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Finite Element Methods · Chapter 9 of 10 · Advanced

Dynamics and Modal Analysis

Add mass and the structure can vibrate. A new mass matrix joins the stiffness, and an eigenvalue problem returns the natural frequencies and mode shapes that decide whether a design will resonate.

Readiness check

This chapter adds inertia. Tick only what you can do closed-notes.

Recall the natural frequency √(k/m).
Set up an eigenvalue problem.
Compute a 2-by-2 determinant.
Recall what a mode shape is.
Convert rad/s to Hz.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit vibration in Dynamics, Chapter 10.

3 or more weak itemsReview the stiffness method in Chapter 2.

The core idea

Dynamics adds a mass matrix to the stiffness: Mü + Ku = F. Free vibration turns this into an eigenvalue problem whose roots are the squared natural frequencies and whose vectors are the mode shapes.

Mü + Ku = F(t)(K − ω²M)φ = 0det(K − ω²M) = 0 gives the frequencies

A static analysis ignores inertia, but a vibrating structure cannot. The finite element method builds a mass matrix M alongside the stiffness K, and the equation of motion becomes Mü + Ku = F. With no forcing, the structure vibrates freely, and assuming harmonic motion turns the equation into the generalised eigenvalue problem (K − ω²M)φ = 0. Its eigenvalues ω² are the squared natural frequencies, the speeds at which the structure wants to oscillate, and its eigenvectors φ are the mode shapes, the patterns of motion at each frequency. For a single degree of freedom this reduces to the familiar √(k/m); for many degrees of freedom it gives a frequency and shape for each. Knowing these tells a designer which excitation frequencies to avoid.

The skill works when: you assemble M and K and solve det(K − ω²M) = 0 for the frequencies and shapes.

The skill breaks down when: mass is omitted, so resonance is never predicted.

The concept. A two-mass system has two modes: a lower frequency where the masses move together, and a higher one where they move oppositely. Each natural frequency has its own mode shape.

The skills, taught in order

Five skills set the dynamic equation, the mass matrix, the eigenvalue problem, the frequencies, and the mode shapes.

9.1 The dynamic equation

Including inertia gives the equation of motion Mü + Ku = F, where M is the mass matrix, K the stiffness, and ü the nodal accelerations. The static problem is the special case with no acceleration. Damping, when included, adds a Cu̇ term.

9.2 The mass matrix

The mass matrix can be lumped, placing each element's mass at its nodes to give a diagonal matrix, or consistent, derived from the same shape functions as the stiffness. Lumped masses are simpler and often adequate; consistent masses are more accurate for higher modes.

9.3 The eigenvalue problem

Free, undamped vibration assumes harmonic motion u = φ sin ωt, which substituted into Mü + Ku = 0 gives (K − ω²M)φ = 0. A nontrivial solution requires det(K − ω²M) = 0, the characteristic equation for the natural frequencies.

9.4 Natural frequencies

The roots ω² of the characteristic equation are the squared natural frequencies; there are as many as degrees of freedom. The lowest, the fundamental frequency, usually matters most, since it is the easiest to excite. Frequencies convert to hertz by f = ω/2π.

9.5 Mode shapes

Each natural frequency has a mode shape φ, the relative pattern of nodal motion when the structure vibrates at that frequency. Mode shapes show where motion concentrates, guiding where to stiffen or add mass to shift a frequency away from an excitation.

Engineering connection: matching these natural frequencies against operating speeds is how rotating machinery and structures are kept clear of resonance.

Worked example 1: a single natural frequency

A single-degree-of-freedom model has stiffness k = 1×10⁵ N/m and mass m = 10 kg. Find its natural frequency in rad/s and in hertz from the eigenvalue problem.

Figure 1. For one degree of freedom the eigenvalue problem reduces to k − ω²m = 0, the classic natural frequency √(k/m).

ProblemFind the natural frequency of the system in Figure 1.
Given / findk = 1×10⁵ N/m, m = 10 kg. Find ω_n and f.
AssumptionsSingle degree of freedom; undamped free vibration.
ModelThe eigenvalue problem (k − ω²m)φ = 0 gives ω² = k/m.
Equationsω_n = √(k/m)f = ω_n/2π
Solveω_n = √(1×10⁵/10) = √(10⁴) = 100 rad/s. f = 100/2π = 15.9 Hz.
CheckThe units of √(N/m / kg) = √(1/s²) = 1/s = rad/s, as required. A stiffer or lighter system would vibrate faster.
ConclusionFor one degree of freedom the eigenvalue problem is just √(k/m). The same logic, in matrix form, gives every natural frequency of a large model.

Result. ω_n = 100 rad/s, f = 15.9 Hz.

