Page 155 - The Mechatronics Handbook

3. A first restriction on a multiport constitutive relation requires that the causal output at any port
is given by

∂E x()
y i = Φ si x() = -------------- (9.6)
∂x i

where F si () is a single-valued function.
4. A second restriction on a multiport constitutive relation requires that the constitutive relations
obey Maxwell reciprocity, or

2
∂ E x()
∂y i ∂y j
------- = ---------------- = ------- (9.7)
∂x j ∂x j ∂x i ∂x i
Deriving Constitutive Relations
The first restriction on the constitutive relations, Eq. (9.6), establishes how constitutive relations can be
derived for a multiport if an energy function can be formulated. This restriction forms the basis for a
method used in many practical applications to find constitutive relationships from energy functions (e.g.,
strain-energy, electromechanics, etc.). In these methods, it is assumed that at least one of the constitutive
relations for an energy-storing multiport is given. Then, the energy function is formed using Eq. (9.5)
where, after interchanging the integral and sum,

n
º ∫
x ( ) ∑ ∫ y i x i = ∫ y 1 x 1 ++ y n xd (9.8)
=
d
d
i=1
Presume that y 1 is a known function of the states, y 1 = Φ si (x). Since the element is conservative, any
energetic state can be reached via a convenient path where dx i = 0 for all i except i = 1. This allows the
determination of E(x).
To illustrate, consider the simple case of a rack and pinion system, shown in Fig. 9.27. The pinion has
rotational inertia, J, about its axis of rotation, and the rack has mass, m. The kinetic co-energy is easily
formulated here, considering that the pinion angular velocity, ω, and the rack velocity, V, are constrained
by the relationship V = Rω, where R is the pinion base radius. If this basic subsystem is modeled directly,
it will be found that one of the inertia elements (pinion, rack) will be in derivative causality. Say, it is
desired to connect to this system through the rotational port, T - ω. To form a single-port I element that
2
2
includes the rack, form the kinetic co-energy as T = T(ω, V) = Jω /2 + mV /2. Use the constraint relation
2
2
to write, T = T(ω) = (J + mR )ω /2. To find the constitutive relation for this 1-port rotational I element,
2
2
let h = ∂T(ω)/∂ω = (J + mR )ω, where we can now define an equivalent rotational inertia as J eq = J + mR .
Pinion
Dependent
J I:J I:m I:J eq
R
T T T
T, ω 1 T 1 1 I:Jeq
ω ω V ω ω
Rack
m
(a) (b) (c)

FIGURE 9.27 (a) Rack and pinion subsystem with torque input. (b) Direct model, showing dependent mass.
(c) Equivalent model, derived using energy principles.

©2002 CRC Press LLC