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

The dynamic behavior of structural components over broad frequency ranges, particularly thick beams under different constraints, is important in many engineering applications where reduced dimensional modeling is required for design. Applications are aerospace structures, mechanical systems and civil infrastructure. The rigid cross-section assumption in Euler-Bernoulli and even third-order beam theories cannot accurately capture the effects of stress-free or finite surface conditions and higher-order stress distribution under dynamic situations. While some higher-order beam theories satisfy shear stress boundary conditions, they do not fully account for normal stress. The higher-order beam theory employed in this study 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 consistent with the necessary stress variation throughout the thickness, which is particularly relevant for thick structures.

There remains a lack of comprehensive comparison between different reduced-order models, particularly in terms of their accuracy in predicting wave dispersion characteristics and dynamic deformation mode shapes in the short and long wavelength limits to evaluate the acceptability of specific models in specific applications. Also, the choice of beam theory directly influences these properties. This study compares four different theories: Euler-Bernoulli, Timoshenko, Third-order shear, and proposed higher-order theory with surface constraints. 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 predicted by 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: one is based on the two-dimensional Fast Fourier Transform (2D FFT), and the other uses particle displacement vector plots. In the first approach, a time-varying excitation is applied to the beam with a specific tonal 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 modes at 300kHz frequency as an example, which is better predicted using the higher-order beam theory. 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.

In summary, this thesis presents a comparative study of various beam theories to highlight the importance of higher-order beam theories where relevant physics needs to be captured. The dynamic effects are relevant in applications in vibrating machinery, dynamic contact effects, bearings, and advanced contact force-based testing like resonance and force microscopy.