Theoretical Modelling of Can-spring Mechanism Using Bond Graph Teoretično modeliranje mehanizma vzmeti v loncu z uporabo

Helical compression springs are extensively used for the combed cotton sliver handling system in spinning preparatory sections in the textile industry. Storage can-spring stiff ness decreases with time due to prolonged fatigue loading. It has been found that older storage can-springs of reduced spring stiff ness deform non-uniformly during combed sliver deposition and withdrawal. In order to produce consistent quality of an intermediate delicate product like combed sliver, sliver stresses should be monitored meticulously at the time of sliver deposition and withdrawal from storage cans. The present research is an attempt to study the dynamics of the can-spring mechanism used for sliver storage through the bond graph approach. The paper records the fi rst stage of the work, which is concerned with establishing the relationship between spring stiff ness and sliver forces. Bond graph modelling of the can-spring mechanism is one of thebest-suited approaches to study the present research problem due to its characteristic features.

Tekstilec, 2020, 63(1), [68][69][70][71][72][73][74][75][76] Theoretical Modelling of Can-spring Mechanism Using Bond Graph used for a uniform deposition and withdrawal of sliver to avoid any undue stretching of sliver, which is very weak. Combed cotton draft ed sliver is a rope-like structure and most of the fi bres are oriented parallel to the sliver axis. Due to very low interfi bre cohesion, the combed sliver is susceptible to undesirable stretching at the time of sliver deposition at the draw-frame and during sliver withdrawal in subsequent machine processes. Th e can-springs used for sliver storage are subjected to repeated reversals of loading over a prolonged period of time resulting in their degradation due to fatigue; consequently, with the passage of time, their stiff ness reduces [3][4]. Th erefore, an older can-spring will defl ect more in comparison to a newer one made of the same material, against the same applied load. In the case of low can-spring stiff ness, there are chances of sliver stretching due to its own weight in the unsupported region, as shown in Figure 1. (L + L1) is the sliver length contributing in sliver stretching during processing. In consequence, the combed sliver, roving and resultant combed yarn quality characteristics are found signifi cantly infl uenced [5][6][7][8]. Th e optimum storage can-spring pressure should be maintained for smoother operations [9]. Th e non-uniform deformation of a can-spring may severely infl uence sliver quality at the time of deposition and withdrawal of sliver on subsequent machine passages. Modelling the dynamics of the can-spring sliver deposition/withdrawal system is necessary to develop the understanding of the system and to study the eff ects of variation of can-spring stiff ness, rate of deposition and withdrawal of sliver, etc. on the quality of sliver and yarn. Th e variation of mass on the top plate due to the deposition or withdrawal of sliver makes the task of developing the model relatively challenging. A review of the literature reveals that modelling such a system with variation in its inertia has not been carried out yet.

Materials and methods
Combed cotton sliver of 14.22 ktex produced from an extra-long variety MCU-5 from the south Indian states was used for this study. A bond graph model for the can-spring sliver deposition/withdrawal system was developed systematically from fi rst principles. Th e model explains the transactions of power and understanding of the cause-effect relationships between the interacting subsystems. Th e model facilitates the study of the eff ect of sliver can-spring stiff ness and the rate of sliver deposition or withdrawal on the forces experienced by the combed sliver and top plate through simulations and analysis.

Brief introduction of bond graph technique
A bond graph is a unifi ed approach to the modelling of physical system dynamics. Th e bond graph approach was originally developed and presented by H. M. Paynter at MIT in 1959. Bond graph is a pictorial or graphical representation of the dynamics of a physical system based on power transactions between component elements and subsystems [10][11][12]. Th e cause and eff ect relationships, or causality, are represented elegantly through the causal strokes on the bonds. Th e bond graph approach has numerous advantages over conventional simulation. It provides a graphical representation of the model of dynamics and systematic extraction of system equations algorithmically [11]. In a physical system, only the exchange and conversion of energy in diff erent forms take place. Power (P) is the change in energy (E) with respect to time (t) and can be expressed as: Th e power transacted using bonds through the parts of a bond graph can be expressed as the product of eff ort (e) and fl ow (f) variables which are functions of time (t):

System elements used in bond graph modelling approach
Th e bond graph elements are categorised as Sources (or active elements), Passive elements, Converters and Junctions [10].

