I. Operational Path Analysis
Conventional TPA is proven to be reliable but its application requires large effort with the main bottleneck remaining the huge measurement time to build the full data model. For this reason the industry is constantly seeking for simpler and faster methods. One such method is Operational Path Analysis (OPA).
1. Description
OPA requires only operational measurements of the path references (body-side mount accelerations, pressures close to vibrating surfaces, nozzles and apertures, etc.) and target response(s). It is very time-efficient but does not deliver any information on the physical loads and suffers from limitations leading to false path contributions. OPA is based on the idea of the transmissibility calculation principle. The goal of the method is to use only operational data to derive TPA-like results without the need for additional measurements. This is achieved by using a different model in which the target response is formulated using operational responses measured at the load locations instead of the loads themselves. Note that this implementation does not yield physical loads at all.
2. Critical review
Since the mathematical formulation of OPA is similar to that of TPA, there is often confusion in the terminology, which might unfortunately lead to an incorrect interpretation on the meaning of the OPA results. TPA and OPA models are fundamentally different despite their apparent similarity. First of all, as opposed to the TPA model, the OPA model is not causal. Instead of a load-response relationship OPA is based on a response-response relationship, this means that while in TPA one can draw a conclusion as to what effect a certain path has on the total response, in OPA one can only talk about a similarity, a “co-existence” between the target and the input responses. Furthermore, OPA calculations use transmissibilities instead of NTF’s which are not system characteristics but depend on the loading conditions.
The OPA method itself has four critical elements:
- Cross-coupling between the path references. Due to the system’s modal behavior, a single force in one of the mounts causes vibrations at all path references. This cross-coupling easily leads to a false identification of significant paths and to wrong engineering decisions.
- Numerical conditioning problems when estimating transmissibilities from operational data. These problems lead to unreliable transmissibility estimates in many cases.
- Potential errors due to missing paths in the analysis. The contributions of missing paths are distributed over the other ones, introducing errors that are hard to recognize as the summed contribution is not affected in most cases.
- A good total OPA synthesis can not be used as a validation argument. Because of the use of transmissibilities, all energy that is put in the system is divided over the chosen number of paths. So the sum of the energy traveling through all paths to the receiver is by definition always equal to the total energy as measured at the receiver location.
Depending on the actual application, these limitations may lead to a false identification of significant transfer paths and prevent the engineer from reaching the right decision and solving the problem.
3. Applicability
In theory the OPA method applies well to solve problems which sources are uncorrelated. In practice, engineers applying the method should be very careful in their analysis with regards to the critical elements mentioned in the previous paragraph.
II. LMS OPAX
1. Description
LMS OPAX is a TPA method for load identification that combines the speed of the operational path method (OPA) and the effectiveness of the conventional TPA methods. The method uses a parametric load models that characterizes the operational forces and acoustic loads in function of measured path inputs. It is based on simplifications that allow balancing path accuracy and speed of execution.
First, operational measurements are performed. These can be a single run-up or run-down or several of these measurements at different conditions (e.g. various throttles, gears, etc.). It all depends for which condition(s) the TPA model must be developed. During the operational measurements, all mount acceleration and pressure inputs and all responses at the target point(s) and extra indicators are measured synchronously. Order envelopes (amplitude and phase in function of rotational speed) are then tracked for all measured input and response channels. Strictly, only the orders of interest must be processed. However, the more orders are used for identifying the parametric load models, the more robust the model parameters can be estimated and the more accurate the path contributions can be derived.
In a second phase, FRF’s are measured between the input loads and target response(s). The FRF’s can be measured in a direct or reciprocal way. The use of reciprocal measurements (exciting at the target location(s), measuring the response at the interfaces) has two advantages: (i) only one excitation is needed per target point while the direct approach requires one excitation per input load; (ii) the limited space at the path inputs can lead to direction errors in the direct FRF measurements of up to 10 dB. In case additional indicators are used, the FRF’s from the inputs to the indicators must also be measured. It is to be noticed that the sequence of phases 1 and 2 may be changed.
