Longitudinal vibrations of a beam with a moving load

International Applied Mechanics -     

Natural transverse vibrations of a rotating rod

We study the natural transverse vibration frequencies and modes of a rod rotating about an axis fixed at an end of the rod. The cases of low, moderately high, and asymptotically high angular velocities are considered. The case of a homogeneous rod with clamped left and free right end is considered in detail. A new constructive algorithm based on the notion of “sagittary function” is used to find the dependences of the natural frequencies and mode shapes on the angular velocity for lower vibration modes. We establish evolution to the model corresponding to vibrations of a rapidly rotating thread subjected to the centrifugal inertial forces. It is shown that the natural frequencies grow practically linearly with increasing angular rotation velocity. The results obtained can be of interest in technical applications, e.g., when studying vibrations of sensor elements in high-precision instruments or of rapidly rotating elongated mechanism elements (turbine or propeller blades, etc).     

Determination of the stress intensity factors at the ends of two collinear transverse shear cracks under steady vibrations

International Applied Mechanics -     

Longitudinal vibrations of arbitrary non-uniform rods

Free and steady-state forced longitudinal vibrations of non-uniform rods are investigated by an iteration method, which results in a series solution. The series obtained are convergent and linearly independent. Its convergence is verified by convergence tests, its linear independence confirmed by the nonzero value of the corresponding Wronski determinant. Then, the solution obtained is an exact one reducible to a classical solution for the case of uniform rods. In order to verify the method, two examples are presented as an application of the proposed method. The results obtained are equivalent to the method in literature. In contrast to the proposed method capable of dealing with arbitrary non-uniform rods in principle, the method in literature is confined to work on special cases.     

Equation of small transverse vibrations of an elastic pipeline filled with a transported fluid

The integro-differential equation of small transverse vibrations of a rectilinear elastic pipeline filled with a transported fluid is obtained. The pipeline vibrations are described in the linear setting in the beam approximation. The mutual dynamic influence of motions of the pipeline and the filling fluid is taken into account. A complete trigonometric series method is presented for solving problems with various boundary and initial conditions for the pipeline deflection.     

Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks

The propagation of time-harmonic waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x = md, y = lh(m, l = 0, ±1, ±2,…). The wave motion is polarized in the z-direction and propagates in the y-direction (normal to the cracks). The theory of Floquet or Bloch waves, together with an appropriate Green's function and the condition of vanishing traction on the crack faces leads to a system of singular integral equations, which provides the basis for the derivation of an exact dispersion equation. Numerical results are presented for the wave number as a function of the frequency. The frequency spectrum shows a pattern of passing and stopping bands. The exact results are compared with the frequency spectrum according to a simplified theory which considers the arrays of collinear cracks in the planes y = lh (l= 0 ±1, ±2,…) as planes of homogeneous transmission and reflection. Good agreement is observed between exact and approximate results.     

含基体横向损伤的黏弹性板的蠕变后屈曲分析

基于Schapery三维黏弹性损伤本构关系,引入沈为和Kachanov损伤演化方程,建立了基体横向损伤的纤维单一方向铺设黏弹性板的损伤模型;应用von Karman板理论,导出了考虑损伤效应的具初始挠度的纤维单一方向铺设黏弹性矩形板的非线性压屈平衡方程. 对未知变量在空间上采用差分法离散,时间上采用增量算法和Newton-Cotes积分法离散,控制方程被迭代求解. 算例中讨论了损伤以及有关参数对黏弹性复合材料板后屈曲行为的影响,且与已有文献的结果进行了比较. 数值结果表明：随着外载荷或者初始挠度的增大,板后屈曲趋于稳定时的挠度就愈大,损伤的影响愈明显;而随着长宽比的增大,板后屈曲趋于稳定时的挠度愈小,损伤的影响却随之增大.     

General transverse vibrations equations for a circular cylindrical viscoelastic shell

Moscow Structural Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 26, No. 4, pp. 41–49, April, 1990.     

Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass

In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlin- ear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of mul- tiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequen- cies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated.