me 2401 mechatronics
TRANSCRIPT
ME 2401 MECHATRONICS UNIT III
SYSTEM MODELS AND CONTROLLERS
Building blocks of Mechanical, Electrical, Fluid and Thermal Systems, Rotational – Transnational Systems , Electromechanical Systems – Hydraulic – Mechanical Systems. Continuous and discrete process Controllers – Control Mode – Two – Step mode –Proportional Mode – Derivative Mode – Integral Mode – PID Controllers – Digital Controllers Velocity Control – Adaptive Control – Digital Logic Control – Micro Processors Control.
System Models
Mathematical ModelsMechanical System Building Blocks
Electrical System Building BlocksFluid System Building Blocks
Thermal Systems Building Blocks
Mathematical Models
• Think how systems behave with time when subject to some disturbances.
• In order to understand the behaviour of systems, mathematical models are required.
• Mathematical models are equations which describe the relationship between the input and output of a system.
• The basis for any mathematical model is provided by the fundamental physical laws that govern the behaviour of the system.
Building Blocks• Systems can be made up from a range of building blocks.• Each building block is considered to have a single property or
function.• Example: an electric circuit system which is made up from
blocks which represent the behaviour of resistance, capacitance, and inductor, respectively.
• By combining these building blocks a variety of electrical circuit systems can be built up and the overall input-output relationship can be obtained.
• A system built in this way is called a lumped parameter system.
BUILDING BLOCKS - MECHANICAL SYSTEM
• Basic building block: spring, dashpots, and masses.
• Springs represent the stiffness of a system
• Dashpots represent the forces opposing motion, for example
frictional or damping effects.
• Masses represent the inertia or resistance to acceleration.
• Mechanical systems does not have to be really made up of
springs, dashpots, and masses but have the properties of
stiffness, damping, and inertia.
• All these building blocks may be considered to have a force
as an input and displacement as an output.
Stiffness of a Spring• Stiffness of a spring is described as the relationship between
the force F used to extend or compress a spring and the resulting extension or compression x.
• In the case of spring where the extension or compression is proportional to the force (linear spring): F = kx, where k is a constant, the bigger the value of k the greater the forces have to be to stretch or compress the spring and so the greater the stiffness.
SpringF x
Translational Spring, k (N)
Fa(t)
x(t)
t
tsa
a
s
as
sa
dttvktF
dt
tdF
kdt
tdxtv
tFk
tx
txktF
tx
tv
tF
0
)()(
)(1)()(
)(1
)(
)()(
(m) )(position Linear
(m/sec) )(ocity Linear vel
Newtonin )( force Appied a
Rotational Spring, ks (N-m-sec/rad)
Fa(t)
(t)
t
tsa
a
s
as
ma
dttktT
dt
tdT
kdt
tdt
tTk
t
tBtT
t
t
tT
0
)()(
)(1)()(
)(1
)(
)()(
(rad) )(nt displacemeAngular
(rad/sec) )(locity Angular ve
m)-(N )( torqueAppied a
(t)
ks
Dashpot• The dashpot block represents the types of forces experienced
when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces.
• The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder.
• Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.
Translational Damper, Bv (N-sec)
Fa(t)
x(t)
t
ta
v
mma
am
ma
dttFB
tx
dt
tdxBtvBtF
tFB
tv
tvBtF
tx
tv
tF
0
)(1
)(
)()()(
)(1
)(
)()(
(m) )(position Linear
(m/sec) )(ocity Linear vel
Newtonin )( force Appied a
Bm
Rotational Damper, Bm (N-m-sec/rad)
Fa(t)
(t)
t
ta
m
mma
am
ma
dttTB
t
dt
tdBtBtT
tTB
t
tBtT
t
t
tT
0
a
)(1
)(
)()()(
)(1
)(
)()(
(rad) )(nt displacemeAngular
(rad/sec) )(locity Angular ve
m)-(N )( torqueAppied
(t)
Bm
Mass• The mass exhibits the property that the bigger the mass the
greater the force required to give it a specific acceleration.• The relationship between the force F and acceleration a is
Newton’s second law as shown below.• Energy is needed to stretch the spring, accelerate the mass and
move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot.
2
2
dt
xdm
dt
dvmmaF
MassForce Acceleration
Mechanical Building BlocksBuilding Block Equation Energy representation
TranslationalSpring F = kx E = 0.5 F2/kDashpot F = c dx/dt P = cv2
Mass F = m d2x/dt2 E = 0.5 mv2
RotationalSpring T = k E = 0.5 T2/kDamper T = c d/dt P = c2
Moment of inertia T = J d2/dt2 P = 0.5 J2
Building Mechanical Blocks
• Mathematical model of a machine mounted on the ground
Mass
GroundInput, force
Output, displacement
Fkxdt
dxc
dt
xdm
2
2
Building Mechanical Blocks
• Mathematical model of a rotating a mass
Tkdt
dc
dt
dJ
2
2Torque
Moment of inertia
Torsional resistance
ShaftPhysical situationBlock model
BUILDING BLOCKS - ELECTRICAL SYSTEM
• From Newton’s law or using Lagrange equations of motions, the second-order differential equations of translational-dynamics and torsional-dynamics are found as
dynamics) (Torsional )(
dynamics) onal(Translati )(
2
2
2
2
tTkdt
dB
dt
dj
tFxkdt
dxB
dt
xdm
asm
asv
Electrical System Building Blocks
• The basic building blocks of electrical systems are resistance, inductance and capacitance.
