Dynamic Analysis — CAE Terminology Glossary

Static analysis assumes loads are applied slowly and that acceleration is zero—in other words, it ignores inertial forces from mass times acceleration. But real structures experience vibration and impact, where inertial forces cannot be ignored. That's where dynamic analysis comes in.

Automotive crash testing, seismic response of buildings, drop impact on electronics, engine rotational vibration—anywhere "the load changes with time." In practice, when someone asks "are you sure this load is truly static?" that's when dynamic analysis becomes necessary.

The foundation of structural dynamics is this equation of motion:

$\mathbf{M}$ is the mass matrix (magnitude of inertia), $\mathbf{C}$ is the damping matrix (energy dissipation), and $\mathbf{K}$ is the stiffness matrix (spring stiffness). The right side $\mathbf{F}(t)$ is the time-varying external load. Static analysis is a special case where you set $\mathbf{M}\ddot{\mathbf{u}}$ and $\mathbf{C}\dot{\mathbf{u}}$ to zero.

Good question. In practice, Rayleigh damping $\mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K}$ is commonly used. The coefficients $\alpha$ and $\beta$ are back-calculated from the target damping ratio (typically 1–2% for steel, 5–10% for rubber) at the frequency range of interest. Without experimental data, it's difficult to pin down damping, so it's often a source of uncertainty.

Set external loads and damping to zero, solve $\mathbf{M}\ddot{\mathbf{u}} + \mathbf{K}\mathbf{u} = \mathbf{0}$, and find the natural frequencies (at what Hz does resonance occur?) and mode shapes (how the structure vibrates at each frequency). It's like fingerprinting the structure's vibration characteristics.

For example, if an engine runs at 3000 rpm, the excitation frequency is 50 Hz. If your structure's first natural frequency is close to 50 Hz, resonance will occur and cause failure. So modal analysis is about confirming "does this frequency band contain any natural frequencies?" It's fundamental design practice. In automotive, aerospace, and consumer electronics, modal analysis is always the first dynamic analysis performed.

Mode shape shows where the structure vibrates most. If a car dashboard exhibits vibration, examining the mode shape reveals "this region deforms significantly," pointing to where reinforcement is needed. You can then add ribs to stiffen that area.

Modal analysis finds natural frequencies without any external load. Frequency response analysis (Harmonic Response Analysis) applies a sinusoidal load $\mathbf{F}(t) = \mathbf{F}_0 e^{i\omega t}$ and computes the response amplitude at each frequency. It characterizes the input-output relationship—the transfer function.

Modal superposition first obtains modes from modal analysis, decouples the equations, then solves. It's faster but introduces truncation error from discarding higher modes. The direct method solves the full coupled equations at each frequency step—more accurate but computationally expensive. In practice, use modal superposition for broad frequency ranges, direct method when you need local detail.

Rotor unbalance response is a classic. You sweep through engine RPM to find "at what speed is vibration maximum?" and "do bearing loads stay within limits?" NVH (Noise, Vibration, Harshness) analysis—measuring cabin noise across frequencies—is another typical application.

You integrate the equation of motion in the time domain step-by-step, directly obtaining displacement, velocity, acceleration, and stress at each time $t$. Its strength is handling arbitrary time-history loads—impact, pulse, earthquake records—anything that can't be represented as a sine wave.

A typical rule: if your highest frequency of interest is $f_{max}$, then $\Delta t \le T/20 = 1/(20 f_{max})$. For example, to capture up to 1000 Hz, use $\Delta t \le 0.05$ ms. With explicit methods, the Courant condition further restricts this. In practice, you adjust while checking that load peaks aren't clipped.

Exactly. Phone drop tests, automotive crash simulations—these are classic transient analyses. For collision (milliseconds), you use explicit methods; for earthquakes (tens of seconds), you use implicit methods. That's our next topic.

Use it when the load cannot be defined deterministically as a time history, only statistically via power spectral density (PSD). Output is given as RMS values (and sometimes 3σ estimates). Classic examples: truck bed vibration during transport, rocket liftoff vibration, aircraft response to turbulence—any "we can't predict the exact waveform, only its statistical properties" environment.

Assuming a Gaussian distribution, the 3σ rule (3 times RMS) covers 99.7% of the data—a standard safety envelope in aerospace and standards like MIL-STD and JAXA guidelines. You can also use the Miles equation for single-DOF quick estimates, but multi-DOF systems require proper FEM analysis.

Here's a quick comparison:

For explicit methods, there's a stability limit: the time step must not exceed the minimum element size divided by wave speed:

Exactly. That's why explicit methods become impractical for long-duration phenomena. But for a 10 ms collision, 50,000 steps is manageable. The decision between explicit and implicit hinges on phenomenon duration versus required time resolution.

A standard time integration scheme for implicit methods. Different parameter choices ($\beta$, $\gamma$) yield different characteristics. With $\beta = 1/4$, $\gamma = 1/2$, you get the average acceleration method (unconditionally stable, 2nd order accurate). Nastran uses variants of this internally.

Here's a practical decision tree:

Exactly right. Modal analysis is the foundation of dynamics, with low computational cost. Master it first, understand natural frequencies and mode shapes, then proceed to frequency response, random vibration, or transient analysis as your problem demands. Jumping straight to transient analysis without modal insight leads to confusion interpreting results.