Vibration & Structural Dynamics for CAE
SDOF to Random Vibration

Category: Fundamental Theory | Updated: 2026-03-25 | NovaSolver Contributors

Vibration failures are among the most common and costly in engineering — from fatigue cracks in aircraft structures driven by turbulence, to resonance damage in bridge cables, to fan blade flutter in jet engines. Understanding how structures respond to dynamic loads is the foundation of NVH engineering, seismic qualification, and vibration isolation design.

1. Fundamentals of Vibration

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What's the difference between "natural frequency," "resonance frequency," and "forcing frequency"? I hear all three terms and I'm not sure they're the same thing.

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Three distinct concepts. Natural frequency is a property of the structure itself — the frequency it would vibrate at if disturbed and left alone. Forcing frequency is the frequency of the external load (engine RPM, rotating imbalance, road roughness). Resonance occurs when the forcing frequency equals or closely approaches the natural frequency — at that point the structure vibrates with enormously amplified amplitude. The amplification factor is 1/(2ζ), so for typical 2% structural damping, resonance gives 25× amplification. That's why design rules aim to keep natural frequencies away from operating ranges.

1.1 Free vs. Forced Vibration

Free vibration: System vibrates at its own natural frequency after initial disturbance, with amplitude decaying due to damping. Used in experimental modal testing to identify natural frequencies and mode shapes.

Forced vibration: System subjected to continuous external loading. Steady-state response depends on the ratio of forcing to natural frequency and the damping. Maximum response at resonance.

1.2 Types of Damping

Type	Force model	Usage in CAE
Viscous	$F_d = c\dot{u}$	Standard in FEM; Rayleigh damping is viscous
Structural (hysteretic)	$F_d = ih K u$ (complex stiffness)	Frequency-domain analysis; material damping
Coulomb (dry friction)	$F_d = \mu N\,\text{sgn}(\dot{u})$	Bolted joints, frictional dampers; nonlinear
Aerodynamic	$F_d \propto \dot{u}^2$	Large-amplitude flutter, bridge aerodynamics

2. SDOF System Analysis

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Why do we study SDOF systems if real structures have thousands of DOFs? Is it just a textbook exercise?

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Not at all — SDOF is directly useful because many real-world systems can be approximated as SDOF near a single resonance. A motor on mounts, a sensor on a cantilever, a floor slab responding to foot traffic — each has one dominant mode. More importantly, MDOF analysis decomposes into uncoupled SDOF problems through modal superposition. So once you truly understand SDOF dynamics, you can handle any linear multi-DOF system. Think of each mode as an independent SDOF oscillator: same math, just applied to modal coordinates instead of physical ones.

2.1 Equation of Motion

$$m\ddot{u} + c\dot{u} + ku = f(t)$$

Dividing by mass and defining $\omega_n = \sqrt{k/m}$, $\zeta = c/(2\sqrt{km}) = c/(2m\omega_n)$:

$$\ddot{u} + 2\zeta\omega_n\dot{u} + \omega_n^2 u = f(t)/m$$

2.2 Free Undamped Vibration ($f=0$, $\zeta=0$)

$$u(t) = A\cos(\omega_n t) + B\sin(\omega_n t), \qquad \omega_n = \sqrt{k/m}, \quad f_n = \frac{\omega_n}{2\pi}$$

2.3 Free Damped Vibration

Three cases based on damping ratio ζ:

Underdamped (ζ < 1) — most structures:

$$u(t) = e^{-\zeta\omega_n t}\left[u_0\cos(\omega_d t) + \frac{\dot{u}_0 + \zeta\omega_n u_0}{\omega_d}\sin(\omega_d t)\right]$$ $$\omega_d = \omega_n\sqrt{1-\zeta^2} \quad \text{(damped natural frequency)}$$

