Читать книгу Analysis and Control of Electric Drives - Ned Mohan - Страница 33

We will begin by applying physical laws of motion in their simplest form, starting with linear systems. In Fig. 2-3a, a load of a constant mass M is acted upon by an external force f_e that causes it to move in the linear direction x at speed u = dx/dt.

Fig. 2-3 Motion of a mass M due to the action of forces.

This movement is opposed by the load, represented by a force f_L. The linear momentum associated with the mass is defined as M × u. As shown in Fig. 2-3b, in accordance with Newton’s Law of Motion, the net force f_M(=f_e − f_L) equals the rate of change of momentum, which causes the mass to accelerate:

(2-1)

where a is the acceleration in m/s², which from Eq. (2-1) is

(2-2)

In MKS units, a net force of 1 Newton (or 1 N), acting on a constant mass of 1 kg, results in an acceleration of 1 m/s². Integrating the acceleration with respect to time, we can calculate the speed as

(2-3)

and, integrating the speed with respect to time, we can calculate the position as

(2-4)

where τ is a variable of integration.

The differential work dW done by the mechanism supplying the force f_e is

(2-5)

Power is the time‐rate at which the work is done. Therefore, differentiating both sides of Eq. (2-5) with respect to time t, and assuming that the force f_e remains constant, the power supplied by the mechanism exerting the force f_e is

(2-6)

It takes a finite amount of energy to bring a mass to a speed from rest. Therefore, a moving mass has stored kinetic energy that can be recovered. Note that in the system of Fig. 2-3, the net force f_M(=f_e − f_L) is responsible for accelerating the mass. Therefore, assuming that f_M remains constant, the net power p_M(t) going into accelerating the mass can be calculated by replacing f_e in Eq. (2-6) with f_M:

(2-7)

From Eq. (2-1), substituting f_M as ,

(2-8)

The energy input, which is stored as kinetic energy in the moving mass, can be calculated by integrating both sides of Eq. (2-8) with respect to time. Assuming the initial speed u to be zero at time t = 0, the stored kinetic energy in the mass M can be calculated as

(2-9)

where τ is a variable of integration.