Page 154 - The Mechatronics Handbook

requires analytically solving algebraic relations for the operating point, or by using a graphical approach
as shown in Fig. 9.26(d).
This is a simple example indicating how algebraic loops are detected with a bond graph, and how the
solution requires solving algebraic relations. In complex systems, this might be difficult to achieve.
Sometimes it is possible to introduce or eliminate elements that are “parasitic,” meaning they normally
would be neglected due to their relatively small effect. However, such elements can relieve the causal
bind. While this might resolve the problem, as in the case of derivative causality there are cases where
such a course could introduce numerical stiffness problems. Sometimes a solution is reached by using
energy methods to resolve some of these problems, as shown in the next section.

9.5 Energy Methods for Mechanical System
Model Formulation

This section describes methods for using energy functions to describe basic energy-storing elements in
mechanical systems, as well as a way to describe collections of energy-storing elements in multiport fields.
Energy methods can be used to simplify model development, providing the means for deriving consti-
tutive relations, and also as a basis for eliminating dependent energy storage (see last section). The
introduction of these methods provides a basis for introducing the Lagrange equations in section 9.7 as
a primary approach for system equation derivation or in combination with the bond graph formulation.
Multiport Models

The energy-storing and resistive models introduced in section 9.3 were summarized in Tables 9.2, 9.4,
and 9.5 as multiport elements. In this section, we review how multiport elements can be used in modeling
mechanical systems, and outline methods for deriving the constitutive relations. Naturally, these methods
apply to the single-port elements as well.
An example of a C element with two-ports was shown in Fig. 9.12 as a model for a cantilevered beam
that can have both translational and rotational deflections at its tip. A 2-port is required in this model
because there are two independent ways to store potential energy in the beam. A distinguishing feature in
this example is that the model is based on relationships between efforts and displacement variables (for
this case of a capacitive element). Multiport model elements developed in this way are categorized as
explicit fields to distinguish them from implicit fields [17]. Implicit fields are formed by assembling energy-
storing 1-port elements with junction structure (i.e., 1, 0, and TF elements) to form multiport models.
Explicit fields are often derived using physical laws directly, relying on an understanding of how the
geometric and material properties affect the basic constitutive relation between physical variables. Geom-
etry and material properties always govern the parametric basis of all constitutive relations, and for some
cases these properties may themselves be functions of state. Indeed, these cases require the multiport
description, which finds extensive use in modeling of many practical devices, especially sensors and
actuators. Multiport models should follow a strict energetic basis, as described in the following.
Restrictions on Constitutive Relations

Energy-storing multiports must follow two basic restrictions, which are also useful in guiding the derivation
of energetically-correct constitutive relations. The definition of the energy-storing descriptions summarized
in Tables 9.4 and 9.5 specifies that there exists an energy state function, E = E(x), where x is either a generalized
displacement, q, for capacitive (C) elements or a generalized momentum, p, for inertive (I) elements. For
the multiport energy-storing element, the specification requires the following specifications [2,3].
1. There exists a rate law, = u i , where u i as input specifies integral causality on port i.
i x ˙
2. The energy stored in a multiport is determined by
n
∫
∑
E (x) y i xx i (9.5)
=
d
i=1
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