Page 323 - Mechanical Engineers' Handbook (Volume 2)

314 Mathematical Models of Dynamic Physical Systems

applies to any closed path in the system. For convenience and to ensure the independence
of the resulting equations, continuity is usually applied to the meshes or ‘‘windows’’ of the
graph. A one-part graph with n nodes and b branches will have b n 1 meshes, each
mesh yielding one independent compatibility equation. A planar graph with p separate parts
(resulting from multiport elements) will have b n p independent compatibility equations.
For a closed path q, the compatibility equation is

v 0
q ij
where the summation is taken over all branches (i, j) on the path.
For the system graph depicted in Fig. 6b, the three compatibility equations based on
the meshes are

path 1 → 2 → g → 1: v v m 2 v m 1 0
b
path 1 → 2 → 1: v v 0
k 1 b
path 2 → 3 → g → 2: v v v 0
k 2 s m 2
These equations are all mutually independent and express apparent geometric identities. The
first equation, for example, states that the velocity difference between the ends of the damper
is identically the difference between the velocities of the masses it connects. Compatibility
relations are also known as path, loop, and connectedness relations.

3.4 Analogs and Duals
Taken together, the element laws and system relations are a complete mathematical model
of a system. When expressed in terms of generalized through and across variables, the model
applies not only to the physical system for which it was derived, but also to any physical
system with the same generalized system graph. Different physical systems with the same
generalized model are called analogs. The mechanical rotational, electrical, and fluid analogs
of the mechanical translational system of Fig. 6a are shown in Fig. 7. Note that because the
original system contains an inductive storage element, there is no thermal analog.
Systems of the same physical type but in which the roles of the through variables and
the across variables have been interchanged are called duals. The analog of a dual—or,
equivalently, the dual of an analog—is sometimes called a dualog. The concepts of analogy
and duality can be exploited in many different ways.

4 STANDARD FORMS FOR LINEAR MODELS
The element laws and system relations together constitute a complete mathematical descrip-
tion of a physical system. For a system graph with n nodes, b branches, and s sources, there
will be b s element laws, n 1 continuity equations, and b n 1 compatibility
equations. This is a total of 2b s differential and algebraic equations. For systems com-
posed entirely of linear elements, it is always possible to reduce these 2b s equations to
either of two standard forms. The input/output, or I/O, form is the basis for transform or
so-called classical linear systems analysis. The state-variable form is the basis for state-
variable or so-called modern linear systems analysis.