Modeling of the Inking Apparatus of the Sheet Printing Machine

It is known that the main purpose of the ink printing system is to provide a given thickness of the ink layer on the prints when printing different types of products. Therefore, the ink system can be considered as an object of regulation. Adjustable, that is, the initial, value is the thickness of the layer of ink on the prints, the input is the supply of ink. Changing the type of printed matter requires changing the printing form. This can be considered as a change in the nature of the disturbance and the load on the inking system.

Inking devices of offset printing machines [1] are the most important technological units, the dynamic and static properties of which significantly affect the print quality. Therefore, the urgent task is to develop mathematical methods for researching the work and further model-oriented design of inking devices based on software and hardware.

The purpose of this work is to create a scientific basis for structural and mathematical modeling of a high-pressure and high-sheet offset printing apparatus for printing machines.

To achieve this goal, the following tasks were set:

Modeling of the inking apparatus as a control object associated with the movement and ink flow of the printing process;

Determination of the nutritional condition for the development of a single-capacitive mathematical model of the inking apparatus;

Development of methods for the distribution of ink layers in an inking apparatus for high and sheet offset printing machines.

$B_{i}=\left.\frac{\beta_{i}}{1-\beta_{i}}\right|_{\beta_{i=0,5}}=1$       (2)

When transferring ink from the knurling roller (k) to the plate cylinder (k+1) with a fill factor σ (provided that the relief is level).

1.jpg

Figure 1. Ink flow model: а- baseline; b- ink transfer through the roller; c- pulse power mode; d- continuous power

$B_{k}=\left.\frac{\sigma \beta_{k}}{1-\sigma \beta_{k}}\right|_{\beta_{k=0,5}}=\frac{\sigma}{2-\sigma}$         (3)

When transferring ink from a mold to rubber and from rubber to paper

$B_{k+1}=\frac{\beta_{k+1}}{1-\beta_{k+1}}$; $B_{k+2}=\frac{\beta_{k+2}}{1-\beta_{k+2}}$    (4)

in letterpress– $B_{k+1}=B_{p}=\frac{\beta_{p}}{1-\beta_{p}}$

in offset printing- $B_{k+2}=B_{p}=\frac{\beta_{p}}{1-\beta_{p}}$

To determine the amount of ink supplied to the inking apparatus in the average for one printing cycle and the amount of ink transferred to the print, it is necessary to consider the power conditions of the inking apparatus.

3.1 Determining the power condition of the inking apparatus

Suppose that the power of the ink unit is pulsed by the oscillating transfer roller: T=m$\tau_{0}$ - the power cycle, where - m is the number of printing cycles $\tau_{0}$; γT power pulse duration - the time it takes for the duct cylinder to rotate a certain angle; $\delta_{0}$- pulse amplitude - the thickness of the supplied portion of ink, which is determined by the equation (Figure 1c). 

$\delta_{0}=h_{D}-(1-\beta) h_{0}$       (5)

The amount of ink supplied on average per print cycle is

$g_{0}=\frac{1}{m} \gamma \delta_{0}=\gamma_{0} \delta_{0}=\gamma_{0} h_{D}-(1-\beta) \gamma_{0} h_{0}=$

$=\beta \gamma_{0} h_{0}-(1-\beta) \gamma h_{1}$   (6)

Whence it follows that pulse power with an amplitude $\delta_{0}$, period T and duty cycle γ is equivalent to pulse power with a period $\tau_{0}$ and duty cycle $\gamma_{0}=\gamma /m$ or continuous power supply with a thickness of the supply layer $\gamma_{0} \delta_{0}$ (Figure 1d).

Assuming that the amount of ink entering node i is equal to the amount of ink being drawn from the node, we can write (Figure 1b).

$g_{i}=\beta_{i-1} h_{i-1}-\left(1-\beta_{i}\right) h_{i}=\beta_{i} h_{i}-\left(1-\beta_{i+i}\right) h_{i+1}$      (7)

In stationary mode, the amount of ink g_i passing through each contact node for one printing cycle is constant and equal to the amount of ink supplied to the ink supply g₀ unit on average for one printing cycle and the amount of ink transferred to the print:

$g_{0}=\delta_{0} \gamma_{0}=\ldots=\beta_{i-1} h_{i-1}-\left(1-\beta_{i}\right) h_{i}=$

$=\beta_{i} h_{i}-\left(1-\beta_{i+i}\right) h_{i+i}=\ldots=\sigma h$        (8)

where, h is the thickness of the ink layer on the print.