Source elements
In the bond graph approach, there are two kinds of source elements, i.e. a source of eff ort and a source of fl ow. Th e source of eff ort (S e ) imposes an eff ort on the system irrespective of the fl ow in return which is decided by the system, as shown in Figure 2a. In the same way, the source of fl ow (S f ) imposes a fl ow on the system irrespective of the returning eff ort which is decided by the system, as shown in Figure 2b. Th e cause and eff ect relationship in terms of causality remains fi xed for source elements, as shown below in Figure 2. For a power bond connecting two subsystems, if eff ort is decided by the subsystem at one end, the fl ow will be decided by the subsystem at the other end. Both the eff ort and fl ow associated with a bond cannot be determined by the system at any end. Th e assignment of causality is an algorithmic process. In a bond graph representation, a causal stroke is placed on the bond at the eff ort receiving end. Th e other end of the bond automatically receives the fl ow.

Junction elements
Th ere are two junction elements which connect different parts in a bond graph. Th ese are the 0-junction and 1-junction. Th e 0-junction is an eff ort equalising junction; thus, eff orts in all the bonds connected to it remain the same, as depicted in Figure 3a. Th e fl ow relationship can be expressed as given below: In the case of the 1-junction, the fl ow remains the same in all the bonds connected to it. It acts as a fl ow equalising junction, as shown in Figure 3b. Th e eff orts summing at this junction can be expressed with equation 5: .

Compliance element
In natural or integral causality, the C-element receives the fl ow from the system and gives back eff ort to the system, as shown in Figure 4. Th e variable q associated with the C-element is the generalised displacement. K represents the stiff ness associated with it.

Figure 4: Compliance element: a) representation of output and input; 4b) cause and eff ect relationship for integrally causalled C-element
Th e output of the C-element is eff ort, which is a function of its state variable q, given as:

Inertia element
In its natural or integral causality, the I-element receives eff ort from the system and gives back the fl ow to the system, as shown in Figure 5. Th e state variable

Figure 5: Inertia element: a) representation of output and input; b) cause and eff ect relationship of Ielement
associated with the I-element is the generalised momentum p. M is the mass associated with the Ielement. Th e output of the I-element is fl ow, which is a function of its state variable p: .

Resistive element
Th e R-element is used to represent dissipation. It can be causalled in both ways, depending on the invertibility exhibited by the phenomenon. Th e two types of causality in the case of the R-element are shown in Figure 6.

Gyrator element
Th e GY-element represents the gyrator in a bond graph. It relates the input fl ow to the output eff ort, and vice-versa. Th is fl ow to eff ort and eff ort to fl ow relationship is established using the modulus μ, as shown in Figure 8. Using the power bond and the nine elements of the bond graph, the dynamics of the physical system in any energy domain or a combination of energy domains can be modelled.

Theoretical modelling of can-spring mechanism using bond graph
An attempt was made to study the dynamics of the can-spring mechanism used at the fi nisher drawframe stage for sliver storage. It was presumed that the combed cotton sliver deposited over the top plate has uniform linear density or fi neness and that the weight of sliver deposited per second over the top plate remains constant. For an accurate prediction of the can-spring mechanism, real values of independent variables based on industrial experience were considered for evaluation. Also, an in-depth study of sliver quality characteristics was conducted while selecting sliver linear density, inter-fi bre cohesion and lengthwise sliver uniformity. Th e schematic diagram of the can-spring mechanism is shown in Figure 9. Th e following nomenclature was adopted: M = initial mass of the top plate before sliver deposition takes place. Δm = increment in mass of deposited sliver on the top plate in the time duration Δt. u = velocity of the top plate at time t. v = fl ow velocity of sliver at time t, before depositing on the top plate. q(t) = deformation in can-spring due to applied load of deposited sliver at time t. g = acceleration due to gravity (∑Δm + Δm) = total weight on the top plate at an instance due to small increment of combed sliver weight.
Th e linear momentum of the top plate, along with deposited sliver on it at time t can be expressed as .
Considering that the deposited sliver moves with the same velocity as that of the top plate, aft er the sliver deposition (u + Δu) is equivalent to (v + Δv). On subtracting equation (7) from equation (8), and on dividing both sides by Δt, we get lim , .
Th e rate of change of the linear momentum of the top plate is caused by the forces acting on it and can also be expressed as given in (13) dp dt and By comparing (16) with (12), Th is represents the formulation of dynamics for the can-spring system. Equation (17) is represented graphically as outlined in green in the bond graph of Figure 10.

