Acoustic Radiated Power
Theory and Physics
What is Acoustic Radiation?
Professor, what is acoustic radiation power?
A vibrating structural surface vibrates the surrounding air to produce sound. It's an index that quantifies the sound power.
$p$: sound pressure on the surface, $v_n$: normal direction vibration velocity of the surface, $*$: complex conjugate. It is the integral of the acoustic intensity over the entire surface.
Radiation Efficiency
Does sound always come out if it's vibrating?
Sometimes vibration does not become sound. That is expressed by the radiation efficiency $\sigma_{rad}$.
$\rho c$: characteristic impedance of air, $S$: radiation area, $\langle v_n^2 \rangle$: surface-averaged squared vibration velocity value.
- $\sigma_{rad} = 1$: Perfect radiator (piston vibration)
- $\sigma_{rad} < 1$: Weak radiation (acoustic short-circuit occurs)
- $\sigma_{rad} > 1$: Rare, but can occur during resonance
What is acoustic short-circuit?
When adjacent regions vibrate in opposite phases, the positive pressure from one is canceled by the negative pressure from the other. This is prominent in the modal vibration of plates and is the reason why radiation efficiency is lower at lower frequencies.
Critical Frequency and Radiation
Radiation efficiency changes significantly with frequency:
- $f < f_c$ (Below coincidence frequency): $\sigma_{rad} \ll 1$. Acoustic short-circuit is dominant.
- $f = f_c$: $\sigma_{rad}$ increases sharply.
- $f > f_c$: $\sigma_{rad} \approx 1$. The entire surface radiates efficiently.
Summary
The concept of radiation efficiency originates from Lord Rayleigh's 1877 paper
The theoretical foundation of radiation efficiency dates back to Lord Rayleigh's 1877 work "The Theory of Sound" Volume 2. He derived the radiation impedance using an infinite flat plate "piston model," but its extension to finite structures had to wait until the 20th century. Wallace (1972) analytically obtained the radiation efficiency of a finite rectangular plate, which became the cornerstone of modern structural acoustics theory.
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 enough that acceleration is negligible." It absolutely cannot be omitted in impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. 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 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 modeling "compression"—it sounds like a joke, but it actually happens when coordinate systems rotate 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—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would keep 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: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
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. Be careful of unit system inconsistencies when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| 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
Acoustic Radiation Calculation Methods
How do I calculate acoustic radiation power using FEM?
There are mainly three approaches.
1. FEM Coupled Analysis
Structural FEM + Acoustic FEM coupling. The acoustic domain is modeled with finite elements, and a PML (Perfectly Matched Layer) is placed externally.
- Advantage: Can calculate both near-field and far-field.
- Disadvantage: The acoustic domain mesh becomes large.
2. BEM (Boundary Element Method)
Calculates the external sound field using only the structural surface mesh. Can handle infinite domains directly.
$G$: Free-space Green's function. Calculates the sound pressure field using the surface vibration velocity as a boundary condition.
3. Rayleigh Integral (Plane Approximation)
Assumes a vibrating surface mounted on an infinite baffle:
Convenient for quick estimation of radiation power from flat panels. Computational cost is lower than BEM.
Summary
Near-field Acoustic Holography is a masterpiece of inverse problem analysis
Near-field Acoustic Holography (NAH) is an innovative method published by Maynard et al. in JASA in 1985, capable of reconstructing the vibration distribution of a radiating surface from measurements by an array of sound pressure microphones. NAH was introduced as experimental bench equipment by Ford and BMW's NVH departments in the 1990s to non-contact identify contributing noise sources like engine covers and fuel tanks. It is now miniaturized and commercialized as acoustic cameras (e.g., gfai tech's SoundCam).
Linear Elements (1st-order elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. Beware of shear locking (mitigated by 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 by 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. There are h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates the tangent stiffness matrix every iteration. Shows quadratic convergence within the convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates the tangent stiffness matrix using the 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 the full load at once, applies it in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy of 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 get closer to the correct answer"—starts with a rough answer but improves accuracy 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
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 a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Practical Acoustic Radiation Power Analysis
Typical application examples include automotive engine radiation noise, home appliance motor noise, and transformer hum noise.
Analysis Flow
1. Structural Vibration Analysis — Obtain surface vibration velocity via modal or frequency response analysis.
2. Radiation Power Calculation — Calculate sound field using BEM or Rayleigh integral.
3. Radiation Efficiency Evaluation — Plot $\sigma_{rad}$ for each frequency.
4. Sound Pressure Level — Evaluate sound pressure at observation points in dB(A).
Practical Checklist
Sound Power Level (SWL)
$W_0 = 10^{-12}$ W (reference acoustic power). $L_W = 80$ dB means $W = 10^{-4}$ W (0.1 mW).
Electromagnetic forces dominate the radiated sound of electric motors
Unlike combustion engines, the radiated noise of EV motors is primarily caused by electromagnetic forces due to torque ripple. During the development of Toyota's first-generation Prius (1997), the fact that the motor sound was clearly audible inside the cabin when the engine was off remained unrecognized until late in the design phase. Today, coupled electromagnetic-structural-acoustic analysis using Maxwell stress tensor has become the de facto standard, with combinations like JMAG + ABAQUS + Actran adopted by major domestic suppliers.
Analogy of 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 (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work." If the mesh quality is poor, the results will be a mess no matter how good the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer is far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct."
How to Think About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling real-world constraints is actually the most important step in the entire analysis.
Software Comparison
Tools
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