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Modular Sound and Vibration Engineering by Substructuring
Abstract and Figures
[ Link to PhD defense video:
https://www.youtube.com/watch?v=IEVuF2rJOYs&t=6s ]
This thesis is the result of a 4-year collaboration between the Technical University of Munich and the BMW Group. The goal was to apply substructuring methods to the Noise Vibration Harshness (NVH) engineering needed for integrating electric climate compressors in upcoming vehicles. The compressor is one of the major contributors to the cabin noise in battery electric vehicles (BEVs). An accurate yet practical development process for its vehicle integration is crucial for industry. Specifically, the aim was to simulate the compressor noise in the cabin for different, virtual design variants of the isolation concept. Therefore, the methods from two broader fields were applied: First, the excitation of the compressor was modeled with component transfer path analysis (TPA) methods. Second, the full transfer path from the compressor to the driver’s ear is assembled from multiple subcomponent models, via dynamic substructuring (DS).
For accomplishing the above mentioned goals, different gaps in the current technology have been identified, which will be addressed in this thesis. With frequency based substructuring (FBS), a subclass of DS, it is possible to couple experimental and numerical substructure models in a virtual assembly. For the compressor, it was found that including rigid body models in the transfer path is a valuable addition. The proper formulation and integration of rigid body models in the framework of FBS will be presented. Another bottleneck at the onset of this project, was the proper modeling of rubber bushings in the transfer path. A novel method for experimentally identifying accurate substructure models of rubber isolators was developed. The rotating components in the compressor introduce gyroscopic effects that influence its dynamics. A novel substructuring method for virtually coupling gyroscopic terms to a component could prove that these effects are not relevant for the compressor case. The compressors excitation is described by blocked forces. Applying the blocked forces to the substructured transfer path of the assembly allows to simulate the sound in a virtual prototype. One goal was to make the simulated results audible to non-acoustic experts, which required the creation of sound files. This allowed for a subjective comparison
of different designs at an early development stage. Since the noise predictions with TPA
are typically in the frequency domain, some signal processing is required to create sound
files in the time domain. Different methods for auralization will be compared, which could
not be found in the existing TPA literature. Due to the inverse process for identifying the
blocked forces, measurement noise can be amplified to unacceptably high levels, which are audible in the sound predictions. Regularization methods have the potential to significantly suppress the noise amplification, which is explained and exemplified for blocked force TPA. Additionally, it was found that only the structure-borne sound transmission was not sufficient to describe the compressor noise in the cabin. The compressor is also directly radiating air-borne sound from its housing, which will be included in the NVH model by means of equivalent monopoles. The application examples at the thesis’ end are extending the current state-of-the-art, by showing how the modular vehicle models can be used for early phase, parametric design optimizations on a complex NVH problem.
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... Resulting spectra stemming from the complete runup of the synchronous machine are presented and differences between active and passive part of the structure are discussed. As far the authors are concerned, this work is unique due to the type of the source which is characterised, the present works like [1], [2], [3] or [4] do not deal with the source in form of full scale traction machine or do not focus on elastomeric isolators. Only the work [5] offers a similar perspective. ...
... There are several domains where it is possible to model substructures. These are discussed within works of [6], [7], [1] or [2]. In this study we make use of the method in the frequency domain, so-called frequency-based substructuring. ...
... To characterise the substructures with source attached the in-situ blocked force estimation, which does not require completely free or completely rigid boundary conditions is applied. It uses the FRFs estimated under impulse excitation and determine the equivalent blocked forces when the sensor response under operation is injected as in [11] or in [2]. In Equation 3, f bl is the matrix of the equivalent tran. ...
The vibrational and acoustical performance of synchronous machines, which are commonly used elec-tromotors in traction applications, significantly influences the overall driving experience of vehicles. This study focuses on characterisation of the excitation caused by a permanent magnet synchronous motor (PMSM) installed on a custom test setup using a novel dynamic substructuring method. The theoretical principles of this method, used for the determination of blocked forces at virtual (transfer) points, are briefly explained. Additionally, the requirements for the setup design and measurement procedure are discussed and condensed into compact recommendations, covering the setup design strategy, necessary measurement equipment for this analysis, and practical tips for sensor and impact placement. Considerations are also made for grounding and distortion issues arising from the working principles of the tested synchronous machine. The investigated PMSM was mounted on an optimized test bench using three elastomeric isolators. A load-free run-up of the PMSM was conducted up to 9000 rpm in this configuration, while the NVH spectra were acquired. This operational measurement was evaluated and used for the dynamic substructuring of the system. Impact and operational measurements were performed on both the active and passive side of the rubber engine mounts to syn-thesise equivalent forces between the PMSM and the vibration isolators, representing the excitation of the rubber bushing. Additionally, the blocked forces were established between the isolators and the test bench or possibly the chassis, reflecting the response of the bushing to the motor operation. The synthesised forces and moments are intended for future Finite Element Method (FEM) or flexible Multibody (MB) simulations of the entire system where the focus should be on the elastomeric elements.