Phase 3 is the key step of the method. Parametric load models are estimated, characterizing the operational forces and acoustic loads as a function of the acceleration and pressure path inputs. Its formulation is obtained by substituting these parametric load models in the classical TPA formulation. The parametric models may be any suitable model describing the loads. A priori known relations among parameters (e.g. mount stiffnesses in x- and y-direction are known to be similar, etc.) may be taken into account to reduce the number of parameters to be estimated and obtain a better conditioning. For a given number of operational path inputs, response data and FRF’s, measured in phases 1 and 2, the LMS OPAX formulation gives rise to a linear system of equations that can be solved for the model parameters using conventional mathematical techniques, like for example a Least Squares (LS) estimation approach. It is clear that the more input information is used, i.e. the more orders, targets and indicator responses, the more accurate the model parameter estimations can be. The estimated model parameters may allow determining additional interesting information regarding the system. For example, the ability of estimating the mount stiffness characteristics from TPA measurement data is an interesting additional feature of the method.
In phase 4, the operational input forces and volume accelerations are determined by substituting the obtained model parameter values in the parametric load models. The loads are typically calculated per order.
Finally, once the operational loads are identified, the path contributions can be calculated for each target point, by multiplying the loads with the corresponding FRF. Visualizations of the path contribution results then allow to (i) assess critical paths, orders and frequency regions and (ii) propose modifications of, for example, mount stiffness characteristics, transfer path FRF’s, etc.
2. Applicability
The LMS OPAX method has several advantages:
- It is fast and accurate.
- It balances speed of execution and path accuracy. The more extra indicators used, the more robust the estimations and the better the path accuracy, but the higher the FRF measurement efforts and time.
- Measurement efforts are small in comparison to the traditional inverse load identification technique. Next to the operational measurements of path inputs and target(s), the method requires in many cases only one reciprocal FRF measurement per target point. Only adding extra indicators for improving robustness requires additional FRF measurements.
- The method does not require mount stiffness data.
- The estimation of the parametric load models is numerically stable. Ill-conditioning problems like those in Operational Path Analysis (OPA) hardly occur.
- The estimated model parameters may deliver some extra interesting information. For example, mount stiffness characteristics can be estimated from measurement data.
Figure 1: LMS Test.Lab OPAX simultaneously analyzes acoustic and structural contributions.
When applied to mechanical structure, the method delivers results that are at the same time fast and accurate. LMS OPAX yields the best results when noise and vibration sources are in low/mid frequency range. Due to its nature, the result accuracy is also optimal when the structure under test is built with dominating soft connections.
III. Time-domain TPA
Although TPA models are best assembled and calculated in frequency domain, there are 2 main reasons why it is beneficial to analyze a TPA model in time domain:
- A sound quality analysis or the partial path contributions, either objective (metrics) or subjective (replay) is required
- the operational data contains transient or impulsive phenomena the need to be preserved when assessing the TPA model
In general, the TPA approach is applied to stationary, pseudo-stationary and run-up noise problems such as engine noise, road noise. Of course, there is great interest to apply this methodology to transient NVH problems such as from road bumps, high frequency suspension noise, engine start-stop or all kinds of transient phenomena in secondary noise sources in the vehicle. However, the TPA results mentioned so far are all described in the frequency domain. Transient and impulsive phenomena are therefore changed and even lost when observing path contributions in the frequency domain.
Time-domain TPA is a method where the transfer path NTF’s are presented as filters in the time-domain. The multiplication of a force with a transfer function in the frequency domain becomes in the time domain a convolution of a NTF filter with the time series of a load, resulting in the contribution of that path to the receiver in time domain. And that path can be listened to in the same operational condition as in the original measurement and or further processed with sound quality tools.
