Page 90 - Electric Drives and Electromechanical Systems

Chapter 3 Power transmission and sizing 83

FIG. 3.9 The cross section of a high-performance ball screw, the circulating balls are clearly visible.
The relationship between the rotational and linear speed for both the lead and ball
screw is given by:
V L
u L ¼ (3.12)
L
1
where u L is the rotational speed in rev min , V L is the linear speed in m min 1 and L is
the lead (in metres). The inertia of the complete system is the sum of the screw inertia I s
and the reflected inertia of the load I L ,
(3.13)
I tot ¼ I s þ I L
M s r 2
I s ¼ (3.14)
2
2

L
I L ¼ M L (3.15)
2p
where M L is the load’s mass in kg, M s is the screw’s mass in kg and r is the radius of the
lead screw (in metres). In addition, the static forces, both frictional and the forces
required by the load, need to be converted to a torque at the lead screw’s input.
The torque caused by external forces, F L , will result in a torque requirement of,
LF L
T L ¼ (3.16)
2p
and a possible torque resulting from slideway friction of,
LM L g cos qm
T f ¼ (3.17)
2p
where q is the inclination of the slideway. It has been assumed so far that the efficiency of
the lead screw is 100%. In practice, losses will occur and the lead-screw efficiency, ε; see
Table 3.1, has to be considered, hence
T f þ T L
T required ¼ (3.18)
ε
Linear digital actuators are based on stepper-motor technology, as discussed in
Chapter 8, where the rotor has been modified to form the nut of the lead screw.