TVC dynamics — gimbal servo bandwidth, time delay

3.5.45 · Physics › Guidance, Navigation & Control (GNC)

TVC dynamics HAI kya?

• Second order kyun? Ek servo ek motor (torque) hai jo ek inertia (nozzle mass) ko spring-jaisi load aur damping ke against drive karta hai — mass–spring–damper $\Rightarrow$ 2nd order.
• Pure delay $e^{-s\tau }$ kyun? Sensor sampling, computation, digital-to-analog conversion, aur valve transport lag sab response ko time mein shift karte hain bina uski shape badlaaye.

Second-order servo ko first principles se KAISE derive karte hain

Nozzle ko ek rotational inertia $J$ maano jis par actuator torque $M_a$, damping $c\dot{\delta }$, aur restoring stiffness $k\delta$ (structural + control) act karta hai:

$J\ddot{\delta }+c\dot{\delta }+k\delta =M_a$

Servo commanded aur actual angle ke beech error ke proportional torque command karta hai, $M_a=K({\delta }_c-\delta )$ (ek proportional inner loop). Substitute karo:

$J\ddot{\delta }+c\dot{\delta }+k\delta =K({\delta }_c-\delta )$ $J\ddot{\delta }+c\dot{\delta }+(k+K)\delta =K{\delta }_c$

$J$ se divide karo aur Laplace transform lo (zero initial conditions):

$(s^2+\frac{c}{J}s+\frac{k+K}{J})\delta (s)=\frac{K}{J}{\delta }_c(s)$

Ab standard form se match karke natural frequency aur damping define karo:

Toh pure lag part hai $G_{\text{servo}}(s)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}.$

Transfer function se Bandwidth

$|G(j\omega ){|}^2$ compute karo: $s=j\omega$ substitute karo, toh $s^2=-{\omega }^2$: $|G{|}^2=\frac{{\omega }_n^4}{({\omega }_n^2-{\omega }^2{)}^2+(2\zeta {\omega }_n\omega {)}^2}$

$=1/2$ set karo aur $x=(\omega /{\omega }_n{)}^2$ lo: $(1-x{)}^2+4{\zeta }^{2}x=2\Rightarrow x^2-2x(1-2{\zeta }^2)-1=0$ $x=(1-2{\zeta }^2)+\sqrt{(1-2{\zeta }^2{)}^2+1}$

Time delay KAISE nuksan pahunchaata hai: phase lag

Pure delay $e^{-s\tau }$ frequency $\omega$ par magnitude $|e^{-j\omega \tau }|=1$ aur phase contribute karta hai:

$\angle e^{-j\omega \tau }=-\omega \tau \text{(radians)}$

Worked examples

Common mistakes

Flashcards

Connections

• Second-order systems — natural frequency & damping
• Bode plots & phase margin
• Padé approximation of transport delay
• Structural bending modes & notch filters
• Rigid-body attitude control loop (autopilot)
• Digital control — sampling & computational delay
• Thrust Vector Control geometry & torque

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps