The natural frequency ratio between Timoshenko and Euler effect by variation of parameter

Applied energy principle analysis natural frequency equations of Euler beam and Timoshenko respectively. The expression of natural frequency is calculated by the separation of variables. Consider the natural frequency ratio between the Euler beam and Timoshenko one to analyze the relationship between geometric parameters and frequency ratio that effect by parameters of frequency. Calculations show that the relationship between frequency ratio and geometric parameters present non-linear parabola characteristic. When the increment of geometric parameters is identical, the frequency value of the beam was lowered by both transverse shear effect and rotary inertia whereby non-linear characteristics are mainly determined by the transverse shear effect.