3.2 The study of modeling the inking apparatus and the method of calculating the distribution of layers of ink the inking apparatus

Based on Eq. (8) taking into account the splitting coefficients $B_{i}$ build a model of the distribution of ink layers:

$\frac{\beta_{i} h_{i}}{h}=\frac{\left[1-\left(1-\beta_{i}\right)\right] h_{i}}{h}=$

$=\sigma\left[1+\frac{1}{B_{i+1}}\left(1+\frac{1}{B_{i+2}}\left(1+\ldots+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right) \cdots\right)\right)\right)\right]$ (9)

At $\beta_{i}$=0,5 (i=1,2,…k) formula (9) gets a simple form

$\frac{\beta_{i} h_{i}}{h}=\sigma(k-i)+\frac{c_{0}}{2}(2-\sigma)$    (10)

where, $C_{0}=2 / \beta_{p}$, for letterpress machines ($\beta_{k+1}=\beta_{p}$; $1+\frac{1}{\beta_{k+2}} \equiv 1$)

$C_{0}=2 \frac{2+\beta_{p}}{1-\beta_{p}}$-for offset printing machines ($\beta_{k+1}$=0,5; $\beta_{k+2}=\beta_{p}$)

Ink distribution in inking devices under conditions

$h_{i}=\beta_{i} h_{i}+\left(1-\beta_{i}\right) h_{i} ; \left(1-\beta_{i}\right) h_{i}=\frac{1}{B_{i}} \beta_{i} h$     (11)

$h_{i}^{\alpha v e}=\frac{1}{2 \pi}\left[\alpha_{i} \beta_{i-1} h_{i-1}+\left(1-\alpha_{i}\right)\left(1-\beta_{i}\right) h_{i}\right]$    (12)

$\frac{\beta_{i} h_{i}}{g_{0}}=1+\frac{1}{B_{i+1}}\left(1+\frac{1}{B_{i+2}}\left(1+\ldots+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right) \cdots\right)\right)\right)$   (13)

as well as

$B_{0}=\ldots=B_{i}=\ldots=B_{k-1}=\frac{\beta}{1-\beta}=1; \quad B_{k}=\frac{\sigma}{2-\sigma};$

$B_{k+1}=\frac{\beta_{k+1}}{1-\beta_{k+1}}$;$B_{k+2}=\frac{\beta_{k+2}}{1-\beta_{k+2}}$      (14)

where, $\alpha_{i}$- the displacement coefficient, which characterizes part of the ink path along the periphery of the roller between the contact nodes when the flow is moving in the forward direction; define in the following form:

For letterpress machines:

$\frac{\beta_{k+1} h_{k+1}}{g_{0}}=1$

$\frac{\sigma \beta h_{k}}{g_{0}}=1+\frac{1}{B_{k+1}}$

$\frac{\beta h_{k-1}}{g_{0}}=1+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\right)$

$\frac{\beta h_{i}}{g_{0}}=k-i+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\right)$     (15)

$\frac{\beta \gamma h_{1}}{g_{0}}=k-i+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\right)$

$\frac{\beta \gamma h_{0}}{g_{0}}=k+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\right)$

$\frac{\gamma h_{d}}{g_{0}}=k+1+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\right)$

Also for offset printing machines:

$\frac{\beta_{k+2} h_{k+2}}{g_{0}}=1$

$\frac{\beta_{k+1} h_{k+1}}{g_{0}}=1+\frac{1}{B_{k+2}}$

$\frac{\sigma \beta h_{k}}{g_{0}}=1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right)$

$\frac{\beta h_{k-1}}{g_{0}}=1+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right)\right)$      (16)

$\frac{\beta h_{i}}{g_{0}}=k-i+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right)\right)$

$\frac{\beta \gamma h_{0}}{g_{0}}=k+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right)\right)$

$\frac{\gamma h_{D}}{g_{0}}=k+1+\frac{1}{B_{k}}\left(1+\frac{1}{B_{k+1}}\left(1+\frac{1}{B_{k+2}}\right)\right)$

Based on the data obtained, we construct an ink flow distribution diagram for the baseline (Figure 2). With $\sigma$=0 on all rollers $i \leq k$ a constant layer is established C₀; at i>k  the layer is equal to zero (there is no transfer of ink to the form, since $\sigma$=0). With $\sigma$=1, the distribution diagram is a descending step line with step I. At $\sigma$<1, the step decreases proportionally to the value s. When passing through the node i=k, the diagram undergoes a jump, which is greater, the smaller $\sigma$. Regardless of i₀=k-C₀/2, a constant layer C₀, is always set on the roller C₀, i.e. all distribution curves merge here. The layer C₀ becomes larger in the presence of an offset cylinder and with a decrease in the coefficient of transition of ink to paper $\beta_{p}$.

