modeling final

TUNING OF INJECTION MOLDING MACHINE

CHAPTER 6

MODELING

6.1 Development of Non-linear ModelThe injection molding process can be divided up into at least 4

different parts: the plastification phase, the injection phase, the packing

phase, and the cooling phase. The goal of this work was to simulate injection

and packing phases of the injection molding cycle. Models of the machine's

response in both injection and packing phases were developed to be used

for controller simulation. Modeling of the injection molding process has been

done in the past, but none of the models dealt with all of the non-linearities

associated with a hydraulically driven injection molding machine. The models

developed here are found using the nonlinear equations governing the

systems, so the final system is a set of non linear state equations which

DEPARTMENT OF MACHANICAL ENGINEERING


include such difficulties as a time delay (e-ts), saturation and valve dead zone.

Because of the difficulty involved in solving general non-linear equations, the

equations were put into block diagram form and then modeled using Matlab’s

Simulink program. Using Simulink, the simulated response of the injection

molding model could be compared with the actual response of the injection

molding machine.

6.1.2 Hydraulic System

For both the packing and the injection phases of the process, much of

the hydraulic system remains the same, the difference between the two is the

valve in the pump being used for control. In the injection phase, velocity is

the primary parameter being controlled, therefore the proportional servo flow

valve is used to control the output from the pump. However, for the packing

phase, the pressure exerted by the ram is the controlled parameter, so the

proportional servo pressure relief valve was used to control the output from

the pump. Since controlling hydraulic systems is the main thrust of this

thesis, the majority of the time spent on developing the IM model was spent

on understanding the hydraulics system. The hydraulic components were

divided into 4 sections which were looked at separately: valve, line pressure,

orifice, and actuator. These components were then analyzed for both the

injection and the packing phase, and models were developed for each and

then put in Matlab’s Simulink blocks for the purpose of simulaton.



6.1.3 Injection Phase

The injection phase was developed to model the injection of the

polymer into open air, or a completely empty mold, where the outside

pressure remains at atmospheric pressure.

6.1.3.1 Proportional Flow ValveIn general, valves of this type can be approximated by using Bernoulli

equation:

Constant

Following form can be approximated:

6.1

Where C is a coefficient which includes the fluid mass density and the

area of the restrictor as such: C=/a2. P is the pressure differential between

the inlet and the outlet of the valve. In the case of the pump flow valve,

P=(Psup - Pline). It is assumed that the supply pressure from the pump, Psup, is

constant,

The final term, Cq is a flow coefficient which accounts for valve specific

losses, and must be found experimentally for each valve design. Here in this

work experimental data from the valve manufacturer was used to find a



piece-wise continuous function which described the response of the valve to

input voltage. The term Cq is taken as Qnom for simulink modeling where Qnom

accounts for valve losses and should be found for each valve design from

manufacturer data.

As per manufacture manual the valve has been designed so that

when it is operated for 3<V<8, the response of the valve is relatively linear,

when operation outside of this range, the values become very non-linear. For

input voltages less than 1.71 volts there is no flow through the valve, causing

a dead zone effect to occur, and for voltages greater than 9.1 volts, the valve

is wide open, and there is no change in the output of the valve. A plot of for

which, is in Figure 6.1.

Fig. 6.1 Proportional flow valve response

The final equation for the flow valve (not including the transmission lines)

becomes:

6.2



6.3.2 Line Dynamics / Transmission DynamicsBetween the pump and the hydraulic cylinder for the ram there are

about 15 feet of hydraulic line of various size and material. There are a

number of connections and other restrictions (such as a filter in the line).

Flow is diverted to both the injection manifold and the clamping manifold

(which is not operating while the polymer is being injected.). It was decided

that the dynamics would be small enough, and die out fast enough that they

could be neglected in this model. To account for the effects of the length of

the lines a time delay, , was added into the equation of the line pressure and

the volume of the lines, Vl was used in the calculation of the total volume, Vt.

The time delay between the pump and the actuator could be found by

sending a step input to the flow valve, and the noting the time between the

step input, and when the pressure began to increase could be considered the

time delay caused by the transmission lines. Due to nonavailability of facility

and as experienced by some researchers. It was assumed that the flow valve

dynamics themselves are fast enough to be neglected.

Finally, the equation for the flow from the pump, including the time

delay due to the transmission lines is:

6.3

In addition to the time delay due to the hydraulic lines, the

approximate volume of the lines was also found inductively by measuring the

length and outside diameter of the lines, an approximation of the I.D. was

made from the measured O.D. For fixtures other than straight pipe, the



approximate total length of the piece was used as the length. Using these

approximated length and diameter values, the estimated volume of the

hydraulic lines was calculated. Table 1 Shows all of these calculations.

