2024-02-09

Orginal article appears in: https://www.eliter-packaging.com/newsroom/cartoning-machine.html

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“I am not such a mechanical engineer that has solid academic background from engineering or has any degree in the subject. However, **with the years of experience I have acquired ever since I took over from my father our family business in the field of packaging automation**, whose history traces back to my grandfather, I would like to share with some industrial insights that might be useful for readers who are interested in cartoning machines.

**The guide including all the basic as well as advanced engineering knowledge about designing an automatic cartoning machine**. I hope that all those who from the industry or those who are looking for information about such kind of packaging machine will find this a helpful article.”

**This article is written for:**

- University students majored in mechanical engineering
- Professionals in the packaging automation industry
- People who are looking for cartoning machines on the market
- Random readers who are interested in “what is a cartoning machine?”
- Journalists who are looking for insights & updates in the industry
- etc.

An industrial equipment that is designed to automate the packaging process by simulating the manual process of folding carton blanks, filling cartons with packages or products, and finishing sealing or closure with the loaded carton blanks – such is tha common understanding of what is a cartoning machine, despite the fact that such a machine can be designed in various patterns.

**Cartoning represents a traditional form of packaging where a legacy of progressive evolution has catalyzed significant enhancements**, meawhile carton has been a long-lastingly popular form of secondary packaging. Cartons, commonly known as boxes, have evolved to encompass an array of configurations, typically characterized by rigid or semi-rigid materials, either monolithic or composite, fashioned into small to medium-sized receptacles featuring structural closures at both ends.

Concerning their utility, cartoning methodologies are meticulously crafted to cater to the specialized packaging requisites of a diverse range of products, **including liquids, semi-liquids, powders, granules, bulk items, and various product combinations.** Once assembled and sealed, these packages primarily serve the retail and distribution sectors.

Carton boxes, crafted from cardboard, kraft paper, or recycled materials, **are favored over plastics and metals, sharing many attributes with corrugated cardboard packaging** (also known as case packing) particularly in terms of their processability, storability, transportability, marketability, and recyclability. Moreover, they offer substantial logistical advantages, both for consumers and within industrial supply chain management (SCM). Through cost-effectiveness, **carton packaging facilitates the mechanization and automation of mass production workflows.** Consequently, cartoning machinery has become a ubiquitous industrial apparatus across sectors such as the food industry, pharmaceuticals, cosmetics, hardware, and electronics, solidifying its role as an essential element of automated packaging systems.

*Fig 1. ELITER’s cartoner pepresents the latest technology in the niche*

*Credit: ELITER Packaging Machinery*

**The cartoning machine falls under the scope of multifunctional packaging equipment,** whose trajectory of development has been intricately intertwined with the dynamic evolution of the items to be encases, the structure and style of cartons, and the susbequent cartoning packaging processes that follow.

**A thorough examination of the evolutionary trends in cartoning equipment mandates a comprehensive perspective that considers these interdependent variables.** It is from such a vantage point that research into the prospective development of cartoning machines should be conducted, ensuring that technological advancements align synergistically with the fluctuating paradigms of product packaging requirements, structural carton design, and process engineering methodologies.

As a result of the years of RDI by all the people in the industry, **cartoning packaging now accommodates an assortment of items**, which may include individual or multiple units, comprised of either uniformly shaped or irregular solid items — some accompanied by ancillary elements such as trays and partition panel within the carton.

*Fig 2. Cartoning Machines are now supposed to handle all sort of products*

*Credit: ELITER Packaging Machinery*

Concurrently, there has been an extension into the encapsulation of non-solid materials that are either **pre-filled or filled** in situ, utilizing various packaging vessels like **pouches, tubes, and bottles , or in some cases the bag-in-box packaging.** This diversification necessitates a comprehensive approach to the packaging design and optimization to ensure compatibility and efficiency across the heterogeneous spectrum of contents and packaging configurations.

A current revolutionary trend is the emerging popularity of collective and multi-item cartoning solutions, to name but a few examples, such as the packaging of assorted pastries, or single-use syringes. Aginst and in reponse to these market expectations, the selection of carton types and dimensions is typically predicated on the categorical classification, shape, size, alignment, and combination of the items contained within. However, **in the majority of scenarios, the practice of employing a uniform carton type to package a diverse array of items is still a prevalent approach, **such is to avoid the complexity of changeover, with minimal downtime when switching between menus.

**Selecting the right type of cartoning machine is a essential factor to how smoothly and reliably the products can be encased.**

- Stable items that are suitably positioned in a horizontal orientation are amenable to
**horizontal cartoning processes**, - whereas those items that maintain stability or possess favorable flow characteristics when positioned vertically are more apt for
**vertical cartoning applications**.

*Fig 3. Horizontal cartoning vs. vertical cartoning*

*Credit: ELITER Packaging Machinery*

This delineation acknowledges **the crucial influence of the physical characteristics of the packed items on the cartoning orientation** and illustrates the necessity for adaptive packaging strategies that accommodate the broad spectrum of product geometries and physical properties.

With the rise of retail and shelf ready packaging, cartoning is no longer limited to loading just the products while **also complementary packaging and combined additional items** such as cards or instruction manuals, whereas larger boxes often incorporate supplementary components like trays, pads, separators, partition paperboard or desiccants.

Especially in the pharmaceutical industry and allied sectors, almost all of the cartoning processes require **the presence of folded leftlet of instruction maual within the packaging**, either for the reasons of applicable regulations, or for the effective and proper utilization for medication.

**Pre-printed instructional materials are typically made available in a variety of formats:** single sheet, booklet, or leaflet.

*Fig 4. The different methods to insert leaflet in to carton*

*Credit: ELITER Packaging Machinery*

For single sheets, a dedicated folding apparatus is required, with a variety of folding patterns available for processes as depicted in Figure 2 and Figure 3. As for the methodology of inserting the folded paper or leaflet into the box, a schematic representation is provided in Fig 4. (based on the presupposition of a synchronized forward movement).

This emphasizes the investment in specialized machinery that not only accommodates the particularities of the packaging materials **but also ensures the seamless integration of additional elements into the final packaged product.**

The evolvement of carton structural design is predominantly reflected through various facts which include the selection of material composition, the refinement of dimensions, and the innovation of closure mechanisms.

Cartons are engineered to be prefabricated in a collapsible format (folded at blanks) which facilitates logistical efficiency for storage and transit, as well as compatibility with the mechanized aspects of supply chain management and cartoning procedures.

Commensurate with the size of the carton and its load-bearing capacity, cartons are generally fabricated from cardboard of an appropriate thickness or from single-wall corrugated cardboard. The modern styles and structures of cartons can be categorized into the following two types:

**Standard Cartons:**The conventional carton is manufactured from a single piece of cardboard through a sequence of processing steps such as die-cutting, creasing, folding, and gluing, resulting in a tube-shaped box with a tuck-flap lid, as illustrated in Figure 5.

