Ye Tao

Date of Graduation


Document Type


Degree Type



Statler College of Engineering and Mineral Resources


Mechanical and Aerospace Engineering

Committee Chair

Hung-Liang Roger Chen

Committee Co-Chair

Marvin Cheng

Committee Member

Abdul-Aziz Omar

Committee Member

Hong-Jyh Yang


In real life, boundary conditions of most structural members are neither totally fixed nor completely free. It is crucial to study the effect of boundary conditions on beam vibrations . This thesis focuses on deriving analytical solutions to natural frequencies and mode shapes for Euler-Bernoulli Beams and Timoshenko Beams with various boundary conditions under free vibrations. In addition, Green's function method is employed to solve the close-form expression of deflection curves for forced vibrations of Euler-Bernoulli Beams and Timoshenko Beams.;A direct and general beam model is set up with two different vertical spring constraints kT1, k T2 and two different rotational spring constraints kR1, kR2 attached at the ends of the beam. These end constraints can represent various combinations of boundary conditions of the beam by varying the spring constraints. A general solution for the Timoshenko beam with this various boundary conditions is derived, and to the best of our knowledge, this solution is not available in the literature. Numerical examples are presented to illustrate the effects of the end constraints on the natural frequencies and mode shapes between Euler-Bernoulli beams and Timoshenko beam. The results show that Euler-Bernoulli beams have higher natural frequencies than Timoshenko beams at different modes. The ratio of the natural frequencies for Timoshenko beams to the natural frequency for Euler-Bernoulli beams decreases at higher modes. Natural frequencies at lower modes are more sensitive to boundary constraints than natural frequencies at higher modes.