# Introduction to Engineering System Dynamics

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1 CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are quite complex and an exact analysis of the system is often not possible. However, simplifying assumptions can be made to reduce the system model to an idealized version whose behaviour or performance approximates that of the real system. The process by which a physical system is simplified to obtain a mathematically tractable situation is called modeling. The simplified version of the real system thus obtained is called the mathematical model or quite simply the model of the system. System dynamics deals with the mathematical modelling of dynamic systems and performance analyses of such systems in order to understand the dynamic nature of the system and improving system performance. The definition of several terms, classification of dynamic systems, modeling of dynamic systems, and analysis and design of dynamic systems are presented in this chapter. 0.2 DYNAMIC SYSTEM DEFINITION A system is a combination of parts or elements or components intended to act together to accomplish an objective. A part or element or component is a single functioning unit of a system. For example, an automobile is a system whose elements are the wheels, suspension, carbody, and so forth. A static element is one whose output at any given time depends only on the input at that time while a dynamic element is one whose present output depends on past inputs. In the same way we also speak of static and dynamic systems. A static system contains all static elements while a dynamic system contains at least one dynamic element. In a dynamic system, the output changes with time if it is not in a state of equilibrium. A dynamic system undergoing a time-varying interchange or dissipation of energy among or within its elementary storage or dissipative devices is said to be in a dynamic state. All of the elements in general are called passive, i.e., they are incapable of generating net energy. A dynamic system composed of a finite number of storage elements is said to be lumped or discrete, while a system containing elements, which are dense in physical space, is called continuous. The analytical description of the dynamics of the discrete case is a set of ordinary differential equations, while for the continuous case it is a set of partial differential equations. The analytical formation of a dynamic system depends upon the kinematic or geometric constraints and the physical laws governing the behaviour of the system. 0.3 CLASSIFICATION OF DYNAMIC SYSTEM MODELS In order to deal in an efficient and systematic way with problems involving time dependent behaviour, we must have a description of the objects or processes involved and such a description

2 2 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB is called a model. The model used most frequently is the mathematical model, which is a description in terms of mathematical relations, and represents an idealization of the actual physical system. For describing a dynamic system, these relations will consist of differential or difference equations. Predicting the performance from a model is called analysis. The model's purpose partly determines its form so that the purpose influences the type of analytical techniques used to predict the dynamic system s behaviour. There are many types of analytical techniques available and their applicability depends on the purpose of the analysis. The physical properties, or characteristics, of a dynamic system are known as parameters. In general, real systems are continuous and their parameters distributed. However, in most cases, it is possible to replace the distributed characteristics of a system by discrete ones. In other words, many variables in a physical system are functions of location as well as time. If we ignore the spatial dependence by choosing a single representative value, then the process is called lumping, and the model of a lumped element or system is called a lumped-parameter model. In a dynamic system the independent variable in the model then would be time only. The model will be an ordinary differential equation, which includes time derivatives but not spatial derivatives. If spatial dependence is included then the resulting model is known as a distributed-parameter model in which the independent variables are the spatial coordinates as well as time. It consists of one or a set of partial differential equations containing partial derivatives with respect to the independent variables. Discrete systems are simpler to analyse than distributed ones. Dynamic systems are classified according to their behaviour as either linear or nonlinear. If the dependent variables in the system differential equation(s) appear to the first power only and there are no cross products thereof, then the system is called linear. If there are fractional or higher powers, then the system is non-linear. On the other hand, if the systems contain terms in which the independent variables appear to powers higher than one or two fractional powers, then they are known as systems with variable coefficients. Thus, the presence of a time varying coefficient does not make a model non-linear. Models with constant coefficients are known as time-invariant or stationary models, while those with variable coefficients are time-variant or non-stationary. If there is uncertainty in the value of the model s coefficients or inputs then often, stochastic models are used. In a stochastic model, the inputs and coefficients are described in terms of probability distributions involving their means and variances, etc. 0.4 DISCRETE AND CONTINUOUS SYSTEMS In a discrete time-system, one or more variables can change only at discrete instants of time. In a continuous-time system, the signals involved are continuous in time. The mathematical model of continuous systems often results in a system of differential equations. 0.5 MATHEMATICAL MODELING OF DYNAMIC SYSTEMS A simplified mathematical model of the physical system can determine the overall complex behaviour of the dynamic system. The analysis of a physical system may be summarised by the following four steps: (a) Mathematical Modeling of a Physical System The purpose of the mathematical modeling is to determine the existence and nature of the system, its features and aspects, and the physical elements or components involved in the physical system. Necessary assumptions are made to simplify the modeling. Implicit assumptions are used that include:

