Page 82 - Servo Motors and Industrial Control Theory -

4.6 DC Servo Motors for Very High Performance Requirements 75

the angular position of the load because the transmission mechanism is assumed
to be flexible. The output torque is proportional to the current and the equation of
motion becomes
K I : J s = m 2 θ m + Cs θ+ T r (4.10)
t
m
The term on the left hand side of the above equation is the torque developed by
the rotor. The parameters J , C, T are the rotor inertia, the mechanical damping
m
r
referred to the motor, and the external torque referred to the rotor respectively. The
torque transferred to the transmission mechanism is

T := NT r (4.11)
s

θ m  
T: K= ⋅ −θ (4.12)
s s     o 
 N  
N is the gearbox speed ratio and T , θ , K is the torque applied to the motor, the
s
s
o
output poison, and the stiffness of the transmission mechanism. The equation of the
motion for the load may be written as

T : J s = 2 θ + C s θ + T (4.13)
s 1 o 1 o 1
The parameters J , C , T are the load inertia, the damping ratio of the transmission
l
l
l
mechanism and the external torque applied to the motor.
The above equations are all the differential equations that describe the dynamic
behavior of the DC servo motor in general form. There is now various methods to
study the dynamics of the system and to design a proper controller to achieve the
required performance. One method is to use the principle of superposition and to
find the transfer function of the system in which the output position is related to the
two input variables of voltage and external torque. The proportional, proportional
plus integrator, and lead-lag network may be studied to find an acceptable compro-
mise between the required accuracy and speed of response. For this purpose, the
Nyquist diagram, Bode diagram, or the root locus may be used. It should be noted
that the characteristic equation of the system will be of order of five and if Integral
or lead-lag control strategy is used the order of the system will increase. With classi-
cal control strategy it is not possible to move all the roots to required position on the
s-plane. The reader is encouraged to do the manipulation by substituting numerical
values by obtaining a catalogue from the manufacturer. With numerical values the
reader should be able to design various classical control strategies. It should be
noted that a velocity feedback and position feedback directly from the motor usu-
ally are provided by manufacturers of DC servo motors.
In this section, state variable feedback control strategy is studied. There is no
need to find the overall transfer function. By carefully manipulating the governing
differential equations, they can be converted to a form that it is possible to define