Modal analysis — natural frequencies, mode shapes

Overview

Modal analysis is the process of determining a structure's natural frequencies and mode shapes—the frequencies at which it naturally vibrates and the characteristic patterns of motion at those frequencies. For spacecraft, this is critical: launch vibrations, thruster firings, and attitude maneuvers can excite these natural modes, potentially causing structural failure, instrument misalignment, or mission-critical damage.

Why it matters: If you excite a spacecraft at its natural frequency, resonance occurs—amplitudes grow catastrophically. Knowing the modal properties lets us design structures that keep natural frequencies away from forcing frequencies, add damping strategically, and predict dynamic response.

Connections

Free vibration fundamentals
Multi-degree-of-freedom systems
Structural damping mechanisms
Finite element analysis (FEA)
Launch vehicle load environments
Orthogonality of mode shapes
Frequency response functions

Core Concepts

Intuition Physical Intuition

Imagine plucking a guitar string. It vibrates at specific pitches (frequencies), each with a characteristic shape—first mode is a single hump, second mode has two humps separated by a still point (node). The string "wants" to vibrate in these patterns.

A spacecraft structure is a3D "guitar string." It has multiple natural frequencies (first bending, first torsion, panel modes, etc.) and corresponding mode shapes. Launch vibrations are like plucking it—the structure responds by vibrating in combinations of its natural modes.

The danger: If launch vibrations contain energy at a natural frequency, that mode gets "plucked" hard, and the structure shakes violently (resonance). Modal analysis finds these vulnerable frequencies so we can either stiffen the structure (raise frequencies above forcing range) or add dampers.

Derivation from First Principles

Starting Point: Multi-DOF Free Vibration

Consider a structure discretized into $n$ degrees of freedom (masses connected by springs). The equation of motion without damping or forcing:

[M]\{\ddot{x}\} + [K]\{x\} = \{0\}

where:

$[M]$ : mass matrix (symmetric, positive definite)
$[K]$ : stiffness matrix (symmetric, positive semi-definite)
$\{x\}$ : displacement vector

Why this form? Each row is Newton's second law for one mass: $m_i \ddot{x}_i = -k_{i,i}x_i - k_{i,j}x_j + \ldots$ (spring forces from neighbors).

Assuming Harmonic Solution

For free vibration, assume all points oscillate at the same frequency $\omega$ with fixed relative amplitudes:

\{x(t)\} = \{\phi\} \sin(\omega t + \theta)

Why harmonic? Linear systems with no damping oscillate sinusoidally. $\{\phi\}$ is the mode shape (spatial pattern), $\omega$ is the natural frequency (time pattern).

Substitute into the equation of motion:

\ddot{x} = -\omega^2 \{\phi\} \sin(\omega t + \theta)

[M](-\omega^2 \{\phi\}) \sin(\omega t + \theta) + [K]\{\phi\} \sin(\omega t + \theta) = \{0\}

Cancel $\sin(\omega t + \theta)$ (nonzero):

([K] - \omega^2 [M])\{\phi\} = \{0\}

Why eigenvalue problem? We need nontrivial solutions ( $\{\phi\} \neq \{0\}$ ). This only happens when the determinant is zero—the characteristic polynomial whose roots are $\omega_i^2$ .

Orthogonality of Mode Shapes

For two different modes $i$ and $j$ :

[K]\{\phi_i\} = \omega_i^2 [M]\{\phi_i\}

[K]\{\phi_j\} = \omega_j^2 [M]\{\phi_j\}

Pre-multiply first by $\{\phi_j\}^T$ , second by $\{\phi_i\}^T$ :

\{\phi_j\}^T [K]\{\phi_i\} = \omega_i^2 \{\phi_j\}^T [M]\{\phi_i\}

\{\phi_i\}^T [K]\{\phi_j\} = \omega_j^2 \{\phi_i\}^T [M]\{\phi_j\}

Since $[K]$ and $[M]$ are symmetric, $\{\phi_j\}^T [K]\{\phi_i\} = \{\phi_i\}^T [K]\{\phi_j\}$ . Subtract the equations:

0 = (\omega_i^2 - \omega_j^2) \{\phi_j\}^T [M]\{\phi_i\}

If $\omega_i \neq \omega_j$ , then:

\boxed{\{\phi_j\}^T [M]\{\phi_i\} = 0 \quad \text{(mass orthogonality)}}

Similarly:

\boxed{\{\phi_j\}^T [K]\{\phi_i\} = 0 \quad \text{(stiffness orthogonality)}}

Why orthogonality matters? Mode shapes are independent. We can transform the coupled equations into $n$ independent single-DOF oscillators (modal coordinates).

