Dynamic Simulation Modeling of Inking System Based on Elastohydrodynamic Lubrication

Lubrication means adding a substance with small shearing resistance between the interacting surfaces with relative motions to reduce friction and decrease abrasion of materials. The substance added is called a lubricating agent. Ink used in offset printing acts as the printing medium and is transferred to the substrate, but the ink also plays the role of lubricating the rollers [10].

Rollers in the ink system of an offset printing machine are mainly rotated via gears and a fluid lubrication film is formed by hydrodynamic effect generated from relative motions of the contacting rollers. The contact surface is nearly non-heavy load linear contact friction, so it is considered as generalized elastohydrodynamic lubrication. There is not a large difference between the film thickness and the film thickness of corresponding hydrodynamic lubrication [11].

Hard rollers and soft rollers are arranged alternately in the inking system of an offset printing machine. Pressure between rollers belongs to the middle and low loads and the thickness of ink at the contact position of the rollers should be determined in advance [12]. In this paper, the average film thickness model of rollers was established through discretizing all rollers in 3mm×3mm and by using the deduction of the Грубин equation as the basis.

$h_{0}=1.95\left(a \eta_{0} U\right)^{8 / 11} R^{4 / 11}(E / W)^{1 / 11}$(1)

$\alpha$-pressure viscosity coefficient  

$\eta_{0}$-the oil film viscosity environment  

$R$-equivalent radius     

$U$-entrainment velocity

$E$- equivalent elastic modulus 

$W$-load per unit length

$h_{0}$-central oil film thickness ×

The equation contacts the Reynolds lubrication theory and the Hertz contact theory and provides the average ink thickness in the contact area, and the equation is applicable to heavy load lubrication conditions [12] [13].

2.1 Reynolds equation for ink transfer

Paper From a mathematical point of view, the solution of the Reynolds equation is the main content of fluid lubrication for the purpose of obtaining the distribution regularity of pressure on the fluid lubricating film. Under the isothermal condition, there are seven postulated conditions for the deduction of the Reynolds equation. The remaining two postulated conditions are added for the convenience of simulating calculation in this topic, totaling to nine postulated conditions as follows:

1. Consideration is not taken to the effect of such physical forces as gravity and magnetic force;
2. There is no velocity difference on the contact surface for lubrication and the fluid attached; i.e., non-slip;
3. As the unit of lubricating film thickness is measured in micron size, consideration is not taken to the pressure change in the film thickness direction. The fluid viscosity and density are two constants in the film thickness direction;
4. The curvature radius of the lubricated surface is much larger than the oil film thickness to ignore the change in velocity vector caused by surface curvature;
5. The lubricant is Newtonian fluid;
6. The inertia force of fluid acceleration and centrifugal force of liquid film bending are ignored;
7. The fluid motion is laminar flow, instead of vortex flow and turbulent flow;
8. As the contact length of the ink roller is infinitely great relative to the flow velocity of the printing ink and the oil film thickness, the axial flow of printing ink is ignored;
9. The ink transfer is assumed to be in steady state; i.e., not considering the time-dependent effect.

Figure 1 shows the force diagram along the $x$ direction with the fluid microelement obtained from the ink roller clearance The $x$-axis is in the width direction of ink roller contact, the $y$-axis is in the axial direction of ink roller contact and the $z$-axis is in the thickness direction of oil film. The velocities of the particle in the microelement along the coordinate axes of $x, y$ and $z$ are denoted by $u, v$ and $w$. From the force diagram of fluid microelement along the $x$ direction, the equilibrium equation can be formulated as follows:

figure_1.png

Figure 1. Infinitesimal element fluid force

$p d y d z+\left(\tau_{x}+\frac{\partial \tau_{x}}{\partial z} d z\right) d x d y=\left(p+\frac{\partial p}{\partial x}\right) d y d z+\tau_{x} d x d y$(2)

From the postulated condition the Newton law of viscosity is substituted $\tau_{x}=\eta \frac{\partial u}{\partial z}$ and$\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho w)}{\partial z}=0$ into the formula above. The Reynolds equation of ink transfer after simplification under postulated conditions is as follows:  

$\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{\eta} \frac{\partial \rho}{\partial x}\right)=6 \frac{\partial}{\partial x}\left(\rho h U_{1}+\rho h U_{2}\right)$(3) 

