MATHEMATICAL MODELING OF DYNAMIC SYSTEMS Mechanical Translational System 1. Spring 2. Damper

k

FS(t)

x(t)

xi(t) xo(t)k

k

x(t)

x(t)

c c

xi(t) xo(t)

3. Mass EXAMPLE I Produce the block diagram for the mass-spring system shown below by considering displacement xi as the input variable and displacement xo as the output variable.

m k

xo(t) xi(t)

mF(t)

x(t)

EXAMPLE II Produce the block diagram for the mass-spring-damper system shown below by taking force Fa as the input variable and displacement x as the output variable. Mechanical Rotational System 4. Shaft stiffness Torque applied on the disc:

m

k x (t)

Fa(t)

c

k θi(t)

θo(t)

5. Viscous damper 6. Mass inertia EXAMPLE I Produce the block diagram for the mechanical rotational system shown below if angular displacement θi is taken as the input variable and angular displacement θo is taken as the output variable.

τ(t) θ(t)

J

B θi(t) θo(t)

θo(t)

J k

B

θi(t)

Leverage and Gearing System Lever and gear are mechanical devices used to transfer energy from one section to another by transformation the force, torque, velocity and displacement. The inertia and friction effects are normally small and are neglected. “Walking lever” can be used for summing up mechanical displacement signals. 1. Fixed lever 2. Walking lever

l1

l2

z

y

b

a

l1

l2

z

x

y

3. Gear train Gear ratio: EXAMPLE I Produce the block diagram for the mechanical translational system shown below if displacement xi is the input and displacement xo is the output.

ωA,θA

ωB,θB

τB

τA

rB rA

C

a

b

k2

k1 m

xo

xi

EXAMPLE II Produce the block diagram for the rotational system shown below considering torque τa as the input variable and angular displacement θ as the output variable. Liquid Level System 1. Fluid Resistance, R

The relation between liquid flow rate, q and system’s hydrostatic head, h is not linear. For linear model approximation (if the change in head and flow rate are small), the flow rate is considered proportional to the system head multiply by a proportional constant, 1/R. R is known as fluid resistance in the liquid level system.

τaθ,ω

K

J

r2

r1

B

h1

h2R

q

2. Capacitance, C

Changes in the volume inside the tank are equals to the difference between the inflow rate and the outflow rate. Capacitance is the cross-sectional area of the tank.

Therefore: Consider the liquid level system shown below. The variables to be monitored are the head, h and the flow rate, q. The system parameters are the fluid resistance, R and capacitance, C.

h

qo

qi

C

h qo

qi

C R

EXAMPLE I Produce the block diagram for the liquid level system shown below if inflow rate qi is taken as the input variable and head h2 is taken as the output variable.

qo h1

qi

C1

R1 q

C2

R2 h2

Where, C – thermal capacitance (kcal/ºC) R – thermal resistance (ºCsec/kcal) hi – heat input rate (kcal/sec) hi – heat output rate (kcal/sec) θ - temperature of outflowing liquid (ºC)

Thermal System

Thermal Resistance, Heat Balance Equation, Electrical System 1. Resistance, R 2. Inductance, L

bendalir panas

bendalir dingin

pemanas

penebat

R

VR

L

VL

i

i

3. Capacitance, C EXAMPLE I Produce the block diagram of the electrical system shown below by considering input voltage Vi as the input variable and output voltage Vo as the output variable.

C

VC

i

L

Vi

C

R Vo

DETERMINATION OF TRANSFER FUNCTION USING BLOCK DIAGRAM SIMPLIFICATION

Transfer Function Transfer function for a linear system is the ratio of the Laplace transform of the output variable (response function) to Laplace transform of the input variable (driving function) under the assumption that all the initial conditions are ZERO. The transfer function of a system represents the relationship which describes the dynamic behaviour of the system under study; describing the relationship between the system’s input and output. However the transfer function does not provides information about internal structure and internal behaviour of the system. EXAMPLE I Construct the transfer function for a system represented by the following differential equation: Block Diagram Simplification A block diagram is regularly being used as pictorial representation of a control system. Block diagram depicts the interrelationships that exist among various components within the system. All system variables are linked to each other through functional blocks. A complex block diagram can be simplified using block diagram manipulation rules shown below. A system block diagram normally shows the relationship of important variables only.

(t)Fkxdtdxc

dtxdm A2

2

=++

Block Diagram Algebra

A +

_ +

+

A-B A-B+C

B C

+

_

+

+ C B

A A+C A-B+C

A +

_ +

+

A-B A-B+C

B C

A + _

+ A-B+C

B

C

G1 G2

G1G2

A + _

B

G C A +

_

B

G C

G

G2G1

atau G1 G2

G2 G1

A G

C

C

A G

C

C G

A G

C

A

A G

C

A G1

A +

±

B

G C A +

± B

G C

G1

+

± G

H

A C

GH1Gm

C A

+

+ G1

G2

A C

C A G1 + G1

EXAMPLE II Consider the mechanical translational system represented by the block diagram as shown below. Produce the system transfer function using the block diagram simplification method.

m

k x (t)

Fa(t)

c

Fa(t)

2D1

m

Dc

k

Fa x + _ _

EXAMPLE III Produce the system transfer function using the block diagram simplification method.

R + _

_

G1 G2

H2

H1

+ C

SIGNAL FLOW GRAPH AND MASON’S RULE Beside block diagram, signal flow graph is another way of illustrating a control system. For a complex system, signal flow graph has an advantage over block diagram as Mason’s Rule can be used to determine the transfer function of the system. However, block diagram is more popular for uncomplicated system as it has close representation of the actual physical system. Signal flow graph is consisting of nodes and branches to represent linear relationships. Variables are represented by nodes and each transfer operator (internal relationship function) is a branch which connecting all the nodes. In general, branch is popularly known as gain which represents the relationship between two variables. Some characteristics of the nodes are: 1. Variable at a node is the summation of all the incoming signals into the node and

the node variable is transferred to all outgoing branches. 2. The summing junction is just a node 3. Input node (source node) will only consist of outgoing branch. 4. Output node (sink node) will only consist of incoming branch

EXAMPLE I Draw the signal flow graph for the control system represented by the block diagrams shown below. a) b) c)

R + _

_

G1 G2

H2

H1

+ C

R G1 G2

C

G3R +

_ _G1 G2

H2

G4

+ C

H1

++

Mason’s Rule Transfer function of a control system can be obtained using Mason’s rule given below: where, Pi = ith forward path gain Δ = 1 – (summation of all individual loops gain) + (product of two

non-touching loops) – (product of three non-touching loops) + …..

Δi = the value of Δ with all loop gains touching path Pi are discarded

EXAMPLE II Construct the signal flow graph and derive the system’s transfer function by employing Mason’s rule for the block diagram shown below.

ΔΔP

U(s)Y(s) ii∑=

G1 R

_

H1

G2

+ C + +

EXAMPLE III Using the Mason’s rule, determine the transfer function of each signal flow graphs shown below.

G6

G1 G2 G3 G4 G5

-H1 -H2

x y

EXAMPLE IV Construct the signal flow graph and derive the system’s transfer function using Mason’s rule for the block diagram shown below.

G3R

_+

G1 G2

H1

+ C

H2

++

_ +

G4