Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 87-109, Warsaw 2010 ACTIVE FENCE WITH FLEXIBLE LINK Tomasz Piątkowski University of Technology and Life Sciences, Department of Postal Technology, Bydgoszcz, Poland e-mail: tomasz.piatkowski@utp.edu.pl This paper presents the results of research on a model of the sorting process of a unit load streambymeans of an active fencewith one degree of freedom. In this study, themanipulated loads are treated as bodies of nonlinear elastic-damping properties described by the modified Kelvin model. In investigating the fence structure, one takes into consideration a rigid link and an adjoining flexible link. The equations ofmotion of the fence flexible link, and those of the interacting object were derived using thefinite elementmethod(FEM).Thedevelopedmodelmakes itpossible to assess the influence of the fence structural andoperational parameters on the process of sorting, and evaluate dynamic forces exerted on the manipulated loads. Key words: unit load, handling process, belt conveyor, sorting, active fence 1. Introduction Oneof technical solutions utilised in logistic centres for automateddistribution of streams of small-size unit loads (i.e. mail parcels) into diverse directions of transportation is the application of scrapping devices in the transport system. These devices are allocated along the transporting conveyor, and their execu- ting elements are e.g. fences,whichperformpendulousmotion (Piątkowski and Sempruch, 2002 – Fig.1a). Such a solution (due to its uncomplicated struc- ture) is characterised by good functionality, high strength and reliability of operation, which significantly increases the overall effectiveness of the sorting system. In these solutions, the fence together with its driving system usually constitutes a structure of very high rigidity. Such a property guarantees the realisation of the assumed trajectory of the fence motion, irrespective of the loading forces. However, such a concept does not satisfy the requirements of 88 T. Piątkowski contemporary logistic processes in which one aims at achieving higher and hi- gher speeds of the object transportation at the same timeminimizing dynamic interactions generated while performingmanipulation actions on the objects. The consequence of applying sorting devices of highly rigid structure is the lack of possibility to absorb and dissipate the impact energy released when the load enters into contact with the fence. In such a case, the conditions of obtaining high effectiveness of the sorting process, and preventing the load frommechanical damage, are difficult to be brought together. It is known from the world literature that the mechanical system consi- sting of a belt conveyor andanarm(active rotary fence, or anumber of passive fences) placed above the belt finds its application inmechanisation of the po- sitioning process. This process consists in giving the loads a precisely defined destination position (rotary and translational – Mason, 1999). In the studies pertaining to this application of fences (Akella et al., 2000; Akella andMason, 1998; Berretty et al., 1998;Mason, 1986; StappenandGldberg, 2000), the pro- blemof interaction between the fence and the load is reduced to quasi-statistic events only – the inertia forces are neglected. It is assumed in theseworks that the process of positioning is carried out at a low transportation speed of the conveyor, owing to which the dynamic reaction forces are negligibly small in comparison with the friction forces. The problem of mitigating reaction forces in the sorting process was con- sidered in the work by Piątkowski and Sempruch (2005), where the authors discussed a solution consisting in replacing the classic straight-linear fence with a curvilinear one, modelled upon the Bézier curve. In work by Piątkow- ski (2004), one analyses the application of a pneumatic system in the fence drive, and in Piątkowski and Sempruch (2006) one proposes a rocker mecha- nism and a worm gear joined by a flexible clutch. These examples do not exhaust all possible constructional solutions aimed at decreasing the dynamic reaction forces exerted on the manipulated loads. Softening of impact effects in the sorting process can also be achieved by utilising a fence of adequately chosen constructional rigidity, i.e. the flexible fence presented in Piątkowski andSempruch (2008). The considered fencewas treated there as a beammade entirely of a homogeneous material. For constructional reasons, making and utilising such a fence is troublesome. A more rational solution would be the fence (Fig.1b)which consists