Chapter 1 Translational Modeling

8/22/2019 Chapter 1 Translational Modeling

1/37


2/37


3/37


4/37

Next Two Slides

Assignment Due Next WeekWednesday (3-Feb 10)

Total Questions = 4


5/37

Question:1

A vibrating machine is shown in

figure below. Write down the

differential equations of thesystem.

?

?

100

100

Question:2

Consider the given system which is

the temperature control system.Suppose that the the physical plant

(GP) is a large chamber used to

test devices under various thermal

stress. The thermal chamber

required temperature is 100 0C.

What reference input R(s); is

needed to get the required output

temperature (1000C)? [ The sensor

transfer function is 0.05 Volts/0C]

Where:

GP(s) = Plant transfer function

Gc(s)=Compensator transfer function

H(s)= Sensor transfer function


6/37

?

?

100

100

Question:3

The system given in figure is the

temperature control system. Suppose

that the physical plant (GP) is a large

chamber used to test devices under

various thermal stress. The thermalchamber required temperature is 100

0C. What reference input R(s); is

needed to get the required output

temperature (1000C)? [ The sensor

transfer function is 0.05 Volts/0C]

X1(t)

X2(t)

BK1

K2

f(t) Tire

Wheel

SuspensionSystem

Automobile

M1

M1Question:4

A simplified automobile suspension

system is given in the figure. Write

its differential equations.


7/37

1.5 Electrical Components (page10)

System Modeling


8/37

Mass (M)

LawNewtons;maf

x(t) displacement

f(t) applied force

1.6 (page 25) Translational Mechanical Components

2

2

dt

)t(dxM)t(f

dt

dSwhere

LaplaseXMS)s(F

2

dt

)t(dxv;

dt

)t(dvM)t(f


9/37

x1(t)


sdt

dwhere)XX(SB)s(F

21f(t)

x2(t)

Damping (Friction)

(Shock observer)

f(t)

B

..distwo

dt

)t(dxB

dt

)t(dxB)t(f 21

)t(x)t(xdt

dB)t(f

21

ntdislpacemeonedt

)t(dx

B)t(f1


10/37

Dampers (Friction) Shock Observer

F(s) = BSX

f(t)

x(t)

ntdislpacemeonedt

)t(dxB)t(f 1 : one side is fixed


11/37

x1(t)


ntdislpacemeone)t(xK)t(f1

)XX(K)s(F 21 f(t)

x2(t)

Spring

f(t)

K ..distwo)t(x)t(xK)t(f 21

Hooks

Law


12/37

1.6 Translational Mechanical Components (page 25)

- Mass has only one displacement in one direction.

- Damper (B) & Spring (K) may have displacement in one or both

directions.


13/37


14/37

KBMFFF)s(F

equationaldifferentiXKXSBXSM)s(F 2

Nsec/m

KBSMS)s(F

)s(XFunctionTransfer

2

1

MF

domaintimedt

)t(dxM)t(fm

2

LawNewtonsmaf

dt

dxa;Mmwhere

2

x

F

domainLLXMS)s(Fm 2

K

B

)X(ntdisplacemeoneBSX)s(FB

)X(ntdisplacemeoneKX)s(FK

F


15/37

Mass (M)

KB

f(t) x(t)

How many displacements ?? ONE

How many equations ?? ONE

Number of displacements

=

Number of differential equations

Kdt

)t(dxB

dt

)t(dxM)t(f

2

Nsec/m

KBSMS)s(F

)s(XFunctionTransfer

2

1

Example

XKXSBXSM)s(F 2


16/37


17/37


18/37

Differential equations

)xx(K)s(SxM)s(F s 2111

)xx(K)s(SxM s 12220

M2M1F

K

x1 x2

)XX(KFK 21

1

2

11XSMFM 2

2

20 XSM

)XX(K 120

F(s)

KM FF)s(F 1 KM FF 20


19/37


20/37

3

2

x2

x1

f2=8sint

f1=10

6

5

4

Example (figure D1.16 (page 29) x1 & x2 hence ; equations =2

x2

3 6 f2=8sin7t4

2

x1

5f1=10

F1 = 2S2X1+5X1

F2= 3S2X2+6X2

+ 4S(x1-x2)

+ 4S(x2-x1)

1212

21

54210 x)xx(Sdt

)t(dx

112

2

264378 x)xx(S

dt

)t(dxtsin

Try Figure D1.17(page 31)

E l (fi 1 17 ( 26)


21/37

4 5

27

6 3

x1 x2

f(t)=sint

Example (figure 1.17 (page 26)

Displacements(x1 & x2) =2; number of differential equations =2

0 = 4S2X1+7SX1

4

x1

7

x2

5 3 f(t)=sint

F(s) = 5S2

X2+3SX2

2

6

+6S(x1-x2)+2(x1-x2)

+6S(x2-x1)+2(x2-x1))s(F

s

tsin

1

1

2

Try Figure 1.18

(page 28)

Try Figure 1.18(page 28)


22/37

The force is acting on the cylinder, resulting in the velocities given

below. What is the applied force?

Drill problem: Find the required forces on the damper

ans. F=0.02N


23/37

Next Few slides

Drill problem

Find the differential equations

D ill bl


24/37

Drill problem

D ill bl


25/37

Write the deferential equation describing the motion of the

following system.

Motion of a Mass on a Spring with Damping

Drill problem


26/37

Drill problem


27/37

A simplified automobile suspension system is given in the figure.

Write its differential equations.

Drill problem

Drill problem


28/37

Write the differential equations for the translating system below.

Drill problem

Drill problem


29/37


Drill problem

Drill problem


30/37

Develop the equation relating the input force to the motion

(in terms of x) of the left hand cart for the problem below.

Drill problem

Drill problem


31/37


Drill problem

Drill problem


32/37

Write set of deferential equations describing motion of the system

Mass spring system with two masses.

Drill problem

Drill problem


33/37

Write the differential equations for the system below.

Drill problem


34/37

1


35/37

M

domaintimeK

dt

)t(dxB

dt

)t(dxM)t(f s

2

)xx(K

dt

)t(dxM)t(f s 21

2

1

1

)xx(Kdt

)t(dx

M s 12

2

2

20

1

1

2

stsin

)xx(KxM s 1222

x2(t)

6


36/37


37/37

Review Questions

What do u understand from title of this course?

Examples of open loop systems?

Examples of closed loop systems?

Why system modeling is important?

Examples of time variant systems?

Date post:	08-Aug-2018
Category:	Documents
Upload:	ali-ahmad
View:	220 times
Download:	0 times

Chapter 1 Translational Modeling

Documents