Download - Dynamic systems-analysis-4
Dynamic System AnalysisLecture “4”
Dr. Sameh Farid Saad
Agenda
1. Mathematical Modeling of Pneumatic systems in state space.
2. Mathematical Modeling of Hydraulic servo systems in state space.
3. Modeling of Mixed system.
Pneumatic Systems
• The working medium in a pneumatic device is a compressible fluid, most
commonly air.
• The availability of air is an advantage for pneumatic devices,
because it can be exhausted to the atmosphere at the end of the device’s
work cycle, thus eliminating the need for return lines.
• On the other hand, because of the compressibility of the working fluid,
the response can be slower and more oscillatory than that of hydraulic
systems.
Resistance and Capacitance of Pressure Systems
Modeling of Pressure Systems
Pneumatic Nozzle–Flapper Amplifiers
• Converts displacement into a pressure signal.
• A large power output can be controlled by the
very little power that is needed to position the
flapper.
Pneumatic Relays
• In practice, in a pneumatic controller, a nozzle–flapper amplifier acts as the
first-stage amplifier and a pneumatic relay as the second stage amplifier.
• The pneumatic relay is capable of handling a large quantity of airflow.
Flapper as a Lever
There are two small movements (𝒆 and 𝒚) in opposite directions,
Consider such movements separately and add up the results of
two movements into one displacement 𝒙.
Bellows Acts Like a Spring
𝐹 = 𝑃𝑐𝐴
𝐹 = 𝑘𝑦
𝑃𝑐𝐴 = 𝑘𝑦
𝒀
𝑷𝒄=𝑨
𝒌
Example (1)
Example (2)
Hydraulic Systems
Hydraulic Servo System
Hydraulic Servo System
Neglecting Time constant
(T)
Hydraulic Proportional Controller
Dashpots
The force acting on the piston must balance the spring force.
Hydraulic Proportional-Plus-Integral Control Action
Example (1)
For the aircraft elevator control system, find the transfer function 𝜙 𝑠
𝜃 𝑠
Process Control System
Input device
Reference input
Input potentiometer
output potentiometer
Feedback signal
Error measuring device
Amplifier Motor Gear train
Load
Error Measuring Device
𝒆 = 𝒓 − 𝒄 ⟹ 𝑬 𝒔 = 𝑹 𝒔 − 𝑪(𝒔)• The angular position 𝒓 is the reference input to the system,
• The electric potential of the arm 𝑒𝑟 is proportional to the angular position of the
arm 𝒓.
𝑒𝑟 = 𝐾0𝑟• The output shaft position determines the angular position 𝒄 of the wiper arm of
the output potentiometer.
𝑒𝑐 = 𝐾0𝑐• The potential difference is the error voltage, 𝑒𝑣
𝑒𝑣 = 𝑒𝑟 − 𝑒𝑐= 𝐾0𝑟 − 𝐾0𝑐= 𝐾0(𝑟 − 𝑐)
𝒆𝒗 = 𝑲𝟎𝒆 ⟹ 𝑬𝒗 𝒔 = 𝑲𝟎𝑬(𝒔)
Amplifier
• The error voltage that appears at the potentiometer terminals is amplified by the
amplifier whose gain constant is 𝐾1.
𝑒𝑎 = 𝐾1𝑒𝑣 ⟹𝑬𝒂 𝒔 = 𝑲𝟏𝑬𝒗(𝒔)
• The output voltage of this amplifier is applied to the armature circuit of the dc
motor.
Motor
• For constant field current, the torque developed by the motor is
𝑇 = 𝐾2𝑖𝑎 ⟹ 𝑻 𝒔 = 𝑲𝟐𝑰𝒂(𝒔)
• The induced voltage 𝒆𝒃 is directly proportional to the angular velocity𝒅𝜽
𝒅𝒕
𝑒𝑏 = 𝐾3𝑑𝜃
𝑑𝑡⟹ 𝑬𝒃 𝒔 = 𝑲𝟑𝒔 𝜣(𝒔)
• The differential equation for the armature circuit is
𝐿𝑎𝑑𝑖𝑎𝑑𝑡
+ 𝑅𝑎𝑖𝑎 + 𝑒𝑏 = 𝑒𝑎
• The equation for torque equilibrium is
Gear Train
• We assume that the gear ratio of the gear train is such that the output shaft
rotates n times for each revolution of the motor shaft. Thus,
𝑪(𝑺) = 𝒏𝜣(𝒔)
Block Diagram
ReportDraw the block diagram, and compute the transfer function
𝐺(𝑠) = 𝑋(𝑠)/𝐸1(𝑠)for the position control system
Thanks for your Attention