Chapter 2

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Chapter 2

Mathematical Models of

Systems

(2)

Illustrations

2 - 2

Dr. Soleimani, Department of Mechanical Engineering, IAUN

Six Step Approach to Dynamic System Problems

• Define the system and its components

• Formulate the mathematical model and list the necessary assumptions

• Write the differential equations describing the model

• Solve the equations for the desired output variables

• Examine the solutions and the assumptions

• If necessary, reanalyze or redesign the system

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Illustrations

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Illustrations

2 - 6

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Illustrations

2 - 8

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Illustrations

2 - 10

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Illustrations

2 - 12

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Illustrations

2 - 14

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𝑓 𝑡 𝑑𝑡^𝑡

0

𝑒^−𝑎𝑡𝑓 𝑡 𝐹(𝑠 + 𝑎)

(16)

Illustrations

2 - 16

s -> 0 t -> infinite

Lim s F s(  ( )) Lim f t( ( ))

7. Final-value Theorem

s -> infinite t -> 0

Lim s F s(  ( )) Lim f t( ( )) f 0( )

6. Initial-value Theorem

0



s F s( )



 d

f t( ) t 5. Frequency Integration

F s(  a) f t( ) e ^ ⁽^{a t}^ ⁾

4. Frequency shifting

s F s( ) d

d

 t f t ( )

3. Frequency differentiation f at( ) 2. Time scaling

1 a F s

 a

 

 f t(  T) u t (  T)

1. Time delay

e^ ⁽^{s T}^ ⁾F s( )

Property Time Domain Frequency Domain

Lim s F s(  ( )) Lim f t( ( ))

Lim s F s(  ( )) Lim f t( ( )) f 0( )

0



s F s( )



 d

F s(  a) f t( ) e ^ ⁽^{a t}^ ⁾

s F s( ) d

d

 t f t ( )

1 a F s

 a

 

 f t(  T) u t (  T)

1. Time delay

e^ ⁽^{s T}^ ⁾F s( )

Lim s F s(  ( )) Lim f t( ( ))

Lim s F s(  ( )) Lim f t( ( )) f 0( )

0



s F s( )



 d

F s(  a) f t( ) e ^ ⁽^{a t}^ ⁾

s F s( ) d

d

 t f t ( )

1 a F s

 a

 

 f t(  T) u t (  T)

1. Time delay

e^ ⁽^{s T}^ ⁾F s( )

Lim s F s(  ( )) Lim f t( ( ))

Lim s F s(  ( )) Lim f t( ( )) f 0( )

0



s F s( )



 d

F s(  a) f t( ) e ^ ⁽^{a t}^ ⁾

s F s( ) d

d

 t f t ( )

1 a F s

 a

 

 f t(  T) u t (  T)

1. Time delay

e^ ⁽^{s T}^ ⁾F s( )

(17)

Differential operator

Integral operator

Final value theorem lim

𝑡→∞

𝑦 𝑡 = lim

𝑠→0

𝑠𝑌(𝑠)

lim

𝑡→0

𝑦 𝑡 = lim

𝑠→∞

𝑠𝑌(𝑠)

Initial value theorem

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Illustrations

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Differential Equation:

Laplace Transform:

Transfer Function:

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Illustrations

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Illustrations

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Rearranged Equations:

Matrix Form:

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Solving for V

₁

(s):

Transfer Function:

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Illustrations

2 - 24

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The flux ϕ is proportional to the field current i

_f

:

The motor torque is related linearly to ϕ and armature current i

_a

:

The field current is related to the field voltage as:

The motor torque T

_m

is divided to the load torque T

_L

and the

A. Field Control (i

_a

=constant)

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Illustrations

2 - 26

The load torque T

_L

for rotating inertia :

Rearranging Equations

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The transfer function of the motor-load combination

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Illustrations

2 - 28

B. Armature control (i

_f

=constant)

The relation between voltage and current of armature:

The motor torque is related linearly to armature current i

_a

:

Back electromotive-force voltage:

(29)

Armature current:

Load torque:

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Illustrations

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V₂( )s V₁( )s

1 RCs

V₂( )s

V₁( )s RCs

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Illustrations

2 - 32

( )s V_f( )s

K_m

s J s(   b)



L_fs  R_f



( )s V_a( )s

K_m

s^



R_a  L_as



⁽^{J s}^ ^ ^b⁾  ^KbK_m

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x_o( )t y t( )  x_in( )t X_o( )s

X_in( )s

s²

 s² b

M

 ^^s

 k

 M

For low frequency oscillations, where  _n X_o j

X_in j ^

2 k M

V₂( )s V₁( )s

R₂ R

R₂ R₁  R₂

R₂ 

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Illustrations

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Gear Ratio = n = N1/N2 N₂_L N₁_m

_L n_m

_L n_m

(35)

T s( ) q s( )

1 C_ts Q S 1

 R

 



T T_o T_e = temperature difference due to thermal process C_t = thermal capacitance

= fluid flow rate = constant Q

Y s( ) X s( )

K s Ms(  B) K A k _x

k_p B b A²

k_p

 







k_x x d g d

k_p P d g d g g x P(  ) flow

A = area of piston

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Illustrations

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Block diagram of a DC motor

General block representation of two-input, two output system

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Block diagram of interconnected system

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Illustrations

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Multi input, multi output system (Matrix form)

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Block Diagram Transformations

Original Diagram Equivalent Diagram

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Illustrations

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Block Diagram Transformations

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Block Diagram Transformations

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Illustrations

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Closed-loop transfer function

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Solution:

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Illustrations

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Signal-flow graph of the DC motor

Signal-flow graph of interconnected system

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Illustrations

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Signal-flow graph of two algebraic equation

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Illustrations

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𝑃

_𝑘

= یجوزخ هب یدورو سا مُا 𝑘 زیسم هزهب

Δ = فازگ نانیمزتد

Δ

_𝑘

= مُا 𝑘 زیسم فذح اب فازگ نانیمزتد

Δ = 1 – ( توافتم یاه هقلح همه هزهب عومجم )

+ ( یسامتزیغ یاه هقلح تفج همه هزهب یزضلصاح عومجم )

– ( یسامتزیغ یاه هقلح هس همه هزهب بزضلصاح عومجم )

+ . . .

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Illustrations

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Illustrations

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Illustrations

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