伝達経路解析（TPA）
Theory and Physics
What is TPA?
Professor, what is TPA (Transfer Path Analysis)?
A technique to quantitatively analyze through which paths sound and vibration reach a response point. Essential for formulating NVH countermeasures.
Basic TPA Equation
The sound pressure $p$ at a response point (e.g., driver's ear) is the sum of contributions from all paths:
$H_i$: Transfer Function (NTF: Noise Transfer Function) for the $i$-th path, $F_i$: Input force (acting force) for the $i$-th path.
So you can see the contribution of each path.
Correct. This makes it immediately clear "which mount is dominant" and "which frequency band is problematic."
Path Classification
In automotive NVH, structure-borne paths dominate low frequencies (~500Hz), and air-borne paths dominate high frequencies.
Input Force Identification
Methods to determine input force $F_i$:
1. Direct Measurement Method: Install force sensors at input points. Most accurate, but sensor installation can be difficult.
2. Mount Stiffness Method: $F = k \cdot \Delta x$. Displacement difference across mount × stiffness.
3. Inverse Matrix Method: $\{F\} = [H]^{-1}\{a\}$. Calculate input force inversely from response acceleration.
Summary
The Prototype of TPA Theory Originated from Building Vibration and Noise Countermeasures
The mathematical framework of Transfer Path Analysis (TPA) was formed in the architectural acoustics and mechanical vibration fields during the 1950s-60s. In particular, Möser (German Institute for Building Acoustics) and others developed vibration transmission path models for building structures, which were later productized in the 1980s by Müller-BBM and LMS International as TPA methods specialized for automotive NVH. Current Component TPA (CTPA) is based on the formulation by Helut Müller in a 1999 SAE paper.
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, assuming "forces are applied slowly enough to ignore acceleration". 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. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? 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 $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). 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 tire 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 typical mistake here: getting the load direction wrong. Intending "tension" but applying "compression"—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—intentionally absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would keep 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: 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. 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
Types of TPA
Are there different types of TPA?
Mainly three.
1. Classical TPA
2. OPA (Operational Path Analysis)
3. CAE-TPA (Virtual TPA)
CAE-TPA Implementation
1. Vehicle Body FEM Model — Define input points (mount locations) and response points (driver's ear).
2. NTF Calculation — Apply unit force at each input point, calculate sound pressure at response point via frequency response.
3. Input Force Setting — MBD analysis results or measured data.
4. Contribution Calculation — Calculate $p_i = H_i \cdot F_i$ for each path, compare with sum.
Summary
Inverse Matrix TPA is Practical Even with Over 100 Excitation Points
Classical TPA (inverse matrix method) estimates operating loads as mount interface forces and calculates contribution sound pressure via product with transfer functions (FRF). As the number of excitation points increases, the inverse matrix becomes unstable, so Siemens (formerly LMS) began recommending in-situ TPA around 2015, which estimates transmission paths using only operational data without exciters. This TPA variant (operational TPA) gained attention after reducing test man-hours by 40% compared to conventional methods in BMW chassis development.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but lower stress accuracy. 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 ~2-3x. Recommendation: 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.
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 using initial value or every few iterations. Lower cost per iteration, 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 in small increments rather than all at once. Arc-length method (Riks method) can traverse beyond limit points on 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: 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 at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
TPA in Practice
Main applications: automotive road noise, engine idle vibration, EV motor noise.
Practical Workflow
1. Define the Problem — "Booming at 60km/h" → Identify frequency, speed.
2. List Candidate Paths — Engine mount ×3 directions, suspension bush ×4 locations ×3 directions = dozens of paths.
3. Obtain NTF — FEM or experimental FRF.
4. Obtain Input Forces — Actual vehicle measurement or MBD.
5. Contribution Analysis — Visualize each path's contribution with bar graphs.
6. Formulate Countermeasures — Modify stiffness of dominant path, change mount characteristics, add damping material.
Practical Checklist
Why is phase important?
Cancellation (out-of-phase) can occur between paths. Adding amplitudes alone leads to overestimation. Synthesize as complex numbers is the golden rule.
EV Road Noise is Subjectively "More Noticeable" Than ICE
In electric vehicles (EVs), the absence of engine noise has made road noise transmitted from tire contact patches through the body the main NVH performance challenge. After the Nissan Leaf (2010) market launch, user surveys showed road noise complaints reached 1.8 times that of ICE vehicles. Contribution analysis via TPA revealed the front subframe mount as the maximum contributor. Improvements optimizing the mount rubber's dynamic stiffness to reduce the 500-800Hz band by 5dB were implemented in the 2013 model year.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing visualization). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work". Meshi
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