Excessive vibration levels in structures need to be controlled in order to reduce noise emissions, avoiding fatigue failure and increasing accuracy of high precision devices. It is therefore necessary to analyze the vibration levels in structures in order to understand the transmission path and determine the measures required to control them. There are a number of methods to carry out vibration analysis of structures. Arguably, the most popular method in the low frequencies is the Finite Element Method. The analysis of high frequencies is commonly done using the Statistical Energy Analysis (SEA) or Experimental Statistical Energy Analysis (ESEA) methods. While these methods give accurate results for many types of structures, they have important limitations, especially in the mid frequency range.

　This research proposes a novel method for structural analysis in the high frequency ranges named Experimental Wave Intensity Analysis (EWIA). The formulation of the EWIA method is based on the Wave Intensity Analysis (WIA) method, which is an extension of the SEA method for non-diffuse energy fields. Therefore, this research can be divided into two sections: the first section investigates the WIA formulation in depth. The reason is that the WIA method shows an important potential for practical application. However, since it was proposed, there has been very little work regarding this method and the literature available is very scarce, what makes the method difficult to understand for non-experts. The second section proposes the EWIA method as and extension of the WIA method for its application to complex systems and its expansion into the mid frequency ranges.

　EWIA is an extension of the WIA method that uses experimental data to describe more accurately the structures under analysis. A simplified experimental procedure is also proposed, since acquiring the EWIA experimental data, wave intensity measurements, can be very cumbersome. This procedure consists in using ESEA experimental models (internal and coupling loss factors) obtained through the power injection method instead of wave intensity measurements. Then, semi-experimental transmission coefficients required to describe the coupling between the different subsystems of the structure are estimated from those experimental coupling loss factors at each boundary and frequency band. The characteristics of the EWIA formulation and its stability have also been analysed and compared with the WIA, SEA and ESEA methods.

　The WIA and EWIA methods were applied to a to a simple two-plate system (Fig. 1) in order to validate the latter. Then, the EWIA method was applied to a complex system consisting in a dash-and-floor substructure of the body-in-white of a car (Fig. 2). Figure 3 shows he subdivision of the model into subsystems. The results confirmed that the WIA method could improve SEA results for the case of plate systems tested in the research of this thesis. However, It was found that the stability of the method, evaluated by the sensitivity of the coupling matrix to the operation of matrix inversion, could have important influence on the energy predictions at specific frequencies. The sensitivity is more apparent in the case of using measured internal loss factors (and theoretical transmission coefficients). On the other hand, the SEA method showed to be much more stable and no problems were found when using the same experimental internal loss factors, as shown by the low condition number in Fig. 4.

　The reason for the sensitivity of the WIA method is the great difference between the values of the elements in its coupling matrix. The elements associated to the diffuse energy fields have, in general, much lower values than those associated to the non-diffuse energy fields. This difference can be importantly increased when using experimental internal loss factors due to their variability. Since the SEA coupling matrix includes only the elements of the WIA coupling matrix associated to the diffuse energy field, the SEA method is not affected by the variability of the internal loss factors as much as the WIA method.

　It was reported by Langley and Bercin that the WIA method can improve energy predictions in systems where, due to its configurations, the energy field is not highly diffuse, for example, a row of connected plates, which promotes energy transmission in its longitudinal axis more than in the transversal axis. For the case of box structures, which configuration promotes a diffuse energy field, Bercin finds that WIA does not present important improvements in the energy predictions in comparison to SEA and based the explanation of the results solely on the type of energy field, diffuse or non-diffuse. However, this thesis shows that also in the case of structures with multiple connections that, in principle, can be considered diffuse (for the same reasons as the box structure tested by Bercin), the WIA method can achieve improvement in the energy predictions. This improvement can be explained based on the effect of the energy filtering effect and the non-direct coupling loss factors rather than using a diffuse or non-diffuse energy field explanation. The reason for the improvements is that, the energy filtering effect and non-directly coupling loss factors occur at every connected boundary, no matter the configuration of the system. Since SEA formulation does not take into account the energy filtering effect and the associated non-direct coupling loss factors, the method introduces errors in the energy prediction at each boundary. If two subsystems are separated by a number of boundaries, the error at each boundary is accumulated so that the miss-prediction increases with distance, as found by Langley and Bercin.

　For systems with multiple connections in different directions such as a box like structure, the energy filtering effect does not promote non-diffuse energy fields. However, the energy filtering is present at every boundary and, also in this case, the SEA method will introduce errors at each boundary. This conclusion is more apparent in large structures with multiple connections. In this structures the energy field could, in principle, be considered diffuse but the energy filtered at each boundary between the excited subsystem and a subsystem located several boundaries far from it will become significant, and the WIA method will predict more accurately the energy levels of those subsystems than SEA.