Define modal matrix $[\Phi] = [\{\phi_1\}, \{\phi_2\}, \ldots, \{\phi_n\}]$ (columns are mode shapes).

Transform physical coordinates to modal coordinates $\{q\}$ :

\{x\} = [\Phi]\{q\}

Physical meaning: $q_i(t)$ is how much mode $i$ participates in the total response.

Substitute into equation of motion:

[M][\Phi]\{\ddot{q}\} + [K][\Phi]\{q\} = \{F(t)\}

Pre-multiply by $[\Phi]^T$ :

[\Phi]^T[M][\Phi]\{\ddot{q}\} + [\Phi]^T[K][\Phi]\{q\} = [\Phi]^T\{F(t)\}

By orthogonality, $[\Phi]^T[M][\Phi]$ and $[\Phi]^T[K][\Phi]$ are diagonal matrices:

[\Phi]^T[M][\Phi] = [M^*] = \text{diag}(M_1^*, M_2^*, \ldots) \quad \text{(generalized mass)}

[\Phi]^T[K][\Phi] = [K^*] = \text{diag}(K_1^*, K_2^*, \ldots) \quad \text{(generalized stiffness)}

Each modal equation becomes:

\boxed{M_i^* \ddot{q}_i + K_i^* q_i = Q_i(t)}

where $Q_i = \{\phi_i\}^T\{F\}$ is the modal force.

This is $n$ independent single-DOF oscillators! Each mode vibrates at $\omega_i = \sqrt{K_i^*/M_i^*}$ independently.

Worked Examples

Worked example Example 1: Two-DOF SpringMass System

Given: Two equal masses $m$ connected by springs. First mass attached to wall with spring $k$ , masses connected by spring $k$ , second mass free.

|--[k]--[m1]--[k[m2]

Find: Natural frequencies and mode shapes.

Solution: Step 1: Write equations of motion.

Mass 1: $m\ddot{x}_1 = -kx_1 - k(x_1 - x_2) = -2kx_1 + kx_2$
Mass 2: $m\ddot{x}_2 = -k(x_2 - x_1) = kx_1 - kx_2$

Matrix form:

[M] = \begin{bmatrix} m & 0 \\ 0 & m \end{bmatrix}, \quad [K] = \begin{bmatrix} 2k & -k \\ -k & k \end{bmatrix}

Why these matrices? $M$ diagonal because masses don't affect each other directly. $K$ has off-diagonal terms because springs couple the motion.

Step 2: Solve eigenvalue problem $\det([K] - \omega^2[M]) = 0$ .

\det\begin{bmatrix} 2k - m\omega^2 & -k \\ -k & k - m\omega^2 \end{bmatrix} = 0

(2k - m\omega^2)(k - m\omega^2) - k^2 = 0

2k^2 - 2km\omega^2 - km\omega^2 + m^2\omega^4 - k^2 = 0

m^2\omega^4 - 3km\omega^2 + k^2 = 0

Why expand? Characteristic polynomial in $\omega^2$ .

Quadratic formula: $\omega^2 = \frac{3k \pm \sqrt{9k^2 - 4k^2}}{2m} = \frac{3k \pm \sqrt{5}k}{2m}$

\boxed{\omega_1 = \sqrt{\frac{(3-\sqrt{5})k}{2m}} \approx 0.618\sqrt{k/m}}

\boxed{\omega_2 = \sqrt{\frac{(3+\sqrt{5})k}{2m}} \approx 1.618\sqrt{k/m}}

Step 3: Find mode shapes. For $\omega_1^2 = (3-\sqrt{5})k/(2m)$ :

[K]\{\phi_1\} = \omega_1^2[M]\{\phi_1\}

\begin{bmatrix} 2k & -k \\ -k & k \end{bmatrix} \begin{bmatrix} \phi_{11} \\ \phi_{21} \end{bmatrix} = \frac{(3-\sqrt{5})k}{2} \begin{bmatrix} \phi_{11} \\ \phi_{21} \end{bmatrix}

