear the of beltof the the teeth. belt belt teeth. Itteeth. is thereforeIt It is is therefore therefore critical cr criticalitical to select to to select select the the correct the correct correct belt belt belt tensioning tensioning tensioning for for thefor thethedesigned designed designed system. system. system. The The The tensioning tensioning tensioning in thein in the currentthe current current design design design is achieved is is achieved achieved with with with an adjustablean an adjustable adjustable bolt boltboltmechanism mechanism mechanism (Figures (Figures (Figures3 and 3 3 4and and). 4). 4).

FigureFigure 3. 3. General GeneralGeneral schematic schematic schematic of of the the linear linear drive drive system system with with a a toothed toothed belt. belt.

Figure 4. Schematic of the linear drive system of the work drive unit (WDU) [23]. FigureFigure 4. 4. Schematic Schematic of of the the linear linear drive drive system system of of the the work work drive drive unit unit (WDU) (WDU) [23]. [23]. In order to apply the simplifying solutions, in which the relative belt displacement in InIn order order to to apply apply the the simplifying simplifying solutions, solutions, in in which which the the relative relative belt belt displacement displacement in in relation to the fastened head is 0, belt tensioning with force Ft, which fulﬁlls the following relation to the fastened head is 0, belt tensioning with force Ft, which fulfills the following relationinequality to the resultant fastened from head d’Alembert’s is 0, belt tensioning principle, with is required: force Ft, which fulfills the following inequalityinequality resultant resultant from from d’Alembert’s d’Alembert’s principle, principle, is is required: required: .. . ≥ ( + ) + + Ft𝐹 ≥(𝑚mTB +𝑚mPDx)𝑥 +𝑏bTBx𝑥 +𝑘kTBx𝑥 (10) 𝐹 ≥(𝑚 +𝑚)𝑥 +𝑏𝑥 +𝑘𝑥 (10) ToTo maintain maintain the the rigidity rigidity of of the the coupling, coupling, th th theee force force resulting resulting from from the the belt’s belt’s flexibility flexibility ﬂexibility shouldshouldshould compensate compensate for for the the inertial inertial force force of of the the moving moving component: component:

𝑘𝑥=𝑚 ∙𝑎 (11) k𝑘TBx𝑥=𝑚= mPD ·∙𝑎aPD (11)(11) As the peak value of force Ft is known, we are able to calculate the displacement value As the peak value of force Ft is known, we are able to calculate the displacement value toto be be affected affected on on the the belt belt to to ensure ensure correct correct tensioning: tensioning:

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𝐹 𝑥 = (12) 𝑘 Energies 2021, 14, 317 6 of 15 Once this value is known, the belt tensioning process, together with controlling the correctness of tensioning, can be easily automated. With known parameters of the thread used in the bolt tensioning mechanism (fine thread M10×1 with pitch diameter d2 = 9.35 As the peak value of force Ft is known, we are able to calculate the displacement value mm and pitch Pt = 1 mm) and the coefficient of friction between the bolt and the nut (μ = to be affected on the belt to ensure correct tensioning: 0.2), the moment of friction between the bolt and the nut can be calculated from the equation: F 1 t peak 𝑇 x=max𝐹=𝑑𝑡𝑔(𝛾 +𝜌 ) (13)(12) 2 kTB where:Once this value is known, the belt tensioning process, together with controlling •the lead correctness angle of of helix tensioning, 𝛾=𝑎𝑟𝑐𝑡𝑔 can be easily = 1.95° automated., With known parameters of the × thread used in the bolt tensioning mechanism (ﬁne thread M10 1 with pitch diameter • apparent angle of friction 𝜌 =𝑎𝑟𝑐𝑡𝑔 = 13°. d2 = 9.35 mm and pitch Pt = 1 mm) and the° coefﬁcient of friction between the bolt and the nut (Theµ = selection 0.2), the momentof the toothed of friction belt betweenwith the theselected bolt andpitch the is nutcarried can out be calculatedon the basis from of displacedthe equation: component and maximum assumed acceleration. The design assumes the mass 1 0 of a single perforating head mPD = T90 kg= andF d thetg massγ + ρof the work drive system mWDU =(13) 50 S3 2 t 2 kg. Based on the catalog data, AT10 belts (Pbelt = 10 mm) with widths 32 mm and 25 mm, where: respectively, (Figure 5) were selected for the drive systems. • Inlead order angle to determine of helix γ =thearctg modelP tparameters= 1.95◦, of the belt as an m-k-b system, the man- πd2 ufacturer data provided in the catalog0 were µused as provided◦ in Figures 6 and 7. For the • apparent angle of friction ρ = arctg cos30◦ = 13 . purpose of calculations, it is further required to identify the distance between the belt The selection of the toothed belt with the selected pitch is carried out on the basis wheel axles a, which for the working drive system is 1245 mm and for the perforating of displaced component and maximum assumed acceleration. The design assumes the head drive is 795 mm. It is used to calculate the number of belt teeth on the examined mass of a single perforating head mPD = 90 kg and the mass of the work drive system section (Equation (14)), as well as the actual mass of the belt mbelt (Equation (15)), which is mWDU = 50 kg. Based on the catalog data, AT10 belts (Pbelt = 10 mm) with widths 32 mm usedand 25to mm,determine respectively, the substitute (Figure mass5) were mTB selected according for theto the drive Equation systems. (7).

