12Vibration of Circular Rings and Curved Beams

12.1 INTRODUCTION

The problems of vibration of circular rings and curved beams (or rods) find application in several practical cases. The vibration of circular rings is encountered in an investigation of the frequencies and dynamic response of ring‐stiffened cylinders, such as those encountered in airplane fuselages, circular machine parts, such as gears and pulleys, rotating machines, and stators of electrical machines. The vibration of a curved rod can be categorized into four types when the centerline of an undeformed rod is a plane curve and its plane is a principal plane of the rod at each point. In the first type, flexural vibrations take place in the plane of the ring without undergoing any extension of the centerline of the ring. In the second type, flexural vibrations, involving both displacement at right angles to the plane of the ring and twist, take place. In the third type, the curved rod or ring vibrates in modes similar to the torsional vibrations of a straight rod. In the fourth type, the ring possesses modes of vibration similar to the extensional vibration of a straight rod. It is assumed that the undeformed centerline of the ring has a radius R, the cross‐section of the ring is uniform, and the cross‐sectional dimensions of the ring are small (for a thin ring) compared to the radius of the centerline of the ring. The vibration of curved beams is important in the study of the dynamic behavior of arches.

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