The equations for these torques are given by the following relations, (4a) Tm = Kiq(t) TL = -fe (4b) where K, is the motor torque constant provided by the manufacturer, fe is the reaction force applied to the motor pinion gear by the rack, and r, is the radius of the motor pinion gear. The sum of the moments about the rotor of the DC motor gives the second equation, JmNö, + Bm Nô, = T,m + TL (3) Here, Im is the rotor inertia, Bm is the damping acting on the rotor, Tm and T are the torques on the motor due to the magnetic field and the external load applied via the interactions of the rack and motor pinion gear respectively. The voltage fed back into the circuit, as a result of the motion of wires in a magnetic field is called the back electromotive force, or vBEMf, and can be determined from the equation, VaEmf(t) = NK90, (2) where N is the gear ratio of the planetary gearbox, Kg is the motor speed coefficient, and å, is the rotational velocity of the motor pinion gear. Here, the constants Ra and La represent the armature resistance and inductance, and the variables ia(t), and va(t) represent the motor current draw and the applied voltage respectively.
DC Motor Current By using Kirchhoffs Voltage Law on the circuit diagram in Figure 11, Raia(t) + La “aO + VBEMF(t) = v4(t) (1) dt La Ra Keha Gearbox v,(t) = K,ôÔm Im %3D Pinion Gear Вт DC Motor Rack Figure 11: DC motor schematic diagram relating applied voltage to position of the motor pinion gear, 8, i.
Mathematical Modeling Please complete the modeling of the dynamics.
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