FigureFigure 5. 5. MethodologyMethodology of of the the selection selection of of the the toothed toothed belt on the basis of the manufacturer’s catalogcata- logdata data [28 ]:[28]: red red dot dot and and star star represents represents the the combination combination of mass of mass and and acceleration acceleration for WDUDfor WDUD and PDDand PDDrespectively. respectively.

In order to determine the model parameters of the belt as an m-k-b system, the manufacturer data provided in the catalog were used as provided in Figures6 and7. For the purpose of calculations, it is further required to identify the distance between the belt wheel axles a, which for the working drive system is 1245 mm and for the perforating head drive is 795 mm. It is used to calculate the number of belt teeth on the examined section Energies 2021, 14, x FOR PEER REVIEW 7 of 15

Energies 2021, 14, 317 7 of 15 Figure 6. Belt catalog data including weight, and elasticity [28].

𝑎 (Equation (14)), as well as the actual mass𝑧= of the belt+1 mbelt (Equation (15)), which is used(14) to 𝑃 determine the substitute mass mTB according to the Equation (7). 𝑚 =𝑤 ∙𝑎 (15) a z = + 1 (14) Using the data in the manufacturer’s catalog,Pbelt it is possible to calculate maximum belt Energies 2021, 14, x FOR PEER REVIEWdeformation ε (Equation (16)), and after multiplying it by the distance between the wheel7 of 15 m = w ·a axles, we obtain the threshold belt displacementbelt belt value x. Dividing the allowable tension-(15) ing force value FTadm by displacement x, we obtain belt elasticity kTB.

𝑎 𝑧= +1 (14) 𝑃

𝑚 =𝑤 ∙𝑎 (15) Using the data in the manufacturer’s catalog, it is possible to calculate maximum belt deformation ε (Equation (16)), and after multiplying it by the distance between the wheel axles, we obtain the threshold belt displacement value x. Dividing the allowable tensioning force value FTadm by displacement x, we obtain belt elasticity kTB.

𝐹 𝜀= (16) 𝐶 The major challenge is in determining the b coefficient, which defines the rheological characteristics of the belt. To this end, the information on the maximum distributed load per one tooth for different rotation speeds of the driven wheel is required. This infor- Figure 7. The catalog data for the belt regarding the inﬂuence of rotation speed n (rpm) on the mationFigure 7. is The transformed catalog data into for the the belt characteristics regarding the ofinfluence the tangential of rotation force speed as na (rpm) function on the of the allowable distributed load on a single tooth F (N/cm) together with the characteristic of the beltallowable linear distributed velocity F loadt(v). on a single tooth Ftspectspec (N/cm) together with the characteristic of the tangentialtangential force force as as a a function of the belt linear velocity forfor beltbelt wheelwheel diameterdiameter DD == 128 mm andand belt beltwidth widthW = W 25 = mm 25 mm [28]. [28].

