Q3Control System

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Q.3. A servomechanism is represented by the equation :

\frac{d ^{2} θ}{d t ^{2}} + 10 \frac{d θ}{d t} = 150 E

Where E = (r - θ) is the actuating signal. Calculate the value of damping ratio, undamped frequency of oscillations.

Answer 1

A servomechanism is a feedback control system in which the controlled variable is a mechanical position, velocity, or acceleration (here, angular position θ), and the actuating signal E is generated by comparing this output against a reference command r. This particular equation is the classic model of a DC position servo (e.g., an armature-controlled DC motor driving a load through a gear train, with a potentiometer or synchro pair providing position feedback): the term 10(dθ/dt) represents the combined viscous friction and back-EMF damping torque of the motor-load system (normalized by inertia), while the term 150E represents the net driving torque, proportional to the actuating error signal through the combined motor-torque-constant and amplifier gain (again normalized by inertia).

The servomechanism equation is:

\frac{d ^{2} θ}{d t ^{2}} + 10 \frac{d θ}{d t} = 150 E, E = r - θ

Substituting E = r - θ:

\ddot{θ} + 10 \dot{θ} + 150 θ = 150 r

Comparing the homogeneous (characteristic) part with the standard second-order form s² + 2ζωn s + ωn² = 0, i.e. matching coefficients ẍ + 2ζωnẋ + ωn²x:

ω_{n}^{2} = 150 \Rightarrow ω_{n} = 150 \approx 12.247 rad/s

2 ζ ω_{n} = 10 \Rightarrow ζ = \frac{10}{2 \times 12.247} = \frac{10}{24.495} \approx 0.408

Since 0 < ζ < 1, the system is underdamped, meaning θ(t) will exhibit a decaying oscillatory response to a step change in reference r(t), settling to the final value with some overshoot (approximately 100·e^(-πζ/√(1-ζ²)) % ≈ 100·e^(-1.485) ≈ 22.6% peak overshoot) before stabilizing, with natural (undamped) frequency of oscillation ωn ≈ 12.25 rad/s and damped oscillation frequency ωd = ωn√(1-ζ²) = 12.247×√(1-0.1665) ≈ 11.19 rad/s.

Practical relevance. A damping ratio of ζ ≈ 0.41 places this servo in the moderately underdamped range — noticeably livelier than the 'optimal' engineering range of ζ ≈ 0.6–0.7 typically targeted for position servos, and closer to the more oscillatory end of acceptable designs. In practice, such a system would exhibit a visible, if not excessive, overshoot (≈23%) on step commands, along with several visible oscillation cycles before settling, since the ratio ωd/(ζωn) = 1/tan(cos⁻¹ζ) is relatively large at this damping level. A control engineer reviewing this design would likely recommend increasing the damping term (e.g., by adding tachometer/velocity feedback, which directly adds to the effective 10(dθ/dt) coefficient without altering ωn as strongly as changing the forward gain would) to push ζ closer to the 0.6–0.7 range, reducing overshoot and oscillatory ringing while retaining a fast enough response — a design trade-off that recurs throughout this course whenever a raw, uncompensated servo model is evaluated against desired transient-response specifications.

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