Page 547 - Mechatronics with Experiments

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JWST499-c07
JWST499-Cetinkunt
ELECTROHYDRAULIC MOTION CONTROL SYSTEMS 533
Torque output of the motor (T (t)) is
m
T (t) = ⋅ D (t) ⋅ p (t) (7.417)
m
m
m
m
where p (t) is the pressure differential developed at the motor input–output ports.
m
5. Cylinder motion: speed and flow rate, as well as the force–pressure relationship
(neglecting inertia effects and fluid compressibility)
dy(t)
Q (t) = A ⋅ (7.418)
c
c
dt
F (t) = A ⋅ p (t) − A ⋅ p re (7.419)
c
he
he
re
where y(t) is the cylinder displacement, A is the cylinder cross-sectional area, Q
c c
flow rate into the cylinder, F force generated due to pressure difference on both sides
c
of the cylinder.
6. Leakage flow through a clearance:
Q leak (t) = f(Δp(t)) ≈ K leak ⋅ Δp(t) (7.420)
7. Pressure drop: across pipes and hydraulic hoses obtained from standard tables as
constant pressure drop as a function of the hose parameters, fluid viscosity ( ) and
flow rate (Q)
Δp = f( , Q, l, d, , RF) (7.421)
where l, d, , RF are hose length, hose diameter, bending angle (if hose is not a straight
line), and roughness factor.
Example: Modeling Hydraulic Circuits Consider the hydraulic circuit shown
in Figure 7.92. Part (a) of the figure shows the hydraulic circuit in symbolic form. Part (b)
shows the shape and dimensions of the pipes and fittings which can be used in pressure loss
estimation. Part (c) shows the various operating conditions considered and simulated for
the circuit. In varying complexity, we will analyze this circuit and simulate its operation.
◦
The hydraulic oil is SAE 10W, operating at a nominal temperature of T = 25 C.
The pump, main valve (valve 1), and relief valve (valve 2) are connected to the circuit via
◦
straight pipes, a T-connector, and a right-angle (90 ) elbow connector.
Given hydraulic fluid viscosity, (which is determined by the hydraulic fluid type
and temperature), the geometric parameters (length and diameter, l, d), surface roughness
(roughness factor, RF, of the pipes), and shape of the connectors (straight, T-connector,
right-angle elbow connector), and nominal flow rate; we can estimate the approximate
pressure loss in each section of the pipe,

p loss = f( , Q, l, d, RF, Shape) (7.422)
where for each section of the connectors this formula can be applied, as follows (Fig-
ure 7.93).
Although the pressure losses are not constant, and are a function of the flow rate and
temperature, and hence change as the flow rate and temperature change, it is a reasonable
approximation to use constant values based on nominal flow rate values in the analysis. In a
simulation for very detailed accurate prediction purposes, the dependence of pressure loss in
the lines as a function of flow rate can be modeled and simulated. However, in the analysis
below, we will use the constant pressure drop estimation based on nominal flow rate,
for a given constant nominal operating temperature condition. In a given application, the
hydraulic fluid type and nominal operating temperature are known as constant. The pressure