Foundations — Propulsive forces — thrust misalignment, gimbal angle

This is the toolbox note for the parent topic on thrust misalignment and gimbal angle. We assume you have seen nothing. Every letter, every arrow, every trig word gets built here, in an order where each idea leans only on the ones before it.

0 · The picture we keep coming back to

Before any symbol, look at the rocket itself. It is a long tube. It has a pointy end (the nose) and a fat end (the engine). We will draw an arrow through the middle of the tube — that line is the rocket's spine.

Keep this drawing in your head. Almost every symbol below is just a label pinned onto some part of this picture.

1 · Arrows that mean "how much and which way" — vectors

The picture: an arrow. A short arrow = a little; a long arrow = a lot. Turn the arrow, and you have changed the direction without changing the amount.

Why the topic needs it: a rocket's push has both a size (how hard) and a direction (which way it points). A plain number cannot hold both facts at once. An arrow can. That is the whole reason vectors show up.

2 · The push itself — force and thrust

The picture: a fat arrow leaving the back of the rocket engine, pointing forward (toward the nose). The gas goes backward, the rocket gets shoved forward.

Unit note: we measure force in newtons (N). One kilonewton is $1\text{kN}=1000\text{N}$. The parent uses $T=800\text{kN}=800000\text{N}$.

3 · The balance-point — center of mass (CoM)

The picture: a dot somewhere along the spine of the tube. Not the middle of the length necessarily — it sits wherever the mass is balanced (the heavy fuel end pulls it back).

Why the topic needs it: a force pushing through the CoM only slides the rocket along. A force pushing past the CoM makes it turn. So the CoM is the pivot everything rotates around. Without marking it, we cannot even ask "does the push spin the rocket?"

4 · Where the push acts, and how far off — position vector and lever arm $L$

The picture: an arrow from the dot (CoM) back down the spine to the engine's pivot point.

The picture: a straight ruler laid from the balance-dot to the engine, reading "$L=20\text{m}$".

Why the topic needs it: the further behind the CoM the push acts, the more leverage it has to spin the rocket — like pushing a door far from its hinge versus right next to it. $L$ is that leverage distance.

5 · Which way things point — coordinates and axes

The picture: a cross of two lines through the CoM dot; the horizontal line runs up the tube to the nose ($+x$), the vertical line points out the side ($+y$).

Why the topic needs it: it is hard to do sums with slanted arrows directly. If we split every arrow into an $x$-piece and a $y$-piece, we can add and multiply plain numbers instead. The $x$-piece of thrust is the part that pushes the rocket forward; the $y$-piece is the part that shoves it sideways (and does the spinning).

6 · The tilt — angle $\delta$, and why we need trigonometry

The picture: the engine's thrust arrow no longer lined up with the spine — there's a small wedge-shaped gap between them. That wedge is $\delta$.

Now here is the key question the topic must answer:

If I tilt the thrust arrow by angle $\delta$, how much of it points forward, and how much points sideways?

To answer "how much of a tilted arrow points each way," we need a tool that turns an angle into a ratio of side-lengths. That tool is trigonometry — specifically $\sin$ and $\cos$.

Building $\sin$ and $\cos$ from a triangle

Draw the tilted thrust arrow. Drop it onto the $x$-axis: you get a right-angled triangle. The full arrow has length $T$. The forward side (along the spine) and the sideways side are the two shorter legs.

Multiply through by the arrow's length $T$:

Why $\cos$ for forward and $\sin$ for sideways? When $\delta =0$ (no tilt), all the push is forward: and indeed $\cos 0=1$ (full forward), $\sin 0=0$ (no sideways). As $\delta$ grows, $\cos \delta$ shrinks (less forward) and $\sin \delta$ grows (more sideways). The two functions are exactly the "forward-ness" and "sideways-ness" dials of a tilted arrow.

The small-angle shortcut

For a tiny angle measured in radians, the arrow and its opposite leg are almost the same length, so $\sin \delta \approx \delta ,\cos \delta \approx 1-\frac{1}{2}{\delta }^2.$

Why radians, not degrees? The shortcut $\sin \delta \approx \delta$ is only true when $\delta$ is in radians. In degrees, $\sin (3^{\circ })=0.052$ but the number "3" is nowhere near it. Radians are built so the angle and its sine line up for small tilts — which is why every trig calculation in this topic converts to radians first.

7 · The twist — torque $\tau$

The picture: the sideways part of the thrust, acting at the end of the lever arm $L$, curling the rocket around the CoM dot like a wrench turning a bolt.

Why the topic needs it: force tells you how the rocket slides; torque tells you how it turns. Steering a rocket is all about turning, so torque is the star quantity. The parent's whole result, $\tau =LT\sin \delta$, is just:

$\underset{\text{lever arm}}{\underbrace{L}}\times \underset{\text{sideways push}}{\underbrace{T\sin \delta }}=\text{twist}.$

Distance to the push, times the sideways part of the push. See Torque and Moment of Inertia for the deeper machinery.

8 · Resistance to turning, and how fast it turns — $I$ and $\ddot{\theta }$

The picture: the rocket, initially pointing straight, slowly rotating faster and faster once a torque is applied.

Why the topic needs it: torque tells you the twist, but the pilot cares about how fast the rocket actually turns. Divide the twist by the stubbornness and you get the turning acceleration. This is the link from "engine tilt" to "rocket points somewhere new," the foundation of Thrust Vector Control (TVC) and Attitude Control & Stability.

The prerequisite map

Read it bottom-up: arrows and Newton's law give you thrust; the angle plus trig splits that thrust; the balance-point gives you a lever arm; together they make torque; torque plus inertia gives turning — and that is the parent topic.

Equipment checklist

Test yourself — you should be able to answer each before opening the parent note.