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Introduction to Modeling and Simulations.ppt

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Introduction to Modeling and Simulations.ppt

  1. 1. Chapter 1: Introduction to Dynamic Systems and Control (DSC) • In engineering problems, there is a need to understand and determine the dynamic response of a physical system that may involve several components • These efforts involve modeling, analysis, simulation, and design of physical systems – Building a prototype system and conducting experiments/tests is not feasible and/or is too expensive for preliminary design – Mathematical modeling and analysis of engineering systems greatly aid the design process
  2. 2. DSC: Definitions • System: A combination of components acting together to perform a specified objective. The components or interacting elements have cause-and-effect (or input/output) relationships. We will investigate mechanical, electrical, fluid, and mixed systems. • Dynamic system: The current output variables of a system depend on the initial conditions (or stored energy) of the system and/or the previous input variables. The dynamic variables of the system (e.g., displacement, velocity, voltage, pressure, etc) vary with time. • Modeling: The process of applying the appropriate fundamental physical laws in order to derive mathematical equations that adequately describe the physics of the engineering system. • Mathematical models: A mathematical description of a system’s behavior, usually a set of differential equations for a dynamic system
  3. 3. DSC: Definitions (2) • Simulation: The process of obtaining the system’s dynamic response by numerically solving the governing modeling equations. Simulation involves numerical integration of the model’s differential equations and is performed by digital computers and simulation software. • System analysis: The use of analytical calculations or numerical simulation tools to determine the system response in order to assess its performance.
  4. 4. 1.2 Classification of Dynamic Systems • Spatial characteristics – Distributed (PDEs) vs. lumped parameters (ODEs) • Time variable continuity – “Analog” vs. “digital” • Time dependence – Time-varying vs. time-invariant parameters • Superposition property – Linear vs. nonlinear systems Red bold-face: focus of DSC textbook
  5. 5. Spatial Characteristics • Distributed system: infinite number of “internal” variables; system is governed by PDE – Example: continuous twist angle for shaft under external torque • Lumped system: finite number of “internal” variables; system is governed by ODE – Example: lump all inertia, stiffness, etc into single elements; single twist angle of free end of shaft under external torque Red bold-face: focus of DSC textbook
  6. 6. Time Variable Continuity Red bold-face: focus of DSC textbook Continuous-time system (“analog”) Discrete-time system (“digital”)
  7. 7. Time Dependence Red bold-face: focus of DSC textbook • Time-varying system: system parameters change with time – Example: springs/shocks “wear out” over time • Time-invariant system: constant system parameters – Example: spring stiffness remains constant over time – Therefore, identical inputs and initial conditions produce identical dynamic responses every trial
  8. 8. Superposition (Linear vs. Nonlinear) • Linear systems obey the superposition property : 1. If u1 is an input, and y1 = f(u1) is the corresponding output, then ay1 = f(au1) , where a = any constant 2. If y1 = f(u1) and y2 = f(u2) , then y1 + y2 = f(u1+u2) – Nonlinear systems do not obey these properties – All physical systems are nonlinear. However, if we confine the input/output variables to a restricted (nominal) range, then we can replace a nonlinear system with a linear model. – Linear dynamic systems are governed by linear differential equations
  9. 9. Linear and Nonlinear ODEs • Examples of linear ODEs: • Examples of nonlinear ODEs: u x x x 2 9 4       u x t x x 2 ) 3 cos 9 ( 4        Linear time-invariant (LTI) ODE Linear time varying ODE u x x x x 2 9 4       u x x x 2 3 6 2      
  10. 10. 1.3 Modeling Dynamic Systems • Mathematical models are obtained by applying the appropriate laws of physics to each element of a system – Some system parameters (such as damping) may be unknown, or these parameters are often determined through experiments which lead to empirical relations • Engineering judgment must be used to trade model complexity with accuracy of the analysis – Nonlinearities (such as gear backlash) are often ignored in preliminary design studies in order to derive linear models – Sometimes, low-order linear models can be solved analytically – Furthermore, simulations (e.g., MATLAB/Simulink) are easier to construct with low-order linear models and therefore system- analysis time is reduced