of two elements: a rigid one (used for joining the fencewith the frame and the drive systemof the sorting device), and a flexible element of constant bending strength (responsible for softening the effects of impact on the object). The theoretical analysis of application of such a fence is presented in this paper. Active fence with flexible link 89 Fig. 1. Example of manipulator with rotary active fence: (a) scheme of sorting assembly, (b) fence with flexible element; 1 – unit load, 2 – main conveyor, 3 – active fence, 4 – chute; s – width of conveyor, ∆s – distance between load and conveyor border, Rs – position of load front at starting moment of fence operation, Rz – length of fence, ∆c – distance between loads fronts, L – length of manipulator working space, A×B – dimensions of load, φ – load position angle, α – fence deflection angle, v – conveyor transportation speed, ls, lp – lengths of rigid and flexible links of fence, respectively 2. Model of load motion 2.1. Assumptions for developing physical model In the physical model of the sorting process, which is treated as the first stage in formulation of a computermodel, one takes the following assumptions: • the sorting process is assumed to be a planar motion on the plane of main conveyor, • the load has a rectangular shape (in the projection on themain conveyor plane), • friction forces aremodelled according toCoulomb’s law (Kikuuwe et al., 2005), • the resulting friction forces andmoments of friction in the planarmotion are described according to the model proposed in the work by Goyal et al. (1991), • the rigid link of the fence is fixed at one end, and is set in rotary mo- tion αw by the drive system (Fig.2a), • theflexible link is treatedasacontinuous system(Jalili andEsmailzadeh, 2002; Wittbrodt et al., 2006) discretised according to the finite element method (FEM) (Gawroński et al., 1984; Zienkiewicz, 2000) implemented in theMatlab environment, 90 T. Piątkowski • the transversal cross-section of the flexible link of the fence has constant bending strength, which is achieved by giving it a variable thickness according to the formula h=h1 √ lp−x lp (2.1) where h1 – initial thickness of the fence flexible link lp – length of the fence flexible link x – coordinate of the flexible link cross-section position (x=0 – point joining rigid link with the flexible one). • two-node beam elements are applied in the discrete model of the fence (Fig.2b), in which every node is subject to translational q3, and rota- tional displacement q5, • the reaction force appearing between the object and the fence is de- scribed based on a modified nonlinear Kelvin’s model (Piątkowski and Sempruch, 2009) N = bpm 0.4 p ḊD 2+kpD 4 (2.2) where D,Ḋ – strain [m] and strain velocity [m/s] of colliding bodies bp,kp – damping [Nskg −0.4m−3] and stiffness [Nm−4] factors mp – mass of the load [kg]. The values of exponents of power functions in the formulae that describe local deformation of the bodies (in Eq. (2.2)) were chosen based on the results of numerical experiments. These were aimed at finding a good mathematical representation for the actual impact course registered in experimental inve- stigations. The factor m0.4p introduced into the model allows us to obtain a constant value of the restitution coefficient (for the assumed parameters bp and kp), irrespective of the masses of bodies used in impact simulation. Ef- fectiveness of the model (Eq. (2.2)) in predicting the course of impact was confirmed by experimental investigations carried out in dynamic load condi- tions: when the object impacts into a deformable and undeformable obstacle (Piątkowski and Sempruch, 2009). In order to simplify the dynamic analysis, the continuous process of sorting was subjected to discretisation – division of the process into individual stages. The kinematic-dynamic characteristics of the load, which result from mutual interaction between the load and the fence, differ significantly in each stage. Active fence with flexible link 91 Fig. 2. Division of fence flexible link into discrete FEMs: (a) discrete model of fence, (b) two-node discrete beammodel; q3,q5 – generalised coordinates (two degrees of freedom: translation and rotation), Q3,Q5 – generalized forces, E – Young’s modulus, G – shear (Kirchhoff’s) modulus, Si = zhi – area of transverse cross-section, ζ – material damping (Giergiel, 1990), ρb –mass density, αw – angular position of fence driving link, ∆l – length of discrete element, ls, lp – length of fence rigid and flexible link, respectively Each characteristic stage of load motion needs an individual mathematical description, and