Mechanics

Mechanics#

Stiff Mechanical System#

The model of a stiff mechanical system is provided in the class motulator.drive.model.MechanicalSystem. The dynamics are governed by

(1)#\[J\frac{\D\omegaM}{\D t} = \tauM - \tauLtot\]

where \(\omegaM\) is the mechanical angular speed of the rotor, \(\tauM\) is the electromagnetic torque, and \(J\) is the total moment of inertia. The total load torque is

(2)#\[\tauLtot = \BL\omegaM + \tauL\]

where \(\tauL = \tauL(t)\) is an external load torque as a function of time and \(\BL\) is the friction coefficient. Figure 1 shows the corresponding block diagram.

A constant friction coefficient \(\BL\) models viscous friction that appears, e.g., due to laminar fluid flow in bearings. The friction coefficient is allowed to depend on the rotor speed, \(\BL = \BL(\omegaM)\). As an example, the quadratic load torque profile is achieved choosing \(\BL = k|\omegaM|\), where \(k\) is a constant. The quadratic load torque appears, e.g., in pumps and fans as well as in vehicles moving at higher speeds due to air resistance.

The mechanical angle \(\thetaM\) of the rotor is related to the mechanical angular speed as

(3)#\[ \frac{\D\thetaM}{\D t} = \omegaM\]
Block diagram of a stiff mechanical system.

Figure 1: Block diagram of a stiff mechanical system.#

Block diagram of a stiff mechanical system.

Figure 1: Block diagram of a stiff mechanical system.#

Two-Mass Mechanical System#

A two-mass mechanical system is modeled in the motulator.drive.model.TwoMassMechanicalSystem class. The dynamics are governed by [Saarakkala and Hinkkanen, 2015]

(4)#\[\begin{split} \JM\frac{\D\omegaM}{\D t} &= \tauM - \tauS \\ \JL\frac{\D\omegaL}{\D t} &= \tauS - \tauL \\ \frac{\D\thetaML}{\D t} &= \omegaM - \omegaL\end{split}\]

where \(\omegaL\) is the angular speed of the load, \(\thetaML = \thetaM - \thetaL\) is the twist angle, \(\JM\) is the moment of inertia of the machine, and \(\JL\) is the moment of inertia of the load. The shaft torque is

(5)#\[ \tauS = \KS\thetaML + \CS(\omegaM - \omegaL)\]

where \(\KS\) is the torsional stiffness of the shaft, and \(\CS\) is the torsional damping of the shaft. Figure 2 shows the block diagram of the system.

See the 2.2-kW IPMSM, 2-mass mechanics, O-V/Hz control example.

Block diagram of a two-mass mechanical system.

Figure 2: Block diagram of a two-mass mechanical system.#

Block diagram of a two-mass mechanical system.

Figure 2: Block diagram of a two-mass mechanical system.#

Externally Specified Rotor Speed#

It is also possible to omit the mechanical dynamics and directly specify the actual rotor speed \(\omegaM\) as a function of time, see the class motulator.drive.model.ExternalRotorSpeed. This feature is typically needed when torque-control mode is studied.

See the example 2.2-kW IM, CVC, torque-control mode.

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