Frequency Sweep and Resonance Evaluation
Theory and Physics
What is Frequency Sweep?
Professor, what is a "frequency sweep"?
It's a method to continuously vary the frequency of an external force from low to high and calculate the response. It's a fundamental approach to obtain the FRF (Frequency Response Function).
So, in essence, it means "calculating sequentially from 1 Hz to 500 Hz"?
Yes. Plotting the response (displacement, acceleration, stress) at each frequency allows you to see the location and magnitude of resonance peaks at a glance.
Frequency Step Design
How to determine the frequency step depends on the nature of the resonance:
| Damping Ratio $\zeta$ | Resonance Peak Half-Power Bandwidth | Required Step |
|---|---|---|
| 0.1% (Very Low Damping) | $\Delta f \approx 0.002 f_n$ | Below 0.1 Hz |
| 1% (Steel Structure) | $\Delta f \approx 0.02 f_n$ | About 1 Hz |
| 5% (RC Structure) | $\Delta f \approx 0.1 f_n$ | About 5 Hz |
| 10% (Seismic Isolation) | $\Delta f \approx 0.2 f_n$ | About 10 Hz |
When damping is small, the peak is sharp, so a fine step is needed, right?
For low-damped steel structures, the step needs to be very fine. For a natural frequency around 100 Hz with $\zeta = 0.5\%$, the step should be below 0.5 Hz. A sweep of 500 points from 1 to 500 Hz with a 1 Hz step might miss the peak.
Logarithmic Step and Mode-Following Step
Efficient stepping methods:
1. Logarithmic Equal Spacing
Place frequencies at equal intervals on a logarithmic scale. Coarse at low frequencies, fine at high frequencies. Common in acoustic systems.
2. Mode-Following Step (Nastran's FREQ4)
Automatically places fine steps around each natural frequency to reliably capture resonance regions. Set using Nastran's FREQ4 card.
3. Adaptive Step
Automatically refines steps near resonance using Abaqus's BIAS parameter. If you specify the number of calculation points, they are automatically concentrated around resonance.
Resonance Evaluation
Resonance evaluation metrics:
| Metric | Definition | Purpose |
|---|---|---|
| Resonance Frequency | Peak position of FRF | Resonance avoidance design |
| Peak Amplitude | Maximum value of FRF | Maximum response evaluation |
| Half-Power Bandwidth | -3 dB width of peak | Damping ratio estimation |
| Phase Change | Approx. 180° change at resonance | Mode confirmation |
Is the FRF peak amplitude the most important in design?
Yes. Peak amplitude × Input force = Maximum response. Whether this maximum response is within allowable limits (displacement limit, acceleration limit, stress limit) is the design judgment.
Summary
Let me organize frequency sweep and resonance evaluation.
Key points:
- Vary frequency to obtain FRF — Identify resonance peaks
- Step size should be less than half-power bandwidth — $\Delta f < \zeta \cdot f_n$
- Efficiency with mode-following step (FREQ4) — Fine only near resonance
- Peak amplitude × Input = Maximum response — Basis for design judgment
- Confirm resonance with phase change — 180° phase jump
Resonance Changes with Sweep Rate
In frequency sweep tests, faster sweep rates (octaves/minute) can cause resonance peaks to appear "false". As shown by CW. de Silva in the 1950s, an apparent shift dependent on sweep rate occurs from the true resonance frequency. The current vibration test standard MIL-STD-810H recommends 4 octaves/minute or less, based on this theoretical background.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly so acceleration is negligible". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire contents" (body force), the force of the tires pushing the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—intentionally absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considers only equilibrium between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extension is needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
Frequency Step Setting
Please teach me how to set the frequency step in each solver.
Nastran
```
$ Equal spacing
FREQ1, 20, 1., 500., 1. $ 1 to 500 Hz, 1 Hz step
$ Mode-following (automatically finer near resonance)
FREQ4, 20, 1., 500., 0.1, 5 $ ±0.1 half-power bandwidth per mode, 5 points
$ Logarithmic equal spacing
FREQ2, 20, 1., 500., 10 $ 1 to 500 Hz, 1/3 octave
```
Abaqus
```
*STEADY STATE DYNAMICS
1., 500., 500, 1. $ 1 to 500 Hz, 500 points, BIAS=1 (equal spacing)
```
BIAS > 1 concentrates on high frequency, BIAS < 1 concentrates on low frequency.
Ansys
```
HARFRQ, 1., 500.
NSUBST, 500 ! 500 steps
```
Nastran's FREQ4 is the smartest stepping method, right?
FREQ4 automatically concentrates calculation points around each natural frequency. It can sometimes achieve equivalent accuracy with 10 times the efficiency of equal spacing. In practice, combining FREQ1 (coarse overall step) + FREQ4 (fine step near resonance) is most effective.
FRF Output and Visualization
Standard FRF display formats:
- Amplitude-Frequency Plot — Logarithmic scale (dB) is common
- Phase-Frequency Plot — 180° jump at resonance
- Nyquist Plot (Real vs. Imaginary Part) — Draws a circle at resonance
- Bode Plot — Displays amplitude and phase in two tiers
What is the dB scale?
$20 \log_{10}(|H|/H_{ref})$. Compresses large amplitude changes for easier viewing. Resonance peaks appear as +40 dB, anti-resonance as -40 dB, etc.
Summary
Let me organize the numerical methods for frequency sweep.
Key points:
- FREQ1 + FREQ4 (Nastran) is most efficient — Equal spacing + mode-following
- Concentrate distribution with BIAS (Abaqus) — Automatically concentrates on frequency bands of interest
- Display FRF in dB scale — Compresses wide dynamic range
- Confirm resonance with Nyquist plot — Draws a circle at resonance
Choosing Between Logarithmic Sweep and Linear Sweep
For low-frequency bands (1–100Hz), logarithmic sweep (constant octave) is standard, allocating equal time to each frequency. Linear sweep is used for high-frequency (1kHz+) precision measurement and electrical characteristic evaluation. Automotive shaker tests (ISO 16750-3) standardize logarithmic sweep at 1 octave/minute from 5–2000Hz. Even for the same test, resonance detection accuracy can differ by up to 3 times between logarithmic and linear.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-Side Nodes)
Can represent curved deformation. Stress accuracy improves significantly but degrees of freedom increase about 2–3 times. Recommended: when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix each iteration. Quadratic convergence within convergence radius but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at initial value or every few iterations. Cost per iteration is low but convergence is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once but in small increments. The arc-length method (Riks method) can trace beyond extremum points in the load-displacement relationship.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
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