this has form of a separate module of the numerical model code. Simulation of the model consists in running the modules in a sequence, according to the position currently taken by the load with respect to the executing elements of the manipulator. Taking into account author’s previous experience in numerical simulation of the sorting process models (Piątkowski and Sempruch, 2002, 2008), it was assumed that the following characteristic stagesmust be taken into account in order to recognise the basic characteristics of load distribution process realised bymeans of an active fence with a flexible link: • oblique impact of the load against the fence, • motion of the load along the fence, • motion of the load along the conveyor border: one of load corners slides along the border of the conveyor, the other one along the fence, • free motion of the load. In the last of the mentioned stages, physical properties of the fence have no influence on behaviour of the load (no contact with the fence). For this reason, we do not present here anymathematical description of the load free motion, assuming that it does not differ from that presented in thework byPiątkowski and Sempruch (2002). 92 T. Piątkowski Four reference coordinate systems were assumed in the determination of the equations of the physical model (Fig.3–Fig.7): • rectangular Oxoyo, connectedwith themanipulator frame, whose origin lies on the fence rotation axis – this one was applied in presentation of the results of load motion simulation, • rectangular Oypzp, connected with the load – applied for determination of the position of the load gravity centre with respect to its geometric centre, • rectangular Ox3x1, connected with a specific discrete element – applied in description of physical properties of discrete elements of the fence, • polar coordinate system, with radius-vector r and polar angle αp – applied in deriving the equations of load motion. 2.2. Oblique impact of load against fence This kind of impact of the load against the fence (Fig.3) takes place when the load (justbefore entering in contactwith the fence) is positionedarbitrarily within the available width of the conveyor. The system of equations of forces andmoments acting on the load and the fence in this stage of the sorting process can be presented as (see Fig.3) Mbq̈+Bbq̇+Kbq=Q mpr̈=Fξ +Pcen−F2cos(αp−q5(N))+N sin(αp− q5(N)) (2.3) mprα̈p =−Fη −PCor+F2 sin(αp−q5(N))+N cos(αp−q5(N)) Ipφ̈=−rCj [ ujN cos ( γj +uj(φ−q5(N)) ) +F2 sin ( γj +uj(φ−q5(N)) )] −T where j – number of the load corner being in contact with the fence, j=1,2,3,4 i – number of node of the fence discrete element being in contact with the load, i=1,2,3, . . . ,nE +1 n – number of degrees of freedom of nodes in the fence discrete model, n=2(nE +1) nE – number of discrete elements Mb,Bb,Kb – matrices of inertia, damping and rigidity of the fence discrete model, of dimensions n×n q – vector of generalised displacements of nodes, of dimensions n×1 Active fence with flexible link 93 Fig. 3. Scheme of forces acting on load during its impact against active fence; 1 – load, 2 – fence Q – vector of generalised forces acting on nodes of discrete elements, of dimensions n×1 q5(N) – angular position of the fence discrete element at its contact point with the load U – according to Eq. (2.22) Q3(i) – external force exerted on the i-th node of the discrete element q5(1) =αw – angular position of the first node of the discrete element mp,Ip – load mass andmass moment of inertia φ – rotation angle of load N – force of reaction between the load and fence in the normal direction of impact – according to Eq. (2.2) PCor,Pcen – Coriolis force and centrifugal force PCor =2mpα̇pṙ Pcen =mpα̇ 2 pr (2.4) 94 T. Piątkowski The first equation in (2.3) is amatrix equation of motion of discrete elements of the fence. The remaining equations describemotion of the load in the polar coordinate system with the radius-vector r and polar angle αp. Depending on which of the load corners Cj is in contact with the fence, the length of the radius-vector rCj connecting the corner Cj (j = 1,2,3,4) with gravity centre of the load Cs is equal to rCj =      √ (0.5A+(3− j)xp)2+(0.5B+(3− j)yp)2 if j=2,4 √ (0.5A− (2− j)xp) 2+(0.5B+(2− j)yp) 2 otherwise (2.5) The position angle of the load gravity centre γj is defined by the relationship γj =          arcsin 0.5B+yp rCj if j=1,2 arcsin 0.5B−yp rCj otherwise (2.6) and the factor uj appearing in (2.3) is defined as uj = { 1 if j=2,4 −1 otherwise (2.7) For the reaction force N between the object and the load be continuously and uniformly transmitted, one assumes