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Resonances of Compact Tapered Inhomogeneous Axially Loaded Shafts

机译：紧凑型锥形不均匀轴向装载轴的共振

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An important technical area is the bending of shafts subjected to an axial load. These shafts could be tapered and made of materials with spatially varying properties (Functionally Graded Material - FGM). Previously the transverse vibrations of such shafts were investigated by the authors assuming the shafts had large slenderness ratios so that Euler-Bernoulli theory could be employed. Here compact shafts are treated necessitating the use of Timoshenko beam theory. For constant axial load case analysis of the effects of both FGMs and tapering on frequencies, the value of the compressive load is chosen to be 80% of the smallest critical (buckling) value for the shafts considered. The equations of motion give rise to two coupled differential equations with variable coefficients. These equations in general do not have analytic solutions and numerical methods must be employed (here using MAPLE) to find the natural frequencies. MAPLE's built-in solver for two-point boundary value problems does not directly provide the eigenvalues. The strategy used is to solve a harmonically forced motion problem. On varying the excitation frequency and observing the mid-span deflection the resonance frequency can be found noting where a change in sign occurs. For example, results for FGM cylindrical and tapered shafts show that for a compact cylindrical beam the resonant frequency obtained differs from the Euler-Bernoulli prediction by 11%, and for a tapered beam by 12%, indicating that the effects of compactness can be significant. Since Timoshenko theory requires a value for the shear coefficient, which is not readily available for FGM beams, a sensitivity study is conducted in order to access the effect of the value on the results. Some effects of axial load variations on frequencies are also presented.
机译：重要的技术领域是经过轴向载荷的轴的弯曲。这些轴可以是锥形的，并且由具有空间变化性质（功能梯度的材料 - FGM）的材料制成。以前，假设轴具有大的细长比，因此研究了这种轴的横向振动，因此可以采用欧拉 - 伯努利理论。在这里，紧密轴需要使用Timoshenko光束理论。对于恒定的轴向载荷壳体分析FGMS和逐渐变细的频率的效果，压缩载荷的值被选择为所考虑的轴的最小临界（屈曲）值的80％。运动方程导致具有可变系数的两个耦合微分方程。这些方程一般没有分析解决方案，必须使用数值方法（这里使用枫树）来找到自然频率。枫木的内置求解器，用于两点边值问题不直接提供特征值。所使用的策略是解决谐波强迫的运动问题。在改变激发频率并观察中跨横向偏转时，可以找到谐振频率注意，其中发生了符号的变化。例如，FGM圆柱形和锥形轴的结果表明，对于紧凑的圆柱形光束，所获得的谐振频率与欧拉-Bernoulli预测的谐振频率不同11％，并且对于锥形光束×12％，表明紧凑性的效果可能是显着的。由于Timoshenko理论需要剪切系数的值，因此不容易获得FGM光束，因此进行灵敏度研究，以便访问该值对结果的影响。还介绍了轴向载荷变化对频率的一些影响。

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