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Modeling pres hyd
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MODULE-1: System Modelling
Part-a: Control Terminology and Transfer Functions;
System Block diagram - simplifying/reduction approach;
Part-b: Modelling simple systems - deriving Transfer Functions - Mechanical
- Electrical
- Electromechanical
- Level control
- Thermal
- Pressure difference
- Hydraulic
- Pneumatic
- Others
Lecture 3c
Simple Pressure difference System
Gas flow Resistance:
Gas flow Capacitance:
q = Gas flow rate = dm/dt
m = Mass of the gas in the vessel (gas-stored) C= capacitance of the gas in the vessel V= Volume of the vessel
---(1)
---(2)
At steady state:
Applying Laplace Transform (assuming zero initial conditions):
---(3)
Substituting in (3)
Large multiplication of force
Hydraulic systems are preferred for continuous control of motion
with significant external loads.
Pressure is transmitted undiminished in an enclosed static fluid!
Pstatic fluid = ρ.g.h
ρ = m/V Fluid density
g = acceleration of gravity
h = depth of fluid
PASCAL’s Principle
Standard symbols used & SI units
Gases:
Liquids:
oi qqdt
dv
oq
hR
h
vC
Constraints R and C in the output flow
Hydraulic Systems
Study of Incompressible fluids such as oil and water
Fluid’s density remains constant despite changes in fluid pressure.
Conservation of mass is equivalent to conservation of volume
Main System Variables:
Pressure,
Mass (m) and
Mass flow rate (qm)
The volume flow rate:
(ρ, the density = constant )
qqm . ---(1)
If q1 and q2 are the total volume inflow and outflow rates in the container, such that
2
1
out
in
21 qqqq outin
V
)( 21
For a container of volume V holding the incompressible fluid of mass m,
qin and qout are the inflow and outflow rates
The conservation of the mass is given by:
---(2) Vqqm outin .)(
---(3)
Simple Hydraulic System
At lower piston, the oil flow-rate is
given by :
Pilot Valve
Power Cylinder
e will cause a displacement in x: e (and x) will move to the right:
Ports I and II will open as shown;
dt
dyAq ..
dtA
qdy
xKq 1
K1 is a constant
At Port-II, the oil flow rate would be:
Find the Transfer function Y/X:
is oil density
A is the piston cross-section area Motion of the Piston:
---(4) ---(5)
Displacement x as a result of adding two small displacements
For the Flapper movement:
yba
ae
ba
bx
Feedback link:
b
yx
ba
ye
xKq 1
dt
dyAq ..
dt
dyAxK ..1
)()(1 sAsYsXK
Hence, equating the two, we get:
Transfer Function: sA
K
sX
sY 1
)(
)(
sA
K1 Y(s) X(s)
---(4)
---(5)
Applying Laplace Transform (assuming zero initial conditions):
---(6)
Proportional Controller
Transfer Function:
ba
a
s
Ks
K
ba
b
sE
sY
1)(
)(
Feedback link: b
yx
ba
ye
Under normal operating conditions, we can
write (8) as |Ka/[s(a+b)]| >>1 pKa
b
sE
sY
)(
)(
)(.)(1.)()(
baasK
sK
ba
bsEsY
---(7)
---(8)
---(9)
Working medium compressible fluid (air/gas);
Slower response than that of hydraulic systems;
Forces are greater than those available from electrical
drive systems;
Quantities used:
— Mass (m),
— Volume (V),
— Pressure (P)
— Temp. (T)
The main variables of pneumatic system are:
Mass flow rate q (a through variable)
pressure P
Pneumatic Systems
(These are analogues of current and voltage in electrical networks)
Ideal Gas Law:
Linearizing
R is similar to turbulent flow resistance;
∆ p is the pressure drop across the component
The compressible Flow/pneumatic Resistance is modeled as,
p is the absolute pressure of the gas with volume V,
m is the mass,
T its absolute temperature, and
Rg the gas constant that depends on the type of gas
TmRpV g
pmqR 2
---(1)
---(2)
---(3)
Pneumatic Proportional Controller
For the Flapper movement:
yba
ae
ba
bx
Bellows act like springs:
zKp
zKp
xKp
c
b
b
3
2
1
xK
KKp
K
Kb
2
31
2
3
ykpA sc .
A is the effective area of bellows
and
kS is the equiv. spring constant
Diaphragm Valve ---(4)
---(5)
---(6)
Transfer Function could be obtained by dividing eqn. (4) by (5) and
applying Laplace transform
s
c
k
A
ba
aK
Kba
b
sE
sP
..1
.
)(
)(
xK
KKpc
2
31
yb
ax
b
bae
ykpA sc .
From eqn. (4)
From eqn. (6)
From eqn. (5)
pc K
sE
sP
)(
)(
Exercise: Aircraft Elevator Control
Automobile systems
-such as hydraulic actuators as the
brake on an automobile
Elevators systems
-such as hydraulic jacks and lifts;
lifting heavy loads in the construction
and mining industry.
Aircraft systems
-such as control flaps of airplanes
Automation and Industrial Robots
Automatic controllers
Guided Missiles etc…
Applications of Hydraulic and Pneumatic systems
Rescue Robot
Next coming up:
- Others combinations…
Hydraulic & Pneumatic Systems
Large forces and torques;
High sped of response;
Availability of both Linear and Rotary Actuators (Hydraulic)
Power is not readily available;
Contaminated oil may cause system failure (Hydraulic);
Leakage;
Expensive;