TheUsing decreasing the data inallowable the manufacturer’s force load catalog,on the belt it is is possible a consequence to calculate of maximumthe allowance belt establisheddeformation fromε (Equation process (16)),dynami andcs. after Therefore, multiplying if the value it by the Ft (v distance = 0) = F betweent0 is assumed the wheel to be staticaxles, and we obtainby calculating the threshold the difference belt displacement between value the obtainedx. Dividing force the value allowable and static tensioning force force value F by displacement x, we obtain belt elasticity k . Ft0 for every pointTadm of measurement, we receive the characteristicTB curve Fb(v). Approximat- ing the characteristic curve to a linear function, we arrive at the belt damping constant bTB FTadm (Figure 8). ε = (16) Cspec

The major challenge is in determining the b coefﬁcient, which deﬁnes the rheological characteristics of the belt. To this end, the information on the maximum distributed load per one tooth for different rotation speeds of the driven wheel is required. This information is transformed into the characteristics of the tangential force as a function of the belt linear velocity Ft(v).

FigureThe 7. The decreasing catalog data allowable for the belt force regarding load on the the influence belt is of a rotation consequence speed n of (rpm) the on allowance the established from process dynamics. Therefore, if the value F (v = 0) = F is assumed to be allowable distributed load on a single tooth Ftspec (N/cm) together witht the characteristict0 of the tangentialstatic and force by calculating as a function the ofdifference the belt linear between velocity the for obtained belt wheel force diameter value D and = 128 static mm force and Ft0 beltfor width every W point = 25 ofmm measurement, [28]. we receive the characteristic curve Fb(v). Approximating the characteristic curve to a linear function, we arrive at the belt damping constant bTB (FigureThe8 ).decreasing allowable force load on the belt is a consequence of the allowance established from process dynamics. Therefore, if the value Ft (v = 0) = Ft0 is assumed to be static and by calculating the difference between the obtained force value and static force Ft0 for every point of measurement, we receive the characteristic curve Fb(v). Approximat- ing the characteristic curve to a linear function, we arrive at the belt damping constant bTB (Figure 8).

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Figure 8. Modeling the coefﬁcientcoefficient of damping bTB forfor the the toothed toothed belt. belt.

