# 1 General framework for TPA

### Introduction

Transfer Path Analysis (TPA) has been a valuable engineering tool for as long as noise and vibrations of products have been of interest. A TPA concerns a product’s actively vibrating components (such as engines, gearing systems, or turbochargers) and the transmission of these vibrations to the connected passive structures. TPA is particularly useful when the actual vibrating mechanisms are too complex to model or measure directly, as it allows to represent a source by forces and vibrations displayed at the interfaces with the passive side. In this way, the source excitations can be separated from the structural/acoustic transfer characteristics, allowing troubleshooting of the dominant vibration transmission paths. The engineer can then anticipate by making changes to either the source itself or the receiving structures connected to it.

With the variety of TPA methods growing over the past years, the need for a clear classification of methods and their pros and cons became inevitable. In 2015, VIBES’ own Dr. Maarten van der Seijs and Dr. Dennis de Klerk, together with Prof. Dr. Daniel Rixen, proposed a general framework to structure the different TPA methods.

For the first time, similarities between well-known classical methods (such as mount stiffness and matrix inversion) and the popular operational TPA method have been made clear. More importantly, all recent approaches dealing with Blocked Forces were included and categorized into the component-based TPA family in a way that helps the engineer choose the right approach, test bench design, or source description for the case at hand. The research paper can be found here.

### The transfer path problem

Let us consider the dynamic system AB as schematically depicted in the following figure.
Two subsystems can be distinguished: an active subsystem A containing an excitation at node 1 and a passive subsystem B comprising the responses of interest at node 3. The subsystems are rigidly interconnected at the interface node 2.

A) Assembly AB

B) Subsystems A and B

For simplicity of derivation, the Degrees of Freedom (DoFs) in this example are restricted to three distinct nodes. These may, however, represent a larger set of DoFs, representing respectively:

• $$\mathbf{u}_{\mathrm{1}}$$ source: internal DoFs belonging to the active component that causes the operational excitation but are unmeasurable in practice;
• $$\mathbf{u}_{\mathrm{2}}$$ interface: coupling DoFs residing on the interface between the active and passive component;
• $$\mathbf{u}_{\mathrm{3}}$$ receiver: response DoFs at locations of interest at the passive component, possibly including acoustic pressures and other physical quantities.

Hence, the example is illustrative for a wide range of practical problems, provided that the structure of interest can be decomposed into an active and passive part. In what follows, all methods assume that the operational excitation at node 1 is unmeasurable in practice, but transmits vibrations through the interfaces at node 2 to receiving locations at node 3. The responses shall then be built up from a certain description of the vibrations measured at the interface (node 1 → 2) and an appropriate set of transfer functions relating these vibrations to the receiving responses (node 2 → 3). The fundamental choices herein dictate to which TPA family the method is classified.

### The TPA workflow

A TPA workflow can typically be subdivided into the following steps:

1. operational measurement on the active component;
2. determination of the passive (sub)system characteristics, often by means of FRFs;
3. determination of the operational interface loads;
4. identification of path contributions.

The steps are shown schematically in the following figure, from left to right, and further categorized into three families. Depending on the TPA method at hand, some or all of these steps may be performed in arbitrary order. The optimization actions that follow from such analyses are generally not considered part of the workflow.

The TPA workflow, depicted stepwise for the three TPA families.

### TPA families

The three families of TPA as identified in the “General framework for Transfer Path Analysis” are listed below. The first two use some notion of force to split up in a source-transmission-receiver, while the transmissibility-based TPA is a response-only approach:

#### Classical TPA

Classical TPA is intended to identify transfer path contributions in existing products. The source excitation is represented by interface forces, which are a property of the assembly they are measured in. Popular ways to obtain the interface forces are the matrix inverse method, the mount stiffness method, or the direct force method, which uses force transducers mounted at the interfaces.

#### Component-based TPA

Component-based TPA is powerful to simulate component vibration levels in new products. The source excitation is characterized by a set of equivalent forces that are an intrinsic property of the active component itself. More popularly, these forces are known as Blocked Forces, as they are the would-be forces (and moments!) when measured against a rigid boundary. Blocked Forces are the perfect means to characterize an active source on a test bench at a supplier and allow the OEM to make NVH predictions for new assemblies by “Substructuring” the components. The Blocked Forces are often obtained in-situ using a matrix inverse procedure, where the test environment may be either a component test bench or the actual vehicle itself.

#### Transmissibility-based TPA

Transmissibility-based TPA is great for troubleshooting dominant sources and paths. Classical TPA and component-based TPA can be tedious processes, as they require FRFs to be measured on several (sub-) assemblies. If one is merely interested in the path contributions of different uncorrelated sources through their interfaces, a transmissibility-based approach such as operational TPA is faster. As these methods use responses only, the insights gained from them are limited to the ranking of sources and their dominant paths.