The standard carton typically features a cross-sectional profile that is square or rectangular. The closure methodologies employed are diverse, comprising tuck-in varieties with or without flaps and closure panels, adhesive attachment either with or without tuck-in panel, and assorted combinations thereof. Tuck-in closures are designed for ease of multiple accesses, while **adhesive methods significantly reinforce the integrity and security of the seal with glue, contributing to the enhancement of tamper-evident properties. **

*Fig 5. Standard Cartons*

*Credit: ELITER Packaging Machinery*

**Atypical cartons:**They are special carton box designs that represent a strategic diversification and augmentation of standard conventional carton box types. The uniqueness of atypical cartons are manifestied in a variety of dimensions and configurations.- They are usually formed as pre-glued carton blanks with rigid frame but hollow in its interior party, completing a space-defining structure within the packaging milieu.
- The cavities of these boxes are engineered to be relatively shallow, resembling tray-like configurations, which serve to amplify the display efficacy of the packaged goods
- In cross-sectional view, the most commonly observed shapes are quadrilateral, though variations such as triangular, polygonal, and other geometrical forms are also utilized to suit specific design objectives.
- The box lids are versatile in design, with options such as detachable fitted components or as an integral part of the box that folds back upon the body. The seal is executed through a myriad of methods including, but not limited to, overlapping perimeters and insertion clasps.

To meet the advancement of packaging structures, **loading techniques in the cartoning packaging has shifted** as well to smarter and more intelligent options including robots, espsecially in the scene of top loading or vertical cartoning applications.

*Fig 6. Delta Robot*

*Credit: ELITER Packaging Machinery*

We have above a picture of a **Delta Robot, which is a 3-axis servo driven mechanism with a 3 Degrees of Freedom (DOFs) that enables the movement to be spatially flexible in a three-axis/dimensional coordination system.** In the case that the motion requires the shift in orientation of the object to be picked and placed, an adjustable linkage, driven and controlled by another servo motor, should be added between the central sections of the stationary and movable platforms, connected through unviersal joints at both ends to allow movements upwards or downwards.

All moving parts of the robot are made from lightweight, high-strength materials, such as aluminum carbon composites, to enhance their high-frequency, high-speed movement performance.

The blow chart shows the control sequence diagram of such a robotic system:

*Fig 7. Control Logix*

*Credit: ELITER Packaging Machinery*

The conventiona way of building a packaging end-of-line is that each set of the machines that fulfills the processes along the line is actually equipment seprated to each other, or say that such a packaging line is integrated of multiple sets of standalone machines. The downsides of such a solution is that, from a probability point of view, may include but not limit to those as follows:

**Increased Failure Rate due to interdependence:**The failure rate calculation of an integrated packaging line with multiple standalone machine connected is actually a multiplication rather than addition – the seral dependence and complex interdependence will result in a higher failure rate – the calculate of which is something like, for example, \( 98\%\times 98 \%= 96.04\%\)**Bottlenecks due to accummulated downtime and set-up complexibility:**Seperated packaaging machines that make up an end of line is nature still a combination of standalone machine that requires individually set-up times, maintenance efforts and repair procedures. The machine downstream will significantly rely on the performance on the upstream ones and in the case the incapability, the whole line will be subject to accumulated bottolnecks all along the procsses.

*Fig 8. Mechanisms along an integrated blister-cartoning system*

*Credit: ELITER Packaging Machinery*

Especially in the case of cartoning machine, which serves for the automation of secondary packaging, they tend to be installed seperately yet connected to the asset owner’s existing equipment. However, in response to the above-mentioned disadvantages, cartoning machine’s evolution has come to a stage that it is now not just considered as a standalone machine, yet supposed to be capable of being designed a mechanism to a integrated packaging system. A typcical example is the blister and cartoning combined packaging system which is getting popular recently typically in the pharmaceutical industry where pharma companies should respond ever more quickly to market changes. [1]

*Fig 9. BEC 200 Blister Line with Integrated Cartoning System*

*Credit: Uhlmann PacSystem – fair use for comments, no infringement intended*

**We will be extending our topic in section with further insights and involve more profound knowledge around machine theory to assist our analysis of cartoning machines.** If you are prepared with knowledge about mechanical engineering, this would be a complex and esoteric section for general readers, however, I will try to organize the content in a intelligible way as possible though some analysis cannot be done in such a manner.

*Fig 10. Mehcanisms of a Continuous End Load Cartoner*

*Credit: ELITER Packaging Machinery*

The analysis will be based on a typical horizontal end-load cartoning machine which is the most common variation of cartoners on the market and widely used among all indsutries.

Single-head carton erection, though still sporadically used on some continuous motion cartoners, has now been replaced bascially by rotary carton feeders with multiple heads. An example as follow is a typical **Internal Meshing Planetary Gear-Type Carton Assembly Mechanism** that primarily consists of a** fixed internal meshing central gear (3)**, three uniformly distributed **planetary gear suction heads (4)**, a **pre-break suction head (5),** and a vacuum system, among others.

*Fig 11. Rotaty Cartoner Assembly Mechanism*

*Credit: ELITER Packaging Machinery*

Rotary carton feeders designed on the basis of planetary gear system follow more or less the smilar kinematics as depicted in Fig 11., despite that fact thy may vary in some slight patterns, they wuold follow the operational sequence as below:

- The suction head will pick carton blanks from the carton magazine consecutively one after another
- The picked carton blank is preliminarily erected by a pre-break mechanism
- The carton is erected between the datum and phase chain lugs on the carton transport assembly
- The vacuum system is cut off and the rotary feeder forwards back to its homeposition and repeats above processes

Regardless of the stereotype that the more heads there are on the rotray feeder, the faster the speed can the cartoner reach, as empirically investigated by industrial casese, the toray feeder with 3 heads is the most suitable option for most of the situations of application, either in terms of efficiency or regarding the cost-effectiveness.

Now that such a design saves engineering complexity and take reasonable use of spatial structure, as well as to facilitate the incorporation of actuation mechanism and vaccum systems so that a dynamic balance can be ideally struck between these appratus. A downside happens when rotary feeder is designed with more heads, especially with the **Internal Meshing Gear Type Feeder** is that concession will be made to acccommodate larger carton sizes, yet in the case of a 3-head rotary feeder, it is a ideal solution to the issue, without compromising to the application of large carton.

Evaluation the right choice between these two gear configuration for designing the rotary carton feeder for cartoning machine is foundamentaly a concern around raw cost, spatial availability and efficiency, ragardless of the fact that the are entirely equivalent in terms of their mechanical functionality.

A common practice is that the internal meshing planetar gear system would be designed relatively larger, heavier and cumbersome, with expected increased manufacturing costs, however, such a configuration is ideally of superior suitability for large-scale and high-continuity cartoning application.

*Fig 12. Internal & External Planetary Gear Configuration: Schematic Diagrams*

*Credit: ELITER Packaging Machinery*

To pave the foundation to our comparative analysis of the 2 configurations of planetary gear systems, it is essential to carry out firstly a kinematic analysis to mathematically defined the motion of the \( P (x,y) \), which in the case of a cartoner’s rotary feeder, is the contact point of the suction plate to the carton blank.