Worked example 2: a two-degree-of-freedom system

Two equal masses m = 1 kg are connected by three springs of stiffness k = 100 N/m (wall, mass, mass, wall), giving K = [200, −100; −100, 200] and M = the identity. Find the two natural frequencies and describe the mode shapes.

Figure 2. The lower mode has both masses moving together; the higher mode has them moving oppositely, stretching the middle spring and raising the frequency.

ProblemFind the two natural frequencies and mode shapes for the system in Figure 2.
Given / findm = 1 kg, k = 100 N/m, K = [200, −100; −100, 200], M = I. Find ω₁, ω₂ and the shapes.
AssumptionsUndamped free vibration; equal masses and springs.
ModelSolve det(K − ω²M) = 0 for ω², then find each eigenvector.
Equationsdet([200 − ω², −100; −100, 200 − ω²]) = 0(200 − ω²)² = 100²
Solve200 − ω² = ±100, so ω² = 100 or 300. ω₁ = √100 = 10 rad/s (φ₁ = φ₂, masses in phase); ω₂ = √300 = 17.3 rad/s (φ₁ = −φ₂, masses out of phase).
CheckThe lower mode does not stretch the middle spring (both masses move together), so it is softer; the higher mode stretches it most, so it is stiffer, consistent with ω₂ > ω₁. Two degrees of freedom give two modes.
ConclusionThe eigenvalue problem returns a frequency and a shape for each degree of freedom. The mode shapes reveal which springs are working hardest at each frequency.

Result. ω₁ = 10 rad/s (in phase), ω₂ = 17.3 rad/s (out of phase).

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Omitting the mass matrix	No frequencies predicted	"Did I build M as well as K?"	Dynamics needs both M and K.
Confusing ω and f	Frequency off by 2π	"Is this rad/s or Hz?"	f = ω/2π.
Expecting one frequency	Higher modes ignored	"How many degrees of freedom?"	There is a natural frequency per DOF.
Ignoring mode shapes	Stiffening the wrong location	"Where does this mode move most?"	Read the mode shape before modifying the design.

Practice ladder

Level 1 · Direct skill

Find the natural frequency of a system with k = 4×10⁴ N/m and m = 4 kg.

Show answer

ω_n = √(k/m) = √(10⁴) = 100 rad/s, or 15.9 Hz.

Level 2 · Mixed concept

For the Worked Example 1 system, what mass would halve the natural frequency to 50 rad/s?

Show answer

ω² = k/m, so halving ω quarters ω², needing four times the mass: m = 40 kg.

Level 3 · Independent problem

For the Worked Example 2 system, what is the fundamental frequency in hertz?

Show answer

f₁ = ω₁/2π = 10/2π = 1.59 Hz, the lowest and most easily excited frequency.

Level 4 · Transfer to real engineering

For a real component, explain how you would use its natural frequencies and mode shapes to avoid resonance with a known excitation.

What good work looks like

Comparing the fundamental frequency to the excitation, and using the mode shape to decide where to add stiffness or mass to shift the frequency clear.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I set up det(K − ω²M) = 0 correctly."

"Give me three systems; I will find their fundamental frequencies."

"Find the natural frequencies for me." Solving the eigenvalue problem yourself is the skill.

"Will this resonate?" Comparing frequencies to the excitation is the point.

Portfolio task

Set up a dynamic FEM for a small system: assemble M and K, solve the eigenvalue problem for the natural frequencies and mode shapes, and identify the fundamental mode.

Must include: a mass and stiffness matrix, natural frequencies, and mode shapes.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the dynamic equation.

Mü + Ku = F, adding the mass matrix to the stiffness.

2. What is the free-vibration eigenvalue problem?

(K − ω²M)φ = 0, from assuming harmonic motion.

3. How are natural frequencies found?

From det(K − ω²M) = 0; there is one per degree of freedom.

4. What is a mode shape?

The eigenvector φ, the relative motion pattern at a natural frequency.

5. Lumped versus consistent mass?

Lumped is diagonal and simple; consistent comes from the shape functions and is more accurate for higher modes.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the single and two-DOF frequencies from a blank page.

+3 daysSolve a new eigenvalue problem.

+7 daysCarry the analysis into convergence and modeling, Chapter 10.

+30 daysReuse modal analysis to keep a design clear of resonance.

Textbook mapping

Item	Mapping
Primary source	Logan, A First Course in the Finite Element Method, Chapter 16 (Structural Dynamics)
Cross-reference	Hutton, Ch. 10 · Dynamics, Ch. 10
Core topics	9.1 Dynamic equation · 9.2 Mass matrix · 9.3 Eigenvalue problem · 9.4 Natural frequencies · 9.5 Mode shapes
Engineering connection	Natural frequencies and modes keep machinery clear of resonance.
Read next	Chapter 10: Convergence, Errors, and Modeling.