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Chapter 4 Theoretical Foundation and Background Material: Modeling of Dynamic Systems

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## Chapter 4 Theoretical Foundation and Background Material: Modeling of Dynamic Systems

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**Chapter 4Theoretical Foundation and Background Material:**Modeling of Dynamic Systems Automatic Control Systems, 9th Edition F. Golnaraghi & B. C. Kuo**0, p. 147**Main Objectives of This Chapter • To introduce modeling of mechanical systems • To introduce modeling of electrical systems • To introduce modeling thermal & fluid systems • To discuss sensors and actuators • To discuss linearization of nonlinear systems • To discuss analogies**1, p. 148**4-1 Introduction to Modeling of Mechanical Systems • Mass: • Translation motion: a motion that takes places along a straight or curved path acceleration a, velocity v, displacement y • Newton’s law of motion: • Force equation:**1, p. 149**Linear Translation Motion • Mass: • Linear spring: • Friction for translation motion: • viscous friction, static friction, Coulomb friction preload tension**1, p. 150**Friction for Translation Motion (a) Viscous friction (b) Static friction (c) Coulomb friction**1, p. 151**Basic Translational Mechanical System**1, p. 152**Example 4-1-1 • Force equation: Transfer function:**1, p. 152**Example 4-1-1 (cont.) • Force equation: • State space form: with initial conditions without initial conditions**1, p. 154**Example 4-1-2 • Force equation: • Transfer function:**1, p. 155**Example 4-1-2 (cont.) • State equations:**1, p. 156**Example 4-1-3 State equations: State variables:**1, p. 157**Rotational Motion • Rotational motions: a motion about a fixed axis angular displacement , angular velocity , angular acceleration • Newton’s law of motion for rotational motion: • Inertia (J): a circular disk or shaft of radius r and mass M • Torque equation: a torque T is applied to a body with inertia J**1, p. 158**Torsional Spring & Friction • Torsional Spring: • Friction for Rotational Motion: • Viscous friction: • Static friction: • Coulomb friction: preload torque**1, p. 159**Basic Rotational Mechanical System**1, p. 158**Example 4-1-4 Torque or moment equation:**1, p. 160**Example 4-1-5 Torque equation:**1, p. 160**Example 4-1-5 (cont.) • Three energy-storage elements: Jm, JL, K 3 state variables • State variables: State equations:**1, p. 161**Conversion between Transiational and Rotational Motions • The equivalent inertia that the motor see: L: the lead of the screw equivalent**1, p. 162**Gear Train**1, p. 163**Gear Train with Friction and Inertia reflecting from gear 2 to gear 1 Figure 4-23**1, p. 164**Backlash and Dead Zone**2, p. 165**4-2 Introduction to Modeling Simple Electrical Systems**2, p. 166**Example 4-2-1 KVL: Current inC: State equations: State variables:**2, p. 166**Example 4-2-1 (cont.) State equations:**2, p. 167**Example 4-2-1 (cont.) • Transfer functions:**2, p. 168**Example 4-2-2 State variables**2, p. 169**Example 4-2-2 (cont.) Transfer functions:**2, p. 170**Example 4-2-3 Find the differential equation of the system.**2, p. 171**Example 4-2-4 Find the differential equation of the system**2, p. 172**Example 4-2-5 Node equation at e1:**3, p. 173**4-3 Modeling of Active Electrical Elements: Operational Amplifiers Idea Op-Amp: • The voltage between the + and terminals is zero. e+ = e, virtual ground or virtual short. • The currents into the + and terminals are zero. input impedance is infinite • The impedance seen looking into the output terminate is zero. the output is an idea voltage source. • eo = A(e+ e),gain A A**3, p. 174**Sum and Difference Negative sum Positive sum Difference**3, p. 174**First-Order Op-Amp Configirations 0 V**3, p. 175**Inverting Op-Amp Transfer Functions**3, p. 176**Table 4-4 (cont.)**3, p. 176**Example 4-3-1 Op-amp realization of PID controller: Proportional: Integral: Derivative: Output: Transfer function:**4, p. 177**4-4 Introduction to Modeling of Thermal Systems Elementary Heat Transfer Properties • Heat transfer is related to the heat flow rateq: • Capacitance (C): storage or (discharge) of heat in a body • The capacitance is related to the change of the body temperatureT with respective to time and the rate of heat flowq: • Three types of heat transfer: conduction, convection, or radiation. volume material density material specific heat**4, p. 178**Conduction • This type of heat transfer happens in solid materials due to a temperature difference between two surfaces. • Heat tends to travel from the hot to the cold region • One-direction heat conduction flow: k: Thermal conductivity related to the material used R: Thermal resistance A: Area normal to thedirection of heat flow x**4, p. 178**Convention • Convention occurs between a solid surface and a fluid exposed to it. h: the coefficient of convention heat transfer T = Tb Tf**4, p. 179**Radiation • The rate of heat transfer through radiation between two separate objects is determined by the Stephan-Boltzmann law, : Stephan-Boltzmann constant**4, p. 179**Table 4-5 Basic Thermal System Properties**4, p. 180**Example 4-4-1 Find the equations of heat transfer process = convention rate:**5, p. 181**4-5 Introduction to Modeling of Fluid Systems Elementary Fluid and Gas Properties • Mass flow rate: : fluid densityq: net fluid flow ratem: net mass flowqi: ingoing fluidqo: outgoing fluid**5, p. 181**Conservation of mass & Fluid Capacitance • Conservation of mass: • Capacitance – Incompressible Fluid:the ratio of the fluid flow rate q to the rate of pressure Mcv: mass of control volumeV: container volume 1. conservation of volume: 2. incompressible fluid:( is constant)**5, p. 182**Examples 4-5-1 & 4-5-2 • Example 4-5-1 The pressure in the tank: • Example 4-5-2 The pressure rate:**5, p. 183**Capacitance – Pneumatic Systems • Capacitance expresses the rate of change of the fluid mass with respect to pressure: • For a constant container: • Polytropic process: (n = 0 ~ : polytropic exponent) • Perfect gas law: (Rg: gas constant)**5, p. 183**General Gas Law • (the mass m is constant in a polytropic process) • For a constant temperature (an isothermal process): • For a constant pressure (an isobaric process): • For a constant volume (an isovolumetric process): • For a reversible adiabatic (an isentropic process): cp: the specific heat of gas at constant pressure cv: the specific heat of gas at constant volume**5, p. 184**Inductance – Incompressible Fluids • Inductance: fluid inertance in relation to the inertia of a moving fluid inside a passage (line or a pipe). • Newton’s second law: Fluid inductance**5, p. 184**Resistance – Incompressible Fluids • As in electrical systems, fluid resistors dissipate energy. • The force resisting the fluid passing through a passage: • Laminar flow: • Turbulent:**5, p. 185**Table 4-6: Resistance for Laminar Flows