Systems Dynamics Dr. Mohammad Kilani Class 1 Introduction

Introduction to System Dynamics  Engineering system dynamics is a discipline that studies the dynamic behavior of various systems, such as mechanical, electrical, fluid, and thermal, either as isolated entities or in their interaction, the case where they are coupled-field (or multiple-field) systems.  This discipline emphasizes that systems belonging to different fields are described by similar mathematical models (expressed most often as differential equations); therefore, the same mathematical apparatus can be utilized for analysis or design.

Systems  A system is a combination of components acting together to perform a specific objective. A component is a single functioning unit of a system.  As an example on a systems, one can give an automobile, a bike, a measuring instrument (voltmeter, ammeter, scale balance, digital balance), hydraulic actuators, pneumatic actuators, etc.  The concept of a system can be extended to abstract dynamic phenomena, such as those encountered in economics, transportation, population growth, and biology.

Example Systems: Weight Scale

Example Systems: Automobile

Example Systems: Water Tank

Example Systems: Shower Hot Cold Flow Valve

Example Systems: Strain Gage

Example Systems: Fired Heater

Heat Exchanger Main Valve Furnace Initial Temperature Rise Flow Rate Adjustment Feed Pump Flow Rate Final Temperature Rise

Modeling using Mathematical Equations  Mathematical models are the set of equations that represent the physical or engineering system of interest.  The set of equations constituting the model are usually based on a few basic engineering principles that are mixed with some empirical information.

Modeling using Mathematical Equations  A number of different approaches are used in mathematical modeling including setups involving simple equations, ordinary differential equations, and partial differential equations.  Setups involving ordinary differential equations include, mechanical translational mechanical rotational systems, fluid flow systems, and electrical networks. In these models, a components is normally modeled using one lumped parameter.  Setups including partial derivatives are commonly found when one is interested in the distribution of a variable in space and time. Example, heat transfer – temperature distribution, fluid flow – velocity distribution, stress – strain distribution, electrostatic potential distribution and electromagnetic potential distribution.

Modeling and Simulation in the Design Process Need Satisfied? Identification of a need Product Specification Evaluate Propose Concept Production Yes No Modify or Update

Modeling and Simulation in the Design Process PrototypeModeling SimulationTesting Need Satisfied? Identification of a need Product Specification Evaluate Propose Concept Production Yes No Modify or Update Evaluate

Modeling and Simulation of Dynamic Systems  The discipline dealing with the mathematical modeling and analysis of systems made of a number of interconnected devices and processes for the purpose of understanding their time- dependent behavior is referred to as Systems Dynamics  While other subjects, such as Newtonian dynamics and electrical circuit theory, also deal with time-dependent behavior, system dynamics emphasizes methods for handling applications containing multiple types of components and processes such as electromechanical devices, electrohydraulic devices, and fluid-thermal processes.

Modeling and Simulation of Dynamic Systems  The goal of system dynamics is to understand the time-dependent behavior of a system of interconnected devices and processes as a whole. The modeling and analysis methods used in system dynamics must be properly selected to reveal how the connections between the system elements affect its overall behavior.  Because systems of interconnected elements often require a control system to work properly, control system design is a major application area in system dynamics.

An Example Control System

Steam Turbine Speed Control System (Watt’s Governor)

Watt’s Governor Block Diagram Desired Speed (Initial Spring Compression) Fly Balls Centrifugal Force Sleeve - Spring System Lever Main Valve Turbine Bevel Gears Turbine Speed Fly Ball Axis Speed Sleeve’s Position Disk’s Position Steam Flow Rate + -

Desired Speed (Initial Spring Compression) Fly Balls Centrifugal Force Σ Lever Main Valve Turbine Bevel Gears Turbine Speed Fly Ball Axis Speed Disk’s Position Steam Flow Rate + - Compr ession Spring Sleeve - Spring System Screw and Nut Displace ment Net force on Sleeve Spring Force Nut Position Nut Angle

Elements of a Measurement System Measured Variable Sensor Variable Conversion Element Signal Processor Use of Measurement at Remote Location Signal Transmission Presentation / Recording Unit Output

Static Systems  A system whose current output depends only on its current input is known as static system.  The output of a static system remains constant if the input does not change. The output changes only when the input changes, and it responds to the input signal instantaneously.  Static systems are sometimes referred to as zero-order systems. l xixi xoxo l1l1 l2l2 xixi xoxo vivi R1R1 R2R2 VoVo

Dynamic Systems  A system is called dynamic if its present response (output) depends on past excitation (input)  In a dynamic system, the output changes with time if the system is not in a state of equilibrium. In this course, we are concerned mostly with dynamic systems. lxixi xoxo m k c xixi xoxo

Dynamic Systems m k c xixi xoxo

Mathematical Modeling of Dynamic Systems  The evaluation phase of the design process for a system allows for predicting the performance of the system before the system is 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 with the time variable being the independent parameter.

Linear, Time-Invariant Differential Equations  A linear, time-invariant differential equation is an equation in which a dependent variable and its derivatives appear as linear combinations.  Since the coefficients of all terms are constant, a linear, time-invariant differential equation is also called a linear, constant-coefficient differential equation.

Linear, Time-Varying Differential Equations  In a linear, time-varying differential equation, the dependent variable and its derivatives appear as linear combinations, but one or more of the coefficients of terms may involve the independent variable.

Nonlinear Differential Equations  An equation is not linear if it contains powers or other functions or products of the dependent variables or its derivatives.  A differential equation is called nonlinear if it is not linear

Linear Systems and Nonlinear Systems  If the differential equations that constitute the model of a system is linear., the system is called a linear system.  The most important property of linear systems is that the principle of superposition is applicable. This principle states that the response produced by simultaneous applications of two different excitations or inputs is the sum of the 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.  In an experimental investigation of a dynamic system, if cause and effect are proportional, thereby implying that the principle of superposition holds, the system can be considered linear.

Linear Systems and Nonlinear Systems  The most important characteristic of nonlinear systems is that the principle of superposition is not applicable. 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 linearize a nonlinear system near the operating condition.

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. (These instants may specify the times at which some physical measurement is performed or the times at which the memory of a digital computer is read out.)  Discrete-time systems that involve digital signals and, possibly, continuous-time signals as well may be described by difference equations after the appropriate discretization of the continuous-time signals. The materials in this course applied mainly to continuous-time systems.