A New Approach to Compute Natural Frequencies and Mode Shapes of Non-Uniform Continuous Bar, Circular Shaft and Beam Vibration

The wave equation governing longitudinal vibration of a bar and torsional vibration of a circular shaft, and the Euler-Bernoulli equation governing transverse vibration of a beam were developed in the eighteenth century. Natural frequencies and mode shapes are easily obtained for uniform or constant spatial parameters (cross sectional area, material property and mass distribution). But, real engineering structures seldom have constant parameters. For non-uniform continuous structure, a large number of papers have been written for more than 100 years since the publication of Kirchhoff’s memoir in 1882. There are analytical solutions only in few cases, and there are approximate numerical methods to deal with other (almost all) cases, most notably Stodola, Holzer and Myklestad methods in addition to Rayleigh-Ritz and finite element methods. This paper presents a novel approach to compute natural frequencies and mode shapes for arbitrary variations of spatial parameters on the basis of linear time-varying system theory. The advantage of this approach is that now it can be claimed that “almost” closed-form solutions are available to find natural frequencies and mode shapes of any non-uniform, linear and one-dimensional continuous structure.

Improvement on Straightness of Micro-Motor Shaft Considering Stress Concentration Caused by the Grooves

Motor shaft is one of the most important parts of motor, which is widely used in industrial field. The straightness of the shaft has a potentially significant effect on motor performance, which is determined by the straightening process in the manufacturing process of the shaft. Considering the grooved characteristics on the micro-motor shaft and using the classical material mechanics principle, the stress and deformation models of the circular shaft with the grooves during the straightening process are established in this paper. By deriving the straighttening stroke-initial deflection model, an accurate and practical relationship between the straightening stroke and the initial deflection was provided. In order to verify the theoretical straightening model, some grooved shaft specimens are designed and manufactured for the straightening experiments. Through the simulation and experimental research, the straightening stroke-initial deflection model for the micro-motor shaft is compared and analyzed, and the correctness and feasibility of the model is verified.