2. PERSPECTIVAS SOBRE LA IED INTENSIVA EN I+D
2.7. Tipos de IED intensiva en I+D
The analysis for the pump (component 1) and pressure regulating valve (component 2) operat-ing with a mathematically defined pump flow variation and no load flow was presented in Chapter 11. The figure shows the addition of a fluid power cylinder and an appropriate control valve. The cylinder is used to raise the load mass mL. The parameters needed to expand the model are shown in Flow area Av through the control valve, shown as component number 3, was input as a time dependent function. This area was held at 0 from time 0 to 0.035 s. The area was then increased linearly from 0 to the maximum value shown in Table 12.1 in the time interval from 0.035 to 0.085 s. After time 0.085 s, the maximum value was maintained.
Oil flow Qpfrom the pump was established as a time dependent function in Section 11.3. This same function was applied to the model established for Figure 12.1. In the analysis considered in Section 11.3, all of the pump oil flow passed through the regulating valve and returned to the oil tank.
When a load is added to a pressure-regulating valve, all of the pump oil flow will flow to the work circuit. Part of the flow will begin to pass through the regulating valve and back to the oil tank when the system load is large enough to develop the regulated pressure. The value used for the load mass mL
about 70% of the regulated pressure. Therefore, all of the pump flow was directed to the cylinder.
The equations that follow are then necessary to complete the model for the entire system. The equation of motion for the cylinder piston and load Substituting the parameters for the example inChapter 11yields:
Consider a complete circuit shown in Figure 12.1.
can be established from the free body diagram shown in Figure 12.2with Table 12.1
, as shown in Table 11.1, was selected to develop a static pressure of
2 1
3 4
Figure 12.1
: Pressure-regulating valve with control valve and cylinder load.π µ
Figure 12.2
: Forces on cylinder piston.Chapter 12 295
Table 12.1
: Characteristics for control valve and hydraulic cylinderCharacteristic Size Units
Load mass, mL 2500 kg
Acceleration of gravity, g 9.81 m/s2
Piston diameter, dp 0.048 m
Coefficient of static friction, µst
0.05
Seal width, `sl 0.0025 m
Valve flow area, Av 3.0E−6 m2
Initial volume, Vo 5.0E−5 m3
Flow coefficient, Cd 0.6
Fluid density, ρ 832 kg/m3
Cylinder pressure, p N/m2
Load flow, QL m3/s
Pump flow, Qp m3/s
Piston motion, y m
Piston velocity, dydt or ˙y m/s
Piston acceleration, ddt2y2 or ¨y m/s2 Pressure rise rate, dpdtor ˙p Pa/s
use of Newton’s Second Law:
App − mLg − πdp`slpµst y˙
| ˙y| = mLy¨
To determine load piston velocity ˙y and position y the equation may be arranged as:
¨ y =
App − mLg − πdp`slpµst
˙ y
| ˙y|
/mL
The piston work area Ap, required in the above equations, is equal to:
Ap= d2pπ 4
A value for the damping coefficient is difficult to determine for a cylinder piston. It is well known from laboratory work, however, that a friction force exists at the piston and rod seals. In the above equation the friction has been modeled as a single value that incorporates piston diameter dp, seal width `sl, load pressure p, and seal friction µ. The friction force always opposes motion of the piston. Therefore, the direction of the friction force must be corrected with the ratio of piston velocity over the absolute velocity
˙ y/ | ˙y|.
Values for the cylinder pressure p can be established with the use of the continuity of flow principle. Flow continuity, as it applies to the cylinder, includes flow into and out of the cylinder barrel volumes. Also, the influence of fluid compressibility and the effect of moving parts must be included.
Flow continuity applied to cylinder volume below the piston may be written as:
Qi= −Apy +˙ V βe
˙ p The volume V is a variable and is expressed as:
V = Apy + Vo
Where flow Qi is flow through the control valve and may be expressed as:
Qi= CdAvr 2 ρ
√p − p2
The continuity equation given above may be arranged as:
˙ p = βe
V (Qi− A ˙y)
This equation is integrated to establish a value of the cylinder pressure p at any time during the solution. Working pressure in the system will at all times develop to the necessary level required to accomplish the desired output work. The function of the pressure-regulating valve, discussed in Section 11.3, is to establish the maximum pressure that can be generated in the system.
The model that has been defined provides the basic equations needed to study the operation of the system. Study of the system allows the specific design to be customized for its intended application.
12.2.1 Example: Solution of Model
Every hy-draulic system simulation must address specific goals. Equations and pa-rameters must then be consistent with achieving the desired goals. The A solution for the general mathematical model developed in Section 12.2 follows with use of information listed in Tables 11.1 and 12.1.
Chapter 12 297
Figure 12.3
: Cylinder piston motion, y, vs. time.Figure 12.4
: Cylinder working pressure, p, vs. time.to examine other effects as needed. Other specific information, needed for the solution, is noted in the example.
Information regarding the operation of the pressure-regulating valve is internal valve pressures p1 and p2.
dependent information on the cylinder piston motion y and cylinder working pressure p.
The type of mathematical model developed in this chapter can be solved with a variety of computer software programs. Solution methods may re-quire the use of mathematical modeling equations or they may allow a graphical method. Graphical methods recognize the existence of the ap-propriate equations that are needed to describe the system. In general, very little variation occurs in results with the use of different programs.
As noted in Section 11.3, the control valve for the cylinder begins to open at time 0.035 s. Therefore, the cylinder piston begins to move upward at that time. The load weight is resting on the cylinder piston at time 0;
therefore, the initial pressure in the cylinder is equal to the value of the load weight divided by the piston area. As the piston begins to accelerate, the cylinder pressure increases. As the cylinder piston approaches steady state motion, the pressure approaches the initial static value.