3 INTRODUCTION TO ENGINEERING SYSTEM DYNAMICS 3 (i) A physical system can be treated as a continuous piece of matter (ii) Newton s laws of motion can be applied by assuming that the earth is an internal frame (iii) Ignore or neglect the relativistic effects All components or elements of the physical system are linear. The resulting mathematical model may be linear or non-linear, depending on the given physical system. Generally speaking, all physical systems exhibit non-linear behaviour. Accurate mathematical modeling of any physical system will lead to non-linear differential equations governing the behaviour of the system. Often, these non-linear differential equations have either no solution or difficult to find a solution. Assumptions are made to linearise a system, which permits quick solutions for practical purposes. The advantages of linear models are the following: (i) their response is proportional to input (ii) superposition is applicable (iii) they closely approximate the behaviour of many dynamic systems (iv) their response characteristics can be obtained from the form of system equations without a detailed solution (v) a closed-form solution is often possible (vi) numerical analysis techniques are well developed, and (vii) they serve as a basis for understanding more complex non-linear system behaviours. It should, however, be noted that in most non-linear problems it is not possible to obtain closed-form analytic solutions for the equations of motion. Therefore, a computer simulation is often used for the response analysis. When analysing the results obtained from the mathematical model, one should realise that the mathematical model is only an approximation to the true or real physical system and therefore the actual behaviour of the system may be different. (b) Formulation of Governing Equations Once the mathematical model is developed, we can apply the basic laws of nature and the principles of dynamics and obtain the differential equations that govern the behaviour of the system. A basic law of nature is a physical law that is applicable to all physical systems irrespective of the material from which the system is constructed. Different materials behave differently under different operating conditions. Constitutive equations provide information about the materials of which a system is made. Application of geometric constraints such as the kinematic relationship between displacement, velocity, and acceleration is often necessary to complete the mathematical modeling of the physical system. The application of geometric constraints is necessary in order to formulate the required boundary and/or initial conditions. The resulting mathematical model may be linear or non-linear, depending upon the behaviour of the elements or components of the dynamic system. (c) Mathematical Solution of the Governing Equations The mathematical modeling of a physical vibrating system results in the formulation of the governing equations of motion. Mathematical modeling of typical systems leads to a system of differential equations of motion. The governing equations of motion of a system are solved to find the response of the system. There are many techniques available for finding the solution, namely, the standard methods for the solution of ordinary differential equations, Laplace transformation methods, matrix methods, and numerical methods. In general, exact analytical solutions are available for many linear dynamic systems, but for only a few non-linear systems. Of course, exact analytical solutions are always preferable to numerical or approximate solutions.

4 4 SOLVING ENGINEERING SYSTEM DYNAMICS PROBLEMS WITH MATLAB (d) Physical Interpretation of the Results The solution of the governing equations of motion for the physical system generally gives the performance. To verify the validity of the model, the predicted performance is compared with the experimental results. The model may have to be refined or a new model is developed and a new prediction compared with the experimental results. Physical interpretation of the results is an important and final step in the analysis procedure. In some situations, this may involve (a) drawing general inferences from the mathematical solution, (b) development of design curves, (c) arrive at a simple arithmetic to arrive at a conclusion (for a typical or specific problem), and (d) recommendations regarding the significance of the results and any changes (if any) required or desirable in the system involved. 0.6 ANALYSIS AND DESIGN OF DYNAMIC SYSTEMS Analysis System analysis means the determination of the behaviour, performance, or response of a system to a given set of inputs that are applied to a given configuration of defined parts, elements or components. The analysis of a dynamic system requires often the development of a mathematical model for each component and combining these models in order to build a model of the complete system. The model should be sufficiently sophisticated to show the significant outputs in order to make use of available methods of analysis. The system parameters in the model can then be varied systematically to obtain a number of solutions in order to make the interpretations and the establishing valuable conclusions Synthesis By system synthesis we mean the establishment of the composition or combination of parts or elements or components such that the system behaves, performs, or responds according to a given set of desired system characteristics or specifications. In analysis, the only unknowns are the system outputs, while in the synthesis the outputs are known and most of the system parts or elements or components are unknown. In general, synthesis procedure is totally mathematical form from the beginning to the conclusion of the design process Design Systems are designed to perform specific tasks. Synthesis is the establishment of the system configuration given the performance specifications while design is the determination of dimensions and other numerical parameters for a given system configuration. System design is the process of determining a system that accomplishes a given task. In general, the design procedure is a trial and error process System Design Procedure It is in general possible to have many designs that can meet the performance specifications. The various steps involved in the design process are summarised below:

5 INTRODUCTION TO ENGINEERING SYSTEM DYNAMICS 5 (a) Conceptual Analyses and Function Structures The first phase of the design process is the method for conceptualising the function needs of the design. This method must be structured yet flexible enough to allow for innovation. The method used to develop the function concepts is referred to as the Function Structure. (b) Mathematical Quantification and Evaluation of the Design The two next phases in the design process must now be considered: these are mathematical quantification and design evaluation. The quantitative connection between the function concepts and configurations is critically important for any design evaluation. Axiomatic Design has allowed the connection of function concepts and configurations. (c) Evaluation of Designs Configurations At this phase in the design process, there are three important considerations. They are: 1. Chose no more than three configurations based on a best selection process consisting of market needs and functional needs. 2. Each of the three configurations is evaluated for functional robustness. 3. As a last phase, the configurations are evaluated for a balance between reliability and economy. The results from the evaluations are used to modify the configurations and further reduce the number of alternatives. In the case of major configurational problems based on omissions of function needs during any of the evaluations, the designer must revisit the function concepts initially selected. All designs must have appropriate function concepts supporting its alternative configurations. (d) Prototyping and Testing The computer tools at the disposal of engineers are impressive. Computer Aided Design (CAD) and Solid Modelling allow the engineer to analyse the design for component tolerance, functional response (animation), reliability, and cost projections. This is also an important step for precisely estimating other economic factors, such as manufacturing costs. The design configuration file can even be downloaded for a CNC machine for rapid prototyping. Testing the actual prototypes is conducted using several Design of Experiment (DOE) techniques. The Taghuchi method is recommended for optimising the design and identifying functional reliability. SUMMARY A brief introduction to engineering system dynamics and the definition of several terms used in dynamic systems, classification of dynamic systems, modeling of dynamic systems, and analysis and design of dynamic systems are presented in this chapter.

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