Study Guide
Introduction to System Dynamics
-
University:
Kentucky State University -
Course:
MAT 172 | Accelerated Contemp. Math Academic year:
2019
-
Views:
118
Pages:
14
Author:
Madelyn Bradshaw
Introduction to System Dynamics
System dynamics deals with the mathematical modelling of
dynamic systems and response analysis of such systems
with a view toward understanding the dynamic nature of
each system and improving the system’s performance.
System. A system is a combination of components acting
together to perform a specific objective. A component is a
single functioning unit of a system.
1 Introduction to System Dynamics
A system is called dynamic if its present output depends on
past input; if its current output depends only on current
input, the system is known as static.
The output of a static system remains constant if the input
does not change. The output changes only when the input
changes. In a dynamic system, the output changes with
time if the system is not in a state of equilibrium.
2 Introduction to System Dynamics
Mathematical models. 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.
For many physical systems, useful mathematical models
are described in terms of differential equations.
3 Introduction to System Dynamics
Linear and nonlinear differential equations.
➢ Linear differential equations
Linear differential equations may be classified as linear,
time-invariant differential equations and linear, timevarying differential equations.
A linear, time-invariant differential equation is an equation
in which a dependent variable and its derivatives appear as
linear combinations.
4 Introduction to System Dynamics
An example of such an equation:
Note that the coefficients of all terms are constant.
In the case of a linear, time-varying differential equation,
the dependent variable and its derivatives appear as linear
combinations, but a coefficient or coefficients of terms
may involve the independent variable.
5 Introduction to System Dynamics
An example of this type of differential equation is:
➢ Nonlinear differential equations
A differential equation is called nonlinear if it is not linear,
that is, the equation contains powers or other functions or
products of the dependent variables or its derivatives.
6 Introduction to System Dynamics
Two examples of nonlinear differential equations are:
and
7 Introduction to System Dynamics
Linear systems and nonlinear systems.
Although physical relationships are often represented by
linear equations, in many instances the actual relationships
may not be quite linear. In fact, a careful study of physical
systems reveals that so-called linear systems are actually
linear only within limited operating ranges.
In general, procedures for finding the solutions of
problems involving such systems are extremely
complicated. Because of the mathematical difficulty
involved, it is frequently necessary to linearise a nonlinear
system near the operating condition.
8 Introduction to System Dynamics
Mathematical modelling of dynamic systems.
➢ Mathematical modelling.
Mathematical modelling involves descriptions of important
system characteristics by sets of equations. By applying
physical laws to a specific system, it may be possible to
develop a mathematical model that describes the dynamics
of the system. Such a model may include unknown
parameters, which must then be evaluated through actual
tests.
9 Introduction to System Dynamics
Sometimes, however, the physical laws governing the
behaviour of a system are not completely defined, and
formulating a mathematical model may be impossible.
If so, an experimental modelling process can be used. In
this process, the system is subjected to a set of known
inputs, and its outputs are measured. Then a mathematical
model is derived from the input-output relationships
obtained.
10 Introduction to System Dynamics
➢ Simplicity of mathematical model versus accuracy of
results of analysis.
In attempting to build a mathematical model, a
compromise must be made between the simplicity of the
model and the accuracy of the results of the analysis.
It is important to note that the results obtained from the
analysis are valid only to the extent that the model
approximates a given physical system.
11 Introduction to System Dynamics
In determining a reasonably simplified model, we must
decide which physical variables and relationships are
negligible and which are crucial to the accuracy of the
model.
Usually, in solving a new problem, it is desirable first to
build a simplified model to obtain a general idea about the
solution. Afterward, a more detailed mathematical model
can be built and used for a more complete analysis.
12 Introduction to System Dynamics
➢ Mathematical modelling procedure
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. To verify the validity of the model, its predicted
performance, obtained by solving the equations of the
model, is compared with experimental results. If the
experimental results deviate from the prediction to a
great extent, the model must be modified.
13 Introduction to System Dynamics
Benefits of teaching mathematical modelling to
engineering students.
1. Helps students solve problems and make connections
between mathematics and the real world.
2. Helps students learn to solve problems using computerbased simulations.
14
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