Clearly, load identification is the crucial part in creating TPA models. For time-domain TPA, all described methods (direct, mount stiffness, matrix inversion, OPAX) can be used for this load identification. In general, in the frequency domain, the load identification is the multiplication of a set of indicators with a frequency source model. By using a FIR filter of this frequency source model, the multiplication translates into a convolution in time domain of this FIR filter with the time series of the indicators. The loads are then identified in the time domain, and can be further convoluted with a FIR filter of a transfer path NTF. The whole frequency domain TPA model can thus also be modeled in time domain, as described in figure 2.
Figure 2: Frequency Domain TPA models described in Time Domain using FIR filters and Convolution.
For transient excitation problems, a stationary test condition is realized by executing repeated tests with identical transient excitations, applying the stationary power spectral analysis methodology. Even while on the road, such conditions can be approximated by driving in a controlled way repeatedly over the same trajectory. Classical spectral averaging and a classical frequency domain analysis can be performed. The TPA results can be visualized in the time domain as long as loads, partial contributions and total responses maintain their phase relations intact.
In case the repeatability of the test is not an option, then true transient techniques have to be applied to derive the required cross-power matrices for the time domain response calculation. Methods such as Short Time Fourier Transform (STFT), Wavelet analysis or multivariate autoregressive models need to be applied. Once cross-power spectra derived, the standard methodology can be used in the derivation of contributions and correlations.
On the measurement side, direct measurement of the loads using strain sensors would offer an interesting alternative to the inverse or differential response procedures.
The time-domain approach offers very interesting perspectives to treat problems involving non-linear components such as mounts. Time-domain TPA based on hybrid models using Multi-body Dynamics models for the loads offer a great potential for transient engine and suspension problems.
Figure 3: With LMS Test.Lab Time Domain TPA, users can listen to partial path contributions.
IV. Multi-reference TPA
Transfer path analysis techniques are not only applicable for single coherent sources (as for instance engine noise), but also for multiple partially-correlated sources (as for road-induced noise in a car).
In order to solve structure-borne road noise problems it is important to understand how the road inputs from the four wheels are transmitted through the suspension components, and through the mounting elements into the car. The wheel inputs are each filtered in a different way, depending upon the dynamics of the transfer. This can influence the effective importance of the different transfer paths, and stimulate specific resonance problems.
When multiple partially-correlated sources are active in a system, multiple reference spectral processing of the operational measurements is required. A totally uncorrelated source can be separated from other sources by single reference crosspower measurements – taking a reference measurement at a point that is descriptive for the source behavior. In many cases, however, it is very difficult to find an adequate reference sensor location – and certainly where the sources are not totally uncorrelated – but partially correlated, or physically not very well separated.
The wheel inputs are always partially correlated, with a degree of correlation depending on the road surface characteristics. Therefore multiple reference crosspower measurements are necessary to describe a road noise problem properly. The number of references must be greater than the number of active sources to be quantified.
Classical transfer path analysis techniques can be used for a contribution analysis of the interior noise problem, only if the multi-source character of road-induced operational problems can be adequately described. However, the multireference crosspower measurements cannot be used directly in a transfer path analysis – decomposition techniques of the multi-reference crosspower data are needed first. Not only road induced noise has a multi-reference character. Other problems can also be tackled by a multi-reference approach, for instance in the study of the combined effect of air-conditioning compressor and engine noise.
Figure 4: Multi-reference transfer path analysis
Partial coherence techniques: an approach with disadvantages
In the past, partial coherence techniques have been used to quantify contributions from each source or wheel input to the car interior pressure. However, due to the partially correlated character of the wheel inputs, partial coherence techniques give rise to an overestimation of the importance of the first wheel input in the analysis – and consequently an underestimation of the other wheel inputs. Also, the result of partial coherence techniques very heavily depends on the sequence in which the references are eliminated. This is not the most appropriate way of decomposing the multi-reference crosspower set.