that adjacent nodes of discrete elements are simultaneously subjected to adequately reduced generalised forces Q3(i) and Q3(i+1), according to the following proportions Q3(i) =N ( 1− dN ∆l ) Q3(i+1) =N dN ∆l (2.8) where ∆l – length of the discrete element, ∆l= lp/nE lp – length of the fence flexible link. Thereduction is carriedout taking intoaccount theprincipleof equilibrium of action and reaction forces and moments, according to geometric relation- ships shown in Fig.4. Assuming that the deflection angle of fence (under the pressure of object – Fig.3, Fig.4), is small, the number of the fence discrete element being in contact with the object can be determined as the integer portion of the qu- Active fence with flexible link 95 Fig. 4. Forces exerted by object on fence otient of R− lp and ∆l (assuming R> lp, and applying theMatlab function ”floor”) i=      floor (R− lp ∆l ) +1 if floor ( R−lp ∆l ) +1 0      Pξext if Fmax >P1ext Fmax Pξext P1ext otherwise      otherwise Fη =                  ηFmax vo √ 1+ ( Tmaxφ̇ Fmaxvo )2 if vo > 0      Pηext if Fmax >P1ext Fmax Pηext P1ext otherwise      otherwise (2.19) T =                Tmax sgn(φ̇) √ 1+ ( Fmaxvo Tmaxφ̇ )2 if |φ̇|> 0 { Text if Tmax > |Text| Tmax sgn(Text) otherwise } otherwise 98 T. Piątkowski F2 =        Nµ2 sgn(wxj) if |wxj|> 0 { P2ext if Nµ2 > |P2ext| Nµ2 sgn(P2ext) otherwise } otherwise where Fmax,Tmax – maximum friction force and maximum friction moment appe- aring in the case of pure translational or pure rotational motion of the load Fmax =mpgµ1 Tmax = mpg S µ1 ∫ S rS dS (2.20) g – acceleration of gravity µ1,µ2 – friction coefficients of the load with respect to carrying surfaces of the conveyor and fence, respectively S – area of contact between the load and friction surface dS – infinitesimal friction surface rS – distance between an infinitesimal friction surface dS and the gravity centre of load P1ext – resultant external force exerted by the fence on the object P1ext = √ P2 ξext +P2ηext (2.21) Pξext,Pηext – components of the force P1ext Pξext =N[µ2 sgn(wxj)cos(αp−q5(N))− sin(αp− q5(N))] (2.22) Pηext =N[(µ2 sgn(wxj)sin(αp− q5(N))+cos(αp−q5(N))] Text –moment of external forces exerted on the object Text =−rCjN[uj cos(γj+uj(φ−q5(N)))+µ2 sgn(wxj)sin(γj+uj(φ−q5(N)))] (2.23) P2ext – external force, tangent to the fence, exerted by frictional coupling existing between the object and carrying surface of the conveyor P2ext =Fξ cos(αp−q5(N))+Fη sin(αp−q5(N)) (2.24) wxj – speed of sliding of the load corner with respect to the fence wxj = ṙcos(αp−q5(N))−r(α̇p− q̇5(N))sin(αp−q5(N))+ (2.25) +rCj(φ̇− q̇5(N))sin(γj +uj(φ−q5(N))) Active fence with flexible link 99 Thevalues of components ξ and η of thevector of relative friction speed vo of the load gravity centre Cs on the conveyor belt are equal to ξ= vcosαp− ṙ η= v sinαp+rα̇p (2.26) 2.3. Motion of load along fence The stage of motion in which the load slides along the fence (Fig.6) takes placewhenthe load is transportednear theconveyorborderon that side,where the fence ismounted.This stage canalsobea continuation of loadmotion after its oblique impact against the fence – after which the load rotates and takes the position parallel to the fence. Fig. 6. Scheme of forces acting on loadmoving along fence 100 T. Piątkowski Dynamic interactions between the load and the fence can be expressed by equations (according to Fig.6) Mbq̈+Bbq̇+Kbq=Q mpr̈=Fξ −F2cos(αp−q5(N))+N sin(αp−q5(N))+Pcen (2.27) (mpr 2+ Ip)α̈p = [F2 sin(αp−q5(N))+N cos(αp−q5(N))−Fη−PCor]r−T Denotations of quantities used in these equations are the same as those in Eqs. (2.3). The reaction forces between the object and the fence are applied at the point C, which is a rectangular projection of the object centre of gravity Cs on the wall of load that touches the fence. The components of external force Pξext,Pηext exerted on the object (which are required for determination of the force of friction between the object and the conveyor, according to (2.19)1,2,3 take the form Pξext =N[µ2 sgn(wx)cos(αp−q5(N))− sin(αp−q5(N))] (2.28) Pηext =N[µ2 sgn(wx)sin(αp− q5(N))+cos(αp−q5(N))]− T r The sliding speed of the object with respect to the fence, which appears in (2.19)4, is wx = ṙcos(αp−q5(N))−r(α̇p− q̇5(N))sin(αp−q5(N)) (2.29) The coordinates of the point C, necessary for determination of the distance R from the i-th node of the fence discrete element being in contact with the object, are equal to (according to Fig.6) xoC = rcosαp+ 1 2 (B+yp)sinq5(N) (2.30) yoC = r sinαp− 1 2 (B+yp)cosq5(N) 2.4. Motion of loadwith one corner sliding along border of conveyor, and another one along fence In the case