　It was also found that the application of WIA to complex systems is greatly limited. The reason is that the experimental data required by WIA for its application in complex structures becomes very difficult to acquire. The main reasons are: location of the devices on the structure for both contact and non-contact methods (like optical methods or accelerometer measurements, respectively). That problem is especially true in contact methods where sets of five to sixteen transducers are required to measure the corresponding intensity vectors. Contact methods also present weight problems, since those sets of transducers become relatively heavy and affect the measurements. Moreover, these methods also present contact problems since in very irregular surfaces the perfect contact of every transducer of every set used is not guaranteed.

　Then, It was proved that the proposed simplified EWIA is a valid method for estimation of vibration energy in complex structures for which the WIA method is not accurate. By using ESEA data (coupling and internal loss factors) rather than intensity measurements, the EWIA method is greatly simplified and all the measuring problems presented above are overcome. EWIA estimates the energy transmitted by combining experimental measurements of coupling loss factors and theoretical distributions of the transmission coefficients. While the theoretical distribution at boundaries of real systems could be far from those of real transmission coefficients, this simplification allows the EWIA method to take into account, in some extent, the energy filtering effect neglected by the ESEA method and thus, ESEA results are improved while using exactly the same experimental data.

　The use of theoretical distribution of the transmitted energy can also be justified by the fact that structural intensity is greatly dependent on factors such as system conditions, loading location and so on. Therefore, the intensity field measured in one set of experiments and its corresponding distribution of transmitted energy could greatly differ from the next set of experiments. Thus, the statistical average of those sets might not be representative of any of them (the variance may be very large). For this reason, the average of measured intensity data might become just a coarse approximation and, in that case, the approximation using theoretical distributions could also be defended. Figure 5 shows an example of energy the energy distributions given by WIA (theoretical) and EWIA of a subsystem in the dash-floor structure.

　EWIA can improve WIA results, as shown in Fig. 6. This is the case especially for complex structures where theoretical data greatly differs from experimental measurements. The reason for the improvements is that, while EWIA uses same distribution of the transmitted energies as WIA, in ESEA, those energies are correlated with experimental data (internal and coupling loss factors). Therefore, the energy filtered at each boundary and the non-direct coupling loss factors represent a better approximation than those estimated by WIA using fully theoretical models.

　However, the very important factor of EWIA is the possibility of improvement of ESEA energy predictions since ESEA is valid for complex structures. EWIA can improve ESEA results while using exactly the same experimental data. The reason is that EWIA can estimate the energy filtered at each boundary of the subsystem and can also define the relationship between non-directly connected subsystems through the estimated non-direct coupling loss factors, as shown in Fig. 7.

　Since the energy filtering effect is accumulative as the energy wave crosses successive boundaries, the EWIA and ESEA results for the excited subsystem should be very similar. However, for the other subsystems in the structure, the EWIA method will improve the ESEA energy predictions. It should be noticed that the improvement does not depend solely on the distance between the subsystem calculated and the excited subsystem but also on the value of the coupling loss factor between subsystems. The reason is that, in EWIA, the coupling loss factors determine not only the amount of energy transmitted, as is the case of ESEA, but also the amount of energy filtered. Therefore, even for directly connected subsystems, if the amount energy filtered is important at the connecting boundary, the EWIA method would be expected to give better energy estimations than the SEA method.

　The EWIA method incorporates a new variable named Cnst (Fig. 8). This variable is a proportionality constant that related the experimental and theoretical models of a given structure. In a more simplistic way, the Cnst constant can be thought as a scaling factor applied to the theoretical transmission coefficient. This scaling factor ensures that, assuming a theoretical distribution of the energy at a given boundary, the amount of energy transmitted matches the energy measured in the experiments, which is determined by the experimental coupling loss factors, form with Cnst is derived. For this reason, EWIA can improve WIA predictions since it ensures that the energy transmitted is same as in the real structure. For the same reason, EWIA can improve ESEA predictions, because having the same amount of energy transmitted, EWIA defines its dependency with the angle of transmission by applying theoretical transmission coefficients.

　The Cnst constant can be applied into the EWIA formulation in different ways. While these are equivalent, it was found on the results that important variations could result on the energy predictions depending on the way Cnst is applied. That difference can be justified by considering the accumulative behavior of the energy filtering effect and the dependency of Cnst on the experimental coupling loss factors. Therefore, if Cnst is applied in a given coupling direction of transmission at each boundary between the subsystems of a structure, its value at each boundary will depend on the coupling loss factor in that direction. Consequently, the total energy filtered will depend on the combination of those Cnst values. On the other hand, If the direction selected to apply Cnst is the opposite, its value at each boundary will depend on the coupling loss factor in the opposite direction and, given the variability of experimental data for the two coupling directions of any given boundary, the total amount of energy filtered could greatly differ from that calculated above.

　In conclusion, it is shown that EWIA is an efficient method for improving ESEA predictions while using the same experimental ESEA procedures. Therefore, EWIA does not require complex experimental procedures and the level of simplicity is similar to ESEA. Moreover, predictions of existing ESEA models can be improved using the available ESEA data with no additional manipulation.