Virtual coherence analysis: a better approach for multivariate problems
Virtual coherence analysis is a technique based on singular value decomposition, used to decompose the chosen set of partially-correlated reference signals into their orthogonal constitutive components (the principal components). The crosspower signals at all other measurement locations can then be processed into single reference crosspowers with respect to each of these principal components. These single reference crosspower spectra are called “virtual crosspowers”. When scaling each of the crosspowers with the corresponding principal component autopower, this results in referenced spectra, the so called “virtual referenced spectra”.
Essentially, the process is one of decomposing the multi-reference problem into a number of mutually-independent single reference cases – each describing one part of the global problem. These single reference data can be used as input data to the transfer path analysis. Path contributions can thus be determined for each of the principal components – because phasing between the transfer paths still exists. In order to assess the global effect of one path, RMS summation is used to combine the contributions of each of the principal independent components.
How to select appropriate driving and road conditions
It is important to perform the operational measurements in conditions that allow the system to be described in the best way. At the same time, the operational conditions must be selected such as to reveal the problem of interest. An initial set of measurements allows the selection of optimal driving and road conditions to check additional limitations– for instance the influence of the running engine on the measurements.
In this initial measurement run, the road noise is acquired at a number of receiver locations (for instance a microphone sensor placed at the location of the passengers’ ears), together with a number of acceleration signals picked up at the wheel centers. Driving tests can be performed on different road surfaces and at different speeds. For a given speed condition and a given road surface, the multiple coherence between an acoustical receiver and the 12 wheel center vibrations must have an acceptable level.
Low coherence can reflect non-linear effects (for instance, excitation by very rough road conditions) as well as the existence of other sources that cannot be characterized by the reference sensor pickups, such as aerodynamic noise at higher speeds, or engine noise when the engine was not switched off during the measurement.
How to select appropriate reference sites
The location of the reference sensors is important, in order to completely describes the multi-reference problem. In the case of a road noise study, the reference sensors for the final road tests must not necessarily be located at the wheel centers. On the contrary, better coherence between the acoustical receiver and the references can be reached when the reference sensors are placed on the car body, for instance. By comparing multiple coherences between the receiver and a sub-set of reference signals from an initial road test, and selecting those references that give an acceptable coherence level, one can reduce the number of references step-by-step and end up with an optimal number.
The principal component decomposition can be performed on any selected sub-set of available references. Of course, it is not possible to use mixed physical quantities as references for the principal component decomposition (i.e. acoustical vs. vibrational). However, when both acoustical receivers and mechanical references were taken in each measurement run, and were used as references for the crosspower calculation, it is perfectly possible to calculate the principal component decomposition with the acoustical receivers as the references, or with the mechanical sensors as the references. Using the acoustical receivers as references may be advantageous, because only that part of the vibrational signals which is coherent with the acoustical signals (and thus shows a causal relationship) is taken into consideration.
When a reference selection has been made, it is possible to calculate the virtual crosspowers, referenced virtual spectra, virtual coherences, summed virtual coherences, coherent autopowers, and summed coherent autopowers. The referenced virtual spectra are further used in the transfer path analysis as the operational data to be processed.
Transfer path analysis for multi-reference problems
The referenced virtual spectra are used as the operational data for the transfer path analysis model. The different virtual spectra related to the different independent sources are considered as individual load cases, which can be processed in sequence. The acoustical frequency response functions, the structural frequency response functions and the complex dynamic stiffness of the mounting elements remain to be defined, as with the single reference transfer path analysis.
In order to have access to the RMS summed contributions over the different independent sources, in the post processing phase, it is possible to combine the source contributions into a load case that contains the RMS summed data.
This RMS load case contains RMS summations, over the selected independent sources, of the operational forces in each transfer path, as well as the RMS summations of the contributions to the receivers from the individual transfer paths. These results can be handled, displayed and analyzed in the same way as each of the individual independent source results. A specific feature of the RMS summations is that phase information does not exist.
Figure 5: LMS Test.Lab Multi-Reference Transfer Path Analysis lets users manage large databases of TPA models.
