Chapter 1 Introduction to System Dynamics

Course Instructor

K. Hajlaoui, Ph.D.

Associate Professor,

Mechanical Engineering Dept., IMAM University, 2015

Imam University, 2015 1 , 441همك

2 Imam University, 2014

, 333همك

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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3 Imam University, 2014

, 333همك

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

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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

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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

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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

system Classifications

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system Classifications

Excitation and response of a system

Dynamic systems

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system Classifications

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.

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system Classifications

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.

Dynamic systems are found in all major engineering areas

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system Classifications

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

Dynamic systems are found in all major engineering areas

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system Classifications

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.

Dynamic systems are found in all major engineering areas

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system Classifications

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.

Dynamic systems are found in all major engineering areas

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system Classifications

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system Classifications

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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.

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Mathematical Modelling of DS

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.

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Mathematical Modelling of DS

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.

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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.

Mathematical Modelling of DS

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Mathematical Modelling of DS

Linear systems and nonlinear systems.

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.

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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.

Mathematical Modelling of DS

Linear systems and nonlinear systems.

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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

Mathematical Modelling of DS

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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

Mathematical Modelling of DS

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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

Mathematical Modelling of DS

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Classification of System Models: Summary

Mathematical Modelling of DS

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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.

Mathematical Modelling of DS

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Mathematical Modelling Procedure:

Mathematical Modelling of DS

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.

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ANALYSIS & DESIGN OF DYNAMIC SYSTEMS

Mathematical Modelling of DS

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

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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

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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

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