A sixth dimension mathematical model for a short printing system of parallel structure is designed, which makes it possible to determine different parameters of optical density for the corresponding test scales. Depending on the tone transmission interval and the optical density of the print, a mathematical model to calculate the thickness of the ink on the print was created. The simulation results are presented, the optical density parameters for different test scales are determined, their deviation from the linear tone transmission is specified. Studies have shown the nature of the effect of the ink thickness of the print surface on the optical density, which should be taken into consideration at tone transfer synthesis.
Reproduction of halftone images by printing means in offset printing is carried out by modulation
of ink flows by a raster printing form which is provided by change of the relative area of printing
elements. In most cases, conducting analysis and synthesizing raster tone transfer, the ink thickness
on raster elements is considered to be constant, that is it does not vary with the tone transfer interval,
and the synthesis itself is the synthesis of raster elements areas [
Short ink printing systems do not have mechanisms for zone adjustment of the ink supply, so they
do not provide a uniform thickness of ink on the imprint surface, which limits their use for printing
books and magazines [
Solving the problem of high-quality synthesis and adjusting the transfer plan requires measuring the ink thickness on the imprint surface, which is inaccurate and complicated. Since, to ensure highquality synthesis and correction of tone transfer, the ink thickness should be measured on the print surface, which is a separate task and has certain difficulties, that is why the task to determine optical density of the print depending on the amount of ink on the tone transfer range is relevant for this type of printing systems.
Ink machines with an anilox ink feeder are a new class of devices. Their analysis differs
significantly from traditional objects and systems, because the processes within them are due to the
circulation of direct and reverse ink flows, as well as the modulation of the raster printing plate.
Conducted studies have shown that the ink thickness on the raster print depends on the interval of
tone transfer and is from 20-30%, and in some cases even more, which affects the quality of book and
magazine products and does not meet the standards of regulatory requirements to them [
Provided that a mathematical model, describing a short printing system with an anilox ink input device, is known and describes such dependences as filling the raster elements with ink on the tone transfer interval, which in its turn is reproduced as an image on the plate having the form of a linear scale. Thus, in relative units, let us show a two-parameter description of the tone transfer on the surface of the imprint scale as following: = , if 0 ≤ ≤ 1, ≤ ≤ , where - is an ink thickness for a set tone transfer interval on the surface of an imprint scale, - maximum and minimum parameters of the ink thickness, P is the relative area of the raster scale that corresponds to the level of filling the scale with ink on the tone transfer interval.
Depending on the relative area of filling of the printing element, let’s determine it on the basis of
Yul-Nicholas formula, taking into account the demodulation line (derasterization) [
= − ∙ 10− пл⁄ + (1 − ) ∙ 10− п⁄ , where п Dп - the optical density of the paper, п is Yul-Nicholas index which depends on the linearity of the ⁄ - is the optical density of the full tone area obtained on the print from a short typewriter, raster and can be within the limits [1,3 ≤ ≤ 3,0]. To increase the accuracy of the calculation in production conditions, the specific value of this indicator can be expressed by expression (2), provided that the relative amount of paint is within the limits [0 ≤ ≤ 1].
If in expression (1) and (2) the degree of coverage is linearly changed within the given limits [0.1], it becomes possible to calculate as well as design parameters of an imprint optical density depending on tone transfer range at ink thickness change.
To solve the set problem and research, let’s consider a short printing system of parallel structure of the 6th dimension with anilox ink supply device, the scheme is shown in Figure 1.
In the chamber K, the ink under pressure fills the small cells of the anilox (raster) cylinder A and on the sixth roller delivers a meaasured amount of ink, which is sequentially rolled out and rolled on by the third and the fourth rollers on the printing plate. The ink stream modulated by the raster printing form Ф is transmitted to the offset cylinder O and further to the substrate material. The part of the ink, which was not absorbed by the nonprinting elements, creates untreated ink flows on the rolling up rollers, which causes the circulation of reverse streams of ink that interact with the lines and partially returns back to the ink chamber. (1) (2) = 1 1
, where – is the average value of the thickness of the ink flow at the points of contact of the painted rollers, form and offset cylinders, 0
- т is the thickness of the ink flow at the model input, - is the amplitude value of the ink thickness at the system output (on the imprint), 0- is the thickness of the flow that returns back to the ink chamber, αі, і - is the transmission coefficients of forward and modulated and untreated flows, 1- is transmission of the model output. reverse flows of ink at the exit from the points of contact, 21, 2, 3, 4- are transmissions of 21 = α2 ; 2 = α3 ; 3 = 1 − α3 ; 4 = 1 − α4 ; 1 = 0 ≤ ≤ 1 where – the ink transfer coefficient on the printed material from the offset cylinder.
