Analysis and Control of Electric Drives. Ned Mohan

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2‐4 FRICTION

Friction within the motor and the load acts to oppose rotation. Friction occurs in the bearings that support rotating structures. Moreover, moving objects in air encounter windage or drag. In vehicles, this drag is a major force that must be overcome. Therefore, friction and windage can be considered as opposing forces or torque that must be overcome. The frictional torque is generally nonlinear in nature. We are all familiar with the need for a higher force (or torque) in the beginning (from rest) to set an object in motion. This friction at zero speed is called stiction. Once in motion, the friction may consist of a component called Coulomb friction, which remains independent of speed magnitude (it always opposes rotation), as well as another component called viscous friction, which increases linearly with speed.

In general, the frictional torque T_f in a system consists of all of the aforementioned components. An example is shown in Fig. 2-10; this friction characteristic may be linearized for an approximate analysis by means of the dotted line. With this approximation, the characteristic is similar to that of viscous friction in which

      (2-30)

      where B is the coefficient of viscous friction or viscous damping.

EXAMPLE 2‐6

The aerodynamic drag force in automobiles can be estimated as f_L = 0.046 C_w A u ², where the coefficient 0.046 has the appropriate units, the drag force is in N, C_w is the drag coefficient (a unit‐less quantity), A is the vehicle cross‐sectional area in m², and u is the sum of the vehicle speed and headwind in km/h [1]. If A = 1.8 m² for two vehicles with C_w = 0.3 and C_w = 0.5, respectively, calculate the drag force and the power required to overcome it at the speeds of 50 and 100 km/h.

       Solution

The drag force is f_L = 0.046 C_w Au ² and the power required at the constant speed, from Eq. (2-6), is P = f_L u where the speed is expressed in m/s. Table 2-1 lists the drag force and the power required at various speeds for the two vehicles. Since the drag force F_L depends on the square of the speed, the power depends on the cube of the speed.

      Traveling at 50 km/h, compared to 100 km/h, requires 1/8th of the power, but it takes twice as long to reach the destination. Therefore, the energy required at 50 km/h would be 1/4th that at 100 km/h.

Fig. 2-10 Actual and linearized friction characteristics.

TABLE 2-1 Drag Force and the Power Required

2‐5 TORSIONAL RESONANCES

In Fig. 2-7, the shaft connecting the motor with the load was assumed to be of infinite stiffness, that is, the two were rigidly connected. In reality, any shaft will twist (flex) as it transmits torque from one end to the other. In Fig. 2-11, the torque T_shaft available to be transmitted by the shaft is

(2-31)

      This torque at the load‐end overcomes the load torque and accelerates it,

      (2-32)