Csl3 19 j15

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION
Mathematical Modeling
Mechanical translational system
1

Review
• Transfer Function
– Def.: The ratio of output by input
– Gives the mathematical characteristic of the system being
analyzed
• Laplace transform
– Converting time domain equation to frequency domain
– Time domain – signals defined as function of time
– Frequency domain –
• signals defined as function of frequency
• s=σ+jω
• All signals are a property of frequency rather than representing the
instant of time.
2

Mechanical system modeling
• Translational
• Rotational
Example
Automobile suspension system
Along the road
1. The vertical displacements at the tires act as the motion
excitation to the automobile suspension system
2. Motion consists of a translational motion of the center of
mass
3. Rotational motion about the center of mass 3

Mechanical system modeling
• Mechanical system model components
– Mass – representing the mass of the system to be
modeled
– Spring – represents the repetitive action of the
system
– Damper-represents the friction of the system
4

Translational system
Example 1
5

Element Law Expression Laplace
Spring Spring Force α
change in length
F(t) = K x(t)
K – Stiffness
constant in N/m
F(s) = K X(s)
Viscous Damper
or Dashpot
Force α velocity F(t) = fv v(t)
F(t) =fv (dx/dt)
fv - Friction or
damping coefficient
,Ns/m
F(s) = fvsX(s)
Mass Newton’s second
law
Force α
acceleration
F(t) = M (d2x/dt2) F(s) = Ms2X(s)
Translational system
6

Step 1: Decide input and output
Input variable:
)(tf
Output variable:
)(txMass position
)(txMass velocity
)(txMass acceleration
Applied force
7

Translational systems
Newton’s second law
𝑚𝑎 = 𝐹
𝑀
𝑑2 𝑥(𝑡)
𝑑𝑡2 = −Kx(t)−𝑓𝑣
𝑑𝑥(𝑡)
𝑑𝑡
+𝑓(𝑡)
8

Step 2: The Time and frequency response
representation
dt
tdx
ftKxtf
dt
txd
M v
)(
)()(
)(
2
2

Taking Laplace Transform
𝑀𝑠2
𝑋 𝑠 + 𝑓𝑣 𝑠𝑋 𝑠 + 𝐾𝑋 𝑠 = 𝐹(𝑠)
9

Step3. Write the transfer function
With x(t) as the
output
𝑋(𝑠)
𝐹(𝑠)
=
1
𝑀𝑠2 + 𝑓𝑣 𝑠 + 𝐾
𝑀𝑠2
𝑋 𝑠 + 𝑓𝑣 𝑠𝑋 𝑠 + 𝐾𝑋 𝑠 = 𝐹(𝑠)
Write the equation with velocity as output.
10

Example 2
F(t)
x1(t)
x2(t)
x3(t)
11

Modeling Steps:
1. Decide the input and the output
2. The free body diagram of the mass M (optional)
3. The frequency-domain representation of the forces
4. The transfer function
Example 2
F(t)
x1(t)
x2(t)
x3(t)
In a multi mass system, consider
each mass separately to write the
equation.
12

Equation for Mass M1
2
1
2
1
)(
dt
txd
M
0))()((
))()(()()()(
31
2121111
2
1
3
1


ssXssXf
sXsXKsXKssXfsXsM
v
v
dt
tdx
fv
)(1
1
)(11 txK ))()(( 212 txtxK 
)
)()(
( 31
3
dt
tdx
dt
tdx
fv 
13

Step2: Free Body diagram of Mass M1
(Optional)
ssXfv )(33
)(1 tx
1M )(12 sXk
)(11 sXk
ssXfv )(13
ssXfv )(11
2
11 )( ssXM
)(22 sXk
14

2
2
2
2
)(
dt
txd
M
Equations for Mass M2
)(tF
dt
tdx
fv
)(2
2
 ))()(( 122 txtxK 
)
)()(
( 32
4
dt
tdx
dt
tdx
fv 
)())()((
))()(()()(
32
12222
2
2
4
2
sFssXssXf
sXsXKssXfsXsM
TransformLaplaceTaking
v
v


15

)(2 tx
2M
)(tF
)(22 txK
)(2
.
4 txfv
)(2
.
2 txfv
)(2
..
2 txM
)(3
.
4 txfv
)(1
.
2 txK
Free Body Diagram for Mass M2 (Optional)
16

Equations for Mass M3
2
3
2
3
)(
dt
txd
M )
)()(
( 23
4
dt
tdx
dt
tdx
fv  )
)()(
( 13
3
dt
tdx
dt
tdx
fv 
0))()((
))()(()(
23
133
2
3
4
3


ssXssXf
ssXssXfsXsM
TransformLaplaceTaking
v
v
17

)(3 tx
3M
)(3
.
3 txfv )(3
.
4 txfv
)(3
..
3 txM
)(2
.
4 txfv
)(1
.
3 txfv
The free body diagram-M3
18

Step 2– Mass M1, M2 and M3
)(1 sX
)(1 sX
)(1 sX
)(2 sX
)(2 sX
)(2 sX
)(3 sX
)(3 sX
)(3 sX



 2K 2
11321 sMsfsfKK vv 
 2
2422 sMsfsfK vv  2K  sfv4 )(sF
0
 2
343 sMsfsf vv  0 sfv4 sfv3
Example 2
19

Example 3 – Car suspension system
Automobile suspension system
Along the road
1. The vertical displacements at the tires act as the
motion excitation to the automobile suspension
system
2. Motion consists of a translational motion of the
center of mass
3. Rotational motion about the center of mass 20

1. Assuming that the motion xi at point P is the input to the
system
2. The vertical motion x0 of the body is the output
3. Transfer function = ?
X0(s)/Xi(s)
21

𝑀 𝑥0 = −𝑘 𝑥0 − 𝑥𝑖 − 𝑏( 𝑥0- 𝑥𝑖)
𝑀𝑠2
+ 𝑏𝑠 + 𝑘 𝑋0 𝑠 = (𝑏𝑠 + 𝑘)𝑋𝑖(𝑠)
𝑋0 𝑠
𝑋𝑖(𝑠)
=
𝑏𝑠 + 𝑘
𝑀𝑠2 + 𝑏𝑠 + 𝑘
Transfer Function
22

Next Class
Rotational and electrical system
Mathematical modeling
23

Csl3 19 j15

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Csl3 19 j15