System dynamics ch 1

Chapter 1
Introduction to System Dynamics
Course Instructor
K. Hajlaoui, Ph.D.
Associate Professor,
Mechanical Engineering Dept., IMAM University, 2015
Imam University, 2015
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What is the system?
Definition
-A system is defined as a combination of components
(elements) that act together to perform a certain
objective.
-it contains of interacting components connected
together in such away that the variation in one
component affect the other components
-A component is a single functioning unit of a system
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What is the system?
System and Surroundings
We scope a system by defining its boundary ; this
means choosing which entities are inside the system
and which are outside the boundary- part of the
environment
SYSTEM
BOUNDARY
SURROUNDINGS
3

What is the system?
The environment affects the system through the input
(cause) and systems responds (effect) due to input
System Variables
To every system there corresponds two sets of variables
Input variables
originate outside the system
and are not affected by what
happens in the system
Output variables
are the internal variables that are
used to monitor or regulate the
system. They result from the
interaction of the system with its
environment and are influenced by
the input variables
4

Static Vs. dynamic systems
Static systems
The current output of the system (t, y(t)) depends only on
current input (t,x(t))
-The output of a static system remains constant if the input
does not change
system Classifications
5

Static Vs. dynamic systems
 dynamic systems
 The present output of the system depends on the
current and the previous input;.
The output changes with time even if the input is
constant (time varying); system is not in a equilibrium
state.
 Have a response that is not instantaneously
proportional to the input or disturbance and that may
continue after the input is held constant.
 Dynamic systems can respond to input signals,
disturbance signals, or initial conditions
6

Excitation and response of a system
Dynamic systems
7

Dynamic systems are found in all major engineering areas
 Mechanical systems
 Mechanical systems are concerned with the behavior of matter under
the action of forces/torques (Governed by Newton's 2nd law)
 Systems that possess significant mass, inertia, spring and energy
dissipation (damper) components driven by forces, torques, specified
displacements are considered to be mechanical systems.
Examples
- An automobile is a good example of a dynamic mechanical system. It has a
dynamic response as it speeds up, slows down, or rounds a curve in the road.
- The body and the suspension system of the car have a dynamic response of the
position of the vehicle as it goes over a bump.
8

Electrical systems
 Electrical systems are concerned with the behavior of three
fundamental quantities; charge, current and voltage (Governed by
Kirchoff's laws)
 Electrical systems include circuits with resistive, capacitive, or inductive
components excited by voltage or current. Electronic circuits can include
transistors or amplifiers.
Examples
A television receiver has a dynamic response of the beam that traces the picture
on the screen of the set. The TV tuning circuit, which allows you to select the
desired channel, also has a dynamic response.
9

Thermal systems
Involves heating of objects and transports of thermal energy
Thermal systems have components that provide resistance (conduction,
convection or radiation) and a capacitance (mass a specific heat) when
excited by temperature or heat flow.
Example
A heating system warming a house has a dynamic response as the temperature
rises to meet the set point on the thermostat. Placing a pot of water over a burner
to boil has a dynamic response of the temperature
10

Fluid systems
 Fluid systems Involves connection of fluid flows in tube and tank
 Fluid systems employ orifices, restrictions, control valves, accumulators
(capacitors), long tubes (inductors), and actuators excited by pressure or
fluid flow.
Example
A city water tower has a dynamic response of the height of the water as a function
of the amount of water pumped into the tower and the amount being used by the
citizens.
11

Mixed systems
 Systems which consists of two or more previously mentioned
systems (electromechanical systems, fluid – mechanical systems,
thermo mechanical systems, mechatronics systems…..) with energy
conversion between various components.
12

Mathematical Modelling of DS
Way we need the Modelling ?
 Any attempt to design a system must begin with a
prediction of its performance before the system itself
can be designed in detail or actually built.
Such prediction is based on a mathematical
description of the system's dynamic characteristics.
This mathematical description is called a
mathematical model.
15