when the load transported by the conveyor lies just next to its border (at the side where the fence is mounted), there could exist a situation when one of the load corners slides along the border of the conveyor, and another one along the fence (Fig.7).Then, the load is lead out fromthe stream of loads in the result of impact-free sliding along the fence. Active fence with flexible link 101 Fig. 7. Scheme of forces acting on load whose one corner rubs against border of conveyor, and another one against fence The load and fencemotions can be described by the following equations – according to Fig. 7: Mbq̈+Bbq̇+Kbq=Q mpr̈=Fξ +Pcen+N2 sinαp−F2(2) cosαp+N1 sin(αp− q5(N))+ −F2(1) cos(αp−q5(N)) (2.31) mprα̈p =−Fη−PCor+N2cosαp+F2(2) sinαp+N1cos(αp− q5(N))+ +F2(1) sin(αp−q5(N)) Ipφ̈= rC1[N1cos(q5(N)+γ1−φ)−F2(1) sin(q5(N)+γ1−φ)]+ −rC2[N2cos(γ2+φ)+F2(2) sin(γ2+φ)]−T In the case of static friction between the load and the conveyor, the reaction forces and themoments exerted on the object (required in (2.19)1,2,3) take the form Pξext =−N2 sinαp+F2(2) cosαp−N1 sin(αp− q5(N))+F2(1) cos(αp− q5(N)) Pηext =N2cosαp+F2(2) sinαp+N1cos(αp−q5(N))+F2(1) sin(αp− q5(N)) Text = rC1[N1cos(q5(N)+γ1−φ)−F2(1) sin(q5(N)+γ1−φ)]+ (2.32) −rC2[N2cos(γ2+φ)+F2(2) sin(γ2+φ)] 102 T. Piątkowski The sliding speeds of the object corners, wx1 – referred to the fence at the contact point C1 and wx2 – referred to the conveyor border at the point C2, are given by equations wx1 = ṙcos(αp−q5(N))−r(α̇p− q̇5(N))sin(αp−q5(N))+ +rC1(α̇p− q̇5(N))sin(α+γ1−φ) (2.33) wx2 = ṙcosαp−rα̇p sinαp+rC2φ̇sin(γ2+φ) Due to conditions existing during sorting, these speeds take values greater than zero –whichwas confirmed inpreliminary investigations. For this reason, we assume that only kinetic friction exists at the points Cj (j = 1,2), and F2(j) =N(j)µ2 – according to Eq. (2.19)4. Deformation of the object at the contact point C2 is D=−yoC2 (2.34) where yoC2 – ordinate of the point C2 in the coordinate system Oxoyo. The remaining denotations are the same as those in description of Eqs. (2.3). 3. Numerical investigations In the simulation investigations carried out in this study, we considered two variants of the fence: the first variant pertaining to the fence compo- sed of two links (a rigid one and a flexible one, of dimensions ls = 0.45m, lp = 0.75m, z × h1 = 0.132 × 0.045m), and the second variant – the fence made entirely of a homogeneous material (ls = 0m, lp = 1.2m, z × h1 = 0.132× 0.073m). It was assumed that the flexible links are ma- de of polyamide PA6 – amaterial characterised by high internal damping and highmechanical strength (Basf technical resources (2008)). The proportion of the rigid link length to the length of the flexible one (in the first variant of the fence structure) was assumed taking into account the need for ensuring enough space necessary to easily couple the rigid link to the frame and to the drive system of the sorting device. The maximum relative impact speed in the normal object-fence direction is equivalent to the speed which the object reaches during free fall from the height of H = 0.3m on an undeformable ground – that is the fall treated in thisworkas the admissible hazarddue todynamic reactions exerted on theob- ject. Themechanical properties of the sorted load (bp =3.3·10 5Nskg−0.4m−3; Active fence with flexible link 103 kp = 1.72 · 10 9Nm−4), required in Eq. (2.2), were assumed on the basis of experimental and numerical investigations of impacts of unit loads described in work by Piątkowski and Sempruch (2009). The linear speed of load trans- portation on the main conveyor and the rotational speed of the fence are selected so that reliable sorting of loads and themaximum sorting capacity of the system can be achieved (Piątkowski and Sempruch, 2002). 3.1. Simulation of impact of object against fence The courses of dynamic processes that take place when the object impacts against the fence are presented in Fig.8-Fig.10. The point marked 1 in the graphs (Fig.8) pertains to the moment of impact initiation, while point 2 denotes the end of impact. The impact process was arranged in a special way so as to evoke maximal dynamic reactions. Before the impact, the loadwas positioned and oriented on the conveyor in such a position that during impact initiation the load collided into the part of the fencemost distant from the rotation axis, and the gravity centre of load Cs was on the normal of impact. The transverse dimensions of flexible links of the examined fences were chosen in such a way that, even with the most difficult impact conditions predicted for this experiment (as described above, for the parameters: main conveyor transportation speed v = 2m/s, load mass mp = 15kg, the maxi- mum rotational speed of the fence drive system α̇wmax =1.53rad/s, fence of length Rz =1.2m, time of fence working cycle tc =1.36s), the deflection of fence tip did not exceed approx. 