... A pseudo-inverse has to be used if the system of equations is over-determined, i.e. the vector u 4 contains more channels than the actual number of blocked forces to be computed in f bl 2 . This over-determination is generally recommended [1]. The pseudo-inverse can either be built with least-squares, or with a regularized inverse to suppress the detrimental effects of measurement noise [1]. ...
... This over-determination is generally recommended [1]. The pseudo-inverse can either be built with least-squares, or with a regularized inverse to suppress the detrimental effects of measurement noise [1]. Notice that the blocked forces can be determined either on a testbench or in the vehicle, equation (7). ...
... Nomenclature: A quantity pertaining to source A (⋆) B quantity pertaining to receiver B (⋆) R quantity pertaining to test-bench R (⋆) AB coupled quantity of A and B (⋆)1 ...
Component based TPA allows to obtain a vibration source-description that is independent of the receiver. Typically, blocked forces are determined on a component testbench to predict vibration levels on a receiver. Therefore, impact measurements must be performed. Operational TPA (OTPA) is typically used as a tool for path ranking of different noise sources in a troubleshooting phase. OTPA has the advantage of not requiring laborious impact campaigns but lacks the possibility to predict receiver noise levels from testbench measurements. In this paper, we derive an OTPA method offering this "testbench to vehicle" capability. This novel, component-based OTPA method will be compared to the blocked force TPA method in a numerical test case. The sensitivities of both methods to typically encountered measurement uncertainties are shown. This is achieved by introducing different random errors in a Monte-Carlo simulation. The resulting variance and bias error in the predicted receiver responses are used to evaluate the error sensitivity of both methods.
... The proposed method is exemplified by a long-range battery electric vehicle (BEV) as an industry relevant example based on the previous research [22]. This paper focuses on the structure-borne noise in the passenger cabin caused by the electric compressor of the vehicle air conditioning system. ...
... Therefore, no reference measurement data is available to validate the TPA prediction. Interested readers are referred to [22] for the validation of the total noise level, where the air-borne noise is included. ...
Reducing noise propagation in mechanical structures can be difficult when several components on multiple intersecting transfer paths interact. One way to reduce design complexity caused by the transfer path interaction is to derive component requirements from the system design goal. The component requirements should ensure that the system design goal is reached whenever these requirements are met. In this paper, we present an approach that computes solution-neutral requirements on isolation elements of multiple connected transfer paths, such that each of them can be developed independently and efficiently. This is accomplished by first modeling the system dynamics with Dynamic Substructuring and Transfer Path Analysis. Secondly, the ranges of admissible stiffness of isolation elements are quantitatively expressed as a so-called solution space for each isolation element. The proposed approach is applied to the noise propagation path of an electric compressor in a passenger vehicle.
... The joint admittance matrix Y J can be obtained by inverting the joint dynamic stiffness matrix Z J [36]: ...
... Alternatively, the joint admittance matrix can be obtained by solving the eigenvalue problem −ω 2 M J +K J = 0, yielding n eigenfrequencies ω r and the corresponding eigenvectors φ r . A damping ratio ζ r can be defined for each non-rigid body mode φ r and the mode-superposition method can be used to synthesize the FRFs of the joint admittance matrix [36]: ...