2 Assuming thethe boltbolt accelerationacceleration isis equalequal to to 0.1 0.1 m/s m/s2 and maximum boltbolt speedspeed 55 mm/s,mm/s, we arrive atat thethe maximummaximum forceforce valuevalueF Ftpeaktpeak = 505.1 505.1 N. N. Therefore, Therefore, the belt is to be tensioned until the displacement value byby 1.15 mm is achieved. Calculating this as the torque forfor thethe tensioning bolt, one needs to useuse aa drive with nominal torquetorque 0.630.63 Nm.Nm. To establishestablish thethe equationsequations ofof motionmotion forfor thethe linearlinear drivedrive system,system, thethe generalgeneral formform ofof Lagrangian equations of the second kind were used together with the diagrams provided in Figures3 3 and and4 .4. The kinetic energy in the system under examinationexamination for the workingworking drivedrive systemsystem isis expressed with the below equation: 1 1 1 1 ω 2 1 1 = 2 + 2 + 2 + 0 M3 + 2 + ( − )2 Ek JM31ωM3 JC13ωM3 JP13ωM3 J1P3 𝜔2 1mWDUvWDU 1 mTB vWDU vS3 (17) 2 2 2 2 ip 2 2 𝐸 = 𝐽𝜔 + 𝐽𝜔 + 𝐽𝜔 + 𝐽 + 𝑚𝑣 + 𝑚(𝑣 −𝑣) (17) 2 2 2 2 𝑖 2 2 Considering the ratio of the belt transmission ip = 1 and identical moments of inertia for bothConsidering belt wheels theJ P3ratio= J P3of0 thetogether belt transmission with the dependency ip = 1 and identical between moments linear and of angular inertia 1 velocityfor both vbeltWDU wheels= ωM 3J·P32 D= PJ3P3,′ astogether well as withvS3 = the 0 due dependency to correct beltbetween tension, linear we and arrive angular at the following:velocity 𝑣 =𝜔 ∙ 𝐷, as well as vS3 = 0 due to correct belt tension, we arrive at the 1 1 following: E = (J + J + 2J + (m + m )D )ω 2 (18) k 2 M3 C3 P3 4 TB WDU P3 M3 1 1 ∂E K𝐸 = (𝐽 + 𝐽 +2𝐽 + 1 (𝑚 +𝑚)𝐷)𝜔 (18) =2 (JM3 + JC3 + 2JP3 + 4(mTB + mWDU)DP3)ωM3 (19) ∂ωM3 4 𝜕𝐸 1 d ∂EK =(𝐽 + 𝐽 +2𝐽 + (1𝑚 +𝑚)𝐷)𝜔. (19) 𝜕𝜔 = (JM3 + JC3 + 2JP3 +4 (mTB + mWDU)DP3)ωM3 (20) dt ∂ωM3 4 𝑑 𝜕𝐸 1 Considering that all the=( constituents𝐽 + 𝐽 +2 of the𝐽 analyzed+ (𝑚 +𝑚 system move)𝐷 )𝜔 along a horizontal(20) 𝑑𝑡 𝜕𝜔 4 axis and the height of their centers of gravity remain unchanged during motion, the potentialConsidering energy does that notall the include constituents the energy of the originating analyzed from system gravity, move only along the a elasticity horizontal of theaxis toothed and the belt: height of their centers of gravity remain unchanged during motion, the po- 1 2 tential energy does not include Ethep = energykTB( origxWDUinating− xTB from) gravity, only the elasticity(21) of the toothed belt: 2 ∂Ep If xWDU = xTB therefore Ep = 0, as1 well as the derivative QjP = − ∂ϕ = 0. 𝐸 = 𝑘 (𝑥 −𝑥 ) M3 (21) To describe the mechanical losses in2 the device, it is required to calculate the power of dissipation Φ : R If 𝑥 =𝑥 therefore 𝐸 =0, as well as the derivative 𝑄 =− =0. 1 1 1 ω 2 1 . . 1 =To describe 2the+ mechanical2 + losses0 inM the3 + device, it is required− 2to+ calculate the 2power Φ bM3ωM3 bB3ωM3 bB3 2 bTB xWDU xTB bLG3vWDU (22) 2 2 2 ip 2 2 of dissipation Φ: 1 Considering 1 the kinematic1 𝜔 correlations1 as described above1 and the transmission ratio Φ= 𝑏𝜔 + 𝑏𝜔 + 𝑏 + 𝑏(𝑥 −𝑥 ) + 𝑏𝑣 (22) 2 together with2 the identical2 bearing𝑖 of2 both belt wheels we2 arrive at:

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1 1 Φ = (b + 2b + b D )ω 2 (23) R 2 M3 B3 4 LG3 P3 M3

∂ΦR 1 QjR = − = −(bM3 + 2bB3 + bLG3DP3)ωM3 (24) ∂ωM3 4

Generalized force: Qj = TM3. The dynamic equation for the examined system has the following form: 1 . 1 J + J + 2J + (m + m )D ω = T − (b + 2b + b D )ω (25) M3 C3 P3 4 TB WDU P3 M3 M3 M3 B3 4 LG3 P3 M3 Calculating the drive torque of the BLDC motor, we receive the characteristic curve . TM3 ωM3, ωM3 :

. 1 . 1 T ω , ω = J + J + 2J + (m + m )D ω + (b + 2b + b D )ω (26) M3 M3 M3 M3 C3 P3 4 TB WDU P3 M3 M3 B3 4 LG3 P3 M3

