Page 77 - Modern Control Systems

Section 2.2 Differential Equations of Physical Systems 51

Thus, the angular rate difference is measured across the torsional spring element and is
referred to as an across-variable. These same types of arguments can be made for most
common physical variables (such as force, current, volume, flow rate, etc.). A more
complete discussion on through- and across-variables can be found in [26,27]. A sum-
mary of the through- and across-variables of dynamic systems is given in Table 2.1 [5].
Information concerning the International System (SI) of units associated with the var-
ious variables discussed in this section can be found at the MCS website/ For example,
variables that measure temperature are degrees Kelvin in SI units, and variables that
measure length are meters. Important conversions between SI and English units are
also given at the MCS website. A summary of the describing equations for lumped,

Table 2.1 Summary of Through- and Across-Variables for Physical Systems
Variable Integrated Variable Integrated
Through Through- Across Across-
System Element Variable Element Variable
Electrical Current, i Charge, q Voltage Flux linkage, A 2 i
difference, v 2i
Mechanical Force, F Translational Velocity Displacement
translational momentum, P difference, v 2 \ difference, y 2\
Mechanical Torque, T Angular Angular velocity Angular
rotational momentum, h difference, «21 displacement
difference, 8 2\
Fluid Fluid Volume, V Pressure Pressure
volumetric rate difference, P 2\ momentum, y 2i
of flow, Q
Thermal Heat flow Heat energy, Temperature
rate, q H difference, ^ 2 |

The companion website is found at www.pcarsonhighcred.com/dorf.