Prediction of vibration of rotating damped beams with arbitrary pretwist

A solution procedure for the bending–bending vibration of a rotating damped beam with arbitrary pretwist and an elastically restrained root is derived. The viscous damping is assumed to be proportional to the distributed mass. The general complex system is divided into two subsystems. The physical meanings of the subsystems are studied. The exact complex frequency relations between two viscously damped beams with arbitrary pretwist and elastic root are revealed. The underdamping, critical damping and overdamping systems are analyzed. Moreover, the influence of the parameters on the decay rate, the natural frequencies, the critical damping, and the phenomenon of divergence instability are investigated.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems

The situation of structural elements supporting motors or engines attached to them is usual in technological applications. The operation of machine may introduce severe dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli–Euler single-span beams carrying a number of spring-mass system and Bernoulli–Euler multi-span beams carrying multiple spring-mass systems are plenty, but that of Timoshenko multi-span beams carrying multiple spring-mass systems with axial force effect is fewer. This paper aims at determining the exact solutions for the first five natural frequencies and mode shapes of a Timoshenko multi-span beam subjected to the axial force. The model allows analyzing the influence of the shear and axial force effects and spring-mass systems on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The calculated natural frequencies of Timoshenko multi-span beam by using secant method for non-trivial solution for the different values of axial force are given in tables. The mode shapes are presented in graphs.     

智能悬臂梁横向振动的行波解

介绍智能梁的行波建模方法及其横向振动固有频率的计算方法.行波建模方法主要步骤包括,①求出Timoshenko梁横向振动方程的谐波解.②根据弯曲波的传播特性,给出弯曲波的传递关系;在智能梁上截面尺寸改变处和边界处,根据其连续条件和平衡条件,给出波的反射关系和透射关系.③通过联立智能梁内所有的传递、反射、透射关系,求得整体智能梁的特征方程.文中以智能悬臂梁为算例,通过解析法(包括Timoshenko梁模型和Euler-Bernoulli梁模型)与有限元法得到横向振动频率的比较,验证行波建模方法的有效性.此外,为考虑压电片材料对智能梁整体模型的影响,引入等效弹性模量.     

Parametric instability of spinning twisted Timoshenko beams under compressive axial pulsating loads

The parametric instability on lateral bending vibrations of a spinning pretwisted beam under compressive axial pulsating forces is investigated. Equations of motion of the twisted beam are derived in the spinning twist coordinate frame using the Timoshenko beam theory and applying the Hamilton’s principle. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations with gyroscopic terms. A set of second-order ordinary differential equations with periodic coefficients of Mathieu-Hill type is formed to obtain the boundary frequencies of instability regimes. The influence of twist angle, spinning speed, static component of axial force, aspect ratio and restraint condition on the instability regions of the spinning twisted Timoshenko beam is discussed.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

Vibration analysis of rectangular Mindlin plates with internal line supports using static Timoshenko beam functions

In this paper, the vibration characteristics of rectangular Mindlin plates with internal line supports in one or two directions are studied by using the Rayleigh–Ritz method. The static Timoshenko beam functions are employed as admissible functions, which are composed of a set of transverse deflection functions and a set of rotation-angle functions due to bending. The static Timoshenko beam functions are derived from a point-supported strip of unit width taken from the particular plate under consideration in a direction parallel to the edge of the plate and acted upon by a series of static sinusoidal loads distributed along the length of the strip. It can be seen that the suggested approach is very simple mathematically, and each of the beam functions is only a sine or cosine function plus a polynomial function of not more than the third order. A unified program can be easily prepared, because the changes in boundary conditions, number and locations of internal line supports and thickness ratio of the plate will result in a corresponding change of only the coefficients of the polynomials. Both high accuracy and low computational cost have been verified by convergence and comparison studies. In addition, it can be seen that the admissible functions presented in this paper can also properly describe the discontinuity conditions of the shear forces at the line supports. Therefore, accurate results can be expected for the analysis of dynamic response and internal force distribution of the plate.     