The values of moments of inertia together with the masses of individual system components were derived from the manufacturer’s catalog or the CAD software for 3D modeling. In order to model the coefficient of damping bM3, which defines mechanical losses resulting primarily from friction, the catalog data on drive efficiency were used. Regarding roller guides employed in the design, the guide frictional resistance may constitute only 1/20 to 1/40 of the frictional resistance of a comparable slide rail. This is observable in particular for static friction, which is significantly lower for the roller bearing. Moreover, the difference between the static and dynamic friction is very low; therefore, the problem of the so-called stick-slip effect is not present. Due to such low friction, relatively low powered drive motors are suitable for the guides. The friction resistance of linear guides may vary within a specific range due to load, initial tension, lubricant viscosity, the number of seals, and other factors. The frictional resistance of the guide can be calculated using a formula based on working load and packing friction. Generally speaking, the coefficient of friction for ball-type guides is 0.002–0.003 (without packing), and for roller type guides is 0.001–0.002 (without packing) [29]. Therefore, this constituent can be omitted in the model, but in the literature, we can find some models to determine the dynamics of such machine components [30]. However, for slide rails, these losses need to be accounted for. Similar phenomena are observed in the case of roller bearings. The breakaway torque can be determined based on manufacturer catalog data [31]; however, its value is low in comparison to resistances resulting from inertia. In order to determine the torque parameters of motors M1 and M2 driving the perforating heads, the model developed above was also utilized. the only difference was in regard to the generalized force value Qj, which was calculated using the principle of virtual work, including an additional resistance force caused by the belt adhering to the perforating head or uncut strands being caught on the punch. The drive torque of the motor should enable continued operation of the process in case such resistances occur (sever the strand or separate the belt from the head).

δL TM1·δϕM1 − FRδxPD1 TM1·δϕM1 − FRDP1δϕM1 Qj = = = = TM1 − FRDP1 (27) δqj δϕM1 δϕM1

The design for both drive systems is identical, therefore a single dynamic equation was used for its mathematical description:

. 1 . TM1,2 ωM1,2, ωM1,2 = JM1 + JC1 + 2JP1 + 4 (mTB + mPD)DP1 ωM1,2 (28) 1 + bM1 + 2bB1 + 4 bLG1DP1 ωM1 + FRDP1 EnergiesEnergies 2021 2021, ,14 14, ,x x FOR FOR PEER PEER REVIEW REVIEW 1010 of of 15 15

11 𝑇𝑇 𝜔𝜔 ,𝜔,𝜔 ==𝐽𝐽 ++𝐽𝐽 +2+2𝐽𝐽 ++ ((𝑚𝑚 +𝑚+𝑚 ))𝐷𝐷 𝜔𝜔 ++ ,, ,, ,, 44 ,, (28)(28) Energies 2021, 14, 317 11 10 of 15 +𝑏+𝑏 +2𝑏+2𝑏 ++ 𝑏𝑏 𝐷𝐷 𝜔𝜔 +𝐹+𝐹𝐷𝐷 44 BasedBased on on the the equations equations of of motion motion 26 26 and and 28 28, ,example example drive drive characteristics characteristics for for various various accelerationsaccelerations (Figures (FiguresBased 9 9 and and on 10) the10) and equations and velocities velocities of motion (Figures (Figures 26 and 11 11 28, and and example 12) 12) were were drive derived. derived. characteristics To To obtain obtain for various thethe characteristic characteristicaccelerations which which will will (Figures enable enable9 selectingandselecting 10) andthe the velocitiesproper proper drive drive (Figures for for chosen 11chosen and 12kinematic kinematic) were derived. pa- pa- To obtain the characteristic which will enable selecting the proper drive for chosen kinematic rameters,rameters, the the peak peak torque torque values values were were compar compareded with with the the rotational rotational acceleration acceleration and and pre- pre- parameters, the peak torque values were compared with the rotational acceleration and sented in Figure 13. sented in Figurepresented 13. in Figure 13.

Figure 9.FigureCharacteristicFigure 9. 9. Characteristic Characteristic curves of the curves curves WDU driveof of the the motor WDU WDU torque drive drive for motor motor various torque torque rotational for for various accelerationvarious rotational rotationalε and maximum acceleration acceleration rotational ε ε speed ω =andand 3000 maximum maximum rev/s. rotational rotational speed speed ω ω = = 3000 3000 rev/s. rev/s.

Figure 10. Characteristic curves for the perforating head drive motor torque for various rotational acceleration ε and FigureFigure 10. 10. Characteristic Characteristic curves curves for for the the perforating perforating head head drive drive motor motor torque torque for for various various rotational rotational maximum rotational speed ω = 500 rev/s and assuming uncut strand resistance force of F = 50 N. accelerationacceleration ε ε and and maximum maximum rotational rotational speed speed ω ω = = 500 500 rev/s rev/s and and assuming assumingR uncut uncut strand strand resistance resistance forceforce of of F FRR = = 50 50 N. N.

