Elastic Wave Dispersion Analysis and Mode Shape Investigation of Higher order Beam Theory for Thick Beams

Abstract

The dynamic behavior of structural components, particularly thick beams, is critical in many engineering applications, including aerospace engineering, civil infrastructure, and mechanical systems. The rigid cross-section assumption in Euler-Bernoulli and even third-order beam theories cannot fully model the effects of stress-free surface conditions and higher-order stress variations. While some higher-order beam theories satisfy shear stress boundary conditions, they do not fully account for normal stress continuity. The higher-order beam theory used in this thesis addresses these limitations. It satisfies both shear and normal traction conditions simultaneously. Another problem in guided wave behavior within thick beams is accurately modeling consistent surface or interior dynamics. For this, the transverse displacement is approximated using a trigonometric variation across the thickness, characterized by a fundamental wave vector showing necessary stress variation throughout the thickness, which is particularly relevant for thick structures. Despite advancements in beam theory, there remains a lack of comprehensive comparison between different models, particularly in their accuracy in predicting dispersion characteristics and mode shapes. Also, the choice of beam theory directly influences these properties. Hence, the comparative study of different beam theories is important. This thesis work compares the four beam theories: Euler-Bernoulli, Timoshenko, Third-order, and Higher-order beam theory. The dispersion characteristics of each beam theory are obtained by solving the characteristic equations using the polynomial eigenvalue method, and dispersion curves are plotted to compare wave propagation behavior across different theories. This comparison highlights the limitations of the lower-order theories, especially in their ability to accurately capture the behavior of thick beams, and demonstrates how higher-order theory provides improved predictions of wave behavior. Two numerical validation techniques are employed to validate and investigate higher-order wave modes present in higher-order beam theory: two-dimensional Fast Fourier Transform (2D FFT) and particle displacement vector plots. In the first approach, a time-varying load is applied to the beam model in COMSOL at a specific excitation frequency, and time-domain response data is collected. The 2D FFT is then performed to extract the dominant wave modes. This method generates the flexural and axial mode at 300kHz frequency, which matches the higher-order beam theory model. In the second approach, wave motion is visualized as particle trajectories by plotting displacement components along axial and transverse directions. This method enables the generation of pure wave modes by solving the displacement field directly, eliminating dependencies on boundary conditions and external excitation. This method validates all mode shapes present in the Higher-order beam theory. This thesis presents a comparative study of various beam theories to highlight the importance of higher-order beam theories in analyzing thick beams. Additionally, HBT has valuable applications in vibrating machinery, dynamic contact effects, bearings, and advanced contact testing. In these contexts, accurate beam theories are required to predict responses and estimate material properties effectively. By incorporating HBT, engineers can achieve more reliable and detailed results in these critical applications.