2.jpg

Figure 2. The distribution of layers of ink in the inking apparatus: a- motion scheme; b - distribution diagram in the direction of direct flow

Using the formula (9), the regularity of the ink distribution over the maximum layers in the direction of the direct flow, and the minimum layers in the direction of the reverse flow $\left(1-\beta_{i}\right) h_{i / h}$ are established. Also, for each roller, the middle layer characterizing the amount of ink accumulated on the roller can be calculated. For simplicity of calculation, it is assumed that the path sections of the forward and reverse flows are equal to each other:

$\frac{\widetilde{h}_{l}}{h}=\frac{\beta_{i-1} h_{i-1}+\left(1-\beta_{i}\right) h_{i}}{2 h}$      (17)

Imagine the distribution of the middle layer $\tilde{h}_{1}$/h along the sections of the ink path l_i/l₀ (l₀- the length of the print). The total amount of ink accumulated in the inking apparatus is determined by the amount characterizing the area under the distribution curve:

$N_{0}=\sum_{i=1}^{k+2} \frac{\tilde{h}_{l}}{\mathrm{h}} \frac{l_{i}}{l_{0}}=C_{0} \bar{d} k+\frac{1}{2}\left(2 i_{0}-k\right) \sigma \bar{d} k+\left(2+\frac{3}{\beta_{p}}\right) \sigma$  (18)

The first term of the sum characterizes the capacity of the rollers under the condition of uniform distribution ($\bar{d}$ is the ratio of the average diameter of the roller to the diameter of the plate cylinder); the second term takes into account the uneven distribution; the third is related to the capacity of the plate and offset cylinders.

The proposed method extended to inking devices with branched streams of ink (as shown in Figure 3).

3.jpg

Figure 3. The distribution of ink in the inking apparatus of complex structure: a- motion scheme; b- distribution diagram; DL baseline AL I, AL II - additional lines

For this purpose, a scheme for branching ink flows passing through contact nodes is constructed (the sum of the components is equal to the total flow). A baseline is highlighted through which the ink enters the plate cylinder in the shortest possible way. For the baseline, the ink is distributed taking into account the attenuation of the ink flow due to branches. At the nodes where the ink flow branches, there are supply layers that serve as the beginning of additional lines. All additional lines end at node k+1. The distribution of layers in additional lines gives the necessary number of equations for determining the components of the total flow.

For non-stationary mode, the change in the volume of ink in the stream is characterized by the equation of the single-capacitance model (the time is counted in printing cycles n).

$\frac{d V}{d n}=q_{t}-q_{f}$    (19)

where, q_t=h_t the supply of ink to the site I x I, q_f=h - is the total consumption of ink from the site I x I taking into account the transfer of ink to the print elements of the print q=$\sigma$h and returning the remaining amount of ink q'=(1-$\sigma$)h to the ink tank. In a similar equation by Mill [7], the return of ink q' was not taken into account. Given the amount of ink accumulated in the stream V=N₀h . Where N₀-  is the constant of the ink unit, we obtain the equation of the non-stationary mode.

$N_{0} \frac{d h}{d n}+h=h_{t}\left(n-n_{0}\right)$   (20)

where, n₀- is the transport delay; the expression for the transient response takes the form.

$h(n)=h+\left(h_{t}+h_{0}\right)\left(1-e^{-\frac{n-n_{0}}{N_{0}}}\right)$        (21)

The constant N₀ is determined taking into account the actual distribution of inking the stream in accordance with the formula (18).

The dynamic characteristic (20) of the single-capacitance model allows one to determine the unevenness of the ink layer on the form in the case of pulsed power supply of the inking apparatus h_t(t). Considering the inking apparatus as an open pulse system [15], using the discrete Laplace transform to solve Eq. (20), we find the amplitude of the pulsations with respect to the average layer thickness

$\Delta=\frac{h_{\max }-h_{\min }}{2 h_{\text {ave}}}=\frac{1-e^{-\frac{1-\gamma_{0}}{N_{0}}}}{1+e^{-\frac{1-\gamma_{0}}{N_{0}}}}$     (22)

Given an acceptable value $\Delta_{0}$, you can determine the required value of the constant N₀ of the inking apparatus.