*Fig 13. Trajectory and Kinematics Analysis of Planetary Gear Mechanism*

*Credit: ELITER Packaging Machinery*

**To establish the mathematical model for this mechanism**,

- denote the
**pitch circle radii**of the**central gear**and**planet gear**as \(R\) and \(r\), respectively, - and the eccentricity of point \(P\) as \(e\).
- When the link \(O\) rotates at a constant angular velocity \( \omega \) for a time \(t\), turning through an angle \(θ\),
- the coordinates of that point change to \( P^\text{‘} (x,y)\)

**For internal meshing configuration**, one should take:

$$ x = (R-r) \cdot cos \theta – e\cdot cos(\varphi – \theta) $$

$$ y = (R-r) \cdot sin \theta + e\cdot sin(\varphi – \theta) $$

where,

$$ \theta = \omega \cdot t = \frac {r\cdot \varphi} {R} \,\,\,\text{ or } \,\,\,\varphi = \frac{R \theta}{r} $$

Substituting the value of θ into the above equation gives:

$$ x = (R-r) \cdot cos \theta – e\cdot cos(\frac{R}{r} -1)\cdot \theta \qquad(1)$$

$$ y = (R-r) \cdot sin \theta + e\cdot sin(\frac{R}{r} -1)\cdot \theta \qquad(2)$$

**Similarly, in the case of external meshing planetary gear system:**

$$ x = (R+r) \cdot cos \theta + e\cdot cos(\frac{R}{r}+1)\cdot \theta \qquad(3)$$

$$ y = (R+r) \cdot sin \theta + e\cdot sin(\frac{R}{r} +1)\cdot \theta \qquad(4)$$

Take the derivative of above functions, respectively from (1) to (4), with respect to \(t\),

$$ \frac{dx}{dt} = – (R \mp r ) \cdot \omega \cdot sin\theta \pm (\frac{R}{r}\mp 1) \cdot e \cdot \omega \cdot sin(\frac{R}{r} \mp 1)\theta \qquad(5)$$

$$ \frac{dy}{dt} = (R \mp r ) \cdot \omega \cdot cos\theta + (\frac{R}{r}\mp 1) \cdot e \cdot \omega \cdot cos(\frac{R}{r} \mp 1)\theta \qquad(6) $$

$$ \frac{d^2 x}{dt^2} = -(R \mp r ) \cdot \omega^2 \cdot cos\theta \pm (\frac{R}{r}\mp 1)^2 \cdot e \cdot \omega^2 \cdot cos(\frac{R}{r} \mp 1)\theta \qquad(7) $$

$$ \frac{d^2 y}{dt^2} = -(R \mp r ) \cdot \omega^2 \cdot sin\theta – (\frac{R}{r}\mp 1)^2 \cdot e \cdot \omega^2 \cdot sin(\frac{R}{r} \mp 1)\theta \qquad(8) $$

In above 4 equations (5) ~ (8), the \( \mp \) and \( \pm \) have their upper part for internal meshing configuration and the part beneth for external meshing systems.

Evidently, the velocity and acceleration of the moving point P change according to the superposition of trigonometric functions.

We have acquired as above the motion equations of either the internal or external meshing planetary gear systems. In this section, we extend our discussion to the factors to be considered as well as the analysis to opt for the right option that provides the ideal suitability for carton picking and carton assembly.

Refer to figure 13, suppose a given condition of \( R=3r \) and \( \varphi = 3\theta \), substitue the elements in equations (5), (6), (7), (8) to acquire the motion and trajectory functions of the planetary gear-type rotary carton feeder of a cartoning machine, demonstrated as below:

$$ x=H \cdot cos (\theta_0 + \theta) + e \cdot cos \Delta \varphi $$

$$ y = H \cdot sin(\theta_0 + \theta) – e \cdot sin \Delta \varphi $$

$$ \nu_x = -\omega \cdot [ H \cdot sin (\theta_0 + \theta) + 2e\cdot sin\Delta \varphi ] $$

$$ \nu_y = \omega \cdot [H \cdot cos (\theta_0 + \theta) – 2e\cdot cos\Delta \varphi] $$

where,

$$ \nu = \sqrt {\nu^2_x + \nu^2_y} $$

$$ H = R-r = 2r $$

$$ \Delta\varphi = \varphi – (\theta_0+\theta)= 2\theta – \theta \text{,}\,\,\,\,\,\, \theta_0 = 30^o $$

Let the instantaneous velocity direction angle of the gear suction head, which moves along the cycloidal trajectory of the planetary gear, is \( \beta\); obviously,

$$ tan \beta = \frac {\nu_y} {\nu_x} \text{,} \,\,\,\,\, \text{equivalent to} \,\,\,\, \beta = arctan \frac {\nu_y} {\nu_x} $$

To facilitate our analysis, select

- the teeth number of planetary gear and the central sun gear respectively as \( Z_a = 24 \) and \( Z_b = 72 \)
- the gear module or modulus as \( m=5mm \)
- with the link rod rotating at a speed of \( n=100r/min \)

Calculating with these values denoted, we acquire the following results:

- the pitch circle radius of the planetary gear is \( r=60mm \)
- the pitch circle radius of the sun gear is \( R=180mm \)
- the length of the connecting rod (link rod) is \( H=120mm \)

Then we take the** lengths of the suction head** mounted on the planetary gear respectively as \( e=45mm\), \( e=60mm\), \( e=75mm\), and carry out through plot program to acquire **Motion Trajectory Curve** and **Motion Characteristic Curve.**

*Fig 14. Motion Trajectory Curve and Motion Characteristic Curve*

*Credit: ELITER Packaging Machinery*

The following conclusions get be acquired with our above analysis and plots depicted:

**Value \(e\) affects the rotary feeder’s capability of handling carton sizes:**

The trajectory plotted by the motion of suction plate, which is mounted on the suction head on the rotary feeder, would be shifting in pattern and vary significantly in the degree of concavity of its three sides (measured by the distance between points O and P), depending on the length of suction head selected, denoted here as the \(e\).Evidently from what can be observed from our plots, the large the value \(e\) is, the more capable the rotary header is of handling increased size of carton boxes (regarding its carton length and width of cross-section). A conclusion here of designing the cartoning machine’s rotary feeder is to choose the adequate value so that the machine developed can handle properly the carton blanks and avoid interference between adjacent suction hands during the processes of transferring and assembly.**The relation between \( e\) and \(r \) affects the instantaneous motion direct and then the erection reliability:**

The points A, B, C of the hypocyclodial curve respectively stand for the picking-up points, pre-break points and the station of erecting the carton blank, where finite abrupt changes is spotted in acceleration to varying degrees.- As it can be observed from the motion characteristics curve,
**when \(e\) equals \(r \),**the instantenrous velocity of these points is zero while the motion direction points directly towards the center \( O_1\). For example, at point A, \( \beta = \theta_0 = 3o^o \), the condition of zero velocity occurs. In the other two cases, the instantaneous velocity of the planetary gear suction head passing through each sharp point is greater than zero. - When \( e \lt r \), the instantaneous velocity is with a counterclockwise motion direction
- When \( e \gt r\), the instantaneous velocity is with a clockwise motion direction

**Such a shift in instantaneous motion direction can compromis to the reliability of carton erection.**- As it can be observed from the motion characteristics curve,
- On the basis of above analysis, \( e \le r \) would be a an ideal option to strike a balance between all the factors to concern. Review our plot when \(e=60mm\) and \(e=45mm\), the blank at moment of being picked out from the magazine should have its superior part picked outwards slightly inclined to acquire a smoother operation process.