First row: $2k\phi_{11} - k\phi_{21} = \frac{(3-\sqrt{5})k}{2}\phi_{11}$

\phi_{21} = \left(2 - \frac{3-\sqrt{5}}{2}\right)\phi_{11} = \frac{1+\sqrt{5}}{2}\phi_{11} \approx 1.618\phi_{11}

Normalize (set $\phi_{11} = 1$ ):

\boxed{\{\phi_1\} = \begin{bmatrix} 1 \\ 1.618 \end{bmatrix} \quad \text{(both masses move in-phase, mass2 larger amplitude)}}

For $\omega_2^2$ , similarly:

\boxed{\{\phi_2\} = \begin{bmatrix} 1 \\ -0.618 \end{bmatrix} \quad \text{(masses move out-of-phase)}}

Physical interpretation:

Mode 1 (lower frequency): both masses move together, "breathing" mode
Mode 2 (higher frequency): masses move opposite, "scissor" mode, center spring stretches/compresses more → higher stiffness → higher frequency

Worked example Example 2: Cantilever Beam Modal Analysis

Given: Uniform cantilever beam, length $L$ , mass per unit length $\mu$ , bending stiffness $EI$ .

Find: First natural frequency and mode shape (continuous system).

Solution:

Step 1: Beam vibration PDE (Euler-Bernoulli):

EI \frac{\partial^4 w}{\partial x^4} + \mu \frac{\partial^2 w}{\partial t^2} = 0

Why this PDE? Bending moment $M = EI\frac{\partial^2 w}{\partial x^2}$ , shear force balance gives fourth derivative.

Step 2: Assume separation of variables $w(x,t) = \phi(x)\sin(\omega t)$ .

EI\frac{d^4\phi}{dx^4} - \mu\omega^2\phi = 0

\beta^4 = \frac{\mu\omega^2}{EI}

\frac{d^4\phi}{dx^4} - \beta^4\phi = 0

General solution: $\phi(x) = A\cos(\beta x) + B\sin(\beta + C\cosh(\beta x) + D\sinh(\beta x)$

Step 3: Apply boundary conditions (cantilever: fixed at $x=0$ , free at $x=L$ ).

$\phi(0) = 0$ (no displacement)
$\phi'(0) = 0$ (no rotation)
$\phi''(L) = 0$ (no moment)
$\phi'''(L) = 0$ (no shear)

After algebra (standard beam problem), characteristic equation:

\cos(\beta L)\cosh(\beta L) = -1

\text{First root: } \beta_1 L \approx 1.875

\boxed{\omega_1 = \beta_1^2\sqrt{\frac{EI}{\mu}} = 3.516\frac{EI}{\mu L^4} \quad \text{[rad/s]}}

Mode shape (normalized):

\boxed{\phi_1(x) = \cosh(\beta_1 x) - \cos(\beta_1 x) - \sigma_1(\sinh(\beta_1 x) - \sin(\beta_1 x)}

where $\sigma_1 = \frac{\cosh(\beta_1 L) + \cos(\beta_1 L)}{\sinh(\beta_1 L) + \sin(\beta_1 L)} \approx 0.7341$

Physical interpretation: First mode is a single smooth curve, maximum deflection at the free end, characteristic "cantilever bend."

Common Pitfalls

Common mistake Mistake 1: Confusing Natural Frequency with Forcing Frequency

Wrong thinking: "The spacecraft vibrates at the engine's frequency during launch."

Why it feels right: External forces do impose their frequency on the structure.

The truth: Natural frequencies are intrinsic properties (like a bell's pitch)—they exist whether or not the structure is forced. Forcing frequency is external. Resonance happens when forcing frequency matches a natural frequency.

Steel-man: You're right that the structure responds to forcing frequency. But at resonance, even a small force at $\omega_n$ causes huge amplitudes because the mode is "in tune." Away from $\omega_n$ , response is smaller.

Fix: Natural frequency = property of structure alone. Forcing frequency = external input. Match them → disaster.

Common mistake Mistake 2: Ignoring Mode Shape Normalization

Wrong thinking: "Mode shapes have definite magnitudes— $\phi_1 = 2$ means2 meters displacement."

Why it feels right: Mode shapes are displacement patterns, so units should be length.

The truth: Mode shapes are relative patterns, not absolute displacements. We can multiply a mode shape by any constant and it's still a valid solution (eigenvectors are only unique up to scaling). Typically normalized so $\{\phi\}^T[M]\{\phi\} = 1$ (unit modal mass).

