RED DE DRENAJE

Q= Vpω;

T= VpP; (4.21)

where V _p is the volume of fl uid displaced per unit radian of rotation.

4.2.6 Gyrator Elements

Gyrator elements are not very common in hydraulic circuits except for a few special cases such as centrifugal pumps and reaction turbines.

4.2.7 Bond Graph Models of Hydraulic Systems

For fl ows that are almost incompressible, the hydraulic circuits can be treated in a manner similar to the electric circuit if volume fl ow is con-served. The bond graph representation may be achieved algorithmically in a way quite similar to mechanical and electrical systems. The steps to be followed are

1. For each distinct pressure, establish a 0-junction.

2. Insert the component models between appropriate 0-junction pairs using 1-junctions; add pressure and volume fl ow sources.

3. Assign power directions.

4. Defi ne all pressures relative to reference (usually atmospheric) pressure, and eliminate the reference 0-junction (atmospheric pressure) and its bonds.

5. Simplify the bond graph (using the standard rules of simplifi cation that were previously discussed).

A

FIGURE 4.6

Schematic of a hydraulic piston.

Examples 4.1–4.5 discuss bond graph based models of hydraulic systems.

EXAMPLE 4.1

This example shows a simple system where fl uid is traveling through a pipe and fi lling a tank. The fl uid input is treated as a constant effort source. The inertia of the fl uid in the pipe, the viscous losses, and the capacitance of the tank all play a role. Points a, b, and c show locations with different pressures.

We start drawing the bond graph by assigning 0 junction to represent each of these three points. The other elements are connected to 1 junctions, which are connected to two adjacent 0 junctions (as per the algorithm). The bond graph representation can, therefore, be written fi rst as shown in Figure 4.7, and the simplifi ed bond graph, obtained by removing the 0 junction representing the atmospheric pressure (point c), is shown in Figure 4.8. The I and R elements are connected using the same 1 junction since the fl ow rate through the tube between points a and b is the same. The inertia of the fl uid in this region expe-riences the resistance from the wall.

FIGURE 4.7

Schematic for Example 4.1 and its initial bond graph.

ρgA_t

EXAMPLE 4.2

This example shows fl uid fl owing through a pipe that is used to store fl uid in two different tanks (Figure 4.9). The pressure and fl ow rate at the entry point FIGURE 4.8

Final simplifi ed bond graph for Example 4.1.

Se

Schematic for Example 4.2 and its initial and fi nal bond graph representation.

R

and exit are shown in the picture. The schematic is followed by the initial model that was drawn up using the list of rules listed before and then the fi nal simplifi ed model. The inertia elements represent the inertia of the fl uid in the three pipe sections. These can be eventually ignored in many situations, e.g. if the pipes are short. The R elements represent the pipe resistance for the three sections, and the C elements are used to capture the energy stored in the tanks.

The source of effort represents the pressure at the inlet.

EXAMPLE 4.3

Figure 4.10 shows a source of steady fl ow that is supplying two tanks. There is a pipe that connects the two tanks, and there is a leaky drain through which the fl uid is fl owing out of the second tank. The bond graph for the model is shown right below. The I and the R1 element represents the inertia of the fl uid in the tube and tube resistance respectively. The resistance for the drain is represented by the R2 element in the model. The two Cs represent the capacitances of the two tanks, and the source of fl ow represents the fl uid input as a fl ow source.

EXAMPLE 4.4

Figure 4.11 shows a schematic for a needle and a plunger that is used to push the fl uid through the needle at a constant rate. The many different pressure points in the system are shown. The plunger itself works as a transformer transforming

I I2

R R1

R R2 C

C1 C

C2

1 1 Junction 2

0 0 Junction 2 Sf

Sf1

0 0 Junction 1

Flow out of a plug drain

FIGURE 4.10

Schematic for Example 4.3 and its bond graph representation.

force and velocity of motion to pressure and volume fl ow rate. The different elements in the model are the wall resistance for the section 1–2, the inertia of the section 1–2, the resistance and inertia of the section 3–4, and the Bernoulli resistance for dimension change in region 2–3. The fi nal simplifi ed bond graph is shown in Figure 4.12.

EXAMPLE 4.5

Figure 4.13 shows a generic hydraulic system that has most of the basic components in the system. The bond graph representation that accompanies the system is a generic bond graph showing how different aspects of the system can be accounted for in a bond graph representation. It accounts for two types of resistances, the wall resistance for the fl ow-through tubes and Bernoulli

1 I

PlungerTF

ForceSe 0

Atmospheric pressureSe

0 1

0 1

I

Resistance 3 4R Bernoulli resistanceR

Resistance 1 2R 0

1 2 3 4

Inertia 1 2 Inertia 3 4

FIGURE 4.11

Schematic for Example 4.4 and its initial bond graph.

R Resistance 1 2

R Bernoulli resistance

Resistance 3 4R

Atmospheric pressureSe 0

Se TF

Plunger Force

I

1 Inertia 1 2

I Inertia 3 4

FIGURE 4.12

Simplifi ed bond graph of Example 4.4.

resistance due to change in fl ow path cross-section, use of valves, bends, and so on in the line. The inertias of different fl uid sections are also accounted for.

The storage tanks are treated as capacitive elements, and the effect of gravity due to height differences is taken into account through the effort sources. This generic bond graph representation will be useful in developing bond graph representation of specifi c hydraulic circuits because it shows how all the com-mon features can be accounted for.

Several other examples of components in hydraulic systems will be discussed in the context of our discussion on hydraulic actuators in a later chapter.