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Rewinding machines are used in the paper converting industrial sector to unroll paper veils from large source reels and wind them around smaller rolls, for the production of paper logs to be used in household activities. Many different types of rewinding machine exist, and they are mostly based on stapling devices which automatically unroll a large coil of paper, roll it up around a new log, stop the veil allowing it to be torn off, unload the completed paper log, and reload a new paper core to be rolled up. Winding units generally include a primary and a secondary roller, supplying the desired rotational speed for winding, and a pressure unit consisting of a pressure roller in contact with the paper log being rolled up, which assures adequate containment and avoids the paper log to escape. All known systems have some mechanical limitations, mainly due to the contact of the pressure roller with the paper log, causing instabilities and vibrations, particularly when working with soft paper veils. This limits the maximum possible winding speed. The aim of this paper is it to present a detailed kinematic analysis of a novel pressure unit which overcomes the aforementioned limitation.

Rewinding machines are used in the paper converting industrial sector to unroll large reels of paper veils and wind them around smaller logs for the commercial sector, e.g., paper rolls for household keeping. In its most common configuration, a rewinding machine comprises a drive unit which supplies the paper veil, a winding unit for winding the paper veil supplied by the drive unit around a winding core, a primary and a secondary winding roller rotating in contact with the core being rolled up, and a pressure unit equipped with a pressure roller in contact with the paper log. During the winding phase, the peripheral speed of the rollers is substantially equal to the winding speed of the veil.

In its classical configuration, the pressure unit consists of one single pressure roller which is accomplishing a circular reciprocating motion around a fixed axis of rotation [

As far as the authors are aware, except the work presented in [

Figure

Computer-aided model of the main components in a paper rewinding machine.

The different steps related to the paper log being rolled up are illustrated in Figure

Snapshots of the system in motion: (a) initial phase involving the paper winding core, (b) paper log being rolled up in contact with one pressure roller, (c-d) paper log being rolled up in contact with two pressure rollers, (e) paper log being completed, and (f) paper log being ejected.

In the following analysis, it is assumed that the system has nominal geometry, i.e., no geometrical or assembly error is considered. Well-established methods for kinematic calibration [

From a kinematic point of view, the system in Figure

Schematic drawing of the system.

The aim of the present analysis is to determine the trajectory of the paper log being rolled up. This is then used in the next section to determine the trajectory of the pressure unit according to the requirements for stability, mentioned in the previous section.

From a kinematic point of view, the motion of the system and its component trajectories are expressed as a function of the increasing radius of the paper log being rolled up, _{L}, which is a function of the paper length wrapped per unit time, i.e., the winding speed, _{Li} is the initial radius of the paper log, which coincides with the paper winding core, _{Lf} is the final radius of the completed paper log and

With reference to the snapshots in Figure

Characteristic phases of the system motion: (a) Phase I; (b) Phase II; (c) Phase III; (d) Phase IV; (e) Phase V; (f) Phase VI.

During Phase I in Figure

During Phase II in Figure

During Phase III in Figure _{1}_{2} in Figure

During Phase IV in Figure

During Phase V in Figure

In the final Phase VI in Figure

During Phases I to IV in Figure

To perform a mathematical analysis, the following parameters are defined, according to Figure _{1} (_{2}) is the radius of the lower (upper) winding roller; _{1}_{2} between the rotational axes of the lower and upper winding rollers; _{1} (_{2}) is the variable distance between the axis of rotation of the lower (upper) winding roller and the instantaneous axis of the rotation of the paper log, _{1}_{2}; and _{1} (_{2}) is the variable angle between the horizontal axis and the segment _{1}_{2}_{1}.

Geometric parameters of the system.

According to the system geometry, the paper log is initially being moved along the line perpendicular to the segment _{1}_{2}.

To ensure continuity of motion, it is required that the paper log is kept in contact with both winding rollers, from the initial diameter to the final diameter.

The following geometric constraints must thus be assured during motion:

Squaring and adding equation (

The ambiguity in the sign in equation (

Substituting equation (

The condition in equation (

The Cartesian trajectory of the paper log axis F, in the reference frame

Once the final diameter of the paper log has been reached, the paper log is disengaged from the upper winding roller due to the proper motion of the pressure unit. Until the contact of the paper log with the lower winding roller is assured, the law of motion of the paper log will follow equation (_{1} is selected based on how quickly the product is being ejected. It is worth to note that if it is necessary to temporarily stop the rotation of the pressure unit in the final winding phase, it is sufficient to keep _{1} constant during the desired duration. As an example, Figure

Example of a trajectory (dashed line) of the paper log being rolled up.