Actual displacement comes from modal coordinates: $x(t) = \phi_i q_i(t)$ , where $q_i(t)$ has the actual amplitude.

Fix: Mode shapes show shape only. Amplitude comes from modal coordinates, determined by initial conditions or forcing.

Active Recall Practice

Recall Explain Like I'm Twelve (Feynman Technique)

Imagine you have a slinky hanging from your hand. If you gently tap the bottom, it bounces up and down in a specific wiggly pattern—that's its mode shape. And it bounces at a certain speed (how many bounces per second)—that's its natural frequency.

Now imagine the slinky isn't just one spring, but a whole jungle gym made of springs and masses connected together. If you tap it, different parts wigle in complicated patterns. But here's the cool part: there are special patterns where every part moves together like a dance routine, all at the same rhythm. These are the natural modes.

A spacecraft is like that jungle gym. When the rocket shakes it during launch, it's like tapping the jungle gym. The spacecraft wants to shake in its favorite patterns (modes). If the rocket shakes at exactly the right rhythm (the natural frequency), the spacecraft shakes harder and harder until things break—that's resonance.

Engineers do modal analysis to figure out all the favorite patterns and rhythms, so they can make sure the rocket never shakes at those exact rhythms, keeping the spacecraft safe.

Flashcards

#flashcards/physics

What is a natural frequency? :: An intrinsic frequency at which a structure vibrates freely after being disturbed, with no external forcing. It depends only on mass and stiffness properties.

What is a mode shape?

The spatial pattern of relative displacements throughout a structure when it vibrates at a particular natural frequency. It is dimensionless and normalized.

Why do mode shapes form an orthogonal set?

Because mass and stiffness matrices are symmetric and positive definite. Mathematically, for

i \neq j

\{\phi_j\}^T[M]\{\phi_i\} = 0

and

\{\phi_j\}^T[K]\{\phi_i\} = 0

What is resonance in structural dynamics?

When forcing frequency matches a natural frequency (

\omega_\text{forcing} = \omega_n

), causing amplitude to grow without bound (undamped) or to very large values (lightly damped), potentially leading to structural failure.

How do you decouple equations of motion in modal analysis?

Transform physical coordinates

\{x\}

to modal coordinates

\{q\}

using the modal matrix:

\{x\} = [\Phi]\{q\}

. This diagonalizes the system into

n

independent single-DOF oscillators.

What is generalized mass for mode

i

The effective mass of the structure when vibrating in mode

i

, calculated as

M_i^* = \{\phi_i\}^T[M]\{\phi_i\}

. Used to compute natural frequency:

\omega_i = \sqrt{K_i^*/M_i^*}

Why are higher modes less important in spacecraft dynamics?

Because typical forcing functions (launch vibrations, thrusters) have most energy at low frequencies, and higher modes have smaller modal participation factors. First10-50 modes typically capture >95% of response.

What does the characteristic equation

\det([K] - \omega^2[M]) = 0

represent?

The condition for nontrivial solutions to the eigenvalue problem. Its roots are

\omega_i^2

, the squared natural frequencies. For an

n

-DOF system, this is an

n

th order polynomial.

How does stiffness affect natural frequency?

Natural frequency increases with square root of stiffness:

\omega = \sqrt{k/m}

. Doubling stiffness increases frequency by

\sqrt{2} \approx 1.41×

, moving it away from low-frequency launch loads.

What is modal truncation?

Keeping only the first

m

modes (where

m < n

) in modal analysis, discarding higher modes that contribute negligibly to response. Typically retain modes up to 2× the highest forcing frequency.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Modal analysis matlab hai jisse hum structure ke natural frequencies aur mode shapes nikalte hain—yani wo frequencies jahan par structure khud se vibrate karta hai, aur us vibration ka pattern kya hai. Spacecraft ke liye ye bahut zaroori hai kyunki launch ke time, thruster firing, ya koi bhi maneuver structure ko shake kar sakta hai.Agar forcing frequency (jaise rocket

Modal analysis — natural frequencies, mode shapes

Overview

Connections

Core Concepts

Derivation from First Principles

Starting Point: Multi-DOF Free Vibration

Assuming Harmonic Solution

Orthogonality of Mode Shapes

Worked Examples

Common Pitfalls

Active Recall Practice

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Spacecraft Structures & Systems Engineering

Connections