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