It is also important to note that a kink is present in the trajectory of motion when the paper log disengages from the upper winding roller. Such kink is due to the different trajectory just before and after said disengagement.

In this section, the variation of the joint angles in A and B is related to the paper log being rolled up, i.e., to its instantaneous radius and thus to the winding speed according to equations (

The required trajectory is designed so that the pressure unit is able to convoy the paper log roll-up and return back to the initial configuration, ready for the next paper log engagement.

The required motion must thus (1) guarantee a smooth trajectory for both the main and secondary arms of the pressure unit, (2) avoid collisions with the other rollers located in the workspace, and (3) guarantee an optimal orientation of the line of pressure between the pressure roller(s) and the paper log.

In order to determine the kinematics of the pressure unit, i.e., the variation of the joints in A and B, it is important to identify two cases, according to the condition where one or two pressure rollers are in contact with the paper log. The first case is discussed in Section

The pressure unit and its main parameters for this case are illustrated in Figure

(a) Main geometry and (b) close-up of the pressure unit in case of one pressure roller in contact with the paper log.

The main arm of the pressure unit is rotating around the fixed axis A in Figure _{x} and _{y} respect to the frame of reference _{1} from axis A. The distance between axis B and the axis of rotation of each of the pressure rollers is constant and labelled as _{2} in Figure

The pressure unit has two degrees-of-freedom, and the corresponding generalized coordinates are labelled as the angles _{1} and _{2} in Figure

An offset parameter

Being C the desired location of the centre of the first engaging pressure roller, as detailed in Figure _{p} is the radius of each pressure roller, and the closure equations of the linkage in the

Squaring and adding equation (

As noted before, the sign ambiguity in equation (

Substituting equation (

The Cartesian coordinates of point B and the centres D and E of the pressure rollers in Figure

The following conditions are to be avoided.

When the pressure unit is approaching the upper roller, the following condition is required:

Should equation (_{r} is a safety parameter providing an adequate distance tolerance among the rollers, and

Geometry for the condition of collision between the pressure roller and the winding rollers.

When the pressure unit is approaching the lower roller, the following condition is required:

Should equation (

Since the pressure unit is rotating clockwise, when the pressure roller [D] is approaching the engagement location, the following condition should be satisfied:

In this case, should equation (

The equations in Section

Due to the increase in the paper log diameter, the pressure unit rotates and the pressure roller [D] approaches the paper log. The contact between the pressure roller [D] and the paper log is achieved when

In this case, the paper log is constrained by four rollers, two winding rollers and two pressure rollers. This is illustrated in Figure

Configuration where the paper log is in contact with two pressure rollers.

To assure the best possible containment for the paper log being rolled up, the configuration of the pressure unit is selected such as the pressure rollers [C] and [D] in Figure _{3}, representing the varying distance between point B and the centre of the paper log,

Equation (

The closure equations of the linkage are then determined as

As detailed in Section

Substituting equation (

In the case in which it is desired to anticipate the engagement of the second pressure roller with the paper log (e.g., to prevent paper log instability), equation (_{La}, at which the early engagement begins, is to be defined, as well as a value for the paper log radius, _{Lb}, at which the final engagement of the second pressure roller with the paper log occurs. It is worth noting that, during the phase from _{La} to _{Lb}, it is not possible to assure the optimal condition for the line of pressure between the pressure roller and the paper log, as described in Section

Once the values for _{La} and _{Lb} have been specified, the value of _{La}, to a final value of_{Lb}, where _{xa} and _{xb} are the coordinates of the position of the paper log axis at radius _{Lb}, the coordinates _{xb} and _{yb} are determined on the base of the corresponding configuration achieved by the system with four rollers in contact, as detailed in Section

The optimal condition for ejection of the paper log is achieved when the angle of the tangent to the pressure roller [C] and the paper log is less than that of the tangent to the lower winding roller and the paper log so that the following relation is satisfied:

In this case, the paper log is smoothly released without the need to disengage the pressure unit, which is then free to return back to its initial position for the next engagement. Such a motion can be defined by a polynomial blending trajectory, which assures the required conditions for continuity and smoothness. For each joint angle _{1} and _{2}, the initial boundary conditions are given by equations (_{2} equal to –120° respect to the previous initial configuration.