Instead of the complex rotary feeder that is based on planetary gear configuration, carton assembly mechanisms on an intermittent cartoning machine follows a simpler deisgn pattern. Such mechanism is **still widely used on low-end, low-speed and entry-level cartoning machines and is available for a speed of up to 80 cartons per minute as maximum**.

*Fig 15. Intermittent Carton Assembly*

*Credit: ELITER Packaging Machinery*

Each working cycle of this kind of mechanism begins by utilizing the swinging vacuum suction head 2 to individually extract carton blanks from beneath the inclined storage from carton magazine, which are preliminarily unfolded as they pass between the steel guide rails 3 and 8, following which the reciprocating action of the push rod 7 inserts the carton into the main conveyor transport chains, thereby completing the box opening process.

A downside of such mechanism is that the carton opening relies foundamentally on the guide rails and it is challenging to increase the working speed without risking damage to the carton boxes, as higher speeds can lead to carton damage.

*Fig 16. Intermittent Carton Assembly Mechanism*

*Credit: ELITER Packaging Machinery*

The carton transport mechanism is usually the principle mechanical system or the mechanism that is driven by the primary axis of the cartoning machine, where erected cartons are transported all along the path flow during which they are loaded, folded with flaps, closed and discharged. There are usually pairs of lug chains between the phase lug and detum lug of which the cartons are secured.

The carton transport mechanism should be able to fit cartons with different length and width, the purpose of which is realized by adjust horizontally the distance between the datum and phase lugs and then the distance between the 2 mounting plates the are paralleled to each other to fit different carton length.

The blow figure shows the mechanical design of the carton transport mechanism and how the adjustament structure is realized, where the numbers stand for the following parts:

**1 and 2:**handwheel**3:**barrel cam loaders**4:**Plastic chains of bucket conveyor**5:**Lugs of bucket conveyor chains**6 and 7:**sprockets**8:**cam divider

*Fig 17. Carton Transport Mehcnism on a Continuous Cartoning Machine*

*Credit: ELITER Packaging Machinery*

The longitudinal spacing in question is determined by the length of the carton boxes being inserted. By rotating the handwheel 14, the translation support plate 13 and the associated set of parallel sprockets 12 connected to it can be adjusted. This alteration changes their **axial distance relative to another set of parallel sprockets 11**, which in turn adjusts the **longitudinal spacing of the discrete clamping plates, **so that the machine can accommodate cartons of different length.

Parts and sections indicated by numbers in the figure are:

**9 and 14:**handwheels**10:**lugs of carton transport lug chains**11 and 12:**sprockets**13:**mounting plate**15:**cam divider

*Fig 18. Carton Transport Mechanism – Length Changeover*

*Credit: ELITER Packaging Machinery*

A cartoning machine’s carton transport mechanism is usually with several pairs of sprockets with lug chains. On the mounting plate of the path flow’s each side there would be seperatly a pair of sprocket-lug-chain combination among which one is datum and another is phase lug chains. The adjustment to fit carton of different width is quite simple by rotating the handwheel 9 to move simultaneous the datum lug chains on each side thus to achieve the changeover.

One would design the carton transport mechanisms where the datum and phase chains on each side of mounting plates are synchronized or that they may also be seperated and be adjusted individually.

In the section we supposing a case that they are seperated and individually adjusted, so as to look into the technical advantages and disadvantages of desining the carton transport mechanism in such a way.

- Taking figure 17 as an example. The lugs on the chains have their width as \(a \) and base width as \(c\), supposing that whole lug’s overall width should be as same as chain’s pitch and take it as \( t \ge a+c \)
- Take the pitch of lugs as \( S_\varphi \), and corerspondingly the number of chain pitch as \( x_\varphi \), and a whole lug chain’s overall number of lugs as \( w_0\), and numbers of lugs as \( x_o \)
- The spacing from driving sprocket’s center to that of driven sprocket is taken as \( A_o \). All operation stations along the lug chains, where carton are present during operation, is taken as \( w\)
- Rotating handwheel 2 to moving the datum lug chians, the width of operational station to load carton is denoted as \(B\), while the width of no-loaded space is \(e\)

Designing the carton transport mehcniam should also following some empirical practices and principles to improve the transmission efficiency, listed as follows:

- Apply driving and driven sprockets with the same teeth number, take as \( z \), can achieve better stability during operation. \( z\) is supposed to be the larger the better but not to such an extent that it will hinder the smooth operation and result in chains coming off from the sprockets.
- To ameliorate the impact during high-speed operation, chians should be opt for with adequately shorter pitch.
- For the convenience of chain linking and to facilitate uniform wear of the chain links, it is best to use chains with an even number of links in combination with sprockets that have an odd number of teeth.

Bearing in mind the denoted elements and the engineering principles, we hereby supposing two cases and types of approch to designing the carton transport mechanism.

Designing the datum and phase lug chains with the same perimeter is one of the most comon approach to carton transport mechanism. Chains linked to the sprockets in phase position is fixed while the datum chains and sprockets are flexible in terms of adjustment by moving which the spacing between lugs is altered to fit cartons with different width.

Refer to Figure 19 and the superior part which refers to type A configuration, denote the indicated sections and elements as: \( B \), \( w\), \( a\), \( c\), \( t\), \( z\), then \(e\).

*Fig 19. Carton Transport Chains Configurations*

*Credit: ELITER Packaging Machinery*

Referring to the geometric relationships of the basic elements of the mechanism as shown in Figure 19, the following formula can be directly written out:

$$ x_\varphi \cdot t = B + 2(a+c)+e $$

so that

$$ x_\varphi = \frac {B+2(a+c)+e} {t} \qquad(9) $$

In the case that the result contains decimal, one should take the nearest whole number value, and denote it as \(x^\text{‘}_\varphi \), then the pitch of lugs on a single chain is:

$$ S_\varphi = x^\text{,}_\varphi \cdot t \qquad(10) $$

so that

$$ e^\text{,}=S_\varphi -B-2(a+c) \qquad(11)$$

let the central spacing of driving sprockets as \(A_o\) and the number of chain links shoud be

$$ x_a=w x^\text{,}_\varphi + m_1 \qquad(12) $$

where \(m_1\) is a pending element that stands for the travel by which the lug chain should be adjusted for the changeover.