The dynamic properties of assembled structures are governed by the substructure dynamics as well as the dynamics of the joints that are part of the assembly. It can be challenging to describe the physical interactions within the joints analytically, as slight modifications, such as static preload, temperature, etc. can lead to significant changes in the assembly's dynamic properties. Therefore, characterizing the dynamic properties of joints typically involves experimental testing and subsequent model updating. In this paper, a machine-learning-based approach to joint identification is proposed that utilizes a physics-based computational model of the joint. The idea is to combine the computational model of the joint with dynamic substructuring techniques to train the machine-learning model. The flexibility of dynamic substructuring permits the enforcement of compatibility and equilibrium conditions between the component models from the experimental and numerical domains, facilitating the development of machine-learning models that can predict the dynamic properties of joints. The proposed approach provides an accurate data-driven method for joint identification in real structures, while reducing the number of measurements needed for the identification. The approach permits the identification of a full 12-DoF joint, enabling the coupling of 3D dynamic models of substructures. Compared to the standard decoupling approach, no spurious peaks are present in the reconstructed assembly response. The proposed approach is validated numerically and experimentally by reconstructing the assembly response and comparing the results with known assembly dynamics.
... For this pur-pose, they were freely suspended in an acoustic test bench and equipped with acceleration sensors, and then a controlled run-up was measured. The blocked forces of both compressors were determined inversely from their transfer function and operational vibrations [5]. FIGURE 3 (a) shows the L2 norm of the translational blocked forces of project and series compressor during run-up. ...
... It was evaluated using an airborne sound TPA method. The airborne TPA principle can be seen as the uncorrelated equivalent of the component based TPA for the structure-borne noise, [5,6]. An acoustic surrogate source is used for source description. ...
Dynamic substructuring methods allow experimental and simulative models to be coupled with one another. Substructuring can defuse the problem of missing model availability in concept phases, which is quite common, and the interior acoustics can be assessed at an early stage. To this end, Magna and VIBES Technology optimized the noise isolation concept of the electric climate compressor in the concept phase of a vehicle development project. It could be proven that the initial isolation design would have exceeded the interior noise target by 20 dB. Optimization of the substructured model made it possible to accomplish the noise target without additional decoupling or masses.
... For this pur-pose, they were freely suspended in an acoustic test bench and equipped with acceleration sensors, and then a controlled run-up was measured. The blocked forces of both compressors were determined inversely from their transfer function and operational vibrations [5]. FIGURE 3 (a) shows the L2 norm of the translational blocked forces of project and series compressor during run-up. ...
... It was evaluated using an airborne sound TPA method. The airborne TPA principle can be seen as the uncorrelated equivalent of the component based TPA for the structure-borne noise, [5,6]. An acoustic surrogate source is used for source description. ...
Methoden der dynamischen Substrukturierung erlauben es, flexibel experimentelle und simulative Modelle miteinander zu koppeln. Das oft vorhandene Problem fehlender Modellverfügbarkeiten in Konzeptphasen wird durch Substrukturierung entschärft, und die frühzeitige Bewertung der Innenraumakustik wird ermöglicht. Magna und VIBES Technology haben so die Entkopplung des elektrischen Klimakompressors schon in der Konzeptphase auf das Innengeräusch optimiert. Es konnte nachgewiesen werden, dass das initiale Lagerungskonzept zu 20 dB Zielüberschreitung im Innenraum geführt hätte. Eine Optimierung am substrukturierten Modell erlaubte es, das Lagerkonzept ohne zusätzliche Entkopplung ins Ziel zu bringen.
... A common practice in inverse problems with test-based data is to perform a singular value decomposition (SVD) on the matrix to be inverted, with the idea of reducing the effect of noise or small artifacts in the data (Haeussler, 2021;Wernsen, 2017). In the remainder of this section, we illustrate that this method's effect is only limited. ...
Selecting the proper set of Degrees of Freedom is essential in inverse (Blocked) Force calculation. Including too many Degrees of Freedom in the computation can lead to overfitting, resulting in inaccurate force estimations and poor prediction quality. The discrepancy arises from errors within the dataset, such as measurement noise or other artifacts. This paper presents a solution to the overfitting problem, introducing the X-DoF procedure to automatically identify the relevant subset of Blocked Force Degrees of Freedom. Its effectiveness is showcased through numerical and experimental validation and compared against regularization techniques.
... Given that the condition number of the matrix to be inverted is high, this can lead to a major amplification of the error in the results. To address this issue, regularization techniques are proposed, such as singular value truncation or Tikhonov regularization [18]. ...
In noise and vibration engineering, a structure’s passive dynamic properties are often evaluated in terms of its frequency response functions (FRFs). The typical FRF measurement campaign consists of controlled structure excitation and the capturing of its response. However, exploiting operational excitation for FRF acquisition is not feasible, with many sources simply too complex to model or measure directly. In the present paper, an alternative approach is proposed for an indirect FRF estimation of a system in operation, in which the source is characterized independently of the final assembly. To overcome the issue of the unmeasurable excitation force, transfer path analysis (TPA) methods are proposed. TPA replicates the source excitations using the set of equivalent or pseudo forces that are an inherent property of the source. The assembly’s FRFs are then evaluated on the basis of receiver responses and pre-determined pseudo forces for independent operational load cases at the source. Thus, a single source description can be applied to estimate the FRFs of any assembly with an identical source and arbitrary passive side.
Transfer Path Analysis (TPA) is a test-based methodology used to analyse the propagation of noise and vibration in complex systems. In this paper we present a covariance based framework for the propagation of experimental uncertainty in classical, blocked force, and component-based TPA procedures. The presence of both complex and correlated uncertainty is acknowledged through a bivariate description of the underlying uncertainty. The framework is summarised by a series of equations that propagate uncertainty through the various stages of a TPA procedure i.e. inverse source characterisation, dynamic sub-structuring, and forward response prediction. The uncertainty associated with rank ordering of source contributions is also addressed. To demonstrate the proposed framework a numerical simulation is presented, the results of which are compared against Monte-Carlo methods with good agreement obtained. An experimental study is also presented, where a blocked force TPA is performed on an electric steering system. The proposed uncertainty framework requires no additional experimental effort over and above what is performed in a standard TPA and may therefore be readily implemented into current TPA practices.
A proper acquisition of FRFs is a prerequisite for a successful implementation of experimental substructuring techniques in the frequency domain. In this context, the study of uncertainty associated with measurements is of particular interest due to the precision standards required by industrial practice. This paper aims to provide a practical and reliable methodology for the quantification and propagation of the random measurement uncertainty in Frequency Based Substructuring applications. Extending previous studies, the framework presents a covariance-based approach for quantifying the complex-valued random uncertainty on measured FRFs and analytical methods for propagating it through interface modeling and substructures coupling approaches. The assumptions underlying the correct application of the method are investigated. An optimal number of impacts for an appropriate Gaussianity of the FRF distribution is computed based on empirical data. Experimental testing reveals encouraging results in the validation of the small error approximation. Considerable correlation effects between FRFs are found, although their impact on the coupled FRFs uncertainty seems to be limited. The methodology is applied to an experimental example.
In the research project QUATTRO, different methods to measure Rotational Degree of Freedom (RDOF) structural dynamics data have been studied and developed. In the present paper, the indirect, added mass, technique is reviewed. The method uses multiple accelerometer measurements of translational degree of freedom vibrations on an added mass to derive the corresponding rotational degree of freedom functions at the connection point. A special sensor was developed hereto. The technique was applied to the analysis of the subframe of a car engine. The use of experimental RDOF data is evaluated for several engineering applications. These include the prediction of the coupled behaviour of the subframe connected to some simple modifications, the assessment of the relative importance of RDOF source contributions at the connection of the subframe to the car body and the updating of a FE model of the subframe.
In this contribution we examine the combination of frequency based substructuring (FBS) and component based Transfer Path Analysis (TPA) for enabling an acoustic design optimization in the early product development stage. The classical TPA approach is mostly used as trouble-shooting-tool for physically already existing prototypes. Here, the transmission of vibrations is characterized by interface forces acting between source and receiver. However, changing the design, e.g. by making the receiver more flexible, will inevitably change these interface forces. Therefore, they are not suited as a source description during the development of design improvements. Recently, there was a renewed interest in so called component based TPA formulations, which rely on "blocked forces" for describing the source excitation. Theoretically, the blocked forces do not depend on a specific receiver design, which allows for an optimization of the latter. For optimizing the receiver structure it is desirable to combine different substructures that form the final receiver path, and model each substructure with the most appropriate modeling approach. Frequency based substructuring (FBS) permits coupling analytical, numerical or experimental models with each other to obtain the transfer functions of the final assembly. Experimental models are typically used for structures that are hard to simulate accurately, e.g. due to high modal density, complicated material models with unknown parameters, friction in joints, etc. Numerical models may be used for designing and optimizing the source support that is used for vibration isolation in the final assembly. In this contribution we show a realization of an acoustic optimization using component based TPA for the source description. We investigate the proper formulation of the cost function and the applicability of different optimization algorithms available. The FBS assembly is explained in this contribution and the optimization process is laid out. In a future publication the method will be applied to a larger optimization example including the FBS assembly of experimental and numerical substructures. The results will be validated with experimental data gathered on the optimized designs.
Dynamic substructuring methods serve as a powerful tool in the analysis of modern complex systems. The coupling of substructures has been successful with analytically obtained results. However, substructuring with experimentally obtained data remains challenging. One of the main problems associated with experimental substructuring is the coupling of the rotational degrees of freedom (RDoFs). A promising method where RDoFs are included implicitly is the virtual point transformation. Even though the transformation has been successfully used in the substructuring process, it is still highly susceptible to inaccuracies in the sensor sensitivity and positioning. In this paper an expansion to the virtual point transformation is proposed, which enables the projection of a directly measured rotation response on the interface deformation modes. A novel formulation of the weighting matrix is introduced to consistently include the measured rotations in the transformation. The proposed expansion is demonstrated on a numerical model of a simple beam-like structure and compared with the standard transformation. The effects of inaccuracies in the sensor sensitivity and placement on the overall quality of both transformation are analysed with a global sensitivity analysis. Finally, an experimental validation of the proposed expansion is carried out on a steel beam.
The evaluation of structure-borne sound power is an effective analysis tool for the analysis of the structure-borne sound transmission in mechanical structures like vehicles, but also for the characterization of components. However in the case of measurement-based methods in particular the phase-accuracy becomes challenging for lightly damped systems vibrating at non-resonant frequencies. In order to enable the measurement of power quantities for such cases a method is discussed incorporating a frequency-dependent phase correction. The latter is predetermined in a small test setup solely for the impedance heads as these introduce the largest phase errors in the measurement chain. The evaluated impedance heads are used for the evaluation of power quantities at an example structure afterwards. Applying the phase correction a significant improvement of the resulting power quantities is finally achieved.
Commercial off-the-shelf rubber isolators often come with no additional information other than the static stiffness in three translational directions. Hydraulic testing machines can be used to obtain frequency dependent dynamic stiffnesses of rubber isolators in translational degrees of freedom (DoF). Alternatively, dynamic substructuring based methods can be used, which can additionally identify the dynamic stiffness in rotational DoF while requiring only standard vibration testing equipment. Results of two substructuring methods will be compared to those from a hydraulic machine. Both of the presented methods use locally rigid fixtures, mounted to the bottom and top of the isolators. Frequency based substructuring (FBS) requires knowing the fixtures dynamics to decouple them. Inverse substructuring, also called in-situ decoupling, does not require knowing the fixtures dynamics, but is assuming negligible mass and a special stiffness matrix topology of the rubber isolator. Both methods produce accurate results for translational DoF up to the kilo Hertz range, which is confirmed by comparison to measurements on the hydraulic machine. However, FBS does not rely on specific assumptions about the isolator, like inverse substructuring. The limits of inverse substructuring's underlying assumptions are shown theoretically and in the measurements presented here. We propose two extensions to compensate for the assumptions and present their results. Nevertheless, the rubber model obtained with the FBS decoupling can provide better results when used in an assembly. This is illustrated by testing the experimental rubber element models, obtained with either method, in a substructuring prediction of coupled frequency response functions (FRFs) and comparing that to reference measurements.
Dynamic Substructuring is a method that combines models for the various parts of a structure to estimate the dynamic response or other properties of the assembled structure. The substructure models may be analytical models such as finite element models, or they may be derived from measurements. This book reviews the most common state-of-the art methods for substructuring and model reduction and presents a framework that encompasses most method, highlighting their similarities and differences. For example, popular methods such as Component Mode Synthesis, Hurty/Craig-Bampton, and the Rubin methods, which are popular within finite element software, are reviewed. Similarly, experimental-to-analytical substructuring methods such as impedance/frequency response based substructuring, modal substructuring and the transmission simulator method are presented. The overarching mathematical concepts are reviewed, as well as practical details needed to implement the methods. Various examples are presented to elucidate the methods, ranging from academic examples such as spring-mass systems, which serve to clarify the concepts, to real industrial case studies involving automotive and aerospace structures. The wealth of examples presented reveal both the potential and limitations of the methods.