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FigureFigure 11. 11. CharacteristicCharacteristic curve curve of the of WDU the WDUdrive mo drivetor torque motor for torque rotational for rotational acceleration acceleration ε = 500 FigureFigure 11. 11. Characteristic Characteristic curve curve of of the the WDU WDU drive drive mo motortor torque torque for for rotational rotational acceleration acceleration ε ε= =500 500 rev/s2 2and various2 maximum rotational speed ω. rev/sεrev/s= 5002 and and rev/s various variousand maximum maximum various maximum rotational rotational rotational speed speed ω ω. speed. ω.

FigureFigure 12. 12. CharacteristicCharacteristic curve curve for the perforatingperforating headhead drivedrivemotor motor torque torque for for rotational rotational acceleration accelera- FigureFigure 12. 12. Characteristic Characteristic curve curve for for the the pe perforatingrforating head head drive drive motor motor torque torque for for rotational rotational accelera- accelera- tion ε = 100 rev/s2 2 2and various maximum rotational speed ω and assuming uncut strand resistance tionεtion= ε 100 ε= =100 rev/s100 rev/s rev/sand2 and and various various various maximum maximum maximum rotational rotational rotational speed speed speedω and ω ω and assumingand assuming assuming uncut uncut uncut strand strand strand resistance resistance resistance force force of FRR = 50 N. forceofforce FR of =of F50 FR = N. =50 50 N. N.

Figure 13. The influence of the rotational acceleration ε on the peak head/WDU drive motor FigureFigure 13. 13.13. The The influence inﬂuenceinfluence of ofof the thethe rotational rotationalrotational acceleration accelerationacceleration εε εonon on thethe the peakpeak peak head/WDUhead/WDU head/WDU drive drivedrive motor motormotor torque torque for maximum rotational speed ω = 500 rev/s (for head drive) or ω = 3000 rev/s (for WDU torquefortorque maximum for for maximum maximum rotational rotational rotational speed speedω speed= 500 ω ω rev/s= =500 500 rev/s (for rev/s head (for (for head drive) head drive) drive) or ω =or or 3000 ω ω = =3000 rev/s 3000 rev/s (forrev/s WDU(for (for WDU WDU drive). drive). drive).drive).

Energies 2021, 14, x FOR PEER REVIEW 12 of 15

4. Discussion Based on the obtained results presented in the previous section, the analysis of the influence of the kinematic parameters (rotational speed and acceleration) on the selected motor torque can be performed. By combining these results, it is possible to calculate the Energies 2021, 14, 317 amount of energy used in the acceleration phase, which can be used to evaluate the power 12 of 15 consumption of the machine. As can be observed in Figures 9 and 10, by increasing the acceleration and maintaining constant maximal rotational speed, the maximum torque rises sharply. The signifi- 4. Discussion cance of this growth is caused by the inertia of the mass of the punching dies. After achieving the maximumBased constant on the speed, obtained the static results torque presented falls down in the to previous 10 times section, lower values the analysis of the (for the greatestinfluence value ofof theacceleration). kinematic This parameters shows the (rotational importance speed of andreducing acceleration) the mass on of the selected the componentsmotor intorque machines can bethat performed. perform high-speed By combining processes. these results, Comparing it is possible both ana- to calculate the lyzed drive systems,amount ofthe energy static usedtorque in is the much acceleration higher for phase, a drive which of the can punching be used to die evaluate than the power for the driveconsumption of the work driving of the machine. unit. It is caused by the drag from the uncut fibers of the perforated belt, whichAs can was be observedtaken into in consideration Figures9 and in10 the, by model increasing derivation. the acceleration If we analyze and maintaining constant maximal rotational speed, the maximum torque rises sharply. The significance the energy used in the acceleration phase for both cases (Figure 14), which can be esti- of this growth is caused by the inertia of the mass of the punching dies. After achieving mated as a product of peak torque Tmax, maximum angular velocity ωmax, and time of ac- the maximum constant speed, the static torque falls down to 10 times lower values (for celeration ta (Equation (29)) [32], it remains almost constant, but for very low acceleration the greatest value of acceleration). This shows the importance of reducing the mass of the (below 100 rev/s2), it starts to grow. This implies that working at very low speeds is not components in machines that perform high-speed processes. Comparing both analyzed recommended, taking into account the power consumption of the machine, especially drive systems, the static torque is much higher for a drive of the punching die than for since it will require greater time to go the same positioning distance as for higher speeds. the drive of the work driving unit. It is caused by the drag from the uncut fibers of the perforated belt, which𝐸=𝑇 was taken∙𝜔 into∙𝑡 consideration in the model derivation.(29) If we analyze the energy used in the acceleration phase for both cases (Figure 14), which can be For varying maximum rotational speed (Figures 11 and 12), the value of the peak estimated as a product of peak torque Tmax, maximum angular velocity ωmax, and time of torque is always the same regardless of the speed. However, it affects the time of acceler- acceleration ta (Equation (29)) [32], it remains almost constant, but for very low acceleration ation as it decreases(below 100 when rev/s the2), torque it starts value to grow. gets This lower. implies If we that analyze working the atenergy very lowused speeds is not during the accelerationrecommended, phase taking in the into function account of the maximum power consumption rotational speed of the for machine, a constant especially since rotational acceleration,it will require for greater both discussed time to go drive the sameunits positioning(Figure 15), distancethe non-linear as for highercorrela- speeds. tions are obtained. It implies that the power consumption can be reduced by rational lim-