6.3.3 line pressureThe pressure in the line is found by summing the flow through a

control volume. The control volume is the hydraulic line and its associated

components discussed in the previous section. It has an estimated volume

V1. The sum of the flows is simply the flow into the hydraulic line from the

pump and out of the line into the actuator. The total sum of the flow is equal

to the change in volume of the control volume, plus the fluid capacitance

times the change in pressure.

6.4



Where is Bulk Modulus

Term specified by the volume of the fluid divide by its bulk modulus, ,

measures the compressibility of the fluid.

It was assumed that the volume of the lines remained constant, so

dVl/dt=0. Thus the equation becomes:

6.5

By dividing by and integrating, the equation for the pressure in the

line is found to be:

6.6

There is a pressure relief valve installed in the pump which is

discussed in section 5.2.2. This valve is modeled in flow model as a

saturation of the pressure in the line. The saturation is designed so that

pressures below zero are equal to zero, and any pressure above the

maximum pressure set by the relief valve remains at the maximum pressure.

Including the pressure relief valve, the final equation used to model the

pressure in the lines is:



6.7

6.3.4 OrificeThe hydraulic fluid flows from the hydraulic lines into the cylinders, but

what is physically happening in that connection is unknown. Between the

hydraulic line and the actuator there is a small connector restriction that

causes a pressure drop between the line pressure and pressure in the

actuator. This unknown pressure drop or restrictor is modeled as an orifice.

The equation for an orifice comes from Bernoulli's equation for steady,

incompressible pipe flow:

6.8

When the energy on both sides of the orifice are equated, the

equation for the orifice becomes:

6.6

Canceling the appropriate terms, the equation is simplified to:



6.10

Using the minor head loss due to a change in area, with flow

substituted for velocity to give:

6.11

Where, K is the loss coefficient, Q is the flow through the orifice, and A

is the area of the orifice. Solving this for Q, the equation for the flow through

the orifice becomes:

6.12

Replacing by Cv equation becomes:

6.13

Where, Cv, the orifice coefficient, was approximated.

6.3.5 ActuatorA schematic of the actuator can be seen in Figure 5.6 located in

section 5.6.1. There are two cylinders, which work together to transfer the



hydraulic pressure into injection ram motion. From the figure, it can be seen

that as volume A is filled, the ram moves forward, injecting polymer into the

mold. When this is reversed and pressure is placed on side B, and side A is

released to the Tank, the ram moves backwards. In addition to this

mechanism for moving the ram, there was also a hydraulic motor (not shown

here) which rotated the auger when the polymer is being plasticised. This

work did not involve the plasticising of the polymer to be injected, the

hydraulic motor will not be analyzed here.

To determine the pressure which was created in the actuator, a

balance of flow in the front cylinder was done. The total flow into the cylinder,

Q2, would have to equal the sum of the change in volume, the change in

pressure and the flow leaking past the seals as such:

or 6.14

Where Aa is the area of the actuator cylinder, is the velocity of the

ram, Where Va is the volume of the actuator cylinder (this term is a function

of the ram position). Also is the bulk modulus and C1 is the leakage

coefficient, which is modeled as constant. Solving this equation for the

derivative of the pressure in the actuator, Pa, gives:



or 6.15

Integrating the entire equation leads to an equation for the pressure in

the actuator:

or 6.16

as is change in volume

6.3.6 Ram PositionThe ram position during the injection phase of the molding cycle is

found using a force balance equation: F=ma. In this case, the sum of the

forces is F=Factuator-Ffriction, which results in:



6.17

Where the friction term Ff incorporates a number of non-linearities

and is:

of which two were identified and modeled in the simulation were:

1. The directional nature of coulomb friction poses a problem because,

depending on the direction of the movement of the ram, the coulomb friction

acting on it will be of a constant amplitude, but opposite sign, causing a

discontinuity at zero velocity. The nonlinear Simulink models accounts for

this by taking the force calculated for the friction and multiplying it by the sign

of the velocity.

2. At very low velocities the force needed to overcome the static

friction is higher than the kinetic frictional force, stick-slip frictional behavior

can cause problems such as chattering or speed hunting. D.,Zheng. [2] Modeled this stick-slip type friction. His model implies that for small velocities

(less than Dv) the system can be assumed to have no movement. The

velocity required for the object to begin moving is called the breakaway

velocity (Dv), and once that is reached, the friction force is reduced, as static

friction is no longer acting. The Simulink model incorporates this idea by

having a function which inputs the current velocity, and the breakaway

velocity, if the velocity is less than Dv, the output velocity is zero, otherwise, it

is the same as the input velocity. The friction function uses the input velocity



to determine if the frictional force should include both the static and kinetic

friction terms, or just the kinetic friction term.

