Vibration of a rotating beam with tip mass

The vibration frequency of a rotating beam with tip mass is investigated. The finite element method is used, a third order polynomial being assumed for the variation of the lateral displacement. The effects of the root radius, the setting angle and the tip mass are incorporated into the finite element model. The results are compared with results from previous authors utilizing Myklestad and extended Galerkin methods. The results show that the setting angle has a significant effect on the first mode frequencies but not on the high frequencies. The tip mass tends to depress the frequencies at low speeds of rotation but it tends to increase the frequencies at high speeds of rotation. The results of this work have applications in wind turbine rotors, helicopter rotors, etc., and the method used here can be extended to investigate the vibration frequency of flexible blade auto cooling fans.     

Vibration frequency of a curved blade with weighted edge

The natural vibration frequencies and mode shapes of a curved cylindrical blade with a weighted edge are investigated. A finite element method is used, in which curved cylindrical shell finite elements are utilized to model the blade. The weighted edge is modelled as a beam with its stiffness and mass added into the stiffness and mass of the blade. Vibration frequencies and mode shapes for blades with different boundary conditions and with different radii of curvature are obtained. Finite element results are compared with experimental results.     

Vibration analysis of a new curved beam element

The common practice in developing a locking-free curved beam element is to ensure that its interpolation functions of displacement explicitly satisfy the inextensible bending mode condition for the membrane locking-free instead of the rigid body modes. In this paper, we study the impact of this practice on the dynamic characteristics of a finite element by conducting vibration analysis using our newly developed three-node locking-free curved beam element. In this case, the inextensible bending mode condition is satisfied explicitly, while the rigid body modes are satisfied implicitly to 4th-order accuracy. Numerical and experimental examples show that with the newly developed curved beam element, developed by using the implicit representation of a rigid body mode condition, it is possible to recover the rigid body modes of curved beams with low and medium slenderness ratios. This is even true for cases involving a half-circular element and the vibration of the curved beam is predicted with high accuracy.     

Vibration frequencies for a non-uniform beam with end mass

This study deals with the determination of natural frequencies of a non-uniform cantilever beam which carries a concentrated mass at the free end. The effect of the rotary inertia of the end mass has been included. Numerical results for the first five eigenfrequencies are presented for a wide range of values of the beam dimensions and the concentrated mass.     

Comments on “vibration of a rotating beam with tip mass”

In diagnostic applications of machinery vibration or noise data it is convenient to describe the whole process by a set of numbers called discriminants. Five discriminants can be constructed and measured for every vibroacoustical process. Two of them have dimensional natures and give information about process amplitude and frequency. Two others, dimensionless, give information about the amplitude and spectral spread of the process. The last one, also dimensionless, characterizes the time fluctuations of the process, and can be used to detect instability of a running machine. Numerical and experimental examples are presented.     

Steering a multi-MeV positron beam with a curved crystal

We observe positron bending by a crystal lattice, presumably being guided by a channeling phenomenon, deflecting the beam by about 10 milliradian over a length of 1 mm of silicon. This technique may lead to the use of the channeling effect for steering particle beams at energies below 1 GeV, for the purpose of producing beams of low emittance with enhanced stability for medical and biological applications. The text was submitted by the authors in English.     

Free vibration of a rotating non-uniform beam with arbitrary pretwist, an elastically restrained root and a tip mass

The governing differential equations for the coupled bending-bending vibration of a rotating beam with a tip mass, arbitrary pretwist, an elastically restrained root, and rotating at a constant angular velocity, are derived by using Hamilton's principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. The influence of the tip mass, the rotary inertia of the tip mass, the rotating speed, the geometric parameter of the cross-section of the beam, the setting angle, and the pretwist parameters on the natural frequencies are investigated. The difference between the effects of the setting angle on the natural frequencies of pretwisted and unpretwisted beams is revealed.     

Vibration analysis of a beam with PCLD Patch

An analytical approach for vibration response analysis of a beam with single passive constrained layer damping (PCLD) patch is presented. The governing equation of motion of the beam is firstly derived on the basis of an energy approach and the Lagrange equation. The noval contribution is that a third admissible function is introduced to represent the longitudinal displacements of the constraining layer in the PCLD patch when the assumed-modes method is applied for discretizing the governing equation. In conventional analytical approaches, only two admissible functions are used together with a longitudinal static equilibrium equation of a section of base beam or constraining layer. Comparison of the computational results from the proposed analytical approach and the conventional analytical approach as well as a commercial FEM code reveals that the proposed analytical approach can describe the vibration responses of the damped beam more accurately for commonly used viscoelastic material (VEM) layer in the PCLD patch while the conventional analytical approach, in general, overestimates the damping effects of the PCLD patch. The advantages and disadvantages of the proposed analytical approach and conventional analytical approach are discussed through some case studies.     

Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam.     

On the approximate determination of the fundamental frequency of a restrained cantilever beam carrying a tip heavy body

This paper deals with the approximate determination of the fundamental bending eigenfrequency of a restrained cantilevered beam carrying a tip heavy body by the combined use of the Dunkerley's and Southwell's methods. A simple algebraic expression is introduced to calculate the fundamental frequency with satisfactory accuracy for preliminary design purposes.