Critically damped (ζ = 1) — fastest non-oscillatory return to rest:

$$u(t) = (u_0 + (\dot{u}_0 + \omega_n u_0)t)e^{-\omega_n t}$$

Overdamped (ζ > 1) — slow non-oscillatory decay:

$$u(t) = e^{-\zeta\omega_n t}\left[C_1 e^{\omega_n\sqrt{\zeta^2-1}\,t} + C_2 e^{-\omega_n\sqrt{\zeta^2-1}\,t}\right]$$

2.4 Forced Harmonic Response

For $f(t) = F_0\cos(\Omega t)$, the steady-state response is:

$$u(t) = X\cos(\Omega t - \phi), \qquad X = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$$ $$r = \frac{\Omega}{\omega_n}, \qquad \phi = \arctan\!\left(\frac{2\zeta r}{1-r^2}\right)$$

The dimensionless dynamic amplification factor $D$:

$$D = \frac{X}{X_{static}} = \frac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$$

At resonance ($r = 1$): $D_{\max} = \frac{1}{2\zeta}$

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So at resonance with 1% damping, the amplification is 50× the static deflection? That seems extreme.

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Exactly — and 1% is not unrealistically low. Many lightly welded steel structures or reinforced concrete have ζ = 1–2%. The Tacoma Narrows Bridge collapse in 1940 is the classic example: aeroelastic resonance drove oscillations to amplitudes large enough to tear the bridge apart. The response is also frequency-sensitive: you only get that 50× amplification if you stay at exactly resonance. Pass through resonance quickly (as in a spin-up test) and the peak is much lower because there's not enough time for energy to build up. That's the concept behind sweep rate effects in vibration testing.

3. MDOF Systems and Eigenvalue Analysis

For a system with $n$ DOFs, the equations of motion are:

$$\mathbf{M}\ddot{\mathbf{u}} + \mathbf{C}\dot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{f}(t)$$

3.1 Undamped Natural Frequencies and Mode Shapes

Setting $\mathbf{C} = 0$ and $\mathbf{f} = 0$, with $\mathbf{u} = \boldsymbol{\phi}e^{i\omega t}$:

$$(\mathbf{K} - \omega^2\mathbf{M})\boldsymbol{\phi} = \mathbf{0}$$

This generalised eigenvalue problem yields $n$ eigenvalues $\omega_i^2$ and eigenvectors $\boldsymbol{\phi}_i$. Key properties:

$$\boldsymbol{\phi}_i^T\mathbf{M}\boldsymbol{\phi}_j = \delta_{ij} \quad \text{(M-orthonormality)}$$ $$\boldsymbol{\phi}_i^T\mathbf{K}\boldsymbol{\phi}_j = \omega_i^2\delta_{ij} \quad \text{(K-orthogonality)}$$

3.2 Modal Superposition

Using the modal transformation $\mathbf{u} = \boldsymbol{\Phi}\mathbf{q}$ (where $\boldsymbol{\Phi}$ is the modal matrix), the EOM decouples into $n$ independent SDOF equations:

$$\ddot{q}_i + 2\zeta_i\omega_i\dot{q}_i + \omega_i^2 q_i = \boldsymbol{\phi}_i^T\mathbf{f}(t) \equiv p_i(t)$$

Each modal coordinate $q_i$ is an independent SDOF oscillator with its own frequency $\omega_i$ and modal damping $\zeta_i$. Solve each independently, then superpose: $\mathbf{u}(t) = \sum_i \boldsymbol{\phi}_i q_i(t)$.

