Graduation Semester and Year

Spring 2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Aerospace Engineering

Department

Mechanical and Aerospace Engineering

First Advisor

Dr. B. P. Wang

Abstract

In recent years, several studies have investigated methods for calculating anti-resonant frequencies. However, no numerical study has applied eigenvalue matrix formulation to tapered beams. Moreover, analytical solutions remain essential for validating numerical results. Accordingly, the first part of this research is to investigate the anti-resonant frequency of a cantilever tapered beam via multimodal methods: finite element modeling, analytical modeling and experiment methods.

In engineering practice, anti-resonant frequencies can be used to optimize damped vibration absorber and bandwidth. However, limited research has been conducted on the bandwidth between two fixed point frequencies for continuous systems. Accordingly, the second part of this work aims to minimize the resonant amplitude and obtain a maximum bandwidth of low amplitude vibrations of tapered beams. For practical structural vibration problems, not only is the reduction of vibration amplitude important, but the sensitivity of vibration characteristics is also critical. Existing mode superposition methods for computing eigenvector derivatives require all eigenvector data, while algebraic methods require both eigenvalues and eigenvectors for the mode of interest. In addition, the CVM method and Nelson’s approach involve more complex procedures. Therefore, the third part of this paper develops a new, simple pseudo-inverse method for computing eigenvector sensitivity. This method requires only the eigenvector of the mode of interest and has the simplest procedure among existing approaches. Furthermore, a residual mode acceleration method is developed to overcome modal truncation errors caused by retaining only a limited number of vibration modes.

In the first part, finite element modeling the cantilever tapered beam is performed, and the matrix formulation for calculating anti-resonant frequencies is derived. Analysis of numerical anti-resonant frequencies of the tapered beam under harmonic base excitation is carried out by ANSYS APDL and MATLAB using eigenvalue matrix formulation. Furthermore, analytical modeling of tapered beam under harmonic excitation is developed based on the Euler-Bernoulli beam theory, and vibration equations are derived. Anti-resonant frequencies are then solved analytically under clamped-free boundary conditions using the uniform beam solution form approximately. Finally, design and conduct the sine sweep experiment to obtain the frequency response by piezoelectric contact accelerometer and vibration controller. Comparisons among numerical, analytical and experiment results demonstrate that the numerical method for solving anti-resonant frequencies is effective.

In the second part, a matrix method is developed to calculate fixed point frequencies based on the numerical method for anti-resonant frequencies. Furthermore, the optimal spring constant and damping of a damped vibration absorber for the cantilever tapered beam under harmonic force excitation are determined by Den Hartog’s optimization strategy to minimize the resonant amplitude. The influence of absorber location on resonant amplitude of tail point of cantilever tapered beam is also investigated. The asymptotic free end of beam is identified as the optimal absorber location at which we significantly reduce resonant amplitude for tail point and obtain a maximum band width of low amplitude vibrations. Finally, the robustness of the tapered beam with the previously optimized absorber is evaluated by simulating small perturbations in absorber mass , spring and damper . The corresponding small percentage changes in resonant amplitude indicate the robustness of the system.

In the third part, a new pseudo-inverse method is developed to calculate eigenvector sensitivity. The method is validated using a 5-DOF spring system and compared with existing approaches, including the algebraic method and mode superposition method. Results show that the proposed method is effective and computationally efficient. Furthermore, a beam optimization problem is further used for validation, in which a uniform beam is designed to minimize weight while satisfying constraints on the natural frequency and mode shapes. The optimization is performed using the MATLAB fmincon function, both with and without gradient. Incorporating sensitivities computed by the pseudo-inverse method significantly improves convergence speed. Finally, based on the classical mode acceleration method, a residual mode acceleration method is developed to reduce modal truncation errors by incorporating higher-mode contributions through a residual term, which is computed by subtracting the lower mode from the static displacement. Transient analysis of a 4-DOF spring system and a beam system demonstrate the effectiveness of the proposed method compared with results obtained using ode45 and the mode superposition method.

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