Differential equations for motors and rotating bodies

　・In Japanese
＜Premise knowledge＞
　・DC motor
　・Scilab

■Differential equations for motors and rotating bodies

Explain the principle of the DC motor here. Calculate differential equations of motors and rotating bodies and simulate their behavior.

The relational expression between voltage and current is as follows. (ω: Motor rotation speed, Ke: Back electromotive force constant)



The generated torque T of the motor is as follows. (Kt: torque constant)



The torque applied to the motor can be expressed as follows from inertia J and angular velocity ω. Details here.



Eliminate T by substituting equation (2) into equation (3)

■Design a model of a motor and a rotating body in Scilab

Express the relationship between the voltage and the rotation speed of the disc in Scilab. (1) is transformed as follows.



Formula (4) is described as follows in the block diagram.



Integrating di(t)/dt gives i(t), which can be expressed as follows.



Next, expression (5) is expressed in a block diagram as follows.



Integrating dω(t)/dt gives ω(t), which can be expressed as follows.



Now it is possible to connect Figures 1 and 2 as follows



With this, we were able to express the rotation speed of the motor with respect to the voltage. This is described in Scilab as follows.

■Simulation results of motion of motor and rotating body

Simulate the above model. The parameters are as follows.

・R=5.7 (Ω) :armature resistance
・L=0.2 (H) :inductance
・Jm=0.00011 (kg・m^2) :armature moment of inertia
・Ji=0.0013 (kg・m^2) :moment of inertia of rotating body
・Kt=0.072(N・m/A) :torque constant
・Ke=0.0716(V/(rad/s)) :back electromotive force constant

　Here J=Jm+Ji.

■When obtaining the voltage from the angular velocity of the motor

We will not use it this time, but when calculating the voltage from the angular velocity, substitute the formula (5) into the formula (1).

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