Welcome to elearning session on CONTROL ENGINEERING ME 55

1
Welcome to e-learning session onCONTROL
ENGINEERING (ME 55)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
2
ByDr. B.K. Sridhara HeadDepartment of
Mechanical EngineeringThe National Institute of
EngineeringMysore 570 008
3
Session 7 CHAPTER II MATHEMATICAL
MODELING (Continued)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
4
Recap of Session VI

• Mathematical Model for Electrical Systems
• Resistance, Capacitance and Inductance
• R-L Circuit, R-L-C Circuit
• Analogous System and Analog Quantities

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
5
Mathematical Modeling of Hydraulic Thermal
Systems

• Can be modeled by using generalized definitions
  of
• Resistance, Inductance Capacitance
• Generalized Definitions

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
6
Generalized Definitions
Resistance Resistance is that which opposes the
flow. It is defined as the change in potential
required to cause a unit change of flow rate.
By this definition R Change in Potential /
Change in flow rate
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
7
Electrical Systems Electrical Resistance
Potential involved is the voltage difference and
the charge is the quantity that flows
e voltage in volts i Current dq/dt, in
Coulombs/sec (Amperes) R e/i - in ohms, if
the relation is linear
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
8
Thermal Systems Thermal Resistance
Potential involved is the temperature difference
and the heat is the quantity that flows R
Change in Potential / Change in flow rate
Ambient temperature, Ta0 C
Body at elevated temperature, T0 C
Q, heat flow
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
9
dT/dQ
Thermal Resistance, RT T/Q, degrees / kJ/sec
if the relation is linear Inverse of thermal
resistance is called thermal conductance.
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
10
Hydraulic Systems Hydraulic Resistance
Potential involved is the level (head) difference
and the liquid is the quantity that flows R
Change in Potential / Change in flow rate
... ....... .......

Liquid
h
q
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
11
RH Change in liquid level / Change of flow rate
RH h/q, if the relationship is linear as in
the case of laminar flow
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
12
Mechanical Systems Resistance to Motion (Damping)
Potential involved is the force difference and
the displacement is analogous to quantity that
flows R Change in Force / Change in
Displacement
Rm dF/dv F/v (Damping Coefficient)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
13
Generalized Definitions
Capacitance Capacitor is a storing element. It
is defined as the change in quantity contained or
stored for a unit change in a reference variable.
a) Electrical Capacitance
R Quantity that is stored / Change in reference
variable
R Charge stored / Change in voltage
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
14
C q/e, if the relationship is linear
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
15
b) Thermal Capacitance
R Quantity that is stored / Change in reference
variable
R Heat stored / Change in temperature
if the relationship is linear
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
16
c) Hydraulic Capacitance
R Quantity that is stored / Change in reference
variable
R Quantity of liquid stored / Change in head
qin
h
Q
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
17
dQ change in quantity stored
qin
h
Q
A cross sectional area of the tank Q Ah
CTH A
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
18
Generalized Definitions
Inductance Inductance is that which opposes
acceleration. It is defined as the change in
potential required to cause a unit change in
acceleration . a) Electrical Inductance
Potential involved is the voltage that causes
unit change in acceleration L Change in
Potential / Unit change in acceleration
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
19
L
b) Mechanical Inductance
Potential involved is the force that causes unit
change in acceleration L Change in Potential /
Unit change in acceleration
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
20
L Change in Potential / Unit change in
acceleration
d2x /dt2
F
m
By definition L Force/Acceleration
L F/d2x/dt2
L M Mechanical Inductance Mass or MOI
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
21
Hydraulic Systems

• Liquid level systems Closely Related to the
  level of a liquid in a tank encountered in
  Chemical Processing Industries.
• (b)Fluid power systems Hydraulic Pneumatic
  Applications

