Dynamic Modelling and Free Vibration Characteristics Analysis of Rotating Beams

Turbine blades are ideally modeled as cantilever beams on a disc rotating at a constant angular velocity. A study is made to understand the dynamic relationships between a rotating cantilever beam and various factors like hub radius, rotation speed, and slenderness ratio in in-plane vibration (Chordwise motion) and out-of-plane vibration (Flapwise motion). Hub is assumed to be rigid in the study. Using Hamilton’s principle, governing differential equations of movement for free vibration analysis of Euler-Bernoulli beam (EB) and Timoshenko (TB) beam under rotation are derived. The effects of the Gyroscopic couple are taken into account in the equations. The beam model is discretized using the Finite element approach. Derived differential equations are transformed into dimensionless quantities in which dimensionless parameters are identified. Under rotation, it is observed that the natural frequencies increase with the increase in rotational speed for both flapwise and chordwise motions of the beam. An interesting phenomenon is observed in the chordwise motion results, where Natural frequencies veer off at certain rotational speeds and certain modes. Slenderness ratios also influence this phenomenon, which shifts the veer-off region and the tuned angular frequency. Numerical results are obtained for different rotational speeds with various hub radius ratios, and it was observed that hub radius directly influences the natural frequencies of the rotating uniform cantilever beam. A thorough study on the influence of the slenderness ratio showed that, for lower slenderness ratio, frequency veering region occurs at the fundamental natural frequency, but for higher slenderness ratios’ there is a shift in frequency veering region for higher modes.

INVESTIGATION OF THE PLANE VIBRATION OF AN ELASTIC PLATE WITH A KINEMATIC VIBRATION PROTECTION SYSTEM

Creation of vibro-protective devices on rolling contact bearings is widely spread in transportation technology and seismic protection. In this work, mathematical modeling of the oscillation movements of the elastic plate will be considered. The equations of motion for elastic plate on vibration supports bounded by high-order rotation surfaces by the Ostrogradsky-Hamilton principle are obtained. The natural frequencies of elastic plate are determined. It is established that the value of the natural frequencies of elastic plate decreases with increasing height and increases with the width of the bases. The ratio of the natural frequency of the second form to the natural frequency of the first form does not depend on the geometrical parameters of the plate.