0.08m (Fig.8b, Fig.9b, Fig.10b). The results of simulation shown in Fig.8, pertaining to two variants of the fences (partly flexible, and fully flexible one), are almost identical. The lines in graphs in Fig.8a and 8d representing thementioned fence variants (thin lines referring to one-link fence, and thick lines for two-link one) coincide along all their lengths. The point where the object loses its contact with the fence lies out of the main conveyor area (Fig.8a), which guarantees that the load is carried to a new direction of transport. The oscillatory motion of the fence fades down before the fence returns to its initial position (Fig.8b), owing to which the following working cycles are free of disturbances that might result from the transients of previous cycles. The courses of dynamic reactions, presented in Fig. 8c, which appear du- ring the impact between the object and the fence with flexible link (N), and between this object and an undeformable obstacle of infinitemass (Nstiff ) are expressed in terms of relative values of the impact force N and Nstiff , referred 104 T. Piątkowski to the maximum impact force between the object and the fence with flexible link, Nmax. It follows from the analysis of the obtained results that during the impact of the object against the flexible fence one can expect almost threefold decrease of dynamic reactions compared to those appearing when the object impacts against anundeformable, immobile obstacle (the onewhich represents a completely rigid fence joined with a rigid drive system; such a fence exerts most destructive effect on the manipulated loads). In the course of impact, the friction force F1 has, for most of the time, form of kinetic friction – Fig.8d. The stage of static friction lasts a very short time – only approx. 0.003s. For this reason, the chart of the friction force F1 (in the initial course) makes up almost a vertical line. Fig. 8. Simulation results of object-against-fence impact (1 and 2 – beginning and end of impact, respectively): (a) motion trajectory of object gravity centre, (b) deflection of fence tip, (c) course of dynamic reactions during impact of load against fence (N), and against undeformable obstacle of infinite mass (Nstiff ), (d) friction force F1; process parameters: bp =3.3 ·10 5Nskg−0.4m−3, kp =1.72 ·10 9Nm−4 (Piątkowski and Sempruch, 2009), v=2m/s, mp =15kg, µ1k =0.65, µ1s =0.85, µ2k =0.35, µ2s =0.5, nE =20, s=0.7m The effectiveness of softening the effects of impact of the load against the fence is illustrated in Fig.9. Figure 9a pertains to the investigations results obtained for the fence consisting of two links (ls =0.45m, lp =0.75m), while Fig.9b presents similar results for the one-link fence (ls = 0m, lp = 1.2m). Active fence with flexible link 105 Theeffectiveness is expressedbyaquotient of the force arisingwhen the object impacts against an undeformable obstacle of infinite mass (Nstiff max ) to the force of impact for the flexible fence (Nmax).The graphs are drawn in function ofmass of themanipulated object mp and the distance Rbetween the contact point and the fence rotation axis (R=0pertains to the pointwhere the fence is fixed, R=1.2m – to its free end). The greater the mass of the object and the distance between the contact point and the fence axis, the greater the effectiveness of softening of the dynamic reactions arising in the impact (as one could predict). Fig. 9. Quotient of dynamic reaction forces exerted on object in impact against undeformable obstacle of infinite mass to reaction forces in impact against flexible link of fence: (a) fence consisting of rigid link and flexible link, (b) entirely flexible fence The differences between the results obtained in investigation of the rigid- flexible fence, and those for the entirely rigid one, are insignificant. The drop of effectiveness of impact energy dissipation is only observed in the case when the point of contact between the load and the fence comes close to the joint between the rigid link and the flexible one. However, such a loss of the impact- softening effectiveness is not dangerous (as far as safety of the manipulated loads is concerned) – the greatest danger is associated with the impact of the object against the free end of the fence. Similar relations also refer to the obtained deflection of the fence – Fig.10. The increase of load mass mp and distance R results in an increase of de- flection of the fence tip. The discrepancies caused by structural differences between the examined fences are, also in this case, visible only in the vici- nity of the joint between the rigid link and the flexible one (in the two-link fence). 