Mathematical Modelling
Mathematical modelling involves descriptions
of important system characteristics by sets of
equations
A mathematical model usually describes a
system by means of variables.
Usually physical laws are applied to obtain
mathematical model but sometimes
experimental procedures are necessary.
16

Mathematical Modelling
Mathematical model can not represent a
physical system completely.
 Approximations and assumptions restrict the
validity of the model.
For many physical systems, useful
mathematical models are described in terms of
differential equations.
17

Linear systems and nonlinear systems.
Linear systems.
The equations that constitute the model are linear
linear systems can be represented by linear, time-
invariant ordinary differential equations.
 Laplace Transformation technique is a powerful
tool for solving linear (constant-coefficient)
differential equations
 The most important property of linear systems is
that the principle of superposition is applicable.
18

This principle states that the response
produced by simultaneous applications of two
different forcing functions or inputs is the
sum of two individual responses.
Consequently, for linear systems, the
response to several inputs can be calculated
by dealing with one input at a time and then
adding the results.
19

Nonlinear systems,
is described by a nonlinear function.
The principle of superposition is not
applicable. In general, procedures for finding
the solutions of problems involving such
systems are extremely complicated.
20

Continuous-time systems and discrete-time
systems
 Continuous-time systems are systems in which the
signals involved are continuous in time. These systems
may be described by differential equations.
Discrete-time systems are systems in which one or
more variables can change only at discrete instants of
time.
 The materials presented in this course apply
to continuous-time systems
21

Conservative system
the total mechanical energy remains constant,
there are no dissipations present, e.g. simple
harmonic oscillator
Nonconservative (dissipative) system
the total mechanical energy changes due to
dissipations like friction or damping, e.g.
damped harmonic oscillator
22

 Multidimensional system is described by a vector of
functions like
where x is a vector with n components, A is n x n
matrix and B is a constant vector
 One-dimensional system is described by a single
function like
where a,b are constants.
btaxtx
bkaxkx


)()´(
)()1(
BA
BA




)()´(
)()1(
txtx
kxkx
23

Classification of System Models: Summary
24

Remarks on mathematical models.
The model that the engineer is analyzing is an
approximate mathematical description of the physical
system; it is not the physical system itself. In reality,
NO mathematical model can represent any physical
component or system precisely.
Approximations and assumptions are always involved
and restrict the range of validity of the mathematical
model.
 So, in making a prediction about a system's
performance, any approximations and assumptions
involved in the model must be kept in mind.
25

Mathematical Modelling Procedure:
Flowchart for the mathematical
modeling procedure
The procedure can be summarized as follows:
1- Draw a schematic diagram of the system, and define
variables.
2. Using physical laws, write equations for each
component, combine them according to the system
diagram, and obtain a mathematical model.
3. Solve the equations of the model,
4- compared with experimental results.
If the experimental results deviate from the prediction
to a great extent, the model must be modified. A new
model is then derived and a new prediction compared
with experimental results. The process is repeated
until satisfactory agreement is obtained between the
predictions and the experimental results.
26

ANALYSIS & DESIGN OF DYNAMIC SYSTEMS
Analysis. Investigation of the
performance of a system under specified
conditions. The most crucial step is the
mathematical model.
Design. Process of finding a system that
accomplishes a task.
Synthesis. Finding a system which will
perform in a specified way
27

Conclusion
a successful engineer must be able to obtain
a mathematical model of a given system and
predict its performance.
 The engineer must be able to carry out a
thorough performance analysis of the system
before a prototype is constructed.
Dynamic Systems
28

Conclusion
The objective of this course is then
(1) to enable the students to build mathematical
models that closely represent behaviors of
physical systems
(2) to develop system responses to various
inputs so that they can effectively analyze and
design dynamic systems.
Dynamic Systems
29END…….

System dynamics ch 1

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