Product of Inertia. The product of inertia for a differential element dm is defined with respect to a set of two orthogonal planes as the product of the mass of the element and the perpendicular (or shortest) distances from the planes to the element. So, with respect to the y − z and x − z planes (z common axis to these planes), the contribution from the differential element to I xy is dI xy and is given by dI xy = xydm. As for the moments of inertia, by integrating over the entire mass of the body for each combination of planes, the products of inertia are (9.23) The product of inertia can be positive, negative, or zero, depending on the sign of the coordinates used to define the quantity. If either one or both of the orthogonal planes are planes of symmetry for the body, the product of inertia with respect to those planes will be zero. Basically, the mass elements would appear as pairs on each side of these planes. Parallel-Axis and Parallel-Plane Theorems. The parallel-axis theorem can be used to transfer the moment of inertia of a body from an axis passing through its mass center to a parallel axis passing through some other point (see also the section “Kinetic Energy Storage”). Often the moments of inertia are known for axes fixed in the body, as shown in Fig. 9.33(b). If the center of gravity is defined by the coordinates (x G , y G , z G ) in the x, y, z axes, the parallel-axis theorem can be used to find moments of inertia relative to the x, y, z axes, given values based on the body-fixed axes. The relations are where, for example, (I xx ) a is the moment of inertia relative to the x a axis, which passes through the center of gravity. Transferring the products of inertia requires use of the parallel-plane theorem, which provides the relations Inertia Tensor. The rotational dynamics of a rigid body rely on knowledge of the inertial properties, which are completely characterized by nine terms of an inertia tensor, six of which are independent. The inertia tensor is = I xy I yx xy dm m ∫ == I yz I zy yz dm m ∫ == I xz I zx xz dm m ∫ == I xx I xx () a my G 2 z G 2 +()+= I yy I yy () a mx G 2 z G 2 +()+= I zz I zz () a mx G 2 y G 2 +()+= I xy I xy () a mx G y G += I yz I yz () a my G z G += I zx I zx () a mz G x G += I I xx I xy – I xz – I yx – I yy I yz – I zx – I zy – I zz 0066-frame-C09 Page 40 Friday, January 18, 2002 11:00 AM ©2002 CRC Press LLC Product of Inertia. The product of inertia for a differential element dm is defined with respect to a set of two orthogonal planes as the product of the mass of the element and the perpendicular (or shortest) distances from the planes to the element. So, with respect to the y − z and x − z planes (z common axis to these planes), the contribution from the differential element to I xy is dI xy and is given by dI xy = xydm. As for the moments of inertia, by integrating over the entire mass of the body for each combination of planes, the products of inertia are (9.23) The product of inertia can be positive, negative, or zero, depending on the sign of the coordinates used to define the quantity. If either one or both of the orthogonal planes are planes of symmetry for the body, the product of inertia with respect to those planes will be zero. Basically, the mass elements would appear as pairs on each side of these planes. Parallel-Axis and Parallel-Plane Theorems. The parallel-axis theorem can be used to transfer the moment of inertia of a body from an axis passing through its mass center to a parallel axis passing through some other point (see also the section “Kinetic Energy Storage”). Often the moments of inertia are known for axes fixed in the body, as shown in Fig. 9.33(b). If the center of gravity is defined by the coordinates (x G , y G , z G ) in the x, y, z axes, the parallel-axis theorem can be used to find moments of inertia relative to the x, y, z axes, given values based on the body-fixed axes. The relations are where, for example, (I xx ) a is the moment of inertia relative to the x a axis, which passes through the center of gravity. Transferring the products of inertia requires use of the parallel-plane theorem, which provides the relations Inertia Tensor. The rotational dynamics of a rigid body rely on knowledge of the inertial properties, which are completely characterized by nine terms of an inertia tensor, six of which are independent. The inertia tensor is = I xy I yx xy dm m ∫ == I yz I zy yz dm m ∫ == I xz I zx xz dm m ∫ == I xx I xx () a my G 2 z G 2 +()+= I yy I yy () a mx G 2 z G 2 +()+= I zz I zz () a mx G 2 y G 2 +()+= I xy I xy () a mx G y G += I yz I yz () a my G z G += I zx I zx () a mz G x G += I I xx I xy – I xz – I yx – I yy I yz – I zx – I zy – I zz 0066-frame-C09 Page 40 Friday, January 18, 2002 11:00 AM ©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 . I zz 0 066 -frame-C09 Page 40 Friday, January 18 , 2002 11 :00 AM 2002 CRC Press LLC 10 Fluid Power Systems 10 .1 Introduction Fluid Power Systems • Electrohydraulic Control Systems 10 .2. 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. dm m ∫ == I xx I xx () a my G 2 z G 2 +()+= I yy I yy () a mx G 2 z G 2 +()+= I zz I zz () a mx G 2 y G 2 +()+= I xy I xy () a mx G y G += I yz I yz () a my G z G += I zx I zx () a mz G x G += I I xx I xy – I xz – I yx – I yy I yz – I zx – I zy – I zz 0 066 -frame-C09 Page 40 Friday, January 18 , 2002 11 :00 AM 2002 CRC Press LLC Product of Inertia. The product of inertia for a differential