Correspondingly, the total number of chain links in a closed loop of chains shoud be:

$$ x_0 = 2x_a +z=2(wx^\text{,}_\varphi + m_1) + z \qquad(13) $$

Bear in mind our empirical principle and practice of engineering and design that \(x_0\) should be even number while \(z\) should be odd number, refer to formula above we can acquire the range in which falls the number of chain links:

$$ m_1 = \pm 0.5, \pm 1.5, \pm 2.5, \text{…}, \pm (x^\text{,}_\varphi – 0.5) \qquad(14) $$

To opt for an adequate \(m_1 \) among above range, we may refer also to the numbers of lugs to assist our calculation that

$$ w_0 = 2w + \frac {z} {x^\text{,}_\varphi} \qquad(15)$$

In case that the calculation acquires a result with decimal, we may either round up or round down to get a whole number value, denoted as \(w^\text{,}_0 \),

$$ w^\text{,}_0 = \frac {x_0} {x^\text{,}_\varphi} = \frac {1} {x^\text{,}_\varphi} \cdot [2(wx^\text{,}_\varphi + m) + z] \qquad(16)$$

from where the result is acquired as

$$ m1 = \frac {1}{2}\cdot [ (w^\text{,}_0 -2w)x^\text{,}_\varphi – z] \qquad(17) $$

Similarly to calculate the central spacing of sprockets

$$ A_0 = x_a t = (wx^\text{,}_\varphi + m1)\cdot t \qquad(18)$$

as well as the perimeter of a single lug chain, which is

$$ L_0 = x_0 t =[ 2(w x^\text{,}_\varphi + m_1) + z ]\cdot t \qquad(19) $$

Finally, we are able to get the results about the maximum and minimum size range the lug chains and carton transport mechanism can fit in accodance to the carton width

$$ B_\text{min} =0, \,\,\,\, B_\text{max}=S_\Phi – (a+c) \qquad(20) $$

$$ e_\text{max} = B_\text{max}, \,\,\,\, e_\text{min}=-(a+c) \qquad(21) $$

Those who with past experience of desining a cartoning machine may take note that the larger the \(e\) is, the better the system may fit continuous motion, while \(e\) is smaller, it is ideal for intermittent motion.

The second configuration, as shown in the inferior part of figure 19, stands for such a design that chains with different perimeter. Refer as well to the formulas created for type A, we are able to get results and write them as follows:

- Chain link numbers of one of the chains: \( x_\text{a1}=w x^\text{,}_\varphi + m_2 \qquad(22)\)
- Chains link numbers of another chains: \( x_\text{a2} = w x^\text{,}_\varphi + m_2 + n \qquad(23) \)
- Whole number of chain links: \( x_0 = x_\text{a1} + x_\text{a2} + z = 2(w x^\text{,}_\varphi + m+2) + n + z \qquad (24) \)

where adjustment of chain link numbers is

$$ m_2 = 0, \pm 1, \pm 2, \text{…}, \pm (x^\text{,}_\varphi -1) \qquad(25)$$

numbers of chain links added

$$ n=1,3,5,\text{…} $$

then to calculate

$$ w_0 = 2w +\frac{z} {x^\text{,}_\varphi} $$

the result of which should be rounded to whole number value and supplant it into formula (24) together with \(n\)

$$ w^\text{,}_0=\frac {x_0} {x^\text{,}_\varphi} = \frac {1}{x^\text{,}_\varphi} [ 2(wx^\text{,}_\varphi + m_2) + n +z ] \qquad(26)$$

by which

$$ m_2 =\frac{1}{2} [ (w^\text{,}_0)-2w)\cdot x^\text{,}_\varphi -n -z ] \qquad(27) $$

so that to acquire the results as follows:

$$ A_0 = x_\text{a1}t= (wx^\text{,}_\varphi + m_2)\cdot t \qquad(28) $$

$$ L_0 = x_0 t = [2 (wx^\text{,}\varphi + m_2) +n +z]\cdot t \qquad(29) $$

In another of our blogs we have explained the difference between intermitten and continuous motion cartoning machines, where we also covered the structural difference of the loader or pusher on each of these two distinct cartoning machines. Again we give a conclusion as follows:

- In general sense, an intermittent cartoning machine applies usually a simpler mechanism that follows a linear reciprocating trajectory to insert products from the bucket conveyor into cartons assembled on the carton transport mechanism at a pace of one load per cycle and by intermittent motion.
- While on a continuous motion cartoning machines, a series of loaders is mounted on a track where a cam is set following the groove of which the pushers (loaders) will realize the action of inserting products.

*Fig 20. Cam Loaders of Continous Motion Cartoning Machine*

*Credit: ELITER Packaging Machinery*

In this section, we go straight to the mechanical analysis of, specifically, the case of continuous motion cartoning machine and motion analysis of the pushers, which is usually composed of slider, pushrod, and slider block, and slider, shown as in the following figure 21., consisting of the following components:

*1. Guide Rod – Pushrod**2. Load-Bearing Groove**3. Pusher**4. Guide Spring Strip**5. Microswitch**6. Linear Upper Guide Rail**7. Stationary Lower Guide Rail*

*Fig 21. Single Pusher of Barrel Cam Loaders*

*Credit: ELITER Packaging Machiinery*

The mechanical analysis of this component looks to provide an insight into designing tips of how the continuous motion cartoning machine’s inserter can follow a smooth movement.

*Fig 22. Example of Barrel Cam Loader on Continuous Cartoning Machine*

*Credit: Cariba [2]*

The forward cycle or movement of the pusher involves the force analysis between pusher (that consists of pushrod, sliding block, sliding cradle) and loaded products.

- Supposing the total length of pushrod is \(l_0\), while the thickness of the supporting plates on both sides being \(B_2\);
- The spacing between two paralleled pushrods mounted on the loader system is \(e\);
- The sliding cradle that engages dynamically with the pushrod is of thickness \(B_1\);
- The height of the upper pulley relative to the axial plane of the pushrod is \(h\), the pulley is constrained by the fixed rail guide, closely related to the inclined angles a1, a2 of the guide rail’s extension stroke and retraction stroke;
- Supposing the total weight of the pusher as \(G_0\) with its center of gravity located at point O on the axis of the pushrod;
- During the pushing process, the sliding friction force between the loaded product weighted as \(G_w\) and the contact surface of the load-bearing groove is \(F_0\), with the coefficient of friction and limit friction angle being \(f_0\) and \(\varphi_0\);
- the normal pressure and rolling friction force exerted by the fixed guide rail on the pulley are \(N_1\) and \(F_1\), with the coefficient of friction and limit friction angle being \(f_1\) and \(\varphi_1\);
- The interactive forces between the components of the sliding motion pair are \(F_2\), \(F^\text{,}_2\), and \(F^3\), with the coefficients of friction and limit friction angles being \(f_2\) and \(\varphi_2\);