Stiffness matrix of thin-walled curved beam for spatially coupled stability analysis

The exact stiffness matrix for the spatially coupled stability analysis of thin-walled curved beam with non-symmetric cross-section subjected to uniform compression is newly presented. For this, the elastic strain energy including the axially/flexurally/torsionally coupled terms and the potential energy due to the initial stress resultant are introduced. Then, equilibrium equations and force–deformation relationship are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Next, the exact element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the finite element solutions using the Hermitian curved beam elements. In addition, the coupling of symmetric and anti-symmetric modes at the buckling load crossover with change in curvature of beam is investigated.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

Analysis of rotating nonuniform pretwisted beams with an elastically restrained root and a tip mass

Using Hamilton's principle derives the governing differential equations for the coupled bending–bending vibration of a rotating pretwisted beam with an elastically restrained root and a tip mass, subjected to the external transverse forces and rotating at a constant angular velocity. Using the mode expansion method derives the closed-form solutions of the dynamic and static systems. The orthogonal condition for the eigenfunctions of the system with elastic boundary conditions is discovered. The self-adjointness of the system is proved. Moreover, the Green functions of the system are obtained. The symmetric properties of the Green functions are revealed. The frequency response on the steady response of the beam is also investigated.     

Characteristic solutions for the statics of repetitive beam-like trusses

This paper concerns two major points: (1) decomposition of functional solutions for the static response of repetitive pin-jointed beam trusses under end loadings into spectrum of elementary function modes; and (2) a mathematical classification of the last. The governing finite difference equation of statics is written as a single matrix form by considering the stiffness matrix of a representative substructure. It is shown that its general solution can be spanned by only 2R individual modes, where R is the number of degrees of freedom for a typical nodal pattern inside the truss. These modes are divided into two primary classes: transfer and localised. A unique set of “canonical” transfer solutions is found by a method based on Jordan decomposition of the transfer matrix. Also, a technique of constructing transfer matrices for a wide class of trusses is presented. The canonical modes can be further subclassified as exponential, polynomial and quasi-polynomial. The complete set of 2R canonical transfer and localised modes uniquely represents the basic structural response behaviour, and gives a basis for the characteristic (non-harmonic) expansion of static solutions. Several illustrative examples are considered.     

Eigenvalue problems of rotor system with uncertain parameters

A general method for investigating the eigenvalue problems of a rotor system with uncertain parameters is presented in this paper. The recurrence perturbation formulas based on the Riccati transfer matrix method are derived and used for calculating the first- and secondorder perturbations of eigenvalues and their respective eigenvectors for the rotor system with uncertain parameters. In addition, these formulas can be used for investigating the independent, and repeated, as well as the complex eigenvalue problems. The general method is called the Riccati perturbation transfer matrix method (Riccati-PTMM). The formulas for calculating the mean value, variance, and covariance of the eigenvalues and eigenvectors of the rotor system with random parameters are also given. Riccati-PTMM is used for calculating the random eigenvalues of a simply supported Timoshenko beam and a test rotor supported by two oil bearings. The results show that the method is accurate and efficient.     

耦合梁中高频抖动的行波分析与主动控制

为研究中高频扰动下耦合梁结构的动力学响应与主动控制,基于Timoshenko梁理论,考虑梁中转动惯量和剪切变形的影响,采用行波方法分别建立梁结构纵向运动、弯曲运动的单元模型与结点散射模型,进而获得耦合梁的行波动力学模型及其精确的中高频抖动响应;引入“功率流”的分析思想,并以此为目标函数,优化得到了最优控制力的大小与相位,然后对结构施加最优控制力,实现耦合梁结构的功率流主动控制。在此基础上,进行数值仿真分析,并与Euler-Bernoulli梁理论计算结果进行对比。结果表明,采用行波方法计算耦合梁结构的动力学响应准确可靠;Timoshenko梁模型较Euler-Bernoulli梁模型在中、高频段更为精确,且更接近工程实际;功率流主动控制可以明显降低耦合梁结构全频域内的抖动,验证了基于行波方法功率流主动控制的正确性与有效性。     

Instability and vibration of a rotating Timoshenko beam with precone

A rotating blade with a precone angle is usually designed, but little literature has investigated the effect of the precone angle on vibration. This paper investigates divergence instability and vibration of a rotating Timoshenko beam with precone and pitch angles. It uses Hamilton's principle to derive the coupled governing differential equations and boundary conditions for a rotating Timoshenko beam. Analytical solution of an inextensional Timoshenko beam without taking into account the Coriolis force effect can be derived. Some simple relations among the parameters of rotating Timoshenko beams are revealed. Based on these relations, one can predict the natural frequencies and parameters of other systems from those of known systems. Moreover, the mechanism of divergence instability (tension buckling) is investigated. Finally, the effects of the parameters on natural frequencies, and the phenomenon of divergence instability are investigated.     

Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform

This study introduces the concept of a differential transform to solve the free vibration problems of a rotating twisted Timoshenko beam under axial loading. First, the concept of differential transform is briefly introduced. Second, taking a differential transform of a Timoshenko beam vibration problem, a set of difference equations is derived. Performing some simple algebraic operations on these equations, we can determine the jth natural-frequency, the closed form series solution of the jth mode shape. Finally, three cases—twist, axial force and rotation—are investigated to illustrate the accuracy and efficiency of the present method.     

An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear analysis.     

Anatomization of hysteresis loops in pure bending of ideal pseudoelastic SMA beams

The results of qualitative analysis of the variation of stress and the phase content distribution in arbitrary symmetric cross-section of the beam are discussed for single bending cycle. The explicit analytical equations for the moment–curvature hysteresis loop are derived. The interrelation between formation of characteristic points on this diagram and movement of the beam planes separating the regions occupied by single phase (austenite or martensite) and their mixture (austenite and martensite) in the course of phase transitions is discussed. It is shown that at such mobile surfaces the gradient of macroscopic stress suffers jump discontinuity. The dimensionless bending diagrams (nomograms) for rectangular beams are constructed. They enable rough quantitative estimation of some basic design parameters without recourse to the computer-aided numerical calculations. One-dimensional Müller–Xu thermodynamic theory of ideal pseudoelasticity is used as the theoretical foundation for the analysis. The theory is briefly discussed and its generalization that accounts for the tension–compression asymmetry effect is presented.     

Eigenanalysis and continuum modelling of an asymmetric beam-like repetitive structure

An asymmetric repetitive pin-jointed structure, based upon a 3-D NASA framework, is analysed using a state variable transfer matrix technique. A conventional transfer matrix cannot be constructed due to the singularity of one partition of the stiffness matrix; instead, a cell (rather than cross-sectional) state vector consisting of displacements only is employed, leading to a generalised eigenvalue problem. The asymmetry of the structure leads to tension–torsion and bending–shear couplings, which may be explained in terms of the tension–shear coupling of a single face of the structure. Equivalent continuum beam properties and coupling coefficients are determined, and the effect of (a)symmetry discussed as a trade between, for example, tension-Poisson's ratio contraction for a symmetric structure, against tension–torsion coupling for the asymmetric.     

A three-dimensional model for the dynamics of micro-endmills including bending, torsional and axial vibrations

Attainable geometric accuracy and surface finish in a micromilling operation depends on predicting and controlling the vibrations of micro-endmills. The specific multi-scale geometry of micro-endmills results in complexities in dynamic behavior, including three-dimensional vibrations, which cannot be accurately captured using one-dimensional (1D) beam models. This paper presents an analytically based three-dimensional (3D) model for micro-endmill dynamics, including actual cross-section and fluted (pretwisted) geometry. The 3D model includes not only bending, but also coupled axial/torsional vibrations. The numerical efficiency is enhanced by modeling the circular cross-sectioned shank and taper sections using 1D beam equations without compromising in model accuracy, while modeling the complex cross-sectioned and pretwisted fluted section using 3D linear elasticity equations. The boundary-value problem for both 1D and 3D models are derived using a variational approach, and the numerical solution for each section is obtained using the spectral-Tchebychev (ST) technique. Subsequently, component mode synthesis is used for joining the individual sections to obtain the dynamic model for the entire tool. The 3D model is validated through modal experimentation, by comparing natural frequencies and mode-shapes, for two-fluted and four-fluted micro-endmills with different geometries. The natural frequencies from the model was seen to be within 2% to those from the experiments for up to 90 kHz frequency. Comparison to numerically intensive, solid-element finite-elements models indicated that the 3D and FE models agree with less than 1% difference in natural frequencies. The 3D-ST model is then used to analyze the effect of geometric parameters on the dynamics of micro-endmills.     

复合材料夹层梁弯曲分析的状态空间法

给出了复合材料夹层梁弯曲刚度和剪切刚度的表达式。导出了其弯曲的状态空间方程，并求出了它的通解，最后，给出了三个例子。本法求解问题简单，明显地优于以往的方法