Page 78 - Modern Control Systems

52 Chapter 2 Mathematical Models of Systems

linear, dynamic elements is given in Table 2.2 [5]. The equations in Table 2.2 are ideal-
ized descriptions and only approximate the actual conditions (for example, when a
linear, lumped approximation is used for a distributed element).

Table 2.2 Summary of Governing Differential Equations for Ideal Elements

Type of Physical Governing Energy £ or
Element Element Equation Power SP Symbol
_ di
( Electrical inductance E = i-Li 2
2
J. dF_
Translational spring «21 = E
k dt ~ 2 k
Inductive storage <
Rotational spring 1 dT III
<»2\ = = - — E =
k dt 2 k
dQ
Fluid inertia E = - 21<?
dt
Electrical capacitance dv 2i i \\C
i = C y 2 i v 2 o—*— o vi
dt
Translational mass F = M dv 2 E = -Mv 2 2 '-sH^Hl-
dt 2 v 2
constant
Capacitive storage ^ Rotational mass T = J da) 2 to, =
dt constant
dP 2i
Fluid capacitance Q = C f E = -C fP^
dt
< Thermal capacitance rdV 2 E = C<5 2
9" 2 9", -
constant
i
R
( Electrical resistance i = ±va n v 2 o— V\A/—*-° v \
/
R
Translational damper F = bv '21 9» = bv 2i 2
v 2
Energy dissipators < Rotational damper T = bco 21 9» = ba) 21 2
T—K> 1 I OG>I
G>2 —>b
9 p
Fluid resistance Q = i » R f Q
= ¥/- P 2 o-AAA/ > ° f |

Thermal resistance d = J** 9 = ^ ¾ R t q
°T 2 o-^VVV > o3"i