Viscoelastic representation of sliver
For the purpose of modelling, the Kelvin-Voigt model, the Maxwell model and the standard model were studied initially. Th e standard model consists of two springs and a dashpot. It is the simplest model that describes both creep and stress relation behaviour of a viscoelastic material. Based on the observed behaviour of the combed sliver, the widely accepted Kelvin representation of the standard model was used. A minor modifi cation was made in the Kelvin representation of the standard model by considering a permanent deformation due to R 2 in the viscous region which appears to be a more suitable representation of the viscoelastic nature of combed sliver, as shown in Figure 11. Th is viscoelastic behaviour was already modelled in the bond graph of Figure 10.

Deriving system equations from bond graph
System equations can be derived algorithmically from the bond graph model as a set of the fi rst order diff erential equations, using the following two equations: What do the elements give to the system?
Th e fi rst order equations obtained from the bond graph model can be conveniently simulated numerically using solvers for ordinary diff erential equations provided by any of the available computational programming soft ware such as Matlab, Scilab etc.

Results and discussion
In order to study the simulated behaviour of the canspring mechanism, some initial conditions were presumed considering the realistic values of parameters during sliver storage and are mentioned in Table 1. Th e linear momentum and displacement of the top plate built up with time and direction was considered as negative. Th is was due to the energy build up in the storage spring through compression, as shown in Figure 12. Similarly, it was observed that aft er a sudden initial impact of the sliver on the top plate, the velocity of the top plate remained almost constant with a small decrease over time. Furthermore, the mass deposited over the top plate increased with time depending on the sliver deposition rate, as shown in Figure 12. Linear stretching remained due to the viscoelastic part of the model q 7 almost unchanged with the passage of time, and the sliver deformation in the elastic region q 8 experienced a sudden increase within a short time. Aft er that, it remained constant, as shown in Figure 13. Moreover, the force experienced by the can-spring increased with an increase in its deformation, as shown in Figure 14. Th e latter agrees with the deformation of the can-spring as   Figure 14. In the case of sliver withdrawal from storage cans, the force experienced by the can-spring gradually reduced due to the decrease in the amount of sliver mass over the top plate with time during sliver withdrawn, as shown in Figure 15.

Conclusion
In order to study the dynamics of the can-spring mechanism, a bond graph model was developed. It was established that sliver deformation in the elastic region q 8 experienced a sudden increase within a short time and aft er that remained constant whereas q 7 remained unchanged. Th e mass deposited over the top plate increased with the increase in time. It was found out that the linear momentum of the top plate and its displacement increased with time. Th e velocity of the top plate showed a peak in short time initially during sliver deposition and remained unchanged aft erwards. Th e force experienced by the storage can-spring also built up as a result of the canspring compression due to the increase in the deposited sliver mass with time. Th e force experienced by the sliver showed an initial sudden increment followed by a downfall and then remained unchanged. It can be inferred that a small undesirable defl ection can deteriorate the sliver structure and can alter the sliver confi guration during draft ing operation. It was found that the linear momentum of the top plate increased in the negative direction over time depending on sliver linear density and sliver deposition rate. However, a more rigorous study is required to study the accurate dynamics of such precise systems as the force and stresses experienced by the sliver are too low due to very low inter-fi bre cohesion. Th e current bond graph model can be used for a further investigation for a more accurate prediction based on the dynamics of the can-spring-sliver system behaviour used for combed sliver handling.