itation of the positioning velocity. On the other Ehand,= Tmax it may·ωmax greatly·ta increase the time of (29) operation, which is a negative effect for high-speed processes.

Figure 14. TheFigure correlation 14. The between correlation the consumed between the energy consumed in the energy acceleration in the phase acceleration and the phase rotational and the acceleration rota- ε for a constant maximumtional rotationalacceleration speed ε forω afor constant WDU (orange)maximum and rotational perforating speed head ω for (blue) WDU drives. (orange) and perforating head (blue) drives. For varying maximum rotational speed (Figures 11 and 12), the value of the peak torque is always the same regardless of the speed. However, it affects the time of acceleration as it decreases when the torque value gets lower. If we analyze the energy used during the acceleration phase in the function of maximum rotational speed for a constant rotational acceleration, for both discussed drive units (Figure 15), the non-linear correlations are obtained. It implies that the power consumption can be reduced by rational limitation of the positioning velocity. On the other hand, it may greatly increase the time of operation, which is a negative effect for high-speed processes. Energies 2021, 14,Energies x FOR PEER2021, REVIEW14, 317 13 of 15 13 of 15

Figure 15. TheFigure correlation 15. The between correlation the consumedbetween the energy consumed in the accelerationenergy in the phase acceleration and the phase rotational andspeed the rota-ω for a constant maximum rotationaltional speed acceleration ω for a constantε for WDU maximum (orange) rotational and perforating acceleration head ε (blue) for WDU drives. (orange) and perforating head (blue) drives. Although it was proved that the rotational acceleration ε does not have a significant Althoughinfluence it was proved on energy that consumption,the rotational itacceleration still has a crucial ε does impact not have on thea significant peak motor torque. As influence on energycan be observedconsumption, in Figure it still 13 has, in botha crucial cases, impact we obtain on the a linearpeak motor positive torque. correlation. Taking As can be observedinto consideration in Figure 13, both in both power cases, consumption we obtain anda linear torque positive necessary correlation. to put the mass into Taking into considerationmotion, it is advantageousboth power consum to selectption the accelerationand torque necessary which lies to in put the middlethe mass of the analyzed into motion, itrange is advantageous to find the compromise to select the (in acceleration the case of which the described lies in the device, middle approx. of the 200 rev/s2). analyzed range to Iffind we the discuss compromise the application (in the of case the presentedof the described model, device, the methodology approx. 200 of its derivation rev/s2). and obtained characteristics, aside from the torque and power consumption estimation, it If we discusscan be the used application to determine of the the presented value of themodel, resistance the methodology force caused of by its the deriva- belt adhering to the tion and obtainedperforating characteristics, head or uncutaside fr strandsom the being torque caught and power on the consumption punch. The value estima- of the resistance tion, it can be forceused canto determine be determined the value by measuring of the resistance the torque force on caused the drive by the and belt comparing adher- the obtained ing to the perforatingcharacteristics head or based uncut on strands the dynamics being caught analytical on model.the punch. Additionally, The value it of can the be also used to resistance forcedetermine can be determined the parameters by meas of theuring positioning the torque regulator on the drive for selected and comparing kinematic parameters. the obtained characteristicsAs was stated based in the introduction,on the dynamics by performinganalytical model. such analysisAdditionally, for adrive it can selection, it is be also used topossible determine to almost the parameters fully exploit of thethemotor positioning and avoid regulator working for withselected the loadkine- below 50% of matic parameters.the rated As was power. stated It will in the also introd enableuction, designing by performing lighter and such more analysis effective for solutions. a drive selection, it is possible to almost fully exploit the