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.     

A sensitivity analysis of the effects of interconnection joint size,flexibility, and inertia on the natural frequencies of Timoshenko frames

The free vibrations of frame structures are influenced by the geometry, stiffness, and inertia of interconnection joints. The effects of generalized joint properties on the natural frequencies and mode shapes are studied for a wide range of natural frequencies by modeling the structure as a Timoshenko continuous system with discretized joints. Dynamic slope-deflection equations are used in the analysis, adapted to the boundary conditions imposed by joints with axial length, axial and rotary stiffness, and inertia. Beam/column axial deformation is also included. Frequency curves are presented for a wide range of beam/column and joint properties to establish the relative importance of model parameters on system free vibrations.     

NON-LINEAR TRANSVERSE VIBRATIONS OF A SIMPLY SUPPORTED BEAM CARRYING CONCENTRATED MASSES

An Euler-Bernoulli beam carrying concentrated masses is considered to be a beam-mass system. The beam is simply supported at both ends. The non-linear equations of motion are derived including stretching due to immovable end conditions. The stretching introduces cubic non-linearities into the equations. Forcing and damping terms are also included. Exact solutions for the natural frequencies are given for the linear problem. For the non-linear problem, an approximate solution using a perturbation method is searched. Non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effect of the positions, magnitudes and number of the masses are investigated.     

Analytical and experimental determination of natural frequencies of double-span beams of non-uniform cross-section and with elastic end constraints

This study deals with the analytical determinations of the fundamental natural frequency of transverse vibrations of a double-span beam with a discontinuous moment of inertia and both ends elastically restrained against rotation. The presence of masses and axial forces is also considered. Comparison of analytical and experimental results is presented as a function of the governing geometric and mechanical parameters. Good agreement is found in all cases from an engineering point of view.     

Vibrations of a timoshenko beam clamped at one end and carrying a finite mass at the other

The present paper deals with an exact solution of the title problem. Modal shapes and natural frequency coefficients are determined for a significant range of the mechanical and geometric parameters that come into play. When the parameter I/AL² (where I is cross-sectional moment of inertia, A is cross-sectional area, and L beam length) approaches zero, the beam dynamic characteristics agree with values already available in the open literature.     

Rapporteur's report,session 3: Other sources of railway noise

A method of finding the flexural and torsional normal modes of beams which have straight stiffness axes is given. The Lanzcos method of minimized iterations, based on the integral equation of beam vibrations, is used to obtain intermediate modes with the distributions along the beam of its mass and rigidity as data. The inertia matrix appropriate to the intermediate modes is theoretically tridiagonal and the stiffness matrix is unit. The dominant eigenvalues and vectors of the inertia matrix give very good approximations to the frequencies and shapes of the graver normal modes. Results of test calculations are given.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Coupled transverse and axial vibratory behaviour of cracked beam with end mass and rotary inertia

In this paper the vibrational behaviour of a cracked cantilever beam carrying end mass and rotary inertia is investigated. The transverse and axial vibrations of the beam are coupled through the crack model. The values of the ratio between the cracked and uncracked beam natural frequencies, the frequency ratio, are examined and are shown to follow well-defined trends with respect to the crack parameters and end mass and rotary inertia. However, the coupling between the transverse and axial vibrations is shown to be weak for the first two modes for moderate values of crack depth ratio. High crack depth ratios appear to increase the coupling effects. Low aspect ratios are expected to show strong coupling effects and further investigation is recommended using Timoshenko beam theory.     

Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory

A general procedure for the determination of the natural frequencies and buckling load for a set of beam system under compressive axial loading is investigated using Timoshenko and high-order shear deformation theory. It is assumed that the set beams of the system are simply supported and continuously joined by a Winkler elastic layer. The model of beam includes the effects of axial loading, shear deformation and rotary inertia. In the special case of identical beams, explicit expressions for the natural frequencies and the critical buckling load are determined using a trigonometric method. The influences of the compressive axial loading and the number of beams in the system on the natural frequencies and the critical buckling load are discussed. These results are of considerable practical interest and have wide application in engineering practice of frameworks.     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

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.     

Modal coupling effects in the free vibration of elastically interconnected beams

The problem of free vibration of a uniform beam elastically interconnected to a cantilevered beam, representing an idealized launch vehicle aeroelastic model in a wind tunnel, is studied. With elementary beam theory modelling, numerical results are obtained for the frequencies, mode shapes and the generalized modal mass of this elastically coupled system, for a range of values of the spring constants and cantilevered beam stiffness and inertia values. The study shows that when the linear springs are supported at the nodal points corresponding to the first free-free beam mode, the modal interaction comes primarily from the rotational spring stiffness. The effect of the linear spring stiffness on the higher model modes is also found to be marginal. However, the rotational stiffness has a significant effect on all the predominantly model modes as it couples the model deformations and the support rod deformations. The study also shows that through the variations in the stiffness or the inertia values of the cantilever beam affect only the predominantly cantilever modes, these variations become important because of the fact that the cantilevered support rod frequencies may come close to, or even cross over, the predominantly model mode frequencies. The results also bring out the fact that shifting of the support points away from the first mode nodal points has a maximum effect only on the first model mode.     

