Simple Harmonic Motion, Resonance, and FEM Modal Analysis

1. The Bridge That Destroyed Itself

On November 7, 1940, the Tacoma Narrows Bridge in Washington State collapsed spectacularly in a moderate 67 km/h wind. The bridge had been open for only four months. The collapse is one of the most famous engineering disasters in history — and it's a textbook lesson in resonance.

The bridge had a long, flat, and relatively narrow deck — aerodynamically prone to vortex shedding. As wind flowed around the deck, alternating vortices were shed alternately from the top and bottom edges (Kármán vortex street), creating an oscillating lift force. When the vortex shedding frequency matched the bridge's torsional natural frequency, the amplitude of oscillation grew without bound — resonance. Within hours, the deck was twisting ±45° and ultimately failed.

2. Simple Harmonic Motion: x = A cos(ωt + φ)

Simple Harmonic Motion (SHM) occurs when a restoring force is proportional to displacement and directed toward equilibrium. The equation of motion for an undamped mass-spring system:

The solution is oscillatory:

Where $A$ is amplitude (m), $\omega_n$ is natural angular frequency (rad/s), and $\varphi$ is initial phase (rad). The velocity and acceleration are:

Key insight: the maximum acceleration = $A\omega_n^2$. For high-frequency vibration with even modest amplitude, accelerations can be enormous — this is why high-frequency vibration of electronics (printed circuit boards, connectors) causes fatigue failure even with sub-millimeter amplitudes.

3. Natural Frequency: ωn = √(k/m)

The natural frequency of an undamped SDOF system:

This simple formula contains enormous insight: to increase natural frequency (stiffen a structure against resonance), increase $k$ or reduce $m$. Both are common design strategies.

4. Damped Oscillation and Damping Ratio

Real structures always have some energy dissipation — material damping, friction at joints, aerodynamic drag. The damped SDOF equation:

The critical damping coefficient: $c_{\text{cr}} = 2\sqrt{km} = 2m\omega_n$. The dimensionless damping ratio:

Solution for underdamped case ($\zeta < 1$):

5. Forced Vibration and Resonance

When an external harmonic force $F_0\cos(\Omega t)$ drives a SDOF system, the steady-state response amplitude is:

Where $r$ is the frequency ratio. The dynamic amplification factor (DAF) is the ratio $X / (F_0/k)$ — how much larger is the dynamic response compared to the static response?

At resonance ($r = 1$, $\Omega = \omega_n$): DAF = $\frac{1}{2\zeta}$. For typical structural damping of 1% ($\zeta = 0.01$): DAF = 50! The dynamic response is 50 times larger than the static response. This is why resonance is so destructive.

Resonance Avoidance Design Rules

• Frequency separation: Keep $f_n$ at least 20–30% away from any expected excitation frequency
• Detuning: Add or remove mass/stiffness to shift $f_n$ away from excitation
• Damping: Add viscoelastic damping layers, tuned mass dampers, or friction dampers
• Isolation: Mount equipment on soft isolators so the transmitted excitation doesn't reach the structure's natural frequency

6. Multi-DOF Systems and Mode Shapes

A real structure has infinitely many DOF (or $N$ DOF in FEM). The undamped free vibration problem becomes an eigenvalue problem:

Where $\omega_i$ are the natural frequencies (eigenvalues) and $\boldsymbol{\phi}_i$ are the mode shapes (eigenvectors). Each mode shape describes the spatial pattern of deformation at that natural frequency.

Modal properties:

• Mass-orthogonality: $\boldsymbol{\phi}_i^T [\mathbf{M}] \boldsymbol{\phi}_j = 0$ for $i \neq j$
• Stiffness-orthogonality: $\boldsymbol{\phi}_i^T [\mathbf{K}] \boldsymbol{\phi}_j = 0$ for $i \neq j$
• Modal mass: $m_i^* = \boldsymbol{\phi}_i^T [\mathbf{M}] \boldsymbol{\phi}_i$
• Modal stiffness: $k_i^* = \boldsymbol{\phi}_i^T [\mathbf{K}] \boldsymbol{\phi}_i = \omega_i^2 m_i^*$

Orthogonality is the mathematical miracle that makes modal analysis work: the coupled MDOF system decomposes into $N$ independent SDOF equations in modal coordinates. Each mode can be solved independently.

7. FEM Modal Analysis

In FEM, natural frequencies and mode shapes are computed by solving the generalized eigenvalue problem numerically. The Lanczos algorithm (used in NASTRAN, Abaqus) is efficient for extracting the lowest $n$ modes of a large sparse system.

Practical Modal Analysis Workflow

1. Build the FEM model — accurately capture mass distribution (concentrated masses, non-structural masses)
2. Run modal analysis — extract the first 20–50 modes (enough to cover the frequency range of interest)
3. Check effective mass — the sum of effective mass fractions across all modes should reach ~90% of total mass; if not, add more modes
4. Inspect mode shapes — identify local modes (single component vibrating), global modes (whole structure), and fluid-structure modes
5. Compare with test (EMA) — experimental modal analysis from hammer or shaker test provides correlation data

Modal Effective Mass

The effective mass participation factor for mode $i$ in direction $j$:

Where $\mathbf{L}_j$ is the unit excitation vector in direction $j$. High effective mass means the mode is important for that excitation direction and should be included in response spectrum analysis.

8. Campbell Diagram for Rotating Machinery

For rotating machinery (engines, turbines, fans), the excitation frequencies are multiples of the rotation speed (engine orders). A Campbell diagram plots natural frequencies (horizontal lines) and engine orders (diagonal lines) against rotation speed:

Where $n$ is the engine order (1×, 2×, etc.) and $N$ is speed in RPM. Crossings of a natural frequency line and an engine order line are potential resonance points. The Campbell diagram helps designers ensure no problematic resonance crossings occur at sustained operating speeds.

For a 4-cylinder 4-stroke engine at 3000 RPM: the primary firing order excitation is the 2nd engine order = 2 × 3000/60 = 100 Hz. Any structural component with a natural frequency near 100 Hz will be excited heavily at cruise speed — a classic NVH design issue.

9. Cross-Topics