遮音性能（透過損失）
Theory and Physics
What is Transmission Loss (TL)?
Professor, what is transmission loss?
It's an index that represents how much of the sound incident on a wall or panel can be prevented from transmitting through.
$W_{in}$: Incident sound power, $W_{tr}$: Transmitted sound power. A larger TL indicates higher sound insulation performance.
Mass Law
Why does making a wall heavier improve sound insulation?
The Mass Law is the most fundamental law of sound insulation.
$m_s$: Surface density [kg/m²], $f$: Frequency [Hz]. Doubling surface density → TL +6dB, Doubling frequency → TL +6dB. This is the "6dB rule of the Mass Law".
So heavier walls are better, right?
That's true for low to mid frequencies, but there is a frequency where the Mass Law breaks down. That is the coincidence frequency.
Coincidence Effect
At the frequency where the wavelength of the panel's bending wave matches the wavelength of the incident sound wave, sound transmission becomes easier.
$c$: Speed of sound, $D$: Bending stiffness $D = \frac{Eh^3}{12(1-\nu^2)}$. At the coincidence frequency, TL dips significantly.
So thinner panels have a higher coincidence frequency, right?
Correct. Halving thickness $h$ → $f_c$ doubles. For a 6mm steel plate, $f_c \approx 2\,\text{kHz}$; for a 3mm aluminum plate, $f_c \approx 4\,\text{kHz}$.
Sound Insulation of Double Walls
Using a double wall (double-leaf wall) can achieve sound insulation performance significantly exceeding the Mass Law.
- Resonant transmission frequency: $f_0 = \frac{1}{2\pi}\sqrt{\frac{\rho c^2}{d}\left(\frac{1}{m_1}+\frac{1}{m_2}\right)}$
- Below $f_0$: Same as a single wall
- Above $f_0$: Sound insulation improves at 12dB/oct (twice the 6dB/oct of a single wall)
$d$: Thickness of the air gap. Placing sound-absorbing material in the air gap can mitigate the dip at $f_0$.
Summary
The Mass Law is a simple yet powerful formula derived by Berger in 1923
The "Mass Law" for sound insulation was formulated in 1923 by the German acoustician E. Berger as TL≈20log₁₀(m·f)−47.5dB (SI units). This formula is still used today as a first approximation in design, but deviations from its derivation assumptions (infinite flat plate, normal incidence) can cause differences of 5-10dB from measured values. The phenomenon where TL drops sharply near the coincidence frequency (coincidence effect) cannot be explained by this formula and was first quantified by wave theory by Cremer (1942).
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 sway 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 is negligible". It absolutely cannot be omitted in impact loading 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 is Hooke's Law $F=kx$, the essence of the stiffness term. Now 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 surface is a force "acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but applying "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 away. That's because the 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 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 inhomogeneities
- 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 load and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful 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
FEM Calculation Method for TL
Please teach me how to calculate transmission loss using FEM.
The basic approach is a coupled model of three domains: incident-side acoustic domain + structural panel + transmitted-side acoustic domain.
1. Incident side: Plane wave incidence. Acoustic FEM or analytical input.
2. Structure: Model the panel with shell/solid elements.
3. Transmitted side: Acoustic FEM. Non-reflecting boundary (PML or impedance boundary).
Calculation of Diffuse Field TL
In experiments, it's measured in a diffuse sound field, right?
Yes. To calculate diffuse field TL with FEM:
- Method 1: Calculate individually for multiple incidence angles (0° to 78°) and average using Paris's formula.
- Method 2: Directly model the diffuse sound field (randomly place acoustic sources).
Method 1 is common. Good agreement is obtained with $\theta_{max} = 78°$ (equivalent to ISO 15186).
SEA (Statistical Energy Analysis)
For high frequencies (above several hundred Hz), SEA is more efficient than FEM.
- Described by the energy balance of each subsystem (panel, air gap, room)
- Modal density and coupling loss factor are key parameters
- Computational cost is extremely low (algebraic equations per frequency band)
Summary
The 2011 revision of ISO 10140 standards revolutionized testing laboratories
The ISO 10140 series of standards for measuring sound insulation of building materials underwent a comprehensive revision in 2010-2011, significantly tightening the requirements for acoustic transmission and reception rooms. Before the revision, differences of up to 8dB for the same sample between different laboratories were problematic. After the revision, requirements for diffusivity in reverberation rooms were added (new standard: DIN EN ISO 10140-5). Prompted by this revision, Japan's JIS A 1416 was also reviewed, and a round-robin test among 8 domestic institutions confirmed measurement reproducibility within 3dB.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Capable of representing 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 for the situation.
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 the tangent stiffness matrix each iteration. Achieves 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, apply it in small increments. The arc-length method (Riks method) can trace beyond limit points on 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 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 method) than to search 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 a "flexible curve"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
TL Analysis in Practice
Typical applications are automotive dash panels, architectural partition walls, and aircraft fuselage panels.
Analysis Flow
1. Identify Target Panel — Shape, material, thickness, constraint conditions
2. Set Frequency Range — Automotive NVH: 20-500Hz, Architectural: 125-4000Hz
3. Build FEM Model — Acoustic mesh should be $\lambda_{min}/6$ or less
4. Set Incidence Conditions — Normal incidence or diffuse incidence (multiple angles)
5. Calculate TL — Ratio of incident power to transmitted power
6. Calculate STC/Rw — Compare with standard values
Practical Checklist
Common Numerical Examples
| Panel | Surface Density [kg/m²] | TL @500Hz [dB] | Coincidence [Hz] |
|---|---|---|---|
| Steel Plate 1.6mm | 12.5 |
Related Topics
なった
詳しく
報告