Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
System Models
Mathematical ModelsMechanical System Building BlocksElectrical System Building Blocks
Fluid System Building BlocksThermal 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 parametersystem.
Mechanical System Building Blocks
• 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.
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 θ rotated is proportional to the torque: T = kθ.
• With a rotary damper a disc is rotated in a fluid and the resistive torque T is proportional to the angular velocity ω.
• 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
JaTdtdccT === ;θω
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
dttdF
kdttdxtv
tFk
tx
txktFtxtv
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)
θ (t)
∫=
==
=
=
t
tsa
a
s
as
ma
dttktT
dttdT
kdttdt
tTk
t
tBtTt
ttT
0
)()(
)(1)()(
)(1)(
)()((rad) )(nt displacemeAngular
(rad/sec) )(locity Angular vem)-(N )( torqueAppied a
ω
θω
θ
θθ
ω
ω (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 = cvor F = c dv/dt.
Translational Damper, Bv (N-sec)
Fa(t)
x(t)
∫=
==
=
=
t
ta
v
mma
am
ma
dttFB
tx
dttdxBtvBtF
tFB
tv
tvBtFtxtv
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)
θ (t)
∫=
==
=
=
t
ta
m
mma
am
ma
dttTB
t
dttdBtBtT
tTB
t
tBtTt
ttT
0
a
)(1)(
)()()(
)(1)(
)()((rad) )(nt displacemeAngular
(rad/sec) )(locity Angular vem)-(N )( torqueAppied
θ
θω
ω
ωθ
ω
ω (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
dtxdm
dtdvmmaF ===
MassForce Acceleration
Mechanical Building Blocks
P = 0.5 Jω2T = J d2θ/dt2Moment of inertiaP = cω2T = c dθ/dtDamper
E = 0.5 T2/kT = kθSpringRotational
E = 0.5 mv2F = m d2x/dt2MassP = cv2F = c dx/dtDashpot
E = 0.5 F2/kF = kxSpringTranslational
Energy representationEquationBuilding Block
Building Mechanical Blocks
• Mathematical model of a machine mounted on the ground
Mass
GroundInput, force
Output, displacement
Fkxdtdxc
dtxdm =++2
2
Building Mechanical Blocks
• Mathematical model of a rotating a mass
Tkdtdc
dtdJ =++ θθθ
2
2
Torque
Moment of inertia
Torsional resistance
ShaftPhysical situation
Block model
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 as
dynamics) (Torsional )(
dynamics) onal(Translati )(
2
2
2
2
tTkdtdB
dtdj
tFxkdtdxB
dtxdm
asm
asv
=++
=++
θθθ
Electrical System Building Blocks
• The basic building blocks of electrical systems are resistance, inductance and capacitance.
2
2
2
21 ; :Capacitor
21 ;1 :Inductor
; :Resistor
CvEdtdvCi
LiEvdtL
i
RiPiRv
==
==
==
∫
Resistance, R (ohm)
v(t) R
i(t)
)(1)(
)()()(Current
)( voltageAppied
tvR
ti
tRitvti
tv
=
=
Inductance, L (H)
v(t) L
i(t)
∫=
=
t
tdttv
Lti
dttdiLtv
titv
0
)(1)(
)()(
)(Current )( voltageAppied
Capacitance, C (F)
v(t) C
i(t)
dttdvCti
dttiC
tv
titv
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.
LCLR
LRs
LCLR
LRs
LCs
LRs
dtdv
Li
LCdtdi
LR
dtid a
122
122
are roots sticcharacteri The
01
11
2
2
2
1
2
2
2
−
+−=
−
−−=
=++
=++
Fluid System Building Blocks• 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 =− 21p1 - p2 is pressure differenceR is the hydraulic resistanceq 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.
dtdpCqq
pgAC
gpdtdp
pgAqq
dtdhAqq
AhV dtdVqq
=−=
=−
=−
==−
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
LgIdtdqIpp
dtdqLp
dtdvALp
dtdvmApp
maAppAppApApFF
==−
===−
=−−=−=−
• 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
ppdCdtdm
Rpp
dtdm
)(1 Inertance
)( eCapacitanc
Resistance
21
21
21
A fluid system
Rpgh
dtdhA
dthpgdC
Rhpgq
Rhpgqhpg-pp
RqppdtdpCqq
+==−
==
=−
=−
)(
;
e)(Resistanc
)(Capacitor
1
221
221
21
q1
h
q2
flow of rate c volumetri theis gravity todueon accelerati theis
density liquid theis
qgp
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
kTTL
AL
TTAkR
TTq −=
−=
Thermal System
L
L
L
TTdtdTRC
RTT
dtdTC
dtdTCq
dtdTCqq
RTTq
=+
−=
==−
−=
;21
qT
TL
resistance thermal theis ecapacitanc theis
flowheat of ratenet theis
RC q