Acoustic-Structure Coupled Frequency Response
Theory and Physics
What is Acoustic-Structure Coupling?
Professor, what is acoustic-structure coupled frequency response?
It is a bidirectional coupled frequency response analysis where structural vibration radiates sound into the air (acoustic field), and conversely, sound pressure exerts force on the structure. Automotive NVH (vehicle interior noise) is its primary application.
Governing Equations of Coupling
Frequency response of the coupled structural-acoustic system:
$[Z_s] = -\omega^2[M_s] + i\omega[C_s] + [K_s]$ is the structural dynamic stiffness.
$[Z_a] = -\omega^2[M_a] + [K_a]$ is the acoustic dynamic stiffness.
$[A]$ is the coupling matrix.
So the structural displacement $\{u\}$ and the acoustic pressure $\{p\}$ are solved simultaneously.
Structure vibrates → panels push air → sound pressure is generated → sound pressure exerts force on the structure → structural vibration changes... This bidirectional coupling is solved by a single system of equations.
NTF (Noise Transfer Function)
NTF = Transfer function of interior sound pressure relative to input force. The most important metric in automotive NVH.
So it evaluates the transmission through the entire path from engine mount input to the sound pressure at the occupant's ear position.
The peak location and magnitude of the NTF directly correlate with the cause of issues like booming noise.
Summary
Key Points:
- Bidirectional coupling: structural vibration → sound pressure → reaction force on structure
- Coupled system solving for displacement $u$ and pressure $p$ simultaneously
- NTF (Noise Transfer Function) is the fundamental metric for NVH
- Prediction of automotive interior noise is the primary application
Coupling Theory Born from Noise Complaints
In the 1960s, the cabin noise of the Boeing 707 was so loud that passengers flooded the company with complaints. In response to this problem, NASA's M.C. Junger and others systematically formalized the mathematical framework for acoustic-structure coupling in their 1972 book 'Sound, Structures, and Their Interaction'. They were the first to rigorously formulate the bidirectional coupling where structural vibration excites air pressure waves, and conversely, sound pressure excites the structure.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? 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, which assumes "forces are applied slowly enough that acceleration can be ignored". 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 is 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" — they are 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 volume" (body force), while the force of the tires pushing on the road surface 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 vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue swaying 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, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Beware of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the 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
Coupled Analysis in FEM
How is coupled frequency response implemented?
Place structural elements (shell/solid) and acoustic elements (like FLUID30) in the same model, and define coupling at the interface.
Nastran
```
SOL 111 $ Modal frequency response
$ Structural elements + CAERO acoustic panels or FLUID elements
```
In Nastran, structural modes and acoustic modes are calculated separately and connected via coupling terms.
Abaqus
```
*TIE, NAME=fsi_interface
acoustic_surface, structural_surface
*STEP
*STEADY STATE DYNAMICS, DIRECT
...
*END STEP
```
Couple structural and acoustic surfaces with TIE. Solve using direct or modal method.
Ansys
```
! Structural elements (SHELL181) + Acoustic elements (FLUID30)
! Define interface with FSI flag
SF, fsi_area, FSI
```
Do the acoustic mesh and structural mesh need to match?
Ideally they should match (shared nodes), but if mesh sizes differ, connect non-matching meshes using TIE constraints or MPC. Typically, structural mesh is fine, acoustic mesh is coarse (acoustic wavelengths are long).
Summary
The Marriage of Finite Elements and Boundary Elements
In the late 1970s, combining FEM and BEM became the mainstream approach for acoustic coupling analysis. The idea of discretizing the structure side with FEM (Finite Element Method) and the fluid side with BEM (Boundary Element Method), then connecting them with a coupling matrix, was formalized in a paper by O. von Estorff in 1990. Even today, Ansys Acoustics and Abaqus/Acoustics adopt this hybrid strategy.
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 Midside Nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (like 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 every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed 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
Instead of applying full load at once, apply in small increments. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
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: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler" — represented by straight line segments, so accuracy is limited. 2nd order elements are like "flexible curves" — can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
Acoustic-Structure Coupling in Practice
The most important analysis technique in automotive NVH development.
Vehicle Interior Noise Prediction Flow
1. Build BIW model (shell elements) + vehicle interior acoustic mesh
2. Natural frequency analysis — structural modes + acoustic modes
3. Coupled frequency response — engine mount input → interior sound pressure (NTF)
4. NTF evaluation — check peak frequencies and sound pressure levels
5. Countermeasures — panel reinforcement, adding damping materials, acoustic absorbers
Practical Checklist
Acoustic-structure coupling is the culmination of NVH, isn't it?
Natural frequency → frequency response → acoustic coupling — all aspects of dynamic analysis are integrated. It is the most important analysis for NVH engineers.
Vehicle Interior Booming Noise is 50–200Hz
Vehicle interior booming noise caused by road noise typically occurs in the 50–200Hz band. In acoustic-structure coupling analysis conducted by Toyota during the development of the Lexus LS600h, it was predicted in advance that adding just 0.1mm thickness to specific body panels would reduce a particular resonance peak by 4dB. This became a milestone for prototype-less development.
Analogy for the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question — which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Easily Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will
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