Geometrical influence of a deposited particle on the performance of bridged carbon nanotube-based mass detectors

A nonlocal continuum-based model is derived to simulate the dynamic behavior of bridged carbon nanotube-based nano-scale mass detectors. The carbon nanotube (CNT) is modeled as an elastic Euler-Bernoulli beam considering von-Kármán type geometric nonlinearity. In order to achieve better accuracy in characterization of the CNTs, the geometrical properties of an attached nano-scale particle are introduced into the model by its moment of inertia with respect to the central axis of the beam. The inter-atomic long-range interactions within the structure of the CNT are incorporated into the model using Eringen's nonlocal elastic field theory. In this model, the mass can be deposited along an arbitrary length of the CNT. After deriving the full nonlinear equations of motion, the natural frequencies and corresponding mode shapes are extracted based on a linear eigenvalue problem analysis. The results show that the geometry of the attached particle has a significant impact on the dynamic behavior of the CNT-based mechanical resonator, especially, for those with small aspect ratios. The developed model and analysis are beneficial for nano-scale mass identification when a CNT-based mechanical resonator is utilized as a small-scale bio-mass sensor and the deposited particles are those, such as proteins, enzymes, cancer cells, DNA and other nano-scale biological objects with different and complex shapes.     

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

Analytical formulation and evaluation for free vibration of naturally curved and twisted beams

An analytical study for free vibration of naturally curved and twisted beams with uniform cross-sectional shapes is carried out using spatial curved beam theory based on the Washizu's static model. In the governing equations of motion of the beams, all displacement functions and the generalized warping coordinate are defined at the centroid axis and also the effects of rotary inertia, transverse shear deformations and torsion-related warping are included in the proposed model. Explicit analytical expressions are derived for the vibrating mode shapes of a curved, bending-torsional-shearing coupled beam under clamped-clamped boundary condition with the help of symbolic computing package Mathematica, and a process of searching is used to determine the natural frequencies. Comparisons of the present results with the FEM results using beam elements in ANSYS code show good accuracy in computation and validity of the model. Further, the present model is used for cylindrical helical springs with circular cross-section fixed at both ends, and the results indicate that the natural frequencies agree well with the theoretical and experimental results available.     

Flexural resonance vibrations of piezoelectric ceramic tubes in Besocke-style scanners

Flexural resonance vibrations of piezoelectric ceramic tubes in Besocke-style scanners with nanometer resolution are studied by using an electro-mechanical coupling Timoshenko beam model.Meanwhile,the effects of friction,the first moment,and moment of inertia induced by mass loads are considered.The predicted resonance frequencies of the ceramic tubes are sensitive to not only the mechanical parameters of the scanners,but also the friction acting on the attached shaking ball and corresponding bending moment on the tubes.The theoretical results are in excellent agreement with the related experimental measurements.This model and corresponding results are applicable for optimizing the structures and performances of the scanners.     

幂函数形式连续变截面梁振动的弯曲固有频率分析*

本文使用微分方程解析法求解变截面梁固有频率。首先,建立变截面梁模型,其中截面面积和惯性矩均按幂次函数变化。得到变截面梁自由振动时挠度的解析表达式,并获得不同边界条件下梁弯曲振动的固有频率方程。其中惯性矩所对应幂指数与截面面积的幂指数的差值为4时,可得自振频率方程的精确形式;而幂指数差值不等于4时,给出近似解法。其次,对4种具体的变截面梁求解不同边界下的自振频率,并与瑞利-里兹法所得的自振频率解比较。验证精确解法结果的正确性,并发现近似解法结果的相对偏差在5%以内。该解析方法较瑞利-里兹法具有能快速求解的特点,且易于分析截面参数对梁固有频率的影响。由算例可得,边界和其他参数不变时,梁的同阶次无量纲自振频率随着幂次指数的增加而增加。几何参数中仅截面形状参数改变时,随着形状参数的增加,梁的同阶次无量纲自振频率随之减小,但固定-自由梁的第一阶自振频率除外。     

Normal modes of a continuous system with quadratic and cubic non-linearities

A power series solution is presented for the free vibrations of simply supported beams resting on elastic foundation having quadratic and cubic non-linearities. The time-dependence is assumed harmonic and the problem is posed as a non-linear eigenvalue problem. The spatial variable is transformed into an independent variable that satisfies the boundary conditions. This permits a power series expansion of the beam motion in terms of the new variable. A recurrence relation is obtained from the governing equation and used in conjunction with the Rayleigh energy principle to compute the natural frequencies. The results show that, for a first order approximation, only the lower frequencies and first mode shape are significantly affected by the cubic non-linearity.     

Dynamic analysis of Bernoulli-Euler piezoelectric nanobeam with electrostatic force

The effect of electrostatic force on the dynamic response of a Bernoulli-Euler piezoelectric nanobeam is analyzed in this paper.The governing equations with the electrostatic stress are derived based on a variational principle.Static bending problem of simply supported and cantilever beam is considered.The influence of the electrostatic force on the first four natural frequencies is discussed.It is shown that when the beam thickness is small,the effect of the electrostatic force is significant.When the beam thickness is large,the electrostatic force is insignificant and can be neglected.The results also indicate that one can adjust the natural frequency of a nanobeam by applying appropriate voltage.     

MODEL-ORDER REDUCTION AND PASS-BAND BASED CALCULATIONS FOR DISORDERED PERIODIC STRUCTURES

This paper is concerned with the dynamics of disordered periodic structures. The free vibration problem is considered. A method akin to the Rayleigh method is presented. This method is particularly suitable for the study of periodic structures as it exploits the nominal periodicity leading to an approximation that greatly reduces the order of the model. The method is used to calculate the natural frequencies and mode shapes for a pass-band by treating the unknown phases between the nominally identical bays as the generalized co-ordinates of the problem. An illustrative example of a cyclically coupled beam model is presented. In spite of a very large reduction in the computational effort, the results obtained are very accurate both for frequencies and mode shapes even when strong mode localization is observed. To test the performance of the proposed approximation further, a situation where two pass-bands are brought close to each other is considered (a coupled beam model having inherent bending-torsion coupling). The method presented here is general in its formulation and has the potential of being used for more complex geometries.