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
22
Liquid Level Systems Divide the flow regimes
in to Laminar flow Re lt 2000 Turbulent flow
Re gt 2000 Re Reynolds Number Laminar flow
Models linear differential equation Turbulent
flow Non- linear differential equations
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
23
Fundamental Relationship
q a v q Volume flow rate m2/sec A Cross
sectional area, m2 v Velocity, m/sec
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
24
Example 1 Obtain a Mathematical Model for the
liquid level system given relating h and q0
qi q0 qs qi inflow rate q0 outflow
rate qs Rate at which the liquid is stored
qi

h
Liquid
q0
c/s area A
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
25
----- (1)
By definition Hydraulic Resistance R dh/dq0
h/q0
----- (a)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
26
Hydraulic capacitance
----- (b)
Substitute (a) and (b) in (1)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
27
qi

h
Liquid
q0
c/s area A
---Model
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
28
Example 2 Obtain a Mathematical Model for the
liquid level system given relating qi and q0
qi q0 qs qi inflow rate q0 outflow
rate qs Rate at which the liquid is stored
qi

h
Liquid
q0
c/s area A
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
29
qi q0 qs
qi q0 A.
----- (a)
We have C A R
R
----- substitute in (a)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
30
qi q0 A.
----- (a)
qi

h
Liquid
q0
c/s area A
---Model
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
31
qi

h
Liquid
q0
c/s area A
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
32
Example 3 Obtain a Mathematical Model for the
twin liquid level system given relating h2 and qi

qi inflow rate to tank I q1 outflow rate
from tank I and inflow rate to tank II q0
outflow rate from tank II h1 head in tank
I h2 head in tank II
qi

h1
Tank I
q1
A1

h2
Tank II
q0
A2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
33
For tank I relating inflow qi and outflow q1 we
have
R1C1
----- (1)
For tank II relating inflow q1 and head h2 we
have
----- (2)
R2C2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
34
To relate h2 and qi eliminate q1 between (1) and
(2)
R1C1
----- (1)
----- (2)
R2C2
From (1) R1 C1 Pq1 q1 qi (2) R2 C2
Ph2 h2 R2q1
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
35
(R1 C1 P 1) q1 qi ---- (3) (R2 C2 P 1)
h2 R2q1 ---- (4)
, substitute in (4)
?From (3)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
36
(R2 C2 P 1) h2 (R1 C1 P 1) R2 qi h2 (R1 R2
C1 C2 P2 R1 C1 P R2 C2 P 1) R2 qi
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
37
Example 4 Obtain a Mathematical Model for the
twin liquid level system given relating q0 and qi

qi inflow rate to tank I q1 outflow rate
from tank I and inflow rate to tank II q0
outflow rate from tank II h1 head in tank
I h2 head in tank II
qi

h1
Tank I
q1
A1

h2
Tank II
q0
A2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
38
For tank I relating inflow qi and outflow q1 we
have
R1C1
----- (1)
For tank II relating inflow q1 and outflow q0 we
have
----- (2)
R2C2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
39
To relate qi and q0 substitute (1) and (2)
R1C1
----- (1)
----- (2)
R2C2
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
40
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
41
Fluid Power Systems

• Oil under pressure is supplied to pressure port
  of the valve
• Valve spool is shown in centre position blocking
  flow to and from all ports
• With the spool centered the piston is stationary
• Displacement of the spool to the right (x),
  permits oil flow to the cylinder causing the
  piston to move the right (y)

Spool Type Actuator (Cylinder)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
42
Fluid Power Systems

• To obtain the mathematical model references are
  chosen as follows.
• x 0, with the spool centered
• Displacement to the right positive.
• y 0, right most position of the piston

Spool Type Actuator (Cylinder)
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
43
For a spool-type hydraulic valve q Cd av --
(1) q Flow rate a Orifice area v
Velocity Cd Co-efficient of discharge
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
44
For a spool-type valve a wx --- (a) Where w
width of the port x displacement Velocity
v
--- (b)
p pressure drop across the orifice ? mass
density
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
45

• q Cd av -- (1)
• Substitute for a and v in --- (1)
• q Cdwx --- (2)
• Cd 0.6 to 0.8

Equation (2) can be simplified by assuming the
pressure drop across the valve as constant, Cd w,
and ? being constants
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
46
q kv x --- (3) Kv valve constant Cd. w
The flow q given by equation (3) produces piston
motion
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
47
q kv x --- (3) Also, we have, for the Actuator
(Piston Cylinder arrangement) q Av A.
dy/dt --- (4) Combining equation (3) and (4)
Model for spool type actuator
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
48
Summary

• Generalized Definitions of Resistance,
  Capacitance and Inductance
• Models of Hydraulic Systems
• Liquid Level System
• Fluid Power System

Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore
49
THANK YOU
Dr. B.K. Sridhara, Head, Department of Mechanical
Engineering, NIE, Mysore