Applications are provided in this section to illustrate the device motion and to provide validation to the algorithm presented in the previous sections.

Three different simulations are performed, according to the geometric parameters listed in Table

Geometric parameters used for simulations.

Parameter | Unit | Case 1 | Case 2 | Case 3 |
---|---|---|---|---|

(mm) | 253.5 | |||

Φ | (rad) | |||

_{1} | (mm) | 97.5 | ||

_{2} | (mm) | 110 | ||

_{Li} | (mm) | 23.2 | ||

_{Lf} | (mm) | 48.8 | 72.0 | 72.0 |

_{La} | (mm) | — | — | 39.5 |

_{Lb} | (mm) | — | — | 46.5 |

_{x} | (mm) | −220.75 | ||

_{y} | (mm) | 160.75 | ||

_{1} | (mm) | 218 | ||

_{2} | (mm) | 49.07 | ||

_{P} | (mm) | 32.5 | ||

_{r} | (mm) | 10 |

Results of the simulations are reported in Figures

Simulation results for Case 1.

Simulation results for Case 2.

Simulation results for Case 3.

The results of the inverse kinematics shown in Figures

In this paper, a detailed kinematic analysis of a novel mechanical pressure unit integrated in a rewinding paper machine is presented. The system is based on a recent patent application, and it allows the creation of paper logs with different diameters by assuring adequate constraints during winding. The novelty of the system relies on the use of a pressure unit with three pressure rollers, respect to standard configurations with one roller only. The proposed system has two degrees-of-freedom, which need to be synchronized in order to adequately restrain and copy the paper roll motion. This allowed to increase the overall winding speed reducing machine downtime and motivated the introduced complexity respect to classical one degree-of-freedom pressure units. The trajectories of the two joint angles of the pressure unit are determined through an inverse kinematic analysis, and the conditions for collision are identified and taken into account in the algorithm. Numerical applications are presented which illustrate the motion of the pressure unit along with the paper log during formation. Some of the different possible scenarios are implemented. The algorithm presented in the paper is validated through the direct kinematics of the system.

Time

Winding speed

Paper thickness

_{L}:

Radius of the paper log at time

_{Li}:

Initial radius of the paper log

_{Li}:

Final radius of the paper log

Paper wrapped on each paper log

_{1}:

Radius of the lower winding roller

_{2}:

Radius of the upper winding roller

_{P}:

Radius of the pressure rollers

Constant distance between the lower and upper winding roller

Constant angle between horizontal axis and the direction of

_{1}:

Variable distance between the lower winding roller and the paper log

_{1}:

Variable angle between horizontal axis and the direction of _{1}

_{2}:

Variable distance between the upper winding roller and the paper log

_{2}:

Variable angle between horizontal axis and the direction of _{2}

Difference between _{1} and _{2}

_{x},

_{y}:

Coordinate

_{1}:

Joint angle of the main arm

_{2}:

Joint angle of the pressure unit

_{1}:

Constant length of the main arm

_{2}:

Constant length of second arms of the pressure unit

_{3}:

Variable distance between the centre of the paper log and secondary pivot joint of the pressure unit

Offset of the pressure roller respect to the paper log at first engagement

_{r}:

Tolerance distance among the rollers

Angular parameter defining early engagement

_{a}:

Angular parameter where early engagement begins

_{b}:

Angular parameter where early engagement ends

_{La}:

Radius of the paper log where early engagement begins

_{Lb}:

Radius of the paper log where early engagement ends.

The data used to support this study are provided within the article.

This research work was performed under a research agreement between the Department of Mechanical, Energy and Management Engineering of the University of Calabria and the private company United Converting Tissue Srl.

The authors declare that they have no conflicts of interest.

The research work related to this article was partially funded by the private company United Converting Tissue Srl, under a research agreement with the Department of Mechanical, Energy, and Management Engineering of the University of Calabria. L. Lupi and G. Lupi would like to acknowledge Mr. Luca Salotti and Mr. Christian Moscardini for the development of the system software.

Supplementary Materials are provided for this paper as follows: S0: animation comparing the proposed system to a traditional one. S1: animation illustrating application Case 1. S2: animation illustrating application Case 2. S3: animation illustrating application Case 3.