Page 92 - The Mechatronics Handbook

0066_Frame_C07 Page 2 Wednesday, January 9, 2002 3:39 PM

for inertial components, ii) Kirchhoff circuit laws for current-charge components, and iii) magnet circuit
laws for magnetic flux devices.
In this chapter we will examine the basic modeling assumptions for inertial, electric, and magnetic
circuits, which are typical of mechatronic systems, and will summarize the dynamic principles and
interactions between the mechanical motion, circuit, and magnetic state variables. We will also illustrate
these principles with a few examples as well as provide some bibliography to more advanced references
in electromechanics.

7.2 Models for Electromechanical Systems

The fundamental equations of motion for physical continua are partial differential equations (PDEs),
which describe dynamic behavior in both time and space. For example, the motions of strings, elastic
beams and plates, fluid flow around and through bodies, as well as magnetic and electric fields require
both spatial and temporal information. These equations include those of elasticity, elastodynamics, the
Navier–Stokes equations of fluid mechanics, and the Maxwell–Faraday equations of electromagnetics.
Electromagnetic field problems may be found in Jackson (1968). Coupled field problems in electric fields
and fluids may be found in Melcher (1980) and problems in magnetic fields and elastic structures may
be found in the monograph by Moon (1984). This short article will only treat solid systems.
Many practical electromechanical devices can be modeled by lumped physical elements such as mass
or inductance. The equations of motion are then integral forms of the basic PDEs and result in coupled
ordinary differential equations (ODEs). This methodology will be explored in this chapter. Where physical
problems have spatial distributions, one can often separate the problem into spatial and temporal parts
called separation of variables. The spatial description is represented by a finite number of spatial or
eigenmodes each of which has its modal amplitude. This method again results in a set of ODEs. Often
these coupled equations can be understood in the context of simple lumped mechanical masses and
electric and magnetic circuits.

7.3 Rigid Body Models

Kinematics of Rigid Bodies
Kinematics is the description of motion in terms of position vectors r, velocities v, acceleration a, rotation
rate vector ω, and generalized coordinates {q k (t)} such as relative angular positions of one part to another
in a machine (Fig. 7.1). In a rigid body one generally specifies the position vector of one point, such as
the center of mass r c , and the velocity of that point, say v c . The angular position of a rigid body is specified
by angle sets call Euler angles. For example, in vehicles there are pitch, roll, and yaw angles (see, e.g.,
Moon, 1999). The angular velocity vector of a rigid body is denoted by ω. The velocity of a point in a
rigid body other than the center of mass, r p = r c + ρ, is given by
v = v + ω × ρ (7.1)
P
c
where the second term is a vector cross product. The angular velocity vector w is a property of the entire
rigid body. In general a rigid body, such as a satellite, has six degrees of freedom. But when machine
elements are modeled as a rigid body, kinematic constraints often limit the number of degrees of freedom.

Constraints and Generalized Coordinates
Machines are often collections of rigid body elements in which each component is constrained to have
one degree of freedom relative to each of its neighbors. For example, in a multi-link robot arm shown
in Fig. 7.2, each rigid link has a revolute degree of freedom. The degrees of freedom of each rigid link
are constrained by bearings, guides, and gearing to have one type of relative motion. Thus, it is convenient

©2002 CRC Press LLC