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