Finite difference analysis of torsional vibrations of pretwisted,rotating, cantilever beams with effects of warping

Theoretical natural frequencies of the first three modes of torsional vibration of pre-twisted, rotating cantilever beams are determined for various thickness and aspect ratios. Conclusions concerning individual and collective effects of warping, pretwist, tension-torsion coupling and tennis racket effect (twist-rotational coupling) terms on the natural frequencies are drawn from numerical results obtained by using a finite difference procedure with first order central differences. The relative importance of structural warping, inertial warping, pretwist, tension-torsion and twist-rotational coupling terms is discussed for various rotational speeds. The accuracy of results obtained by using the finite difference approach is verified by a comparison with the exact solution for specialized simple cases of the equation of motion used in this paper.     

Transverse nonlinear vibrations of a circular spinning disk with a varying rotating speed

We analyze the transverse nonlinear vibrations of a rotating flexible disk subjected to a rotating point force with a periodically varying rotating speed. Based on Hamilton’s principle, the nonlinear governing equations of motion (coupled equations among the radial, tangential and transverse displacements) are derived for the rotating flexible disk. When the in-plane inertia is ignored and a stress function is introduced, the three nonlinearly coupled partial differential equations are reduced to two nonlinearly coupled partial differential equations. According to Galerkin’s approach, a four-degree-of-freedom nonlinear system governing the weakly split resonant modes is derived. The resonant case considered here is 1:1:2:2 internal resonance and a critical speed resonance. The primary parametric resonance for the first-order sin and cos modes and the fundamental parametric resonance for the second-order sin and cos modes are also considered. The method of multiple scales is used to obtain a set of eight-dimensional nonlinear averaged equations. Based on the averaged equations, using numerical simulations, the influence of different parameters on the nonlinear vibrations of the spinning disk is detected. It is concluded that there exist complicated nonlinear behaviors including the periodic, period-n and multi-pulse type chaotic motions for the spinning disk with a varying rotating speed. It is also found that among all parameters, the damping and excitation have great influence on the nonlinear responses of the spinning disk with a varying rotating speed.     

Modal analysis of a rotating multi-packet blade system

A modeling method for the modal analysis of a multi-packet blade system undergoing rotational motion is presented in this paper. Blades are idealized as tapered cantilever beams that are fixed to a rotating disc. The stiffness coupling effects between blades due to the flexibilities of the disc and the shroud are modeled with discrete springs. Hybrid deformation variables are employed to derive the equations of motion. To obtain more general information, the equations of motion are transformed into a dimensionless form in which dimensionless parameters are identified. The effects of the dimensionless parameters and the number of packets on the modal characteristics of the rotating multi-packet blade system are investigated with numerical examples.     

Analytical periodic motions in a parametrically excited, nonlinear rotating blade

The stability and bifurcation analyses of periodic motions in a rotating blade subject to a torsional excitation are investigated. For high speed rotations, cubic geometric nonlinearity and gyroscopic effects of the rotating blade are considered. From the Galerkin method, the partial differential equation of the nonlinear rotating blade is simplified to the ordinary differential equations, and periodic motions and stability of the rotating blade are studied by the generalized harmonic balance method. The analytical and numerical results of periodic solutions are compared. The rich dynamics and co-existing periodic solutions of the nonlinear rotating blades are investigated.     

Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Stability of a rotating pinch

The problem of pinch stabilisation is of interest in connection with plasma confinement and has been the subject matter of various investigations in recent years, (Kruskal andSchwarzschild, Taylor, Shafranov). The theoretical investigations in the past have been restricted to static equilibrium configurations. It is of some interest to explore whether an initial motion of plasma tends to make the pinch more, or less stable than when the fluid velocity is zero. Keeping this in mind let us investigate theaxisymmetric stability of an infinitely long cylindrical plasma column of radiusR rotating uniformly with angular velocityΩ about theZ-axis, the fluid being assumed to the incompressible, inviscid, and of infinite electrical conductivity. The plasma carries a uniform axial current of densityj ₀ and a uniform axial magnetic fieldH ₁. The field outside is taken to beH ₂ inZ-direction and a toroidal component which is continuous across the boundary.     

Forced vibrations of a torsional oscillator with Coulomb friction under a periodically varying normal load

The forced vibration response of a single degree of freedom torsional system with Coulomb friction, under a periodically varying normal load, is studied. First, an enhanced multi-term harmonic balance method is developed to calculate nonlinear responses directly in frequency domain; this should be in general applicable to periodically varying nonlinear systems. Second, a pulse width modulated normal load is approximated by a truncated Fourier series with a reasonable number of harmonic components and utilized for case studies. Finally, the effects of duty ratio on nonlinear frequency responses are examined.     

Analysis of torsional vibrations of an ethane molecule

Based on the solution for the Mathieu equation, we obtained wave functions for the internal rotation of an ethane molecule that satisfy the symmetry properties of the group G₃₆. We calculated the frequency of the principal torsional transition (273 cm⁻¹). The types of symmetry of the energy levels and transition probabilities in IR and Raman spectra are determined. We note drawbacks of divisible permutable-inverse nuclear groups and the groups of molecular symmetry associated with their construction, as well as difficulties that appear when equivalent rotations are used. The possibility of avoiding the application of an extended group is indicated. Belarusian State University, 4, F. Skorina Ave., Minsk, 220050, Belarus. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 64, No. 1, pp. 26–31, January–February, 1997.     

A potential model for torsional vibrations of molecules

The Schrödinger equation with a periodic potential which is the sum of two cosine functions admits simple analytic solutions for the ground and the first few excited states of a rotating body. This potential may be used to describe the torsional oscillations of a molecule with m and 2m-fold barriers.     

