and associated mass moments of inertia are given in Fig. 9.13. General rigid bodies are discussed in section “Inertia Properties.” There are several useful concepts and theorems related to the properties of rigid bodies that can be helpful at this point. First, if the mass moment of inertia is known about an axis through its center of mass (I G ), then Steiner’s theorem (parallel axis theorem) relates this moment of inertia to that about another axis a distance d away by I = I G + md 2 , where m is the mass of the body. It is also possible to build a moment of inertia for composite bodies, in those situations where the individual motion of each body is negligible. A useful concept is the radius of gyration, k, which is the radius of an imaginary cylinder of infinitely small wall thickness having the same mass, m, and the same mass moment of inertia, I, as a body in question, and given by, k = . The radius of gyration can be used to find an equivalent mass for a rolling body, say, using m eq = I/k 2 . Coupling Mechanisms Numerous types of devices serve as couplers or power transforming mechanisms, with the most common being levers, gear trains, scotch yokes, block and tackle, and chain hoists. Ideally, these devices and their analogs in other energy domains are power conserving, and it is useful to represent them using a 2-port model. In such a model element, the power in is equal to the power out, or in terms of effort-flow pairs, e 1 f 1 = e 2 f 2 . It turns out that there are two types of basic devices that can be represented this way, based on the relationship between the power variables on the two ports. For either type, a relationship between two of the variables can usually be identified from geometry or from basic physics of the device. By imposing the restriction that there is an ideal power-conserving transformation inherent in the device, a second relationship is derived. Once one relation is established the device can usually be classified as a transformer or gyrator. It is emphasized that these model elements are used to represent the ideal power-conserving aspects of a device. Losses or dynamic effects are added to model real devices. A device can be modeled as a transformer when e 1 = me 2 and mf 1 = f 2 . In this relation, m is a transformer modulus defined by the device physics to be constant or in some cases a function of states of the system. For example, in a simple gear train the angular velocities can be ideally related by the ratio of pitch radii, and in a slider crank there can be formed a relation between the slider motion and the crank angle. Consequently, the two torques can be related, so the gear train is a transformer. A device can be modeled as a gyrator if e 1 = rf 2 and rf 1 = e 2 , where r is the gyrator modulus. Note that this model can represent FIGURE 9.13 Mass moments of inertia for some common bodies. 2 Jmr Pointmassatradiusr Rod or bar about centroid c c Cylinder about axis c-c (radius r) L 2 12 mL J 2 1 2 Jmr d L 22 (4) 12 m Jdl Short bar about pivot c c Cylindrical shell about axis c -c (inner radius r) 2 Jmr Slender bar case, d=0 If outer radius is R,and not a thin shell, 22 1 2 ( ) JmRr I/m ©2002 CRC Press LLC and associated mass moments of inertia are given in Fig. 9.13. General rigid bodies are discussed in section “Inertia Properties.” There are several useful concepts and theorems related to the properties of rigid bodies that can be helpful at this point. First, if the mass moment of inertia is known about an axis through its center of mass (I G ), then Steiner’s theorem (parallel axis theorem) relates this moment of inertia to that about another axis a distance d away by I = I G + md 2 , where m is the mass of the body. It is also possible to build a moment of inertia for composite bodies, in those situations where the individual motion of each body is negligible. A useful concept is the radius of gyration, k, which is the radius of an imaginary cylinder of infinitely small wall thickness having the same mass, m, and the same mass moment of inertia, I, as a body in question, and given by, k = . The radius of gyration can be used to find an equivalent mass for a rolling body, say, using m eq = I/k 2 . Coupling Mechanisms Numerous types of devices serve as couplers or power transforming mechanisms, with the most common being levers, gear trains, scotch yokes, block and tackle, and chain hoists. Ideally, these devices and their analogs in other energy domains are power conserving, and it is useful to represent them using a 2-port model. In such a model element, the power in is equal to the power out, or in terms of effort-flow pairs, e 1 f 1 = e 2 f 2 . It turns out that there are two types of basic devices that can be represented this way, based on the relationship between the power variables on the two ports. For either type, a relationship between two of the variables can usually be identified from geometry or from basic physics of the device. By imposing the restriction that there is an ideal power-conserving transformation inherent in the device, a second relationship is derived. Once one relation is established the device can usually be classified as a transformer or gyrator. It is emphasized that these model elements are used to represent the ideal power-conserving aspects of a device. Losses or dynamic effects are added to model real devices. A device can be modeled as a transformer when e 1 = me 2 and mf 1 = f 2 . In this relation, m is a transformer modulus defined by the device physics to be constant or in some cases a function of states of the system. For example, in a simple gear train the angular velocities can be ideally related by the ratio of pitch radii, and in a slider crank there can be formed a relation between the slider motion and the crank angle. Consequently, the two torques can be related, so the gear train is a transformer. A device can be modeled as a gyrator if e 1 = rf 2 and rf 1 = e 2 , where r is the gyrator modulus. Note that this model can represent FIGURE 9.13 Mass moments of inertia for some common bodies. 