  11. 11. Modeling Dynamic Systems (2) • Engineers must remember that the results from a model and/or simulation are only approximate and are valid only to the extent of the assumptions used to derive the model • The model must be sufficiently sophisticated to demonstrate the significant features of the dynamic response without becoming too cumbersome for available analysis tools – Higher-order, complex nonlinear models typically require smaller integration time steps to accurately solve the governing differential equations, which increases computer run time – Consequently, there is usually a trade-off between model complexity and analysis time – The validity of a mathematical model can often be verified by comparing the model solution (i.e., simulation results) with experimental results
  12. 12. High-Fidelity Modeling Examples • Shuttle Vehicle Dynamics (SVD) was a computer simulation for analyzing the separation dynamics between the Space Shuttle and its solid rocket boosters – SVD uses mathematical models for aerodynamic forces, propulsion forces, spring and damper forces at interconnection points, etc • Shuttle Avionics Integration Lab (SAIL) at NASA Johnson Space Center was used to simulate the dynamics of the entire Space Shuttle mission profile – SAIL was composed of “hardware in the loop” (such as sensors and cockpit displays) mixed with mathematical models (such as aerodynamic force models, gravity force models) and flight software (such as guidance, navigation, and control functions) – SAIL results compared very well with actual measured flight
  13. 13. Simulation Tools • Simulink is a numerical simulation tool that is part of the MATLAB software package developed by MathWorks – Uses a graphical user interface (GUI) to develop a block diagram representation of dynamic systems. – Simulink is used by engineers in industry and academia. – Constructing system models with Simulink is relatively easy and therefore it is often used to build simple models during the preliminary design stage.
  14. 14. Simulation Tools (2) • Caterpillar developed and uses a computer simulation tool called Dynasty to model and analyze integrated hydraulic systems and hydraulic controls – Dynasty models mechanical components (pistons, linkages, springs), electrical components (servos, solenoids), and fluid components (pressures in hoses, cylinders, accumulators).
  15. 15. Caterpillar’s Dynasty Software Dynasty software lets Caterpillar engineers take new vehicles for test rides long before physical prototypes are available. Taking sharp turns lets user see, for example, whether or not there is sufficient room for wheel dynamics and how the dump body shifts in turns. (article in MachineDesign.com, November 6, 2003, by Paul Dvorak) Dynasty lets users focus on performance modeling, not the underlying math and physics. Users build schematic machines by dragging and dropping components and connecting them together. The software comes with more than 230 components such as engines, controls, electronics, fluids, linkages, body structures, and drivelines. After building a model, the program converts components and associated design and performance data to symbolic equations.
  16. 16. Simulation Tools (3) • Boeing developed and uses a graphical simulation tool called EASY5 which can model complete integrated systems – User can construct integrated systems from menu of sub- systems (mechanical, electrical, or hydraulic) From MCS Software webpage: “Adams and MD Adams provide accurate and efficient multi-body dynamics and motion analysis of 3D mechanical systems. The core package (Adams/View, Adams/Solver, and Adams/PostProcessor) allows you to import geometry from most major CAD systems or to build a solid model of the mechanical system from scratch. Adams models can be integrated with EASY5 controls models directly, or via co-simulation for full multidiscipline analysis.”
  17. 17. 1.4 Objectives • Introduce students to the mathematical modeling of physical systems – Mechanical, electrical, fluid, and thermal systems – Show “real-world” examples from ASME articles, industry • Introduce students to analytical and numerical methods for obtaining a system’s dynamic response to various initial conditions and input functions – Analytical: solving ODEs “by hand” – Numerical: MATLAB and Simulink • Analyze and design feedback control systems in order to achieve a desirable system response
  18. 18. Course Outline • Part 1: Modeling dynamic systems (Chapters 2-5) – Mechanical, electrical, and fluid systems – Learn by studying real-world examples! • Part 2: Dynamic system analysis (Chapters 6-9) – Analytical and numerical methods – Time response, block diagrams, MATLAB/Simulink, frequency response, vibrations – Learn by solving real-world examples (Chapter 11) • Part 3: Introduction to control systems (Chapter 10) – Closed-loop feedback systems, control algorithms, closed-loop response – Real-world examples from industry/research (Chapter 11)

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