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# Wave Based Method for Structural Vibrations of Thick Plates

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## Abstract

The noise and vibration characteristics of newly developed products become increasingly important due to restrictive governmental regulations and customers’ demand for acoustical comfort. A detailed knowledge about the product’s structural behaviour is required in the early design phases to allow for an efficient optimization of the sound and vibration properties. Nowadays, virtual simulation tools are applied to get this information in a time- and cost-efficient way. The flexural vibrations of plates are considered to be one of the most important sources of sound. Therefore, an accurate but simple mathematical model of the plate is required and efficient numerical techniques to solve the resulting governing equations have to be developed. This dissertation addresses the modeling of the structural vibrations of plates and the improvement and extension of an efficient numerical technique called Wave Based Method. The most common mathematical models of plates are the Kirchhoff plate theory and the Mindlin plate theory. While the simpler Kirchhoff plate theory is generally applicable for thin plates and low frequencies, the more complicated Mindlin plate theory can be used for thick plates and higher frequencies. In this work, both models are analysed and their range of validity concerning the plate thickness and excitation frequency is thoroughly examined. The Finite Element Method (FEM) is generally applied to predict the harmonic response of a plate in the low frequency range. Since the computational load of the FEM strongly increases with rising frequencies, alternative calculation methods are needed to get accurate results for plate vibration problems in the so-called mid-frequency range. A deterministic method calledWave Based Method (WBM) is able to tackle problems in the mid-frequency range due to an increased computational efficiency. This dissertation considers the development of the WBM for thick plate vibration problems governed by the Mindlin plate theory. The general methodology of the WBM is specialized for the governing equations of the Mindlin plate theory and a different approach to select the basis functions in the WBM is proposed. Furthermore, new particular solution functions, which are closed-form analytical solutions of an infinite plate under certain excitation types, are presented. The computational performance of the WBM compared to the FEM is investigated through a variety of validation examples and the advantages of the new wave function selection is shown.
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... Among these methods, Wave Based Method (WBM, [21]), a deterministic approach based on the Trefftz method, is an efficient technique to solve mid-frequency dynamic problems. It has been successfully applied to engineering problems, such as acoustic simulations [21][22][23][24], plate bending problems [25][26][27][28][29][30][31][32], 4 vibro-acoustic analyses [33][34][35][36][37], et al. The usage of the WBM for vibro-acoustic coupling analysis has been well established by Desmet [21]. ...
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The steady-state vibro-acoustic behavior of a plate-cavity system under harmonic excitations and static temperature loads is investigated using a Wave Based Method (WBM). The solutions of the governing equations of the vibro-acoustic problem considering thermal effects are derived. The coupled wave based model is constructed based on wave functions and particular solution functions for the acoustic and structural system. The accuracy and efficiency of the WBM are verified by the Finite Element Method (FEM) on a plate-cavity system subjected to harmonic excitations and static temperature loads. Then, numerical simulations are performed to investigate the influence of thermal effects on the vibro-acoustic responses of the system. Further, the design sensitivity is implemented using the WBM. Results show that the developed WBM has a better convergence rate than FEM for the investigated plate-cavity system. Although the temperature-dependent material properties of the cavity have little effect on the plate-cavity resonant peaks of the vibro-acoustic responses, they make the other resonant peaks of sound pressure level responses shift towards higher frequency when the temperature increases. When only the thermal stresses of the structural system are considered, the plate-cavity resonant peaks of the vibro-acoustic responses shift towards lower frequency with increasing temperature. When both kinds of thermal effects are considered, the influence of structural thermal stresses plays a leading role in the plate-cavity resonant peaks of vibro-acoustic responses.
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