2 Jmr Pointmassatradiusr Rod or bar about centroid c c Cylinder about axis c-c (radius r) L 2 12 mL J 2 1 2 Jmr d L 22 (4) 12 m Jdl Short bar about pivot c c Cylindrical shell about axis c -c (inner radius r) 2 Jmr Slender bar case, d=0 If outer radius is R,and not a thin shell, 22 1 2 ( ) JmRr I/m ©2002 CRC Press LLC 10 Fluid Power Systems 10.1 Introduction Fluid Power Systems • Electrohydraulic Control Systems 10.2 Hydraulic Fluids Density • Viscosity • Bulk Modulus 10.3 Hydraulic Control Valves Principle of Valve Control • Hydraulic Control Valves 10.4 Hydraulic Pumps Principles of Pump Operation • Pump Controls and Systems 10.5 Hydraulic Cylinders Cylinder Parameters 10.6 Fluid Power Systems Control System Steady-State Characteristics • System Dynamic Characteristics • E/H System Feedforward-Plus-PID Control • E/H System Generic Fuzzy Control 10.7 Programmable Electrohydraulic Valves 10.1 Introduction Fluid Power Systems A fluid power system uses either liquid or gas to perform desired tasks. Operation of both the liquid systems (hydraulic systems) and the gas systems (pneumatic systems) is based on the same principles. For brevity, we will focus on hydraulic systems only. A fluid power system typically consists of a hydraulic pump, a line relief valve, a proportional direction control valve, and an actuator (Fig. 10.1). Fluid power systems are widely used on aerospace, industrial, and mobile equipment because of their remarkable advantages over other control systems. The major advantages include high power-to-weight ratio, capability of being stalled, reversed, or operated inter- mittently, capability of fast response and acceleration, and reliable operation and long service life. Due to differing tasks and working environments, the characteristics of fluid power systems are different for industrial and mobile applications (Lambeck, 1983). In industrial applications, low noise level is a major concern. Normally, a noise level below 70 dB is desirable and over 80 dB is excessive. Industrial systems commonly operate in the low (below 7 MPa or 1000 psi) to moderate (below 21 MPa or 3000 psi) pressure range. In mobile applications, the size is the premier concern. Therefore, mobile hydraulic systems commonly operate between 14 and 35 MPa (2000–5000 psi). Also, their allowable temperature operating range is usually higher than in industrial applications. Qin Zhang University of Illinois Carroll E. Goering University of Illinois ©2002 CRC Press LLC 11 Electrical Engineering 11.1 Introduction 11.2 Fundamentals of Electric Circuits Electric Power and Sign Convention • Circuit Elements and Their i-v Characteristics • Resistance and Ohm’s Law • Practical Voltage and Current Sources • Measuring Devices 11.3 Resistive Network Analysis The Node Voltage Method • The Mesh Current Method • One-Port Networks and Equivalent Circuits • Nonlinear Circuit Elements 11.4 AC Network Analysis Energy-Storage (Dynamic) Circuit Elements • Time- Dependent Signal Sources • Solution of Circuits Containing Dynamic Elements • Phasors and Impedance 11.1 Introduction The role played by electrical and electronic engineering in mechanical systems has dramatically increased in importance in the past two decades, thanks to advances in integrated circuit electronics and in materials that have permitted the integration of sensing, computing, and actuation technology into industrial systems and consumer products. Examples of this integration revolution, which has been referred to as a new field called Mechatronics , can be found in consumer electronics (auto-focus cameras, printers, microprocessor-controlled appliances), in industrial automation, and in transportation systems, most notably in passenger vehicles. The aim of this chapter is to review and summarize the foundations of electrical engineering for the purpose of providing the practicing mechanical engineer a quick and useful reference to the different fields of electrical engineering. Special emphasis has been placed on those topics that are likely to be relevant to product design. 11.2 Fundamentals of Electric Circuits This section presents the fundamental laws of circuit analysis and serves as the foundation for the study of electrical circuits. The fundamental concepts developed in these first pages will be called on through the chapter. The fundamental electric quantity is charge , and the smallest amount of charge that exists is the charge carried by an electron, equal to (11.1) As you can see, the amount of charge associated with an electron is rather small. This, of course, has to do with the size of the unit we use to measure charge, the coulomb (C), named after Charles Coulomb. However, the definition of the coulomb leads to an appropriate unit when we define electric current, q e 1.602 10 19– coulomb×–= Giorgio Rizzoni Ohio State University ©2002 CRC Press LLC . as the foundation for the study of electrical circuits. The fundamental concepts developed in these first pages will be called on through the chapter. The fundamental electric quantity is charge . course, has to do with the size of the unit we use to measure charge, the coulomb (C), named after Charles Coulomb. However, the definition of the coulomb leads to an appropriate unit when we. Ideally, these devices and their analogs in other energy domains are power conserving, and it is useful to represent them using a 2-port model. In such a model element, the power in is equal to the