Mechanics#

Stiff Mechanical System#

The dynamics of a stiff rotational mechanical system are governed by

(1)#\[\begin{split}J\frac{\mathrm{d}\omega_\mathrm{M}}{\mathrm{d} t} &= \tau_\mathrm{M} - B_\mathrm{L}\omega_\mathrm{M} - \tau_\mathrm{L} \\ \frac{\mathrm{d}\vartheta_\mathrm{M}}{\mathrm{d} t} &= \omega_\mathrm{M}\end{split}\]

where \(\omega_\mathrm{M}\) is the mechanical angular speed of the rotor, \(\vartheta_\mathrm{M}\) is the mechanical angle of the rotor, \(\tau_\mathrm{M}\) is the electromagnetic torque, \(\tau_\mathrm{L}\) is the external load torque as a function of time, \(B_\mathrm{L}\) is the friction coefficient, and \(J\) is the total moment of inertia. The total load torque is

(2)#\[\tau_\mathrm{L,tot} = B_\mathrm{L}\omega_\mathrm{M} + \tau_{\mathrm{L}}\]

A constant friction coefficient \(B_\mathrm{L}\) 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, \(B_\mathrm{L} = B_\mathrm{L}(\omega_\mathrm{M})\). As an example, the quadratic load torque profile is achieved choosing \(B_\mathrm{L} = k|\omega_\mathrm{M}|\), 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 model of a stiff mechanical system is provided in the class motulator.drive.model.StiffMechanicalSystem.

Block diagram of a stiff mechanical system.

Block diagram of a stiff mechanical system.#

Two-Mass Mechanical System#

A two-mass mechanical system can be modeled as

(3)#\[\begin{split}J_\mathrm{M}\frac{\mathrm{d}\omega_\mathrm{M}}{\mathrm{d} t} &= \tau_\mathrm{M} - \tau_\mathrm{S} \\ J_\mathrm{L}\frac{\mathrm{d}\omega_\mathrm{L}}{\mathrm{d} t} &= \tau_\mathrm{S} - \tau_\mathrm{L} \\ \frac{\mathrm{d}\vartheta_\mathrm{ML}}{\mathrm{d} t} &= \omega_\mathrm{M} - \omega_\mathrm{L}\end{split}\]

where \(\omega_\mathrm{L}\) is the angular speed of the load, \(\vartheta_\mathrm{ML}=\vartheta_\mathrm{M}-\vartheta_\mathrm{L}\) is the twist angle, \(J_\mathrm{M}\) is the moment of inertia of the machine, and \(J_\mathrm{L}\) is the moment of inertia of the load. The shaft torque is

(4)#\[\tau_\mathrm{S} = K_\mathrm{S}\vartheta_\mathrm{ML} + C_\mathrm{S}(\omega_\mathrm{M} - \omega_\mathrm{L})\]

where \(K_\mathrm{S}\) is the torsional stiffness of the shaft, and \(C_\mathrm{S}\) is the torsional damping of the shaft. The other quantities correspond to those defined for the stiff mechanical system. A two-mass mechanical system is modeled in the class motulator.drive.model.TwoMassMechanicalSystem. See also the example in 2.2-kW PMSM, 2-mass mechanics.

Block diagram of a two-mass mechanical system.

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 \(\omega_\mathrm{M}\) 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 induction motor, torque-control mode.