Actuators and Mechatronics system Design

Actuators and Mechatronics system Design, Case Study1.pptx

1. System Models
Mathematical Models
Mechanical System Building Blocks
Electrical System Building Blocks
Fluid System Building Blocks
Thermal Systems Building Blocks

2. 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.

3. 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.

4. Mechanical System Building Blocks
•Basic building block: spring, dashpots, and masses.
•Springs represent the stiffness of a system
•Dashpotsrepresent the forces opposing motion, for
example frictional or damping effects.
•Massesrepresent 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.

5. 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 Tis 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 accelerationJaT
dt
d
ccT  ;



6. Stiffness of a Spring
•Stiffness of a spring is described as the relationship
between the force Fused 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, wherek is a constant, the bigger the value of kthe
greater the forces have to be to stretch or compress the
spring and so the greater the stiffness.
Spring
F x

7. Translational Spring, k (N)
F
a(t)
x(t)



t
t
sa
a
s
a
s
sa
dttvktF
dt
tdF
kdt
tdx
tv
tF
k
tx
txktF
tx
tv
tF
0
)()(
)(1)(
)(
)(
1
)(
)()(
(m) )(position Linear
(m/sec) )(ocity Linear vel
Newtonin )( force Appied
a

8. Rotational Spring, k
s(N-m-sec/rad)
F
a(t)
(t)



t
t
sa
a
s
a
s
ma
dttktT
dt
tdT
kdt
td
t
tT
k
t
tBtT
t
t
tT
0
)()(
)(1)(
)(
)(
1
)(
)()(
(rad) )(nt displacemeAngular
(rad/sec) )(locity Angular ve
m)-(N )( torqueAppied
a






 (t)
k
s

9. 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 vof the piston: F = cv
orF = c dv/dt.

10. Translational Damper, B
v(N-sec)
F
a(t)
x(t)



t
t
a
v
mma
a
m
ma
dttF
B
tx
dt
tdx
BtvBtF
tF
B
tv
tvBtF
tx
tv
tF
0
)(
1
)(
)(
)()(
)(
1
)(
)()(
(m) )(position Linear
(m/sec) )(ocity Linear vel
Newtonin )( force Appied
a
B
m

11. Rotational Damper, B
m(N-m-sec/rad)
F
a(t)
(t)



t
t
a
m
mma
a
m
ma
dttT
B
t
dt
td
BtBtT
tT
B
t
tBtT
t
t
tT
0
a
)(
1
)(
)(
)()(
)(
1
)(
)()(
(rad) )(nt displacemeAngular
(rad/sec) )(locity Angular ve
m)-(N )( torqueAppied






 (t)
B
m

12. 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 Fand acceleration ais
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
xd
m
dt
dv
mmaF 
Mass
Force Acceleration

13. Mechanical Building Blocks
Building Block Equation Energy representation
Translational
Spring F = kx E = 0.5 F
2
/k
Dashpot F = c dx/dt P = cv
2
Mass F = m d
2
x/dt
2
E= 0.5 mv
2
Rotational
Spring T = k E = 0.5 T
2
/k
Damper T = c d/dt P = c
2
Moment of inertiaT= J d
2
/dt
2
P = 0.5 J
2

14. Building Mechanical Blocks
•Mathematical model of a
machine mounted on the
ground
Mass
Ground
Input, force
Output, displacementFkx
dt
dx
c
dt
xd
m 
2
2

15. Building Mechanical Blocks
•Mathematical model of a
rotating a massTk
dt
d
c
dt
d
J  

2
2
Torque
Moment of inertia
Torsional resistance
Shaft
Physical situation
Block model

16. Electromechanical Analogies
•From Newton’s law or using Lagrange equations of motions, the second-
order differential equations of translational-dynamics and torsional-
dynamics are found asdynamics) (Torsional )(
dynamics) onal(Translati )(
2
2
2
2
tTk
dt
d
B
dt
d
j
tFxk
dt
dx
B
dt
xd
m
asm
asv





17. 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
CvE
dt
dv
Ci
LiEvdt
L
i
RiPiRv





18. Resistance, R(ohm)
v(t) R
i(t))(
1
)(
)()(
)(Current
)( voltageAppied
tv
R
ti
tRitv
ti
tv



19. Inductance, L(H)
v(t) L
i(t)

t
t
dttv
L
ti
dt
tdi
Ltv
ti
tv
0
)(
1
)(
)(
)(
)(Current
)( voltageAppied

20. Capacitance, C(F)
v(t) C
i(t)dt
tdv
Cti
dtti
C
tv
ti
tv
t
t
)(
)(
)(
1
)(
)(Current
)( voltageAppied
0



21. For a series RLC circuit, find the characteristic equation
and define the analytical relationships between the
characteristic roots and circuitry parameters.LCL
R
L
R
s
LCL
R
L
R
s
LC
s
L
R
s
dt
dv
L
i
LCdt
di
L
R
dt
id
a
1
22
1
22
are roots sticcharacteri The
0
1
11
2
2
2
1
2
2
2

















22. Fluid System Building Blocks
•The basic building blocks of fluid systems are the volumetric rate of
flow qand the pressure difference.
Input Output
Volumetric rate of flow
Pressure difference
Fluid system can be divided into two types: hydraulic and pneumatic.
Hydraulic resistanceis the resistance to flow of liquid as the liquid flow
through valves or changes in pipe diameter takes place.q
Rpp 
21
p
1 -p
2is pressure difference
Ris the hydraulic resistance
qis the volumetric rate of flow

23. •Hydrauliccapacitanceisthetermusedtodescribeenergystoragewitha
liquidwhereitisstoredintheformofpotentialenergy.Aheightofliquidina
containerisoneformofsuchastorage.Forsuchcapacitance,therateof
changeofvolumeVinthecontainer(dV/dt)isequaltothedifference
betweenthevolumetricrateatwhichliquidentersthecontainerq
1andthe
rateatwhichitleavesq
2.dt
dp
Cqq
pg
A
C
gp
dt
dp
pg
A
qq
dt
dh
Aqq
AhV
dt
dV
qq




21
21
21
21
;
gravity) todueon accelerati theis density; liquid is (

;

24. •Hydraulicinertanceistheequivalentofinductanceinelectricalsystemsora
springinmechanicalsystems.Toaccelerateafluidandsoincreaseits
velocityaforceisrequired.
Mass m
F
1=p
1A
F2=p2A
Ldensity theis g andblock theoflength theis
inertance hydraulic theis ;
)(
)(
)(
21
21
21
212121
L
A
Lg
I
dt
dq
Ipp
dt
dq
Lp
dt
dv
ALp
dt
dv
mApp
maApp
AppApApFF





25. •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.





dtpp
Ldt
dm
dt
ppd
C
dt
dm
R
pp
dt
dm
)(
1
Inertance
)(
eCapacitanc
Resistance
21
21
21

26. A fluid systemR
pgh
dt
dh
A
dt
hpgd
C
R
hpg
q
R
hpg
qhpg-pp
Rqpp
dt
dp
Cqq




)(
;
e)(Resistanc
)(Capacitor
1
221
221
21
q
1
h
q
2flow of rate c volumetri theis
gravity todueon accelerati theis
density liquid theis
q
g
p

27. Thermal System Building Blocks
•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
A
L
TT
Ak
R
TT
q





28. Thermal SystemL
L
L
TT
dt
dT
RC
R
TT
dt
dT
C
dt
dT
Cq
dt
dT
Cqq
R
TT
q






;
21
q
T
T
Lresistance thermal theis
ecapacitanc theis
flowheat of ratenet theis
R
C
q