Las fusiones y adquisiciones transnacionales como modo de

The mathematical model, which has been established for the system, was Input flow Q_p(t) to the system was started at time equal to zero at 0.2E−3 m³/s. At time equal to 0.1 s the flow was increased linearly to a value of 0.2133E−3 m³/s at 2 s.

It should be noted that selection of input values requires some care and judgement. The performance of the actuator, i.e., the extension and rate of extension, were fixed by machine operating requirements. The static pressure required to raise the load was easily calculated. These values were used to estimate the fixed and variable orifices in the flow regulator valve.

First estimates of the preload and the spring rate could also be obtained from static considerations. It had been decided that the flows in the two branches of the circuit should be approximately equal. After obtaining the preliminary estimates, the system was simulated and certain values adjusted until the system could be integrated over the desired time interval. It was found that the simulation was quite sensitive to the spring rate value. The value of k = 2700 N/m was the lowest value that allowed operation without integration failure or the valve closing prematurely. Results are shown in The plotted results show the valve motion x, actuator motion y, pressure and flow at various points in the circuit. It will be observed that the system performs quite well A generic flowchart is presented in Figure 13.10 showing the calculation

Figures 13.11 to 13.14for the 2 s time interval.

solved with parameter values as shown in Table 13.5.

Chapter 13 323

BEGIN SUBROUTINE

CALCULATE ALL RESISTANCE VALUES

CONSOLIDATE THE RESISTANCE VALUES

CALCULATE THE SYSTEM PRESSURE FROM THE CONSOLIDATED RESISTANCE

CALCULATE FLOWS

FORMULATE THE DIFFERENTIAL EQUATION SET

END SUBROUTINE

PASS SYTEM CHARACTERISTICS AND INITIAL VALUES

EVALUATE THE PUMP FLOW DRIVING FUNCTION

START

MAIN PROGRAM

READ IN SYSTEM CHARACTERISTICS

READ IN INITIAL VALUES OF VARIABLES

ESTABLISH THE SIMULATION PERIOD

CALL THE DIFFERENTIAL EQUATION SOLVER

WRITE THE RESULTS

END MAIN PROGRAM

Figure 13.10

: Program flowchart for unsteady flow example.

even at this lowest spring rate. The plot of the load pressure pL vs. time,

p and system pressure ps are increasing.

Variations in the valve spring rate k, and its preload F , provide the (Figure 13.13) is essentially constant even when the pump flow Q

Table 13.5

: Values and appropriate units applied for the example

Characteristic Value Units

Variable load orifice area, Az

0.903E−6 m²

Valve end area, Av 0.129E−3 m²

Valve mass, m_v 0.023 kg

Fixed valve orifice, Ao 6.45E−6 m² Valve flow gradient, w 2.5E−3 m²/m

Valve spring rate, k 2700 N/m

Valve spring preload, F 32 N

Orifice flow coefficient, Cd 0.6

Length of flow slot, `0 4.0E−3 m Oil volume in valve, V_c 33E−6 m³

Oil mass density, ρ 855 kg/m³

Acceleration of gravity, g 9.8 m/s² Oil bulk modulus, βe 1.172E+9 N/m² Cylinder piston diameter, d 48.3E−3 m Piston seal width, `_sl 2.5E−3 m Coefficient of static friction

(seal), µst

0.05

Cylinder load mass, m_L 2140 kg

= 2700 N/m allowed little margin for unexpected changes in actuator per-formance because the valve spool excursion was nearly 85% of the allowed maximum. This opening could be reduced by increasing the valve spring rate increased to k = 5000 N/m. As expected, the valve opens less at the end of the simulation. The actuator extension is slightly greater because the orifice area, Ax, is larger. A benefit of the stiffer spring is that the system pressure is slightly reduced. The pressure in the actuator remains constant and unchanged. Minor disadvantages of using the stiffer spring are that the actuator flow rises slightly towards the end of the simulation and the flow to the upper branch is slightly less. This reduction is a consequence of the lower system pressure.

rate. Figures 13.15 to13.18show the system performance with the spring

Chapter 13 325

! "

Figure 13.11

: Flow division example, valve spool motion, x, vs. time, t.

