Foundations — Modal analysis — natural frequencies, mode shapes

Before you can read the parent note Modal Analysis, every squiggle in its equations must mean something to you. This page builds each one from nothing — plain words first, then a picture, then why the topic cannot live without it. Read top to bottom; each block uses only things defined above it.

1. A "degree of freedom" (DOF)

The picture. Look at the two beads below. To know the whole picture at any instant you must state two numbers: how far bead 1 has moved and how far bead 2 has moved. That "how many numbers" count is the number of DOF, here $2$.

Why the topic needs it. A real spacecraft is a smooth continuous body with infinitely many points — infinitely many DOF. We cannot compute with infinity, so Finite element analysis (FEA) chops it into a finite number $n$ of DOF. Everything downstream ($n$ frequencies, $n$ shapes) is counted in DOF.

2. Displacement $x$ and the displacement vector $\{x\}$

When there are several DOF we stack all their displacements into one tall column of numbers — a vector — and write it with curly braces:

Why the topic needs it. Newton's law will be written once for the whole structure using $\{x\}$, instead of $n$ separate scribbles.

3. Velocity $\dot{x}$ and acceleration $\ddot{x}$ — the dots

The picture. On a graph of $x$ against time, $\dot{x}$ is the steepness of the curve and $\ddot{x}$ is how quickly that steepness itself changes — the curviness.

Why the topic needs it. Newton's law is $\text{force}=\text{mass}\times \text{acceleration}$, so acceleration $\ddot{x}$ must appear. Stacking all accelerations gives $\{\ddot{x}\}$.

4. Stiffness $k$ and the spring force

The minus sign matters. The force always points back toward rest: pull right ($x>0$) and it pulls left (negative force). That restoring push is what makes things wobble instead of drifting away.

Why the topic needs it. A spacecraft's metal panels and struts behave like stiff springs. Stiffness sets how fast it wants to wobble — high stiffness pushes the natural frequency up, away from the dangerous launch frequencies described in Launch vehicle load environments.

5. Mass matrix $[M]$ and stiffness matrix $[K]$

When springs connect several masses, each mass feels forces from its own springs and its neighbours'. Writing one Newton equation per mass and stacking them, two grids of numbers appear. Square brackets $[]$ flag a matrix — a grid, rows × columns.

For the two-bead example the parent note uses:

Two words the parent leans on:

Why the topic needs it. The entire equation of motion $[M]\{\ddot{x}\}+[K]\{x\}=\{0\}$ is built from exactly these two grids. Symmetry later hands us the beautiful Orthogonality of mode shapes result for free.

6. Frequency: $\omega$, $f$, and the wobble clock

The subscript $n$ in ${\omega }_n$ just means natural — "the frequency the structure picks by itself when left alone," no external pushing.

Why the topic needs it. The one thing modal analysis exists to compute is this list of natural frequencies, so we may keep the forcing frequencies of launch far away from them.

7. The harmonic (sine) motion $\sin (\omega t+\theta )$

Why the topic needs it. Assuming this exact time-shape is the trick that turns the differential equation into a plain algebra problem (the eigenvalue problem), because a sine differentiated twice is just $-{\omega }^2$ times itself.

8. Mode shape $\{\varphi \}$

The picture. For the two-bead system the parent finds two shapes:

• $\{{\varphi }_1\}=\left[\begin{array}{c}1\\ 1.618\end{array}\right]$ — both beads swing the same way (in-phase), bead 2 further.
• $\{{\varphi }_2\}=\left[\begin{array}{c}1\\ -0.618\end{array}\right]$ — beads swing opposite ways (out-of-phase).

Why the topic needs it. The pair $({\omega }_i,\{{\varphi }_i\})$ is mode $i$. Knowing the shape tells engineers which part of the spacecraft moves most, so instruments needing to stay still can be placed at the calm points (nodes).

9. Transpose $\{{\}}^T$ and the determinant $\det$

Two operations show up in the derivation; here is what each does.

Why the topic needs it. Setting the determinant to zero is the machine that spits out the natural frequencies; the transpose is the tool that later decouples the equations into simple one-DOF oscillators.

Prerequisite map

Equipment checklist

Cover the right side and answer out loud; reveal to check.