Page 51 - Electric Drives and Electromechanical Systems

44 Electric Drives and Electromechanical Systems

2.1.4 Acceleration without an external load

If a motor, with a suitable drive, is capable of supplying a peak torque of T peak , the
acceleration, a L , of a load with an inertia I L , through a gear train of ratio n:1 is given by;
T peak
a L ¼
I L (2.9)
n I d þ 2
n
The term in parentheses is the total inertia referred to the motor; I L includes the
inertia of the load, and the sum of inertias of the gears, shafts, and couplings rotating at
the system’s output speed; I d includes the inertias of the motor’s rotor, connecting
shafts, gears, and couplings rotating at the motor’s output speed. In the case of a belt
drive, the inertias of the belt, pulleys, and idlers must be included and referred to the
correct side of the speed changer. The optimum gear ratio, n*, can be determined from
Eq. (2.9), by equating dT peak /dn ¼ 0, which on solving gives,
s ffiffiffiffi
I L
(2.10)
n ¼
I d
Therefore, the inertia on the input side of the gearing has to be equal to the reflected
inertia of the output side to give a maximum acceleration of the load of,
T peak
a L ¼ (2.11)
2I d n
The value of a L is the load acceleration, hence the acceleration of the motor will be n*
times greater. The acceleration parameters of a motor should be considered during its
selection, which in practice will be limited by the motor’s construction, particularly if the
motor is brushed and a cooling fan is fitted. Since the acceleration torque is a function of
the motor current, the actual acceleration rate will be limited by the current limit on the
drive. This needs to be carefully considered when the system is being commissioned.

2.1.5 Acceleration with an applied external load
When an external load, T L , is applied to an accelerating load (for example, the cutting
force in a machine-tool application), the load’s peak acceleration is given by,
T peak e T L =n
a L ¼
I L (2.12)
n I d þ 2
n
This value is lower than that given by Eq. (2.9) for an identical unloaded system.
The optimum gear ratio for an application, where the load is accelerating with a
constant applied load, can be determined from this equation, giving,

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
I L a L T L
(2.13)
n ¼
I d a L