3. Another part of the friction in a fluid system is the viscous friction.

Since the velocities are rather low, and polymers heated to their proper

temperatures are much less viscous than cool polymers, it was decided that

viscous friction should be neglected in the model. The final model for friction

was:

= signum(Vram)*Fc 6.18

Where Vram is zero for values less than Dv.

Putting this into the equation 6.17:

6.19

By integrating this equation (adding an additional pole at zero in the

Laplace domain transfer function) the equation for the ram position was

found to be:

6.20

6.3.7 Combination of All ComponentsPutting these 5 components together resulted in the Matlab Simulink

block diagram shown in Figure 16.



Simulink model



6.1.4 Packing PhaseThe packing phase was developed to model the situation were the

machine is packing against an already solid mold. In this case the major

source of compliance is the compressibility of the hydraulic fluid. The

following alterations were made to the injection phase model for the packing

phase: the pressure relief valve was added, the ram was fixed, and the

orifice coefficient was changed.

6.1.4.1 Pressure Relief ValveThe pressure relief valve is a feedback compensated solenoid

proportional relief valve which can be controlled by sending a 0 to 10 V input

signal to an amplifier card. This amplifier converts the signal into a

pulsewidth modulated signal which is sent to the valve, much like in the

proportional flow valve. For modeling purpose a relationship between voltage

input and pressure is required as the proportional relief valve is operated by

sending an input signal to an amplifier card which modulates the signal and

passes it to the valve for appropriate functioning where there may be some

caliberation error. For which pressure reading of relief valve set at constant

voltages of both 2V and 5V flow rates (corresponding to a flow of 0.00005

l/sec and 0.0006 l/sec respectively). Recorded over the range of 0.844V to

9.91V are taken from manufacturer manual.

he calibration between the input signal to the amplifier card, and the

resulting steady state hydraulic pressure was determined experimentally as

such:

1. The flow valve was set to a constant flow by sending it a constant

voltage from the computer. Tests were run with voltages of both 2 V and 5 V

(corresponding to a flow of 0.00005 l/sec and 0.0006 l/sec respectively).



2. The voltage to the relief valve was set at a constant voltage, the

steady state voltage reading from the pressure transducer was then

recorded. In general the transient response of the pressure looked like an

under-damped 2nd order system, with a time delay.

3. The pressure was then increased in increments which loosely

corresponded to a pressure increase of 100 psi (as determined by the

microprocessor's open loop voltage commands to the amplifier), and steady

state voltage was recorded again.

4. This process was done for both 2V and 5V flow rates, over the

range of 0.844V to 9.91 V, corresponding to a pressure range of 1.07e6 to

1.75e7 Pa. The results of these tests can be seen in Table 2.

Using the results, a plot of the pressure response of the valve from 0-

9.9 V was created. This function was then simplified into two linear

continuous functions which were used as the calibration between voltage

input and steady state pressure in both the modeling and the experimental

controller as well.

The functions used are as follows:

Plots of the experimental results and the linearized functions are found

in Figure 17.



From the tests run, it was clear that the pressure could be controlled more

accurately at low pressures when the flow rate was lower (the lower flow rate

in the tests was around 0.00005 l/sec which corresponded to a 2 V input),

however, the machine is unable to create pressures above 3.4e6 Pa with this

low flow rate. Since the cavity pressure of the packing phase is normally

around 8e6 Pa for this size mold, the machine could not be operated at such



a low flow rate. In order to be able to generate higher pressures, the flow rate

had to be set higher, such as the 0.0006 l/sec corresponding to a 5 V input.

At this higher flow rate, the machine was able to generate pressures as high

as those generated with the flow valve completely open (10 V output), and

still have acceptable control of low pressures.

The transient response of the pressure valve was modeled by a second

order transfer function, the coefficients of the transfer function were

determined empirically using plots of the experimental pressure response.

The damping coefficient and natural frequency chosen from the experimental

results were tweaked using the final model to make the simulated pressure

react in the same way that the experimental pressure did.



4.1.4.2 Ram Position / Orifice CoefficientBy fixing the ram position, the assumption is made that there is no

flow of hydraulic fluid in to the cylinders any longer. The affect of the orifice

will change as the flow goes to zero. To model this, the orifice coefficient was

reduced until it was nearly zero (ie. there was not flow through the orifice)

this gave the results closest to the experimental pressure trace.

4.1.4.3 Combination of All ComponentsMaking the appropriate changes to the injection phase block diagram

resulted in the following packing phase Matlab Simulink block diagram

(Figure 18).

Simulink model


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