2.2 Hertz contact in ink system

The contact relation of two parallel rollers can be simplified into the relation between equivalent cylinders and the plane. It can be known from Hertz pressure that the larger the distance is from the center of contact width, namely, when the x-axis value is smaller than -b or larger than b, the smaller is the pressure, gradually decreasing to 0 [14]. It can be known from the extremum principle that must exist in the bearing area when. Then the equation is integrated and the boundary conditions are substituted into the equation to obtain the integration constant and further to obtain the first order Reynolds equation, namely:

$\frac{d p}{d x}=12 \eta U \frac{h-h^{*}}{h^{3}}$(4) 

$U=\left(U_{1}+U_{2}\right) / 2$- entrainment speed of two opposite rollers;

$h$- gap function;            

$p$-the pressure of the contact area;

$U$-Entrainment velocity;

2.3 Film thickness equation in gaps of rollers

The average ink film thickness in the contact area is calculated according to the deduction thought of the Грубин equation and the following three steps:

1.  Defining the induction pressure to obtain the one-dimension Reynolds equation considering the pressure-viscosity relationship;
2.  Nondimensionalization of the Reynolds equation;
3.  Integrating the equation by using heavy load in the entrance area as the boundary conditions [15].

When ink exists between two rollers, the pressure at the contact half-width position is not increased rapidly while a certain transition curve exists to connect the pressure sections in the Hertz area due to hydrodynamic effects of the ink film. If the entrance pressure is set as$p_{x=-b}=p$, then the induction pressure here will be:

$q=\left(1-e^{-\alpha p^{\prime}}\right) / a$(5)

When the non-dimensional coordinate tends to $+\infty$, both $p$ and $q$ are 0. When the non-dimensional coordinate tends to the contact half-width at the entrance, it can be known from the above definition that $p$ is $p$ and $q$ is $q=\left(1-e^{-\alpha p}\right) /a$.

 Then the relation $h_{0}=(4 / 3) h_{\min }$ between the average film thickness and the minimum film thickness is utilized and the non-dimensional boundary conditions are substituted into the above equation to obtain:

$\overline{q}=48\left(\frac{\overline{W}}{2 \pi}\right)^{1 / 2} \overline{U} \int_{1}^{+\infty} \frac{\left[(\sqrt{3}(\overline{x}-1)+1)^{2}-1\right]\left(1-\varepsilon_{\delta}\right)}{\left\{\overline{h}_{0}+\frac{4 \overline{W}}{3 \pi}\left[(\sqrt{3}(\overline{x}-1)+1)^{2}-1\right]\left(1-\varepsilon_{\delta}\right)\right\}}$(6)

$\varphi=\left\{\begin{array}{ll}{\frac{1}{(1-\lambda)^{2}}\left[2\left(1+\frac{2}{\lambda}\right)-\frac{4-\lambda}{\sqrt{1-\lambda}} \ln \frac{(1+\sqrt{1-\lambda})^{2}}{\lambda}\right]} & {\lambda<1} \\ {\frac{1}{(1-\lambda)^{2}}\left[2\left(1+\frac{2}{\lambda}\right)-\frac{4-\lambda}{\sqrt{\lambda-1}}\left(\pi-2 \operatorname{arctg} \frac{1}{\sqrt{\lambda-1}}\right)\right]} & {\lambda>1}\end{array}\right.$(7)

where:

$\varphi=0.26\left(\frac{\overline{W}^{-1.5}}{\overline{G} \overline{U}}\right)\left(1-e^{-0.08 \overline{G} \overline{W}^{-0.5}}\right)$, $\lambda=\pi\left(\frac{\overline{h_{0}}}{\overline{W}}\right)$

The solutions of equation (7) are the analytic solutions of the Reynolds equation obtained here. The values are substituted to obtain the minimum ink film thickness in the gaps of the rollers [16]. The solutions are calculated by using the secant method and the procedures are shown in Fig. (2).

figure_2.png

Figure 2. Traces Calculation flowchart

In this paper, the relations model is established between roller parameters such as pressure and ink thickness in roller gaps by Hertz contact theory. The equation was solved through a program to obtain the minimum ink film thickness in roller’s gaps under given parameters and it studies the influences of different roller parameters, pressure on average film thickness in the roller’s gaps. The analysis yielded the following results. The larger the equivalent radius of rollers is, the larger is the ink film thickness in roller’s gap, and the larger the contact pressure of rollers is, the smaller is ink film thickness is in roller’s gap.

Finally, simulation analysis was carried out on the inking system of the Roland 700 type offset printing machine. The results showed that the larger the printing speed is, the larger is the inking stabilization time.