motor and avoid working with the 5. Conclusions load below 50% of the rated power. It will also enable designing lighter and more effective solutions. The conclusions of the presented research can be divided into two groups based on their impact: the general ones connected with the presented methodology of modeling and 5. Conclusionsits application and the detailed ones connected with the analyzed design solution. First of all, it is worth mentioning that the presented methodology of modeling The conclusions of the presented research can be divided into two groups based on facilitates the analysis of dynamics in the designed device based on the information made their impact: the general ones connected with the presented methodology of modeling available by manufacturers in product catalogs. It can therefore be successfully employed and its application and the detailed ones connected with the analyzed design solution. in the design process of mechatronic devices in order to develop an effective solution with First of all, it is worth mentioning that the presented methodology of modeling facil- optimal drive systems, without oversizing the structure and working with a load below itates the analysis50% ofof the dynamics rated power, in the which designed leads device to lower based efficiency. on the information made available by manufacturersSecondly, in the product analysis catalogs of the timing. It can belttherefore behavior be successfully showed the significanceemployed of determin- in the design processing the suitableof mechatronic belt tensioning devices in in order the belt to develop transmission, an effective which solution can be achieved with based on optimal drivethe systems, dynamic withou modelt oversizing as presented the instructure the paper. and working with a load below 50% of the rated power,Based which on the leads obtained to lower results, efficiency. it was shown that although using greater rotational Secondly,acceleration the analysis will of the cause timing a significant belt behavior increase showed of the the torque significance generated of deter- by the motor (to mining the suitablevalues belt even tensioning ten times in greater the belt than transmission, the static torque), which can which be highlightsachieved based the importance of on the dynamicminimizing model as presented the inertia in of the the paper. mechanical parts, especially the ones used in high-speed process machines. However, limiting the acceleration too much may lead to greater power

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consumption. Although the greater velocity of positioning does not affect the peak torque, it will increase the power consumption in the acceleration phase, but due to the shorter positioning time, it may still have a positive effect on overall exploitation. As discussed, selecting kinematic parameters is crucial and it is required to ﬁnd the compromise without tending to neither maximization nor minimization. Lastly, in the example of the linear drive system utilized in the automatic device for belt perforation, it was demonstrated that Lagrangian equations of the second kind are usable in the process of selecting the mechatronic device based on characteristics obtained from equations of motion. It may also have some additional applications, i.e., in the measurement methods for resistance force or to adjust the positioning regulator parameters. However, it can still be made more detailed e.g., by a more accurate description of the phenomenon of friction in the guides and bearings.

Author Contributions: Conceptualization, D.W. (Dominik Wojtkowiak) and K.T.; methodology, D.W. (Dominik Wojtkowiak) and K.T.; validation, D.W. (Dominik Wojtkowiak), K.T., D.W. (Dominik Wilczyński),J.G. and K.W.; formal analysis, D.W. (Dominik Wojtkowiak), K.T., D.W. (Dominik Wilczyński),J.G. and K.W.; investigation, D.W. (Dominik Wojtkowiak); resources, D.W. (Dominik Wojtkowiak); data curation, D.W. (Dominik Wojtkowiak), J.G. and K.W.; writing—original draft preparation, D.W. (Dominik Wojtkowiak); writing—review and editing, K.T. and D.W. (Dominik Wilczyński);visualization, D.W. (Dominik Wojtkowiak); supervision, K.T.; project administration, D.W. (Dominik Wojtkowiak); funding acquisition, K.T. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing not applicable. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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