Control Systems

Mathematical Models of Physical Systems

By D M. Karkare

BMS Evening College of Engineering

Introduction

A physical system is a collection of physical objects connected together to serve an objective. Examples:- Industrial Plant Governing mechanism of steam turbine Communications satellite orbiting the earth


Contd…

No physical system can be represented in its full physical details and therefore idealizing assumptions are always made for the purpose of analysis and synthesis of the system.

An idealized physical system is called Physical Model.

Once a physical system is obtained, the next step is to obtain a mathematical model which is the mathematical representation of physical model through use of appropriate physical laws.


Contd…

When the mathematical model of a physical system is solved for various input conditions, the results represent the dynamic response of the system. The mathematical model of a system is linear,

if it obeys the principle of superposition and homogeneity


Contd…

This principle implies that if a system model has responses y1(t) and y2(t) to any two points x1(t) and x2(t) respectively,

then the system response to the linear combination of these inputs

1x1(t)+ 2 x2(t) is given by the linear combination of the individual

outputs, 1y1(t)+ 2y2(t) where 1 and 2 are constants


Contd… Mathematical models of physical system are characterized by

differential equations. A mathematical model is linear, if The differential equation describing it has coefficients, which are either functions only of the independent variable or are constants. If the coefficients of the describing differential equations are functions of time, then the model is linear time varying. If the coefficients of the describing differential equations are constants, the model is linear time invariant.


Contd…

Powerful mathematical tools like the Fourier and Laplace transforms are available for use in linear systems.


Differential Equations of Physical System

An ideal element results by making two assumptions: 1) Spatial distribution of the element is ignored and it

is regarded as a point phenomenon. Thus mass which has physical dimensions, is considered at a point and temperature in a room which is distributed out into the whole room space is replaced by a representative temperature as if of a single point in the room.

The process of ignoring the spatial dependence by choosing a representative value is called lumping and the corresponding modeling is known as lumped-parameter modeling as distinguished from the distributed-parameter modeling which accounts for space distribution.


Contd…

2) We shall assume that the variables associated with the elements lie in the range that the element can be described by a simple linear law of

A constant of Proportionality A first order Derivative or A first order Integration


Ideal Elements- Classification of Element Variables

Ideal Element VT

I---------------------------- VA ---------------------------------I + -

1) Through variable VT which sort of passes through the element and so has the same value in at one port and out at the other. Example: Current through an Electrical Resistance 2) Across variable VA which appears across the two terminals of the element. Example: Voltage across an Electrical Resistance


Another Classification of Element Variables

1) Input variable or independent variable (Vi) 2) Output variable or dependent (response) variable (Vo )

Cause – Affect form

Elemental Law Vo=f(Vi)

Vi Vo


Contd…

The Spring Element

x(m), v(m/sec), M(kg), F(newton), K(newton/m), f(newton per m/sec)

x1(t) v1(t)

F(t)

x2(t) v2(t)

K


Mechanical Systems

x(t) v(t)

F(t)

Reference

The Mass Element

x(m), v(m/sec), M(kg), F(newton), K(newton/m), f(newton per m/sec) BMS Evening College of Engineering

Contd…

The Damper Element

x(m), v(m/sec), M(kg), F(newton), K(newton/m), f(newton per m/sec)

v1(t) x1(t)

F(t)

v2(t) x2(t) f


Rotational Elements

The Inertia Element

J

T , Reference

(rad), (rad/sec), J(kg-m2), T(newton-m)

K(newton-m/rad), f(newton-m per rad/sec) BMS Evening College of Engineering

Contd…

The Torsional Spring Element

J

T 1, 1 2, 2

K


K(newton-m/rad), f(newton-m per rad/sec) BMS Evening College of Engineering

Contd…

J

T 1, 1, 2, 2,

The Damper Element f


K(newton-m/rad), f(newton-m per rad/sec)


Mass/inertia and the two kinds of springs are the energy storage elements where in energy can be stored and retrieved without loss and so these are called conservative elements.

Energy stored in these elements is expressed as Mass: E=(1/2)Mv2 = kinetic energy (J); motional energy Inertia: E=(1/2)J2 = kinetic energy (J); motional energy Spring( translatory) E=(1/2)kx2 = Potential Energy (J); deformation energy Spring( torsional) E=(1/2)k2 = Potential Energy (J); deformation energy

Damper is a dissipative element and power it consumes (lost in form of heat ) is given as

P=fv2 (W) = f2 (W)


Friction

The friction exists in physical systems whenever mechanical surfaces are operated in sliding contact. Types: 1) Coulomb Friction Force 2) Viscous Friction Force 3) Stiction


Coulomb Friction Force: The force of sliding friction between dry surfaces. This force is substantially constant.

Viscous Friction Force: The force of friction between

moving surfaces separated by viscous fluid or the force between a solid body and a fluid medium. This force is approximately linearly proportional to velocity over a certain limited velocity range.

Stiction: The force required to initiate motion

between two contacting surfaces. BMS Evening College of Engineering

Translational System

J

M x 0

F

f K

F

x 0

Kx f dx/dt

M

a) A mass-spring dashpot system b) Free body diagram

or


THE END


Control Systems

Documents

Transcript of Control Systems