! " #

Figure 13.12

: Flow division example actuator motion, y, vs. time, t.

!

" #

Figure 13.13

: Flow division example showing system, p_s, valve, p_v, and actuator, p_L, pressures vs. time, t.

! " #

Figure 13.14

: Flow division example showing pump, Q_p, actuator, Q_i, and upper branch, Q_z, flows vs. time, t.

Chapter 13 327

!

Figure 13.15

: Flow division example valve spool motion, x, vs. time, t.

! "

Figure 13.16

: Flow division example actuator motion, y, vs. time, t.

!

" #

Figure 13.17

: Flow division example showing system, p_s, valve, p_v, and actuator, p_L, pressures vs. time, t.

! "

Figure 13.18

: Flow division example showing pump, Q_p, actuator, Q_i, and upper branch, Q_z, flows vs. time, t.

Chapter 13 329

13.7 CONCLUSIONS

The type of mathematical model presented is very useful in sizing param-eters to provide the desired operating characteristics of a flow regulator valve. Many parameter variations could be applied to the model. The major purpose, presented here, is to define the mathematical model and illustrate its usefulness in component design. Many methods are available to solve the equations, a symbolic computer software program was used for the solution presented here [10].

Values for the fluid flow variables Qz, and Qi, and pressure ps at the pump outlet were determined with use of the hydraulic ohm principle.

These values may also be determined with a simultaneous solution of the appropriate equations that contain these parameters. Numerical values de-termined with the simultaneous solution are identical to those calculated with use of the ohm principle.

Because computer solution involves iteration, that is usually small time steps are made towards a final solution, the hydraulic resistance terms can be updated at each iteration step using the current pressure and flow values. It has been found in practice that the small steps generally required to solve the dynamic equations in dx/dt, d ˙x/dt, and dp/dt allow adequate estimation of resistance values without needing intrastep iteration for the resistance terms [4-6].

Numerous combinations of equations are possible. Solution of the equa-tions with the hydraulic ohm method, however, consistently provides the same results as solution by simultaneous methods. Because both methods make use of the same principles from fluid mechanics, consistent answers may be expected.

Values for the flow variables Qz, and Qo can be determined with the simultaneous solution of the four equations that follow:

Q_o= C_dA_or 2

These flow values can then be used to calculate the coefficients in the state variable equations in pressure, displacement, and velocity (Equations 13.30, 13.31, 13.32, 13.33, 13.35, and 13.36).

It may be observed that simultaneous solution of the equations in Qo, Qi, Qz, and Qp is easy in this example because the resistance Rz is in-variant. More complex branches with different components might lead to simultaneous equations in various Q values that would be more difficult to solve. Under such conditions, the hydraulic ohm method may be much easier to apply.

PROBLEMS

13.1 Flow, Q, from the pump divides through the two branches as shown in the figure.

R

R p

Q 2 Q

Q 1

2 1

Write a set of equations, based on conventional fluid flow theory, that can be used to solve for pressure, p, and the flow values, Q₁and Q₂. Neglect line friction loss.

Characteristics of flow division with two fixed orifices

Characteristic Size Units

Upper branch orifice diameter, d₁

0.048 in.

Lower branch orifice diameter, d2

0.040 in.

Orifice flow coefficient (both), Cd

0.6

Oil mass density, ρ 0.78E−4 lb_m· s²/in.⁴

13.2 Use the equations determined in Problem 13.1 to solve for numerical values of pressure, p, and flow values, Q1 and Q2, when pump flow, Q, is equal to 4.00 gpm.

Chapter 13 331 13.3 Use the information given for exercises Problems 13.1 and 13.2 to obtain numerical values for pressure, p, and flow values, Q1 and Q2, with use of the hydraulic ohm method.