Under the forced traction of the chain, the pulleys on the slider cradle allow each pushrod to move forward at a constant pace by passing through the fixed guide rail. When the distance moved by a particular slider support relative to the front support plate of the guide rod is \(b_0\), the related active forces are decomposed along the longitudinal \((x)\) and transverse \((y)\) directions, and the power equation for the forward stroke is established based on the previously illustrated body of forces:

$$ N_\text{1x}=N_1 \cdot (sin a_1 – f_1 \cdot cos a_1) \qquad(30) $$

$$ N_\text{1y}=N_1 \cdot (cos a_1 + f_1 \cdot sin a_1) \qquad(31) $$

Meanwhile,

$$ N_\text{1x}= F_0 + F_2 + F^\text{,}_2 + F^3 \qquad(32) $$

where,

\begin{align}

& F_0 = f_0 \cdot G_w \\

& F_2 = f_2 \cdot \sqrt{ (T_1 + \frac{T_2}{2} – \frac {b_2 G_0} {2b_0} )^2 + (\frac{N_2}{2})^2 }\\

& F_2^\text{,} = f_2 \cdot \sqrt{ (T_1 – \frac{T_2}{2} + \frac {b_2 G_0} {2b_0} )^2 + (\frac{N_2}{2})^2 }\\

& F_3= f_2 \cdot (T_2 + \frac {b_1 G_0} {2b_0} )\\

\end{align}

where

$$

T1= \frac{h}{e} \cdot N_\text{1y} \,\,\,\, , T2=\frac{h}{b_0}\cdot N_\text{1x},\,\,\,\, N_2=N_\text{1y}$$

$$f_0=tan\varphi_0,\,\,\,\, f_1=tan\varphi_1 , \,\,\,\, f_3=tan\varphi_2

$$

As per above analysis, it can be found that wit the loaders pushing forward to insert products into the carton, \( b_0 \) decreases gradually while the forces applied on bearing increases. For which reason we should proceed the analysis when the slider cradle reaches the critical value of \( b_\text{or} = b_1 = \frac {1}{2} \cdot(B_1 + B_2) \), and \( b_2=0 \) that

$$

\begin{align}

F_\text{2t} &= f_2 \cdot \sqrt {(T_1 + \frac{T_2}{2})^2 + (\frac{N_2}{2})^2 } \\

&= k_1 f_2 N_1 \cdot (cosa_1 + f_1 \cdot sina_1) \\

\end{align}

$$

where

$$ k_1 = \frac{1}{2} \cdot \sqrt{ 1+ h^2 \cdot [ \frac{2}{e} + \frac{1}{b_\text{or}} \cdot tan (a_1 – \varphi_1) ]^2 } \qquad(33)$$

$$

\begin{align}

F^\text{,}_\text{2r} & = f_2 \cdot \sqrt { (T_1 – \frac{T_2}{2} )^2 + (\frac{N_2}{2})^2 } \\

& = k_2f_2N_1 \cdot (cosa_1 + f_1 \cdot sina_1) \\

\end{align}

$$

in which,

$$ k_2 = \frac{1}{2} \cdot \sqrt{ 1+h^2 \cdot[ \frac{2}{e} – \frac{1}{b_\text{or}} \cdot tan (a_1 – \varphi_1) ]^2 } \qquad(34) $$

$$ F_\text{3r} = f_2 [G_0 + \frac{hN_1}{b_\text{or}} \cdot (sina_1 – f_1 costa_1) ] $$

Substitue all above results into equation (32) and combined with equation (30), to acquire:

$$ N_1 = \frac {f_0 G_w + f_2 G_0} { (1 – \frac{f_2 h}{b_\text{or}}) \cdot (sina_1 – f_1 cosa_1) – f2 \cdot (k_1 + k_2)\cdot (cosa_1 + f_1 sina_1) } \qquad(35)$$

To achieve the purpose that the mechaism will not get jammed, it is suggested that

$$ (1 – \frac{f_2 h}{b_\text{or} }) \cdot (sina_1 – f_1 cosa_1) \gt f2 \cdot (k_1 + k_2)\cdot (cosa_1 + f_1 sina_1) $$

which can be transformed as:

$$ \frac {sina_1 – f_1 cosa_1} {cosa_1 + f_1 sina_1} \gt \frac {b_\text{or} f_2 (k_1+k_2)} {b_\text{or} – f_2 h} $$

and let

$$ \varphi^\text{,}_2 = arctan\frac{b_\text{or} f_2 (k_1+k_2)}{b_\text{or} – f_2 h} \qquad(36)$$

with which the result is acquired as

$$tan(a_1 – \varphi_1) \gt tan\varphi^\text{,}_2$$

which is equivalent to:

$$a_1 \gt \varphi_1 + \varphi^\text{,}_2 \qquad(37)$$

on the basis of above analysis, given the fact that \(k_1 \gt 0\), and \(k_2 \gt 0\), it is essential that

$$b_\text{or}\gt f_2h \qquad (38) $$

only with which the following equation is achieved

$$\varphi_2 \gt 0 $$

so that equation (36) is valid – as a result of which the design of the barrel cam loader should be confident to makes sure that \( b_\text{or} = b_1 = \frac{1}{2} (B_1 + B_2) \) so that the movement is smooth along th maximum travel distance of slider cradle on the pushrod which is \( S_max = l_0 – (B1 + 2B_2) \)

Refer to equation (35) and start the analysis from the timing the begining of retraction stroke, take \( G_w=0 \) and substitute \( a_2 \) by \(a_1\), and switch the sign of \(T_2\), to calculate the normal pressure of fixed guide rail on the pulley:

$$ N^\text{,}_1 = \frac {f_2 G_0} { (1 + \frac{f_2 h}{b_\text{or}}) \cdot (sina_2 – f_1 cosa_2) – f_2 \cdot (k_1 + k_2)\cdot (cosa_2 + f_1 sina_2) } \qquad (39) $$

Similarly to apply the same methodology in the previous chapter to acquire the condition by which the movement will not face with blockage or jam:

$$ a_2 \gt \varphi_1 + \varphi^\text{,,}_2 \qquad(40)$$

where

$$ \varphi^\text{,,}_2 = arctan \frac {b_\text{or}f_2(k_1+k_2)} {b_\text{or}+f_2 h} \qquad(41) $$

evidently, it is found that \(N^\text{,}_1 \lt N_1\), and \(\varphi^\text{,,}_2 \lt \varphi^\text{,}_2\), and ideally, \( a_2 \le a_1 \).

Supposing that \( S_max \) is a known value, once \(a_1\) and \(a_2\) is confirmed, it is able to confirm the valid travel distance of the barrel cam loader that

for forward stroke: \( L_1 = S_max seca_1 \qquad(42) \)

for retraction stroke: \( L_2 = S_max seca_2 \qquad(43) \)

Let the traction speed of chain as \(v_1\), it is easy to acquire the speed of both forward stroke and retraction stroke as follows:

$$ v_1=v_l cota_1, \,\,\,\,\, v2=v_lcota_2 \qquad(44) $$

Cartons are currently being used in an array of different designs that the closure of them may not necessarily be always the same or follow a fixed pattern. However, among all the carton styles, the most popular and widely accepted cartons are tuck-end ones, especially those of straight tuck-end of reverse tuck-end, the reasons that they are repeatable for closure and access, reliable in strength and ideal for mechanical automation processes that require the packaging to be relatively standard.

