Question

In a position control system employs a servomotor having an inertia of 5.28×10⁻⁶ kg m² (IM) and a viscous damping coefficient of 6.8×10⁻⁵ Nms/radian(βm). The motor is driven by an electronic amplifier that develops a torque of 0.034Nm 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 (Be)
(2) The overall system gain (K)
(3) The undamped natural frequency of the system (ωn)
(4) The damped natural frequency of the system (ωd)
(5) The servoamplifier constant (G)
(6) The velocity lag of the system (θe)n given that the input shaft rotates at a constant angular velocity of 10rev/min.

Answer 1

The differential equation of the system is IM * θ''(t) + βm * θ'(t) + Kp * θ(t) = 0. The critical damping coefficient (Be), overall system gain (K), undamped natural frequency (ωn), damped natural frequency (ωd), servoamplifier constant (G), and velocity lag of the system (θe)n can be calculated using the given formulas and values.

Step-by-step explanation:

The differential equation of the system symbolically can be represented as:

IM * θ''(t) + βm * θ'(t) + Kp * θ(t) = 0

Where:

IM = Inertia of the servomotor (5.28×10⁻⁶ kg m²)

βm = Viscous damping coefficient (6.8×10⁻⁵ Nms/radian)

Kp = Sensitivity of the error-detector (0.5 volts/degree misalignment)

The critical damping coefficient of the system (Be) can be calculated using the following formula:

Be = 2 * √(IM * Kp)

The overall system gain (K) can be calculated using the following formula:

K = Torque current relationship * Gear ratio

The undamped natural frequency of the system (ωn) can be calculated using the following formula:

ωn = √(Kp / IM)

The damped natural frequency of the system (ωd) can be calculated using the following formula:

ωd = √(Kp / IM) * √(1 - ζ²)

The servoamplifier constant (G) can be calculated using the following formula:

G = Torque / Amplifier output current

The velocity lag of the system (θe)n can be calculated using the following formula:

θe = Input angular velocity / Amplifier output current