If to change linearly the coefficient in expressions (3) and (4) in the limits [0 ≤ ≤ 1] for a given constant value of the ink flow thickness at the model input, thus, it is possible to determine the dependence of ink thickness at the output of the ink-printing system. To simplify the task, the method of simulation has been applied, for which, at first, according to the scheme of Fig. 1 and the system of equations (3) a graph of ink flows of the printing system, presented in Figure 2 has been created.
The vertices of the graph marked correspond to the average value of the thickness of the ink
streams at the points of contact of the forming cylinder with the offset and rolling group rollers. The
arcs of the graph marked as α , correspond to the direct and reverse ink flows and depend on the
transfer coefficient at the output of the contact points. Arrows on the arcs indicate the direction of
flows. According to the graph, let’s determine the amplitude of the ink flow thickness at the input of
the model by Maison formula [
∆6 where ∆6 is the determinant of the graph which is cumbersome and therefore not given.
Using the nonlinear dependence (5) we determine the ink thickness on the surface of the imprint scale and depending on the parameters of the printing system by expression (1) we determine a twoparameter tone transfer - the amount of ink transferred on the surface of the imprint scale. 0, = [α6α5α3α2α1 1(1 + α4 + 4) + α6α5 51α4 21α2α1 1] (6) ∆6
Since we determine the optical tone transfer density for a short printing system by expression (2),
taking into account the amount of ink on the surface of this scale. Using object-oriented programming
implemented in the Matlab package: Simulink [
The summation blocks correspond to the graph nodes at the inputs of which direct and reverse ink flows are fed. Gain blocks correspond to the graph arcs, values of the transfer coefficients of the direct and reverse ink flows are set in their dialog boxes. The variable transfer coefficients of the modulated and unmanaged flows are implemented by the expressions (4) and are located in the Subsystem blocks. The thickness of the ink flow at the outlet is set by the Constant unit. At the output of the model, the amplitude value of the thickness on the scale of the imprint is obtained. In the dialog box of Fcn mathematical functions block in accordance with expression (2) there is a program for calculating the optical density. At the bottom of the circuit there is the Ramp unit which generates a linear scale signal within [0 ≤ ≤ 1] limits, which is fed to the Subsystem inputs required to calculate the transfer according to the expression (4). Scope and Display blocks are used to visualize the results of simulation modeling.
The purpose of simulation was to calculate and plot optical density graphs for scales of different types and analyze their properties. The model was adjusted to the nominal parameters of the system (α = = 0,5 , 1 = 0,7 , = 0,8). In the Constant unit, the ink flow thickness at the input of the model 0 = 6мкм, are shown in Figure 4.
At the beginning of the interval, the thickness of ink layer is 1,334 μm, it gradually decreases and at the end of the interval it is 1,143 μm. Therefore, the dependence of the ink thickness on the tone transfer interval is nonlinear. The average value of the ink thickness at the beginning of the range is zero, gradually it increases and at the end of the interval it is equal to 1.334 μm. The convergence of the absolute and average values of the ink thicknesses confirms the reliability of the developed model.
To determine the optical density in the Fcn mathematical functions block the value of the optical density of the die = 3,0 and the value n = 3 have been set. For comparison, the maximum value of the optical density 0 = 2,0 has been set in the Gain unit. The results of simulation modeling of optical density values depending on the tone transfer interval are shown in Figure 5.
For comparison, the figure shows a linear characteristic of the optical density, which is located at the top. Values of the optical density of a short printing system, the maximum value of which at the end of the interval is 2.0. and has the form of a concave curve. It makes it possible to evaluate the tone transfer properties for a short printing system, because the characteristic of the optical density is determined for a linear test raster scale. To do this, let’s determine the deviation of the optical density from the linear one where D0 is a linear characteristic.
= D − D0 (7)
As a result of modeling of the deviation of the optical density of the printing system from the linear, the graph, shown in Figure 6 was obtained.
The deviation of the optical density is a U-shaped curve. The maximum value of the deviation is in
the middle of the tone interval and is equal to -0.4. On the basis of the standardization data, we obtain
that the tolerance for the deviation of the optical density equals E = ± 0.15 [
The constructed mathematical model takes into account the relative change in the amount of ink on the surface of a linear raster scale for a short printing system of parallel structure of the sixth dimension for a two-parameter description of tone reproduction, which expands its capabilities. To increase the accuracy of determining visual optical density of the image obtained on a short printing system, Yul-Nichols demodulation formula was modified, which accounts for the relative change in the ink amount.
The structural scheme of the simulator for calculation of ink thickness and optical density of the short printing system which gives the chance to define its properties has been developed. The results of simulation modeling in the form of graphs of dependence of ink thickness and characteristics of the optical density on the tone transfer interval have been presented and its properties have been determined.
The necessity to adjust the tone transfer was defined, because the value of the optical density of a short printing system is a nonlinear concave curve, the maximum deviation of this characteristic from the linear one is - 0.40, resulting in a significantly brighter image, thus, tone transfer requires adjusting. The results can be used to adjust the images at the stage of preparation for rasterization and production of printing plates.