3.3 Modal Truncation

In practice, only the first $m \ll n$ modes are computed (Lanczos or subspace iteration). The truncated modal model is accurate if:

The retained modes cover the frequency range of the excitation
Static correction (residual mode) accounts for the contribution of truncated high-frequency modes
Modal effective mass: the fraction of total mass $m_i^{eff}/M_\text{total}$ participating in each mode. Retain modes until cumulative participation exceeds 90% in each direction

$$m_i^\text{eff} = \frac{(\boldsymbol{\phi}_i^T\mathbf{M}\mathbf{r})^2}{\boldsymbol{\phi}_i^T\mathbf{M}\boldsymbol{\phi}_i}$$

4. Numerical Time Integration

For nonlinear dynamics or complex loading, time integration is required. The Newmark-β family (covered in the structural mechanics article) includes:

Explicit central difference (β=0, γ=½): Cheap per step but tiny Δt required ($\Delta t < T_{\min}/\pi$, where $T_{\min}$ is the period of the highest mode). Used in LS-DYNA crash, Abaqus/Explicit.
HHT-α method (Hilber-Hughes-Taylor): Unconditionally stable implicit scheme with numerical high-frequency dissipation — the default in Abaqus/Standard dynamic steps. Controlled by parameter α ∈ [-1/3, 0]: α = 0 gives Newmark; α = -1/3 gives maximum numerical damping.

$$\mathbf{M}\ddot{\mathbf{u}}_{n+1} + (1+\alpha)\mathbf{C}\dot{\mathbf{u}}_{n+1} - \alpha\mathbf{C}\dot{\mathbf{u}}_n + (1+\alpha)\mathbf{K}\mathbf{u}_{n+1} - \alpha\mathbf{K}\mathbf{u}_n = (1+\alpha)\mathbf{f}_{n+1} - \alpha\mathbf{f}_n$$

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For a drop test simulation — a product falling from 1 meter — should I use explicit or implicit time integration?

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Explicit is almost always used for drop tests. The reason: contact between the product and the floor is handled much more naturally in explicit codes — no convergence issues in the middle of impact. The event duration is short (milliseconds) and involves highly nonlinear contact and possibly plastic deformation. For this, LS-DYNA or Abaqus/Explicit with a very small time step (typically on the order of microseconds) is the standard industry approach. Implicit would struggle with the contact and might require unrealistically small load steps anyway — so you'd lose the efficiency advantage.

5. Frequency Response Analysis

5.1 Frequency Response Function (FRF)

For harmonic excitation $\mathbf{f}(t) = \hat{\mathbf{f}}e^{i\Omega t}$, the steady-state response is $\mathbf{u}(t) = \hat{\mathbf{u}}e^{i\Omega t}$, giving:

$$(-\Omega^2\mathbf{M} + i\Omega\mathbf{C} + \mathbf{K})\hat{\mathbf{u}} = \hat{\mathbf{f}}$$ $$\mathbf{H}(\Omega) = (-\Omega^2\mathbf{M} + i\Omega\mathbf{C} + \mathbf{K})^{-1}$$

$\mathbf{H}(\Omega)$ is the Frequency Response Function matrix. For each output DOF $j$ and input DOF $k$:

$$H_{jk}(\Omega) = \sum_{r=1}^n \frac{\phi_{jr}\phi_{kr}/m_r}{\omega_r^2 - \Omega^2 + 2i\zeta_r\omega_r\Omega}$$

5.2 Three Types of FRF

FRF Type	Output/Input	Low-freq limit	High-freq limit
Receptance (Compliance)	Displacement / Force	1/k (stiffness line)	1/(-Ω²m) (mass line)
Mobility	Velocity / Force	Slope up	Slope down
Accelerance (Inertance)	Acceleration / Force	Ω²/k	1/m (flat)

5.3 Modal Effective Mass and Participation Factors

When performing base excitation analysis (seismic, shaker table), the modal participation factor Γ_i for excitation direction r:

$$\Gamma_i = \boldsymbol{\phi}_i^T \mathbf{M}\mathbf{r}, \qquad m_i^\text{eff} = \frac{(\boldsymbol{\phi}_i^T\mathbf{M}\mathbf{r})^2}{M_\text{total}}$$

Rule of thumb: when the sum of modal effective masses exceeds 90% of total mass in each direction, the truncated modal model is adequate for most structural response calculations.

