BDU20102 Electromechanical & Control System

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

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

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

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

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

Mechanical Translational System Example 1  System diagram  FREE BODY DIAGRAM (FBD) !!!

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

Mechanical Translational System ; Example

Mechanical System on Aircraft ??

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, 

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

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

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

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

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

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

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

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

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