Thrust vector control — single-gimbal, dual-gimbal; TVC angles

1. Setup and coordinate frame

The gimbal point is the mechanical pivot. The moment arm $\ell$ is the distance from the CoM to that pivot along $x_b$ .

Figure — Thrust vector control — single-gimbal, dual-gimbal; TVC angles

2. Deriving the control torque from scratch

We derive the moment produced by tilting the thrust. Do it in 2D first (single-gimbal), then generalize.

Formula Single-gimbal torque — derivation

Let thrust magnitude be $T$ , deflected by angle $\delta$ from the vehicle axis in one plane.

Step 1 — resolve thrust. In body axes the thrust vector is $\vec{F} = T\big(\cos\delta,\ \sin\delta\big)$ Why this step? A rotation of a vector of length $T$ by $\delta$ has axial component $T\cos\delta$ and transverse component $T\sin\delta$ — just trigonometry of the tilt.

Step 2 — position of the force. The force acts at the gimbal, at $\vec r = (-\ell,\ 0)$ relative to CoM. Why? Torque needs the point of application; the pivot is a distance $\ell$ behind the CoM.

Step 3 — torque = $\vec r \times \vec F$ (2D cross product gives scalar $z$ -component): $M = r_x F_y - r_y F_x = (-\ell)(T\sin\delta) - 0 = -\ell\,T\sin\delta$ The sign just says the torque points to restore/rotate consistently; the magnitude is what matters: $\boxed{M = \ell\,T\sin\delta}$

Step 4 — small-angle simplification. Gimbals move only a few degrees, so $\sin\delta \approx \delta$ : $\boxed{M \approx \ell\,T\,\delta}$ Why? For $\delta \le \sim 8^\circ$ , $\sin\delta \approx \delta$ to within ~1%. This linear form is what the autopilot uses.

3. Thrust loss — the cost of steering

This is the key trade: torque is linear in $\delta$ , thrust loss is quadratic in $\delta$ — a fabulous deal at small angles.

4. Single-gimbal vs Dual-gimbal

Formula Dual-gimbal torque components

Deflect in pitch by $\delta_p$ (about $y_b$ ) and yaw by $\delta_y$ (about $z_b$ ). The transverse thrust components are $T\sin\delta_y$ (in $y$ ) and $T\sin\delta_p$ (in $z$ ). With arm $\ell$ : $M_\text{pitch} \approx \ell T \delta_p, \qquad M_\text{yaw} \approx \ell T \delta_y$ Total deflection magnitude (what the actuator/gimbal limit constrains): $\delta_\text{tot} = \sqrt{\delta_p^2 + \delta_y^2} \le \delta_\text{max}$ Why the square root? The two deflections are components of one physical tilt vector; the geometric tilt is their vector sum. The mechanical gimbal can only reach so far off-center, so it's $\delta_\text{tot}$ (not $\delta_p$ , $\delta_y$ separately) that hits the limit.

5. Worked examples

Worked example Example 1 — required gimbal angle for a commanded angular acceleration

A rocket has $T=800\text{ kN}$ , moment arm $\ell=6\text{ m}$ , pitch moment of inertia $I=1.2\times10^{5}\ \text{kg·m}^2$ . The autopilot wants angular acceleration $\dot\omega = 0.5\ \text{rad/s}^2$ . Find gimbal angle $\delta$ .

Step 1: required torque $M = I\dot\omega = 1.2\times10^5 \times 0.5 = 6.0\times10^4$ N·m. Why? Newton's rotational law $M=I\dot\omega$ : torque produces angular acceleration.
Step 2: invert $M=\ell T\delta$ : $\delta = \dfrac{M}{\ell T}=\dfrac{6.0\times10^4}{6\times 8\times10^5}=0.0125$ rad. Why? Small-angle law lets us solve linearly.
Step 3: convert: $\delta = 0.0125\times \tfrac{180}{\pi}\approx 0.72^\circ$ . Tiny! Why this matters: confirms only small gimbal angles are needed → light actuators, small thrust loss.

6. Common mistakes (steel-manned)

7. Active recall

Recall Quick self-test (cover the answers)

What physical quantity does TVC change? → the direction of the thrust vector.
Exact torque formula? → $M=\ell T\sin\delta$ .
Why is steering "cheap"? → torque is $\mathcal O(\delta)$ but thrust loss is $\mathcal O(\delta^2)$ .
Difference single vs dual gimbal? → 1 angle/1 plane vs 2 angles (pitch+yaw)/2 planes.
Why can't one centered engine control roll? → zero moment arm about roll axis.
Combined gimbal limit rule? → $\sqrt{\delta_p^2+\delta_y^2}\le\delta_\text{max}$ .