Figure 1: Illustration of simply supported L-plate system

Figure 2: Dash-Floor model from body in white of a car

Figure 3: Subdivision of dash-floor model into subsystems

Figure 4: Comparison of matrix condition between EWIA, ESEA and SEA and WIA using experimental internal loss factors for the L-shape system

Figure 5: Comparison between the energy functions of WIA and EWIA model of a two-plate system at 2500 Hz using three Fourier components.

Subsystem 2

Figure 6: Comparison of measured energy, WIA using experimental ILF (WIA#1) and WIA using constant ILF(WIA#2) and EWIA for L-plate structure

Figure 7: Energy prediction comparisons between measured energies, ESEA and EWIA predictions for Subsystem 8

Figure 8: EWIA proportionality constant, Cnst, oin both coupling directions L-shape structure

Cnst 12: coupling from Subsystem 1 to Subsystem 2

Cnst 21: coupling from Subsystem 2 to Subsystem 1

　本論文は，「Experimental Method for Improvement of Structural Vibrations Analysis at High Frequency(高周波域の構造振動の実験的予測手法に関する研究)」と題し，8章より構成されている．

　高周波数域での構造振動解析にはSEA(統計的エネルギー解析法)がよく用いられるが、拡散場であること、モード密度が大きいこと、要素のつながりが弱結合であること、などの前提条件があり、それを満たさないと結果の精度が悪くなる。特に周波数が低くなるにつれて、特定のモードが支配的な振動場になると拡散場の仮定が満たされなくなるので、中周波域への適用では誤差が大きくなる。このため、結合部での曲げ波の方向性を考慮にいれるような手法が検討されており、その一つにWIA(Wave Intensity Analysis、波動インテンシティ解析法)があるものの、理論的定式化がなされたのみで、適用例がほとんどない。本論文では、このような背景のもと、実験データを用いてWIAをベースとした解析を行うEWIA(実験的WIA)を提案し、その有効性について論じている。

　第1章「Introduction」では，研究の背景，目的を述べ，これまでに提案された手法を概説し、本論文の構成をしめしている．

　第2章「Literature Review」では、過去の研究事例を紹介し、現状の技術レベルと本論文の位置づけを示している。

　第3章「Wave Intensity Analysis」では，Prof. Langleyによって提案されたWIAの定式化を示し、新たにその意味と意義に関して検討を加え、新しい知見を得ている。具体的には、WIAは境界におけるエネルギの伝達を指向性を持ったものとして定式化されるが、それはSEAの考え方の一部に指向性の考慮の部分を付加したものと位置づけ、WIAとSEAの類似点と相違点を理論的な式展開において示している。さらに非直接結合損失(Non-direct Coupling Loss Factor)という概念を導入し、SEAでは考慮されない非直接結合要素間の関係を示し、それにより拡散性を仮定したSEAでの等方性エネルギ分布という扱いではフィルタ化されてくる部分をうまく表すことができると示している。このようにWIAとSEAの違いを類似点と相違点という形で表現することで、従来あまり適用がなされていないWIAの意味をより明確にすることができている。

　第4章「Experimental Wave Intensity Analysis」では，前章の検討でより位置づけが明確になったWIAを、実際の解析へ展開できるように、実験データをもとにするアプローチとして、新たにEWIAという概念を提案し、その詳細について述べている。SEAでも、複雑な構造物への適用を考えると理論だけでパラメータ設定が難しく、実験的に決めていくESEAが多く用いられているが、WIAでもそれに相当するEWIAが考えられる。しかしながら、エネルギーを直接求めようとすると非常に複雑な振動インテンシティ計測が必要になり、あまり現実的でない。本論文では、WIA-SEAとの対比を考慮しつつ、ESEAに対応するものとしてEWIAを位置づけ、いくつかの新しい提案を盛り込み、実際の手順を具体的に示している。そこでは、インテンシティではなく、加速度計測データを用い、実験的な係数を定義して、それらから実効的なパラーメタ導出とエネルギー伝達の算出が可能である。

　第5章「EWIA Application to Simple and Complex Systems」では，前章で提案したEWIAの検証と有効性確認のために、2枚の平板要素と車両フロア部へのEWIA適用を述べている。2要素系では、SEAでも高周波では十分な精度の結果が得られるので、EWIAに際立った精度向上が見込めるわけではないものの、非拡散場の影響が顕著になる周波数帯ではEWIAによる良好な結果が得られている。また、複雑な形状の構造物の例として対象とした車両フロア部については、EWIAの利点が十分確認され、方法の妥当性と有効性が確認されている。

　第6章「Summary and Conclusions」では，本論文の成果をまとめ、結論、今後の展望を述べている。

　以上要するに，本論文は，高周波域での構造振動の解析法として、従来よく用いられるSEAの限界を回避できる有効な手法として、WIAに注目し、その理論的意味を再検討した上で、SEAとの類似性と相違を明確に位置づけ、さらに実験的手法としてEWIAの提案と検証を行っているものである。本論文で得られた成果は、工学技術への貢献が非常に大であり、機械工学の発展に寄与することが大きい。

　よって本論文は博士(工学)の学位請求論文として合格と認められる．