1. BDU20102 Electromechanical & Control System

2. CHAPTER 2 : MATHEMATICAL MODELING
Math Model  Transfer Function  To derive transfer function of various dynamic systems ;  Mechanical System ; Translational & Rotational  Electrical System  Electromechanical System  Liquid level system How??  Use physic laws

3. Mechanical Translational System
The components ;  Spring  Damper  Mass
Spring  To absorb energy  Able to sustain tension and compression  The governing law : Hook’s Law F =kx F : the force (N) k : spring stiffness (unit??) x : spring displacement (m)
The schematic
The properties

4. Mechanical Translational System
Damper  To dissipate/release energy.  A piston in cylinder contained with oil.  The equation, F : the force (N) C : damping coefficient (unit??) dx/dt : rate of piston’s displaacement (m/s)
The schematic

5. Mechanical Translational System
Mass  If a force is applied on mass, the governing law is, F : the force (N) M : the mass (kg) d2x/dt2 : the acceleration (m/s2)
The schematic

6. Mechanical Translational System
Example 1 Derive the math model relating the force (input) and the displacement (output) of the body.

7. Mechanical Translational System
Example  System diagram  FREE BODY DIAGRAM (FBD) !!!

8. Mechanical Translational System
Example 2 Derive the math model relating the force (input) and the displacement (output) of the body.

9. Mechanical Translational System ; Example

10. Mechanical System on Aircraft ??

11. Mechanical Rotational System
Systems involve fixed-axis rotation
Similar analysis with translational system
Examples: automobiles, radar tracking, rotary actuator.
Translational Rotational Spring  Shaft stiffness, k Damper  Rotational damper, B Mass  Inertia, J Displacement  angular disp,  velocity  angular velocity,  acceleration  angular accel, 

12. Mechanical Rotational System  Elements
Torsional shaft/spring
T is torque
 is angular displacement
K is shaft stiffness
 TK = K K
T, 
Rotational damper
 is angular velocity
B is rotational damper
 TB=BD B
T, 
Inertia
J is inertia
 T = J = JD =JD2
T,  J

13. Mechanical Rotational System
Example 2.1 For the radar tracking system shown in the figure, derive the EOM, draw the block diagram, simply the BD to obtain the TF. The input is torque, T and the output is the base angular displacement, . K

14. Electrical System How to model it ? Derive the transfer function
Basic electrical circuit ; resistor, capacitor, inductor
How to model it ?
Derive the transfer function

15. Electrical System  VR = iR Capacitor Inductor  VL = LDi i = CDVC
Resistor
 VR = iR
Capacitor
i = CDVC
Inductor
 VL = LDi

16. Electrical System : The Laws
1st Kirchoff Law : current
 current flow into junction =  current flow out from junction
2nd Kirchoff’s Law : voltage
 Voltage in a loop = 0

17. Electrical System
Example 2.2 For the given electrical circuit, derive the equations, draw the block diagram, simplfy the BD to obtain the TF.

18. Electrical System
Example 2.3 For the given electrical circuit, derive the equations, draw the block diagram, simplfy the BD to obtain the TF.

19. Electrical System
Example 2.4 For the given electrical circuit, derive the equations, draw the block diagram, simplify the BD to obtain the TF.

20. Electromechanical System
electrical + mechanical system. Consists of dc motor,