Page 53 - Electric Drives and Electromechanical Systems

46 Electric Drives and Electromechanical Systems

through a conventional gearbox. The gear ratio is to be considered the optimum value as
2
defined by Eq. (2.10) As the arm extends the effective load inertia increases from 0.75 to 2 kg m .
The optimum gear ratio, n* can be calculated, using Eq. (2.10). The gear ratio has limiting
values of 19 and 31, given the range of the inertia. To maintain performance at the maximum
inertia the larger gear ratio is selected, hence the required motor torque is:

I max
T ¼ 31a max I m þ ¼ 1:26 Nm
31 2
If the lower gear ratio is selected, the motor torque required to maintain the same acceleration
is 1.4 Nm, hence the system will be possibly overpowered.
nnn

If a constant peak value in the acceleration is required for all conditions, the gear ratio
will have to be optimised for the maximum value of the load inertia. At lower values of
the inertia, the optimum conditions will not be met, although the load can still be
accelerated at the required value.

2.2 Linear systems

From the viewpoint of a drive system, a linear system is normally simpler to analyse than
a rotary system. In such systems a constant acceleration occurs when a constant force
acts on a body of constant mass,
F
€ x ¼ (2.16)
m
where € x is the linear acceleration, F the applied force and m the mass of the object being
accelerated. As with the rotary system, a similar relationship to Eq. (2.1) exists,
F m ¼ F L þ m tot € x þ B L _ x (2.17)
where m tot is the system’s total mass; B L is the damping constant (in Nm 1 s); _ x is the
1
linear velocity (in m s ); F L is the force required to drive the load (in N), including the
external load forces and frictional loads (for example, those caused by any bearings or
other system inefficiencies); F m is the force (in N) developed by a linear motor or a
rotary-to-linear actuator.
The kinetic energy change for a linear system is given by,
2
m tot _ x _ x 2 1

2
DE k ¼ (2.18)
2
for a speed change from _ x 2 to _ x 1 .
2.3 Wheeled systems

In a wheeled vehicle the selection of the drive motors is constrained by two factors.
Can the driven wheels provide the torque to allow the vehicle to undertake its full
range of tasks, included accelerating at the required value and climbing slopes as
required?
Can the required torque be transferred to the surface, without the wheels slipping?