Page 128 - The Mechatronics Handbook

application to basic translational and rotational elements, which characterize a wide class of mechatronic
applications. The underlying basis of mechanical motion (kinematics) is presumed known and not
reviewed here, with more discussion and emphasis placed on a system dynamics perspective. More
advanced applications requiring two- or three-dimensional motion is presented in section 9.6.
Mechanical systems can be conceptualized as rigid and/or elastic bodies that may move relative to one
another, depending on how they are interconnected by components such as joints, dampers, and other
passive devices. This chapter focuses on those systems that can be represented using lumped-parameter
descriptions, wherein bodies are treated as rigid and no dependence on spatial extent need be considered
in the elastic effects. The modeling of mechanical systems in general has reached a fairly high level of
maturity, being based on classical methods rooted in the Newtonian laws of motion. One benefits from
the extensive and overwhelming knowledge base developed to deal with problems ranging from basic
mass-spring systems to complex multibody systems. While the underlying physics are well understood,
there exist many different means and ways to arrive at an end result. This can be especially true when
the need arises to model a multibody system, which requires a considerable investment in methods for
formulating and solving equations of motion. Those applications are not within the scope of this chapter,
and the immediate focus is on modeling basic and moderately complex systems that may be of primary
interest to a mechatronic system designer/analyst.

9.2 Mechanical System Modeling in Mechatronic Systems

Initial steps in modeling any physical system include defining a system boundary, and identifying how
basic components can be partitioned and then put back together. In mechanical systems, these analyses
can often be facilitated by identifying points in a system that have a distinct velocity. For purposes of
analysis, active forces and moments are “applied” at these points, which could represent energetic inter-
actions at a system boundary. These forces and moments are typically applied by actuators but might
represent other loads applied by the environment.
A mechanical component modeled as a point mass or rigid body is readily identified by its velocity,
and depending on the number of bodies and complexity of motion there is a need to introduce a
coordinate system to formally describe the kinematics (e.g., see [12] or [15]). Through a kinematic
analysis, additional (relative) velocities can be identified that indicate the connection with and motion
of additional mechanical components such as springs, dampers, and/or actuators. The interconnection
of mechanical components can generally have a dependence on geometry. Indeed, it is dependence of
mechanical systems on geometry that complicates analysis in many cases and requires special consider-
ation, especially when handling complex systems.
A preliminary description of a mechanical system should also account for any constraints on the
motional states, which may be functions of time or of the states themselves. The dynamics of mechanical
systems depends, in many practical cases, on the effect of constraints. Quantifying and accounting for
constraints is of paramount importance, especially in multibody dynamics, and there are different schools
of thought on how to develop models. Ultimately, the decision on a particular approach depends on the
application needs as well as on personal preference.
It turns out that a fairly large class of systems can be understood and modeled by first understanding
basic one-dimensional translation and fixed-axis rotation. These systems can be modeled using methods
consistent with those used to study other systems, such as those of an electric or hydraulic type. Fur-
thermore, building interconnected mechatronic system models is facilitated, and it is usually easier for
a system analyst to conceptualize and analyze these models.
In summary, once an understanding of (a) the system components and their interconnections (includ-
ing dependence on geometry), (b) applied forces/torques, and (c) the role of constraints, is developed,
dynamic equations fundamentally due to Newton can be formulated. The rest of this section introduces
the selection of physical variables consistent with a power flow and energy-based approach to modeling
basic mechanical translational and rotational systems. In doing so, a bond graph approach [28,3,17] is
introduced for developing models of mechanical systems. This provides a basis for introducing the

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