Page 322 - Mechanical Engineers' Handbook (Volume 2)

3 System Structure and Interconnection Laws 313

ƒ bv bv 21
b
b
The reference direction for the through variable is determined by the convention that power
flow P ƒ v into an element is positive. Referring to Fig. 6a, when v 21 is positive, the
b b
b
damper is in compression. Therefore, ƒ must be positive for compressive forces in order to
b
obey the sign convention for power. By similar reasoning, tensile forces will be negative.
3.3 System Relations
The structure of a system gives rise to two sets of interconnection laws or system relations.
Continuity relations apply to through variables and compatibility relations apply to across
variables. The interpretation of system relations for various physical systems is given in
Table 3.
Continuity is a general expression of dynamic equilibrium. In terms of the system graph,
continuity states that the algebraic sum of all through variables entering a given node must
be zero. Continuity applies at each node in the graph. For a graph with n nodes, continuity
gives rise to n continuity equations, n 1 of which are independent. For node i, the con-
tinuity equation is
ƒ 0

j ij
where the sum is taken over all branches (i, j) incident on i.
For the system graph depicted in Fig. 6b, the four continuity equations are

node 1: ƒ ƒ ƒ 0
k 1 b m 1
node 2: ƒ ƒ ƒ ƒ 0
k 2 k 1 b m 2
node 3: ƒ ƒ 0
s
k 2
node g: ƒ ƒ ƒ 0
m 1 m 2 s
Only three of these four equations are independent. Note, also, that the equations for nodes
1–3 could have been obtained from the conventional free-body diagrams shown in Fig. 6c,
where ƒ and ƒ are the D’Alembert forces associated with the pure masses. Continuity
m 1 m 2
relations are also known as vertex, node, flow, and equilibrium relations.
Compatibility expresses the fact that the magnitudes of all across variables are scalar
quantities. In terms of the system graph, compatibility states that the algebraic sum of the
across-variable differences around any closed path in the graph must be zero. Compatibility

Table 3 System Relations for Various Systems

System Continuity Compatibility
Mechanical Newton’s first and third laws Geometrical constraints
(conservation of momentum) (distance is a scalar)
Electrical Kirchhoff’s current law Kirchhoff’s voltage
(conservation of charge) law (potential is a
scalar)
Fluid Conservation of matter Pressure is a scalar
Thermal Conservation of energy Temperature is a scalar