In a position control system employs a servomotor having an inertia of 5.28 x 10⁻⁶ kg m² (IM) and a viscous damping coefficient of 6.8 x 10⁻⁵ Nms/radian (βₘ). The motor is driven by an electronic amplifier that develops a torque of 0.034 Nm for 200 mA amplifier output current, the torque current relationship being linear. The motor drives the output shaft through a 100:1 reduction gear, the gearing and the shaft are considered to have negligible inertia. The error-detector has a sensitivity of 0.5 volts/degree misalignment (Kₚ). The non-dimensional damping ratio is 0.4 (ζ).
Display the differential equation of the system symbolically, showing the motion of the output shaft and hence calculate:
1. The critical damping coefficient of the system (β)
2. The overall system gain (K)