106 T. Piątkowski Fig. 10. Deflection of free end of fence: (a) fence consisting of rigid link and flexible link, (b) entirely flexible fence 3.2. Simulation of object motion along fence The course of sorting process for the object which, before scrapping, lies near the border of the conveyor (∆s=0) is shown in Fig.11. In the investiga- tions, we examined two variants of fences, whose structural parameters were consistent with the parameters used when deriving the data for Fig.8-Fig.10. Reference mark 1 indicates the moment of initiation of contact between the object and the fence, and reference mark 2 – the end of this contact. Thediscrepancies between the test results obtained for the fence consisting of two links (rigid one, and flexible one) and those for the fence consisting of one (flexible) link are insignificant – similarly as it was in the cases presented in Fig.8-Fig.10. The two graphs, the one drawn in thick line (pertaining to the fence consisting of a rigid link and a flexible one), and the one drawn in thin line (pertaining to the entirely flexible fence) do not overlap only in Fig.11b illustrating the course of deflection of the fence tip, and in Fig.11c depicting the course of dynamic interactions between the object and the fence. A complete coincidence of these two graphs can be seen in Fig.11a (represen- ting trajectory of the object motion), and in Fig.11d (where the course of the friction force F1 is presented). 4. Conclusions In this study, the author presented a proposal for themodelling of the process of sorting of unit load streams transported on belt conveyors by means of an active fence consisting of two links: a rigid andaflexible one. In thedescription of dynamic properties of the fence flexible link, the finite elementmethodwas Active fence with flexible link 107 Fig. 11. Results of simulation of object motion along fence (1 – initiation of contact between object and fence, 2 – beginning of free motion of object): (a) motion trajectory of gravity centre of object, (b) deflection of fence tip, (c) courses of dynamic interactions between object and fence (N), and between object and undeformable obstacle of infinite mass (Nstiff ), (d) friction force F1; process parameters as in Fig.8; Nmax –maximum force of impact of object against flexible fence applied. The reaction forces arising between the manipulated object and the fence were described with the use of a nonlinear Kelvin’s model. The developed model of the sorting process makes it possible to examine the influence of structural and operational parameters of the fence on the magnitude of forces exerted on the manipulated objects and on the course of motion of the object and the fence. It follows from the performed simulation tests that by application of a rigid-flexible fence (consisting of two links, a rigid and a flexible one, with the length of the rigid link even as large as 1/3 of the fence total length), one can obtain similar effects of reducing the dynamic overloads exerted on the manipulated objects and a similar course of the distribution process as in the case when an entirely-flexible fence is applied. This work has been financed from the funds for science as a Research Project in the years 2008-2010. 108 T. Piątkowski References 1. Akella S., Huang W.H., Lynch K.M., Mason M.T., 2000, Parts feeding on a conveyor with a one joint robot,Algorithmica, 26, 313-344 2. 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Zienkiewicz O.C., 2000, The Finite Element Method, Butterworth- Heinemann, Oxford, UK Zastawa aktywna z członem podatnym Streszczenie W artykule przedstawiono wyniki badań modelu procesu sortowania strumienia ładunków jednostkowych zastawą aktywną o jednym stopniu swobody. Podczas ba- dań manipulowane ładunki traktowane są jako ciała o nieliniowych właściwościach sprężysto-tłumiących.W konstrukcji zastawy uwzględniono człon sztywny i połączo- ny z nim człon podatny. Równania ruchu członu podatnego zastawy i wchodzącego znimw interakcje obiektuwyprowadzonowoparciuometodę elementówskończonych (MES). Odpowiednie kształtowanie parametrów konstrukcyjnych członu podatnego zastawy pozwala wpływać na złagodzenie oddziaływań dynamicznych wywieranych na sortowane ładunki. Manuscript received March 2, 2009; accepted for print May 11, 2009