2
2
2
2
1 ; :Capacitor
2
1 ;
1 :Inductor
; :Resistor
CvEdt
dvCi
LiEvdtL
i
RiPiRv
Resistance, R (ohm)
v(t) R
i(t)
)(1
)(
)()(
)(Current
)( voltageAppied
tvR
ti
tRitv
ti
tv
Inductance, L (H)
v(t) L
i(t)
t
t
dttvL
ti
dt
tdiLtv
ti
tv
0
)(1
)(
)()(
)(Current
)( voltageAppied
Capacitance, C (F)
v(t) C
i(t)
dt
tdvCti
dttiC
tv
ti
tv
t
t
)()(
)(1
)(
)(Current
)( voltageAppied
0
For a series RLC circuit, find the characteristic equation and define the analytical relationships between the characteristic roots and circuitry parameters.
LCL
R
L
Rs
LCL
R
L
Rs
LCs
L
Rs
dt
dv
Li
LCdt
di
L
R
dt
id a
1
22
1
22
are roots sticcharacteri The
01
11
2
2
2
1
2
2
2
BUILDING BLOCKS – FLUID SYSTEM• The basic building blocks of fluid systems are the volumetric rate of flow q and
the pressure difference.
Input Output
Volumetric rate of flow Pressure difference
Fluid system can be divided into two types: hydraulic and pneumatic.Hydraulic resistance is the resistance to flow of liquid as the liquid flow through valves or changes in pipe diameter takes place.
qRpp 21
p1 - p2 is pressure differenceR is the hydraulic resistance
q is the volumetric rate of flow
Hydraulic capacitance is the term used to describe energy storage with a liquid where it is stored in the form of potential energy. A height of liquid in a container is one form of such a storage. For such capacitance, the rate of change of volume V in the container (dV / dt) is equal to the difference between the volumetric rate at which liquid enters the container q1 and the rate at which it leaves q2.
dt
dpCqq
pg
AC
gp
dt
dp
pg
Aqq
dt
dhAqq
AhV dt
dVqq
21
21
21
21
;
gravity) todueon accelerati theis density; liquid is (
;
Hydraulic inertance is the equivalent of inductance in electrical systems or a spring in mechanical systems. To accelerate a fluid and so increase its velocity a force is required.
Mass mF1=p1A
F2=p2A
L
density theis g andblock theoflength theis
inertance hydraulic theis ;
)(
)(
)(
21
21
21
212121
LA
LgI
dt
dqIpp
dt
dqLp
dt
dvALp
dt
dvmApp
maApp
AppApApFF
With pneumatic systems the three basic buildings blocks are as with hydraulic systems, resistance, capacitance, and inertance. However, gasses differ from liquids in being compressible.
dtppLdt
dmdt
ppdC
dt
dmR
pp
dt
dm
)(1
Inertance
)( eCapacitanc
Resistance
21
21
21
A fluid system
R
pgh
dt
dhA
dt
hpgdC
R
hpgq
R
hpgqhpg-pp
Rqppdt
dpCqq
)(
;
e)(Resistanc
)(Capacitor
1
221
221
21
q1
h
q2
flow of rate c volumetri theis
gravity todueon accelerati theis
density liquid theis
q
g
p
BUILDING BLOCKS - THERMAL SYSTEM
• There are only two basic building blocks for thermal systems: resistance and capacitance.
• There is a net flow of heat between two points if there is a temperature difference between them.
• The value of the resistance depends on the mode of heat transfer.
tyconductivi thermal theis
. and are re temperatuheat which t points ebetween th material oflength theis
conducted being isheat hich the through wmaterial theof area sectional Cross:
21
1212
k
TTL
AL
TTAk
R
TTq
Thermal System
L
L
L
TTdt
dTRC
R
TT
dt
dTC
dt
dTCq
dt
dTCqq
R
TTq
;21
qT
TL
resistance thermal theis
ecapacitanc theis
flowheat of ratenet theis
R
C
q
Rotational Systems• The mass, spring, and dashpot are the basic building blocks for mechanical
systems where forces and straight line displacements are involved without any rotation.
• If rotation is involved, then the equivalent three building blocks are a torsional spring, a rotary damper and the moment of inertia (i.e. the inertia of a rotating mass).
• With a torsional spring the angle rotated is proportional to the torque: T = k.
• With a rotary damper a disc is rotated in a fluid and the resistive torque T is proportional to the angular velocity .
• The moment of inertia block exhibit the property that the greater the moment of inertia J the greater the torque needed to produce an angular acceleration
JaTdt
dccT ;
TRANSNATIONAL SYSTEM
ELECTROMECHANICAL SYSTEM
HYDRAULIC SYSTEM
MECHANICAL SYSTEM
CONTINUOUS PROCESS CONTROLLERS
DISCRETE PROCESSCONTROLLERS
CONTROL MODE
TWOSTEP MODE
PROPORTIONAL MODE
DERIVATIVE MODE
INTEGRAL MODE
PIDCONTROLLERS
DIGITALCONTROLLERS
VELOCITY CONTROL
ADAPTIVE CONTROL
DIGITAL LOGIC CONTROL
MICROPROCESSORS CONTROL.