Ned Mohan - Analysis and Control of Electric Drives

In many systems, a linear motion is achieved by using a rotating‐type motor, as shown in Fig. 2-14.

Fig. 2-14 Combination of rotary and linear motion.

In such a system, the angular and the linear speeds are related by the radius r of the drum:

(2-35)

To accelerate the mass M in Fig. 2-14, in the presence of an opposing force f L, the force f applied to the mass, from Eq. (2-1), must be

(2-36)

This force is delivered by the motor in the form of a torque T , which is related to f , using Eq. (2-35), as

(2-37)

Therefore, the electromagnetic torque required from the motor is

(2-38)

In the vehicle of Example 2-6with C w= 0.5, assume that each wheel is powered by its own electric motor that is directly coupled to it. If the wheel diameter is 60 cm, calculate the torque and the power required from each motor to overcome the drag force when the vehicle is traveling at a speed of 100 km/h.

In Example 2-6, the vehicle with C w= 0.5 presented a drag force f L= 413.7 N at the speed u = 100 km/h. The force required from each of the four motors is . Therefore, the torque required from each motor is

From Eq. (2-35),

Therefore, the power required from each motor is

For matching speeds, Fig. 2-15shows a gear mechanism where the shafts are assumed to be of infinite stiffness, and the masses of the gears are ignored. We will further assume that there is no power loss in the gears. Both gears must have the same linear speed at the point of contact. Therefore, their angular speeds are related by their respective radii r 1and r 2such that

(2-39)

and

(2-40)

Combining Eqs. (2-39)and (2-40),

(2-41)

where T 1and T 2are the torques at the ends of the gear mechanism, as shown in Fig. 2-15. Expressing T 1and T 2in terms of T emand T Lin Eq. (2-41),

(2-42)

From Eq. (2-42), the electromagnetic torque required from the motor is

(2-43)

where the equivalent inertia at the motor side is

(2-44)

Fig. 2-15 Gear mechanism for coupling the motor to the load.

Equation (2-43)shows that the electromagnetic torque required from the motor to accelerate a motor‐load combination depends on the gear ratio. In a basically inertial load where T Lcan be assumed to be negligible, T emcan be minimized, for a given load‐acceleration , by selecting an optimum gear ratio ( r 1/ r 2) opt.. The derivation of the optimum gear ratio shows that the load inertia “seen” by the motor should equal the motor inertia, that is, in Eq. (2-44),

(2-45a)

and, consequently,

(2-45b)

With the optimum gear ratio, in Eq. (2-43), using T L= 0, and using Eq. (2-41),

(2-46)

Similar calculations can be made for other types of coupling mechanisms (see homework problems).

Load torques normally act to oppose rotation. In practice, loads can be classified into the following categories [2]:

1 Centrifugal (Squared) Torque

2 Constant Torque

3 Squared Power

4 Constant Power

Centrifugal loads, such as fans and blowers, require torque that varies with speed 2, and load power that varies with speed 3. Similarly, wind turbines produce torque that is proportional to speed 2.

In constant‐torque loads such as conveyors, hoists, cranes, and elevators, torque remains constant with speed, and the load power varies linearly with speed. In squared‐power loads such as compressors and rollers, the torque varies linearly with speed and the load power varies with speed 2. In constant‐power loads, such as winders and unwinders, the torque beyond a certain speed range varies inversely with speed and the load power remains constant with speed.