13.4 Leakage input flow, Q = Q1+ Q2, passes through the spool valve shown in the figure.

ACTUATOR CONNECTIONS p

LEAKAGE FLOW TO DRAIN Q1

p 1 TO DRAIN

p 2 LEAKAGE FLOWTO DRAIN Q 2 FROM PUMP d ps

The oil flows past the two spool lands in annular passages as, Q₁and Q₂. Write a set of equations, based on conventional fluid flow theory, that can be used to solve for pressure, p_s, and the flow values, Q₁and Q₂.

Characteristics of flow division in a spool valve

Characteristic Size Units

Spool diameter, d_sp 20.0 mm

Land length, `1 93.0 mm

Land length, `₂ 73.0 mm

Spool radial clearance (cen-tered)

0.025 mm

13.5 Use the equations determined in Problem 13.4 to solve for numerical values of pressure, ps, and flow values, Q1 and Q2 when the leakage input flow, Q = Q1+ Q2, is equal to 0.01 L/s.

13.6 Use the information given for exercises Problems 13.4 and 13.5 to obtain numerical values for pressure, p_s, and leakage flow values, Q₁ and Q2, with use of the hydraulic ohm method.

13.7 Model the flow regulator valve discussed in this chapter with any available simulation method and compare the results to the published results.

13.8 A manufacturing machine includes a fluid power system with a motor and cylinder operating in parallel.

Determine the time, t, in seconds that is required before the cylinder piston begins to move for the conditions given in the table. The motor torque, T , increases with time, t, seconds. Determine the pressure, p, that has developed in the system for the time, t, calculated for the previous part of the question. Determine the initial velocity, ˙x, for the cylinder piston for the given pump flow, Q, and with motor speed, n, equal to 500 rpm when the piston begins to move.

Characteristics of a manufacturing machine system

Characteristic Size Units

Cylinder piston area, Ap 1800 mm²

Cylinder rod side area, A_r 1290 mm²

Cylinder load force, F 4500 N

Motor displacement, D_m 72.0 mL/rev

Motor discharge pressure, pr 695 kPa Motor output torque, T 10.0 t^1.4 N · m Pump flow rate 0 < t < 2.1, Q 0.496 t L/s Pump flow rate 2.1 < t, Q 1.0 L/s

REFERENCES

1. Esposito, A., 1969, ”A Simplified Method for Analyzing Circuits by Analogy”, Machine Design, October, pp. 173-177.

Chapter 13 333 2. The Lee Company, 1987, Lee Technical Hydraulic Handbook, The Lee

Company, Westbrook, CN, pp. 367-397.

3. Del Toro, V., 1972, Principles of Electrical Engineering, 2nd ed.

Prentice-Hall, Inc., Englewood Cliffs, NJ.

4. Gassman, M. P., 1978, ”Prediction of Pressure and Flow Values for a Hydraulic System”, Proceedings of the 34th National conference on Fluid Power, NFPA, Philadelphia, PA, pp. 377-382.

5. Gassman, M. P., 1992, ”Fluid Power System Flow Distribution and Components Analysis”, SAE Paper 921686, SAE, Milwaukee, WI.

6. Gassman, M. P., 1993, ”Use of the Hydraulic Ohm to Determine Flow Distribution”, SAE Paper 932489, SAE, Milwaukee, WI.

7. L. Dodge, 1968, ”How to Compute and Combine Fluid Flow Resis-tances in Components, Part 1”, Hydraulics and Pneumatics, Septem-ber, p. 118.

8. L. Dodge, 1968, ”How to Compute and Combine Fluid Flow Resis-tances in Components, Part 2”, Hydraulics and Pneumatics, Novem-ber, p. 98.

9. Martin, H., 1995, The Design of Hydraulic Components and Systems, Ellis Horwood, New York, NY, pp. 204-225.

10. MathSoft, Inc., 2001, ”Mathcad, User’s Guide with Reference Man-ual”, Cambridge, MA.

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