Recall Feynman: explain to a 12-year-old

Imagine pushing a shopping cart from behind with a stick. If you push straight through the middle, it goes straight. If you push a little off to the side, the cart turns. A rocket does the same: the flame is its "push." By tilting the engine a tiny bit, the push points slightly sideways and spins the rocket to point where we want. Tilting only a few degrees is enough because the engine is very strong and sits far behind the rocket's balance point — so a small tilt makes a big turn while barely slowing the rocket down.

Flashcards

What does Thrust Vector Control physically change about the thrust?

Its direction (the thrust vector is tilted), not primarily its magnitude.

Exact single-gimbal control torque formula?

M=\ell T\sin\delta

, with

\ell

= CoM-to-gimbal arm,

T

= thrust,

\delta

= deflection.

Small-angle form of the control torque?

M\approx \ell T\delta

(since

\sin\delta\approx\delta

Why is TVC steering cheap in thrust?

Torque grows linearly (

\propto\delta

) but thrust loss

1-\cos\delta\approx\delta^2/2

grows quadratically, so small angles give lots of torque for tiny loss.

Axial (useful) thrust when gimbaled by

\delta

T\cos\delta

Single-gimbal vs dual-gimbal?

Single = 1 pivot axis, 1 angle, torque in one plane; dual = 2 orthogonal axes, angles

\delta_p,\delta_y

, torque in pitch and yaw.

How do dual-gimbal deflections combine against the hard limit?

As a vector magnitude

\sqrt{\delta_p^2+\delta_y^2}\le\delta_\text{max}

Gimbal angle for commanded angular acceleration

\dot\omega

\delta=\dfrac{I\dot\omega}{\ell T}

(from

M=I\dot\omega=\ell T\delta

Why can a single centered gimbaled engine not control roll?

Its thrust line lies along/near the roll axis → moment arm ≈ 0 → no roll torque; roll needs canted engines/differential gimbal/RCS.

At what deflection is forward thrust maximum?

\delta=0

; any gimbal reduces axial thrust to

T\cos\delta

Connections

Rigid Body Rotational Dynamics — $M=I\dot\omega$ underlies the actuator command.
Torque and Moment Arm — $\vec M=\vec r\times\vec F$ is the whole basis of TVC.
Attitude Control Autopilot — computes commanded $\delta$ from attitude error.
Center of Mass Migration — $\ell$ changes as propellant burns, altering gain.
Reaction Control System (RCS) — complements TVC (roll, coast phases).
Small-Angle Approximation — justifies the linear $M\approx\ell T\delta$ .
Rocket Thrust Equation — where $T$ comes from.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab rocket dheere chal raha hota hai (ya space mein hota hai) tab hawa hoti hi nahi ki fins se steer karo. Toh trick ye hai — engine ko hi thoda tilt kar do. Jab thrust seedha CoM se nahi jaata, toh ek torque ban jaata hai aur rocket ghoom jaata hai. Isi ko Thrust Vector Control (TVC) bolte hain: hum thrust ka direction change karte hain, uski power nahi. Formula simple hai: $M=\ell T\sin\delta$ , jahan $\ell$ = CoM se gimbal tak ki doori, $T$ = thrust, $\delta$ = tilt angle. Chhote angle par $M\approx \ell T\delta$ .

Sabse mast baat: steering bahut sasta hai. Torque toh $\delta$ ke seedhe proportional badhta hai, lekin forward thrust ka nuksaan sirf $\delta^2/2$ (quadratic) hota hai. Matlab $8^\circ$ tilt par sirf ~1% thrust loss, lekin turning force kaafi mil jaata hai. Isliye actual gimbal angles bahut chhote hote hain — kabhi ek do degree hi.

Single-gimbal ka matlab engine ek hi axis par ghoomta hai — sirf pitch ya yaw. Dual-gimbal mein do axis, do angle $\delta_p$ aur $\delta_y$ , toh pitch aur yaw dono control ho jaate hain ek hi engine se. Par yaad rakho — gimbal ki hard limit dono ke vector sum par lagti hai: $\sqrt{\delta_p^2+\delta_y^2}\le\delta_\text{max}$ , seedha add nahi karte. Aur ek centered engine se roll control nahi hota, kyunki roll axis par moment arm zero hota hai — uske liye canted engine ya RCS chahiye. Bas yahi core hai, exam mein $\delta=\frac{I\dot\omega}{\ell T}$ nikaalna aana chahiye.

Thrust vector control — single-gimbal, dual-gimbal; TVC angles

1. Setup and coordinate frame

2. Deriving the control torque from scratch

3. Thrust loss — the cost of steering

4. Single-gimbal vs Dual-gimbal

5. Worked examples

6. Common mistakes (steel-manned)

7. Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Guidance, Navigation & Control (GNC)

Connections