Torsional vibrations of non-uniform rotating blades with attachment flexibility

The torsional vibrations of non-uniform pretwisted rotating blades are studied by using finite element methods based on both the Rayleigh-Ritz and Galerkin formulations. The apparent differences between the matrices obtained from these formulations are explained and, as obtained by using three different orders of elements, results are presented for blades with flexibly attached roots and for a non-uniform blade representative of a bearingless rotor. A parametric study is carried out to resolve a controversy regarding the relative importance of certain terms in the equations of motion of pretwisted rotating blades. In Appendix I, an exact solution is presented for the torsional vibrations of flexibly attached rotating blades with piecewise constant inertia and elastic properties, which serves as a benchmark solution for the finite element results.     

Variation in the stability of a rotating blade disk with a local crack defect

The effect of a near root local blade crack on the stability of a grouped blade disk is investigated in this paper. A bladed disk comprised of periodically shrouded blades is used to simulate the coupled periodic structure. The blade crack is modeled using the local flexibility with coupling terms. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations in the rotating blades has also been considered. Using the Galerkin's method, the imperturbation equations of a bladed disk in which one of the blades is cracked, subject to fluctuations in the rotation speed, can be derived. Employing the multiple scales method, the boundaries of the instability zones in the mistuned turbo blade system are approximated. Numerical results indicate that an additional unstable zone is introduced near the localization frequency and the regions of unstable zones are varied with the crack size and fluctuations in disk speed.     

Linear and nonlinear vibrations analysis of viscoelastic sandwich beams

In this paper, numerical models are proposed for linear and nonlinear vibrations analyses of viscoelastic sandwich beams with various viscoelastic frequency dependent laws using the finite element based solution. Real and various complex eigenmodes approaches are investigated as Galerkin bases. Based on harmonic balance method, simplified and general approaches are developed for nonlinear vibration analysis. Analytical frequency-amplitude and phase-amplitude relationships are elaborated based on the numerically computed complex eigenmodes. The equivalent loss factors and frequencies as well as the forced harmonic response and phase curves are performed for sandwich beams with various boundary conditions and frequency dependent viscoelastic laws.     

Visualization of aerodynamic noise source around a rotating fan blade

The purpose of this study is to understand the aerodynamic noise source distribution around a rotating fan blade by measuring the noise signal and velocity field around the blade. The local noise-level distribution over the fan blade is measured by microphone arrays, and the flow field is visualized by smoke and phase-averaged PIV measurement. The noise source distribution is examined by cross-correlation analysis between noise signal and velocity fluctuation. It is found that the noise source is located near the rotating fan blade, especially around leading and trailing edges. The separation and reattachment of flow are observed near the leading edge, and the tip vortices and vortex shedding are found near the trailing edge. The cross-correlation distribution of the noise signal and the radial velocity fluctuation shows large magnitude in the correlated regions, which indicates the noise generation by the formation of vortex structure around the blade.     

Coupled longitudinal and bending vibrations of a rotating shaft with an open crack

The coupling of longitudinal and bending vibrations of a rotating shaft, due to an open transverse surface crack is investigated. The assumption of the open crack leads to a system with behaviour similar to that of a rotor with dissimilar moments of inertia along two perpendicular directions. The local flexibility due to the presence of the crack can be represented by way of a 6×6 matrix for six degrees of freedom in a short shaft element which includes the crack. This matrix has off-diagonal terms which cause coupling along the directions which are indicated by these terms. Here shear is not considered and three degrees of freedom are used: bending in the two main directions and extension. This leads to a 3×3 stiffness matrix with coupling terms. The undamped free and forced coupled vibration are first considered. The coupling is investigated and the effects of unbalance and gravity are examined. Then damped coupled vibration is considered for free and forced vibration. The existence of coupling between longitudinal and bending vibration due to the crack is a very useful property which, together with the sub-critical resonance due to crack, can form a basis for crack identification in rotating shafts. New and interesting phenomena of coupled transverse and longitudinal motion are presented and discussed.     

Forced torsional vibrations of a cylindrical rod connected to an elastic half-space

An analysis is presented of the forced torsional vibrations of a cylindrical rod connected to an elastic half-space under the condition that the circumferential displacement at the free end of the rod, where the disturbing moment is applied, varies proportionally with the distance from the rod axis. Both the Pochhammer-Chree and elementary theory are utilized. The response curve of the rotational amplitude at the free end of the rod, obtained from the Pochhammer-Chree theory, and that obtained from the elementary theory almost coincide with each other except in the case where the rod and the half-space are of the same material. The amplitude attenuation is largest when the rod and the half-space are of the same material. The maximum values of half-space displacement distribution at the interface lie within the rod edge.     

Forced vibrations of oscillators with a purely nonlinear power-form restoring force

Harmonically excited oscillators with non-negative real-power geometric nonlinearities and no linear term in the restoring force are considered. Perturbation approaches are developed for the cases of weak and strong nonlinearity. Frequency-amplitude equations are derived for an arbitrary value of the non-negative real power of the restoring force as well as analytical expressions for the steady-state response at the frequency of excitation. It is shown that the system response is of a softening type for the powers lower than unity and of a hardening type for the powers higher than unity. Frequency-response curves of the antisymmetric (constant force) oscillator, the restoring force of which has a zero power, are also discussed. Comparisons with numerical results are presented for confirmation of the analytical results obtained.     

A review of rotating blade noise technology

Rotating blade noise is a topic of wide concern because it is the source of a variety of noise problems. These concerns range from safety, on the one hand, to community acceptability on the other.