A classic tuck-in station on a cartonig machine is as shown in figure 23., that similar but slightly varied mechanisms are mounted on the two sides along the carton transport mechanism together with a series of guide rails that support and guide the closure panel and tuck flap of cartons.

The mechanisms on the two sides follow basically the same operational sequence including:

- the folding of closure panel as a whole
- the folding of tuck flap
- the insertion and closure o tuck flap
- the further securing of inserted tuck-flap (neglected on some entry-level cartoning machines)

Despite the fact that cartons are designed with distinctive sizes and styles while they need automated packaging solutions, tuck-in is still the optimal option for carton packaging now that it is to which that the automation can follow a given systematical sequence to accomplish the purpose.

The key factor to the study of tuck-in mechanism is the analysis of motional relation between cartons, the combination of closure panel and tuck flap, and the machine’s flap tucker, under the given conditions of carton sizes, flap shapes, and design of flap tuckers.

A general prerequisite and guideline of designing the tuck-in station should follow the below concept:

**Timing:**when the parallel double crank the drives the flap tucker turns perpendicular to the horizontal baseline \(f -f \) of the carton transport mechanism, and simultaneous when the flap tuckers are at their critical position that insert the tuck flaps in the the deepest point into the carton,**What to do:**it is essential to adjust the lug chains so that the centerlines of the carton’s body should coincide with that of the flap tucker,**The result:**as a result of which the trajectory of the tuck folder’s insertion and withdrawal relative to the centerline of the box body will have complete symmetry.

Nevertheless, in actual situation, **the frictions between flap tuckers and cartons as well as the surface of the cartoning machine may cause the deviation of flaps when cartons are moving forward** along the carton transport path. This would happen in case that the flap tucker does not insert the tuck flap into the carton at an adequate depth. An ideal circumstance is that when the flap tucker withdraw from the cartons, the cut outs at the two sides of the closure panel shoud have get in contact with the dust flaps so that the whole carton stays stable during the motion, **given which fact the analysis of the mechanism should have the importance attached to the analysis of the depth that tuck flaps are pushed into the carton box**.

The mechanism sechematic below shows the details of tuck folding and tuck-in station of a cartoning machine where they numbered components are:

- Carton
- Lugs on carton transport mechanism
- Dust flap folder driven by cam and crank
- Rail guides for dust flaps
- Rail guides for closure panel
- Rail guides for closure panel
- Guides for tuck flaps
- Tucker for tuck flaps and tuck-in
- Guides for tuck flaps
- Parallel double crank

*Fig 23. Mechanism Schematic of Tuck Folder Mechanism*

*Credit: ELITER Packaging Machinery*

In accordance to above description, supposing a carton with width \(B\), width of tuck flap as \(D\), pitch of carton transport mechanism as \(S_varphi\), speed of the carton transport mechanism of \(v_1\), the improvement of tuck folder mechanism shoud meet the following requirements:

- The flap tuckers should push and insert the tuck flap to a depth that is slightly larger than the width of the tuck flap, denoted as \(H \gt D \)
- The instantaneous direction at which the tuck flaps are inserted into the carton is determined by the magnitude of initial insertion angle \( \theta_m \). While the movement of cartons is contrary to the motion of tuck folders, friction force is expected both longitudinal and lateral between the two. When reflected in the lateral component of speed at the tip of the tuck folder, this results in a continuous change process that goes from relative lagging, to complete synchronization, to relative leading. When the tuck folder’s rotational angle \( \theta \) equals 0, its lateral component of speed becomes equal to the translational circular velocity, and we take \(v_a\) (the velocity of the tucker) to be greater than \(v_l\) (the longitudinal velocity). In such a scenario, the tab is subjected to the tuck folder’s control, which allows the corresponding insertion action to be completed.
- The insertion point of the flap tucker possesses a degree of flexibility, and coupled with the aforementioned lateral play, it is determined that \( W \gt B \). From a practical standpoint, this width should not be too narrow. Whilst the front end should be rounded to facilitate guiding into the box.

To proove above assemption, we need to carry out the analysis mathematically to provide proof to its validity.

*Fig 24. Kinematic Analysis Diagram of Tuck Folder*

*Credit: ELITER Packaging Machinery*

Supposing the rotational speed of the double crank is \(n_d\), which is equivalent to the output of the cartoning machine, equals to how many cartons per minute, the angular velocity

$$ w_d = \frac{\pi n_d}{30} $$

is determined by

$$ v_d=\frac{\pi \rho n_d}{30}= \rho w_d \qquad(45) $$

$$ v_1=\frac{S_\varphi n_d}{60} = \frac{S_\varphi w_d}{2 \pi} \qquad(46)$$

$$ \rho \gt \frac{S_\varphi}{2\pi} \qquad(47)$$

when \( v_d \gt v_1 \)

And when \( \rho \gt H \),

$$ cos \theta _m = \frac {\rho – H}{\rho} = 1- \frac{H}{\rho} \gt 0 \qquad(48)$$

So that the initial insertion angle is confirmed that \( \theta_m \lt 90^°\). Denote the \( H\) as known, \( \theta_m \) will follow simultaneous the increase or decrease of \( \rho \).

Let \( \theta_r \) stand for the phase angle between tuckers and the cartons, and it is essential that

$$ cos \theta_r = \frac {v_1}{v_d} = \frac {S_\varphi}{2\pi \rho} \qquad(49) $$

$$ \theta_r \lt \theta_m \lt 90^° ,\,\,\,\, cos\theta_r \gt cos\theta_m \gt 0 $$

transcripted into

$$ \frac{S_\varphi}{2\pi \rho} \gt 1- \frac{H}{\rho} \gt 0 $$

or alternatively

$$ \frac{S_\varphi}{2\pi \rho} \gt \rho – H \gt 0 $$

Solve the equation with (47) simultaneously, the result is

$$ \rho \gt \frac{S_\varphi}{2\pi} \gt \rho -H \gt 0 \qquad(50) $$

Take avail of equation (50) to get value of \( \rho \), is the prerequisite on which that the tuck folder can effectively accomplish the tuck-in function. Evidently, \(\rho\) is dependent on the value of \( S_\varphi \) and \(H\), yet still allowing further improvement with regard to the value taken.

The following study will focuse on the qualitative and quantitative analysis on relative motion relationship between the carton, tuck flaps, and the tuck folder, based on the diagram displayed in fig 25.

*Fig 25. Motion Trajectory Between Tucker and Tuck Flap*

*Credit: ELITER Packaging Machinery*

To simplify the derivation and calculation, and to facilitate the examination of interference conditions between components, it is assumed that the tuck flap is not constrained by the boundaries of the box but moves together with the tucker.

*Fig 26. Tuck Folding and Tuck-In Process on Cartoner*

*Credit: Cariba*

Let the insertion point of the tuck folder into the moving box be denoted as \( a_m\). At this moment, the distance between the centerlines of the tuckerand the box is \( \Delta m>0 \) (with leftward deviation being positive and rightward deviation being negative); when the tucker moves in a positive circular translation clockwise from the insertion point, it undergoes a rotation angle of the following result, with respect to the coordinate system \(x~y\) over time:

$$\theta = \theta_m – w_d t$$

from which the result is acquired as

$$ t=\frac{\theta_m – \theta}{w_d} $$

respectively, the lateral displacement of the tuker’s tip with respect to point \( a_m\) on the box is

$$ \begin{align}

\Delta S_\text{mt} &= v_1 t – \int^t_0 v_d cos\theta dt \qquad(51) \\

&= \rho [ (\theta_m – \theta)cos\theta_r – sin\theta_m + sin\theta ]

\end{align}$$

take \( \theta = 0 \) and substitute that in above formula, and now that \( \Delta m + \Delta S_\text{mt} = 0 \), it is calculated that

$$ \Delta m = \rho(sin\theta_m – \theta_m cos\theta_r)= \rho sin\theta_m – \frac{S_\varphi \theta_m}{2\pi} \qquad(52)$$

which indicates that \Delta m is related only to \( \rho \), \(H\) and \(S_\varphi \).

Similarly, refer to formula (51) it is not something challenging to get the lateral displacement of tucker’s tip from point \( a_m\) to \(a_r\), which is

$$ \Delta S_\text{mr} =\rho [ (\theta_m – \theta)cos\theta_r – sin\theta_m + sin\theta_r ] \qquad(53)$$

then as well the lateral displacement from \(a_r\) to \(a_0\)

$$ \Delta S_\text{r0}=\Delta m + \Delta S_\text{mr} = \rho (sin\theta_r – \theta_r cos\theta_r) \qquad(54) $$

A conclusion can be reached that \Delta S_\text{mr} is also only related to \( \rho \), \(H\) and \(S_\varphi \), by which it is suggested to take value reasonably of \(\Delta m\) and \(\Delta S_\text{mr}\), to achieve an ideal motion trajectory of tuck folder that it will not insert the tuck flaps too much into the carton.

In order to take advantage of Figure 25, with its detailed solid lines with arrows, to broadly reflect the step-by-step relative motion trajectories of the tucker’s tip and the left corner point of the tuck flap as they respectively insert into the box from points \(a_m\) and \(a\), it is necessary to derive the longitudinal displacement corresponding to the lateral displacement \(\Delta S_\text{mr}\) of the tucker’s tip and tuck flap. This is essential in order to reasonably determine the structural form and geometric dimensions of the tucker and tuck flap, for which

$$\Delta h_\text{mr}= \rho (cos\theta_r – cos\theta_m) = H – \rho (1-cos\theta_r) \qquad (55) $$

$$ \Delta h^,_\text{mr}=\Delta h_\text{mr} – (H-D) = D- \rho (1-cos\theta_r) \qquad(56) $$

subsequently, when the spacing bettwen the centerlines of tucker and tuck flap retrives to \( \Delta m\) from \( \Delta m + \Delta S_\text{mr}\), let the longitudinal displacement of the tucker’s tip from \(a_m\) to \(a^,_m\), and the left corner point of the tuck flap from \( a_m \) and \(a^,_m\) to \( \overline{a}_m \) and \( \overline{a}^,_m \), denoted as vector \( \Delta h_\text{mm} \) and \(\Delta h^,_\text{mm} \), respectively. The corresponding rotation angle of the tucker is \( \theta^,_m (\theta_r \gt \theta^,_m \gt 0) \), derived from which,

$$ \Delta h_\text{mm}= \rho (cos\theta^,-m – cos\theta_m) = H – \rho (1-cos\theta^,_m) \qquad (57) $$

$$ \Delta h^,_\text{mm}=\Delta h_\text{mm} – (H-D) = D- \rho (1-cos\theta^,_m) \qquad(58) $$

Regarding the calculation of the value for \( \theta^,_m \), considering the process of the tucker’s tip moving from \(a_m\) to \(a_r\), and then from \(a_r\) to \(\overline{a}_m\), the lateral displacements are \( \Delta S_\text{mr}\) and \(\Delta S_\text{rm} \), respectively. Since their magnitudes are equal but the directions of displacement are opposite, therefore,

$$\Delta S_\text{mr} + \Delta S_\text{rm} = 0$$

refer to formular (51), get the result that

$$ \Delta_\text{rm}= \rho [(\theta_r – \theta^,_m)cos\theta_r – sin\theta_r + sin\theta^,_m ] $$

which, when combined with formular (53)

$$\theta_m cos \theta_r – sin\theta_m = \theta^,_m cos\theta_r – sin \theta^,_m $$

transcripted as

$$ S_m \theta^,_m -2 \pi \rho sin\theta^,_m = S_\varphi \theta_m – 2\pi\rho sin\theta_m $$

In the case that \(S_\varphi\), \(\rho\) and \(\theta_m\) is known value, take

$$ K=S_\varphi \theta_m – 2\pi\rho sin\theta_m $$

then

$$ S_\varphi \theta^,_m – 2\pi \rho sin^,_m = K \qquad(59) $$

For the above equation, the intersection point of the two lines can be drawn using the analytic graphical method, and the abscissa of this intersection point is \( \theta^,_m\). Subsequently, by relating to other relevant values, one can solve for \( \Delta h_\text{mm} \) and \(\Delta h^,_\text{mm} \).

Furthermore, referring to Figure 25, once the coordinates of point \( a^,_r\) \( (\Delta S_\text{mr}, \Delta h^,_\text{mr}) \) have been determined, it is possible to measure the fillet radius \(r\) of the tuck flap, with the requirement that \(\Delta h^,_\text{mm} \le r \lt D\). **This ensures that when the tuck flap is inserted, the tuck flap does not interfere with the inner wall of the box, and the sealing of the box is secure and reliable.**

Despite the fact that cartoners have already evolved into a mature market section of relatively lower entry barrier, companies that costantly flocking into the market as well as those who have been dedicating to cartoning automation for decades have cleaved some new outles of cartoning automation and machines’s operational sequence and patterns are now not limited only to end-load or vertical-loading and assembly from folding cartons.

**The trend modular design together with the development of robotics have also changed the way how a cartoning equipment is defined**, especially with those industrial leading automation companies who count on distinct approach to automaton solutions for cartons, for example, a top-load system combined with carton formers, linear product transport systems, robotic pick and place systems, etc., totally different from a convention sense of rigidly developed mechanical or servo-driven cartoner.

- [1] BEC 200 blister line: Greater flexibility in pharmaceutical packaging: Uhlmann Pac-Systeme, https://www.uhlmann.de/products/blister-lines/blister-line-bec-200/
- [2] Cariba S.R.L.,C300 Astucciatrice orizzontale, https://www.cariba.it/gamma-macchine/astucciatrici-orizzontali/astucciatrice-orizzontale-c300/

The article cited several pictures from 3rd party businesses for purposes of criticism, comment, education and research. The action does not intent any kind of infringement and is with mindset of fair use. We express tribute to 3rd parties from which we have cited the content who are Uhlmann Pac-Systeme of Germany and Cariba S.R.L. of Italy.