6. Random Vibration (PSD, RMS)

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I have a satellite component that must survive launch vibration. The specification gives me a Power Spectral Density (PSD) profile in g²/Hz. How do I use that in FEM?

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Random vibration analysis in FEM is essentially a frequency-domain method. You first compute the frequency response functions of your structure (the FRF matrix), then multiply by the input PSD to get the output PSD at any point. Integrating the output PSD over frequency gives the mean-square response, and the square root gives the RMS stress or acceleration. For fatigue assessment, you apply the 3σ criterion: peak stress = 3 × RMS stress with 99.73% probability. That's your design load. In Ansys Mechanical, this is the "Random Vibration" analysis type — you link it directly to a Modal analysis.

6.1 Power Spectral Density

For a stationary random process $x(t)$, the PSD $S_x(f)$ is the Fourier transform of the autocorrelation function:

$$S_x(f) = \lim_{T\to\infty}\frac{1}{T}|\hat{x}(f)|^2$$

The mean-square (variance) of the process:

$$\sigma_x^2 = \int_0^\infty S_x(f)\, df$$

Units of PSD: [units²/Hz]. For base acceleration: [g²/Hz] or [(m/s²)²/Hz].

6.2 Response PSD and Propagation Through Structure

For a SDOF system with FRF $H(f)$ and input PSD $S_{\text{in}}(f)$, the output PSD:

$$S_{\text{out}}(f) = |H(f)|^2 S_{\text{in}}(f)$$

For an MDOF system (in modal coordinates):

$$\sigma_\text{response}^2 = \int_0^\infty |H(f)|^2 S_{\text{in}}(f)\, df$$

6.3 Miles Equation (SDOF Approximation)

For a lightly damped SDOF system under broadband white noise input $W_0$ [g²/Hz]:

$$\sigma_{\ddot{u}} = \sqrt{\frac{\pi f_n W_0}{2\zeta}} \approx \sqrt{\frac{\pi f_n W_0}{2\zeta}}$$ $$G_{\text{RMS}} = \sqrt{\frac{\pi}{2}\,f_n\,\text{PSD}(f_n)/\zeta}$$

Miles equation is widely used in aerospace for quick estimates. For example, a 200 Hz bracket with ζ = 2% in a flat PSD environment of 0.04 g²/Hz: G_RMS = √(π/2 × 200 × 0.04 / 0.02) ≈ 17.7 g.

6.4 3σ Design Criterion

For a Gaussian random process, the probability of exceeding ±3σ is only 0.27%. The design loads for fatigue and yield assessment:

$$\sigma_{\text{peak}} = 3\sigma_{\text{RMS}}, \qquad P(|x| > 3\sigma) = 0.0027$$

7. Shock Response Spectrum

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Military qualification specs always mention "shock response spectrum." What is that and how does it relate to regular time-history shock analysis?

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The SRS is a clever compression of shock severity information. Imagine an infinite bank of SDOF oscillators with different natural frequencies, all subject to the same shock pulse. The SRS plots the maximum response of each oscillator as a function of frequency. So it tells you: "for a component with this natural frequency and this damping, the peak acceleration during this shock will be this much." It's more useful than the raw time history for design because you can directly assess whether your structure (which has specific resonances) will survive. MIL-STD-810 and similar standards specify shocks in SRS form, and you validate by reproducing the SRS on a shaker table.

7.1 SRS Definition

$$\text{SRS}(f_n) = \max_{t} |a_{\text{response}}(t; f_n)|$$

The maximum absolute acceleration response of a SDOF oscillator with natural frequency $f_n$ (and typically ζ = 5%) subjected to the base shock acceleration $a_{\text{base}}(t)$.

Primary SRS: Maximum response during the pulse. Residual SRS: Maximum response after the pulse ends. The overall SRS is the maximum of both.

