Initial modeling

The complex model of the pump produced is presented in the following figure. An expert engineer built this model with the intention of providing fine detail in the sketch at the expense of longer simulation time.

This model contains:

• Moving parts: pistons, inlet and outlet valves with their inertias, and friction phenomena were taken into account.

• Elastic end stops of the valves.

• Elastic contact between piston and camshaft.

• Hydraulic volumes.

• Pressure drops due to restrictions.

• Leakage flows in pistons.

This model contains 35 state variables: pressure in volumes, velocity and displacement of inertias and relative displacement in the piston/camshaft contacts.

Figure 41: A complex model of pump

These state variables had to be correctly initialised, and a simulation for 10 revolutions of the pump was carried out in order to reach an equilibrium state. At 1000 rpm the range needed for the simulation was [0, 0.6] sec, the tolerance was 10-7 and the computation time was 35 min with a PC working at 2.8 GHz. State variables were initialized with the final values obtained at the end of this simulation. Eigenvalue analysis performed at intervals of 0.01 seconds revealed a large range of frequencies. You can see below the highest and lowest (non-zero) frequencies at a time of 0.01 seconds.

Figure 42: Highest and lowest frequencies at 0.01 seconds

Relating this to the concept of a proper model, clearly the highest frequencies are outside the range of interest. A frequency of 5.5e8 Hz is a ultra high frequency (UHF).

As expected the activity indexes also show a great variation. The highest activity is due to the load and the pipe system supplying the load. Three hydraulic chambers next to the pistons are also identified as being critical.

Figure 43: Great variation between the activity indexes

Of equal significance are the low activity items.

Figure 44: Low activity items

The methodology was to make a copy of the system (File

Save as) and progressively simplify it. At each stage strategic graphs were plotted and compared with the originals.

The first candidates for removal were the three SPR000A springs (remember you can click on items in the Activity Index List to identify a component in the sketch). Between them they account for about 9e-6%. Why is their activity index so low? Plotting a few graphs revealed that the mass (i.e.the ball in the valve) moves very little and it is only permitted a maximum movement of 1 mm. This means the spring force is almost constant at the preload value. The simplest solution is to replace the springs by a constant force. This created a marginal improvement. Note that there are no states in the springs so that we do not expect a great gain.

Figure 45: Replace the spring with a constant force

The next candidates for removal were the hydraulic chambers BHC11 instances 4, 5 and 6. These were difficult to remove because they supply a state variable pressure to the orifices BHO11 instances 1, 2 and 3.

Figure 46: Hydraulic chambers

However, these orifices also have a very low activity index. In addition we can see that:

• the pressure at each cylinder input is virtually the same as the tank pressure, and,

• the orifices are big (3 mm diameter) for the flow that they carry and hence the pressure drop across each orifice is very small.

Figure 47: Activity index of the orifices

It is very simple to remove these chambers and orifices in one step connecting the tank directly to the pump inlet.

At this point it is worth introducing four very important points:

• The Activity Index facility suggests components or phenomena within components that perhaps could be removed but it does not tell you how to do it.

• It is usually not possible to simply start at the lowest activity and work upwards. As we will see there are sometimes very low activity elements which we cannot remove. This is usually because the adjacent components cannot be connected.

• Often a threshhold such as 0.1% or even 1% can be used. We try to remove components from the bottom of the list upwards until the sum is just below the threshold. In the current example the sum of the activity index of all the components removed was below the 0.1%.

• Sometimes it is possible to remove groups of components simultaneously.

The methodology was applied again respecting the points above. We do not give instance numbers because these change as components are removed. The main steps are:

1. MAS21 submodels in the outlet valves: we replace the elastic endstops with ideal endstops and we reset the corresponding state variables initial values to zero.

2. The three BHO011 orifice submodels between the outlet valves and the pump pistons are low on the lists. We can easily remove them provided we combine the BHC11 chambers. We must correct the dead volume in the single chamber to be the sum of the dead volume in the two chambers. Figure 48: Remove the BHO11 orifices

This removes three more state variables and the simulation now runs significantly faster. At this point we start a new run and consider the resulting Activity Index values. Figure 49: New Activity Index values

Clearly there are components or elements within components that we would like to remove but it is not always easy. The easiest low activity elements to remove are the three BHO011 submodels. However, to do this it is necessary to do something about the gaps created. Each of these orifices receives a pressure at one port from the single HL000 line submodel and at the other ports from three hydraulic chamber BHC11 submodels. Plotting the pressures (which are state variables) in these chambers, they are virtually identical. Indeed the pressure in HL000 is also virtually the same. Plotting the volume in each hydraulic chamber it varies from 0.3983 to 0.4 cc. Since HL000 has no friction or inertia calculations, it is effectively also a hydraulic chamber. Hence it seemed reasonable to remove orifices but also hydraulic chambers. Taking their total volume to be 1.2 cc, we increase the volume in HL000 to compensate. Figure 50: Remove hydraulic chambers

Figure 51: Reduced model

The model then contained 26 state variables instead of 35. The simulation time for one revolution of the pump, for the reduced model was 30 seconds compared with 150 seconds for the complex model. The gain in performance was important; the removal of less active elements lead to the removal of the highest dynamics of the system which previously slowed down the simulation. As the system was reduced, it was validated against the original model which had been itself validated by experimental measurements.