7.2 Common Shock Environments

Environment	Typical peak acceleration	Duration
Package drop (1 m)	50–150 g	5–15 ms
Pyrotechnic shock (separation)	1,000–10,000 g	0.1–1 ms
MIL-STD-810 Functional Shock	40 g (11 ms sawtooth)	11 ms
Automotive pothole (100 km/h)	50–200 g (at wheel)	2–5 ms
Earthquake (El Centro, 1940)	0.3–0.5 g (peak ground)	30+ s duration

8. Practical Applications

8.1 NVH (Noise, Vibration, Harshness) in Automotive

NVH is the most comprehensive vibration engineering application in consumer products. Key analysis types:

Transfer path analysis (TPA): How vibration from engine/road/axle transmits to the cabin through body structure and mounts
Panel contribution analysis: Which body panels radiate the most interior noise at each frequency
Operational deflection shapes (ODS): How the structure actually deforms during operation, measured by accelerometers
Mode coupling: Body-in-white bending/torsion modes must be above ~25 Hz to avoid low-frequency boom

8.2 Seismic Analysis of Buildings and Equipment

For civil structures and nuclear/industrial equipment, seismic loading is characterized by floor response spectra (for equipment mounted on buildings) or ground response spectra (for the building itself). Methods:

Response Spectrum Method: Modal superposition with SRSS (Square Root of Sum of Squares) or CQC (Complete Quadratic Combination) for mode combination
Time history analysis: Transient analysis using actual or synthetic accelerograms; required for nonlinear behavior

$$u_{\max} \approx \sqrt{\sum_i u_{i,\max}^2} \quad \text{(SRSS)}$$ $$u_{\max} \approx \sqrt{\sum_i\sum_j \rho_{ij} u_{i,\max} u_{j,\max}} \quad \text{(CQC)}$$ $$\rho_{ij} = \frac{8\sqrt{\zeta_i\zeta_j}(\zeta_i+r\zeta_j)r^{3/2}}{(1-r^2)^2+4\zeta_i\zeta_j r(1+r^2)+4(\zeta_i^2+\zeta_j^2)r^2}, \quad r = \omega_j/\omega_i$$

8.3 Vibration Isolation Design

To isolate a sensitive component from base vibration, choose the mount stiffness so the system natural frequency is well below the excitation frequency:

$$T_r = \sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}} \quad \text{(transmissibility)}$$

For $r = \Omega/\omega_n > \sqrt{2}$, transmissibility < 1 — the mount isolates. For $r = 3$ with ζ = 5%: $T_r \approx 0.13$ — 87% isolation. Rule of thumb: set $\omega_n$ at least 3× below the lowest excitation frequency. Adding more damping above √2 actually reduces isolation performance — light damping is better for isolation, heavier damping is needed if passing through resonance during operation.

Key Takeaways

Natural frequency is a structure property; resonance occurs when forcing = natural frequency
SDOF amplification at resonance: $D = 1/(2\zeta)$ — 25× for 2% damping
MDOF modal superposition: decouple into independent SDOF oscillators via eigenvectors
Random vibration: input PSD → FRF² → output PSD → RMS → 3σ peak design load
Miles equation quick estimate: $G_{RMS} = \sqrt{(\pi/2)\,f_n\,W_0/\zeta}$
Vibration isolation works when $\Omega/\omega_n > \sqrt{2}$

Related Interactive Tools

SDOF Frequency Response — Sweep forcing frequency and visualize resonance amplification
SDOF Random Vibration — PSD input → RMS response + 3σ estimate
Euler Buckling Calculator — Critical load and mode shapes

Cross-Topics

Structural Mechanics Fatigue Under Random Vibration Eigenvalue Solvers Structural Analysis Ansys Modal/Harmonic

Written by NovaSolver Contributors (Anonymous Engineers & AI) | CAE Technical Encyclopedia

Vibration & Structural Dynamics for CAESDOF to Random Vibration