Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform

This study investigates the vibration problems of an axially loaded non-uniform spinning twisted Timoshenko beam. First, using the Timoshenko beam theory and Hamilton's principle, we derive the governing equations and boundary conditions of the beam. Secondly, the differential transform method is used to solve these equations with appropriate boundary conditions. Finally, the effects of the twist angle, spinning speed, and axial force on the natural frequencies of a non-uniform Timoshenko beam are investigated and discussed.     

微分变换法在动力学响应分析及系统边界参数识别中的应用研究

引入求解非线性微分方程的微分变换法，将其推广为广义微分变换法。建立求解一般非线性振动微分方程的一般框架，将此方法用于求解著名的Vander pol方程。并且将微分变换法推广到结构边界参数识别，以一个典型的悬臂梁边界参数识别为例，对其进行数值仿真和实验研究，并将此方法的实验研究识别结果与用实测频率响应函数法的识别结果作比较。说明该方法具有良好的工程应用价值。     

弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。     

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.     

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.     

轴向运动功能梯度梁的横向振动

新型非均匀复合材料,功能梯度材料具有防止脱层和减缓热应力等优良性能,将其应用于功能梯度梁的结构有着非常重要的工程应用价值。基于Euler-Bernoulli梁理论和Hamilton原理,建立轴向运动功能梯度梁横向自由振动的运动微分方程,其中假设功能梯度梁的材料特性沿梁厚度方向按各组分材料体积分数的幂函数连续变化;再对运动微分方程和边界条件进行量纲一处理,采用微分求积法对其进行离散化,导出系统的广义复特征方程,然后计算分析轴向运动功能梯度简支梁横向振动复频率的实部和虚部随量纲一轴向运动速度、梯度指标等参数的变化情况,并讨论量纲一轴向运动速度和梯度指标对功能梯度梁的横向振动特性以及失稳形式的影响。     

Forced vibration of composite cylindrical helical rods

The dynamic behavior of composite cylindrical helical rods subjected to time-dependent loads is theoretically investigated in the Laplace domain. The governing equations for naturally twisted and curved spatial laminated rods obtained using Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, the anisotropy of the rod material, effect of the rotary inertia, axial and shear deformations are considered in the formulations. The material of the rod is assumed to be homogeneous, linear elastic and anisotropic. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem accurately. The solutions obtained are transformed to the time domain using an appropriate numerical inverse Laplace transform method. The free vibration is then taken into account as a special case of forced vibration. The results obtained in this study are found to be in a good agreement with those available in the literature.     

Anomalous dynamic elastic-plastic response of a galerkin beam model

A study of the anomalous motion of an elastic—plastic beam under short pulse loading is presented. The geometric nonlinearity due to axial end constraints is taken into account. We apply the Galerkin method to the governing partial differential equation of the transverse motion to obtain a general model of n degrees of freedom (nDoF). The results of elastic—plastic deformation analysis and dynamic response for the 2DoF model of a pin-ended beam are presented. The regular and irregular motions of the 2DoF model for the pin-ended beam are examined by various methods including time history, phase diagram, Lyapunov characteristic exponent and power spectral density.     

Parametric instability of a spinning pretwisted beam under periodic axial force

This paper addresses the parametric instability of a cantilever pretwisted beam rotating around its longitudinal axis under a time-dependent conservative end axial force which contains a steady-state part and a small periodically fluctuating component. This structural element can be used to model fluted cutting tools such as the twist drill bit and the end milling cutter, etc. Using the Euler—Bernoulli beam theory and Hamilton's principle, the present study derives the equation of motion which governs the lateral vibration of a spinning pretwisted beam. Rotary inertia, structural viscous damping and conservative end axial force are included. The Galerkin method is then applied to obtain the associated finite element equations of motion. Due to the existence of the Coriolis force, the resulting finite element equations of motion are transformed into a set of first-order simultaneous differential equations by a special modal analysis procedure. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and expressions for the boundaries of the unstable regions. Numerical results are presented to demonstrate the effects of pretwist angle, spinning speed and steady-state part of the end axial force on the parametric instability regions of the present problem.     

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

Linear free vibration analysis of cross-ply laminated cylindrical helical springs

A linear free vibration analysis of symmetric cross-ply laminated cylindrical helical springs is performed based on the first-order shear deformation theory. Considering the rotary inertia, the shear and axial deformation effects, governing equations of symmetric laminated helical springs made of a linear, homogeneous, and orthotropic material are presented in a straightforward manner based on the classical beam theory. The free vibration equations consisting of 12 scalar ordinary differential equations are solved by the transfer matrix method. The overall transfer matrix of the helix is computed up to any desired accuracy. The soundness of the present results are verified with the reported values which were obtained theoretically and experimentally. After presenting the non-dimensional graphical forms of the free vibrational characteristics of (0°/90°/90°/0°) laminated helical spring made of graphite-epoxy material (AS4/3501-6) with fixed–fixed ends, a non-dimensional parametric study is worked out to examine the effects of the number of active turns, the shear modulus in the 1–2 plane (G₁₂), the ratio of the cylinder diameter to the thickness (D/d), and Young's moduli ratio in 1 and 2 directions (E₁/E₂) on the first six natural frequencies of a uniaxial composite helical spring with clamped-free, clamped-simple, and clamped–clamped ends.     

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.     

Bending and vibration of a discontinuous beam with a curvic coupling under different axial forces

The transverse stiffness and vibration characteristics of discontinuous beams can significantly differ from those of continuous beams given that an abrupt change in stiffness may occur at the interface of the former. In this study, the equations for the deflection curve and vibration frequencies of a simply supported discontinuous beam under axial loads are derived analytically on the basis of boundary, continuity, and deformation compatibility conditions by using equivalent spring models. The equation for the deflection curve is solved using undetermined coefficient methods. The normal function of the transverse vibration equation is obtained by separating variables. The differential equations for the beam that consider moments of inertia, shearing effects, and gyroscopic moments are investigated using the transfer matrix method. The deflection and vibration frequencies of the discontinuous beam are studied under different axial loads and connection spring stiffness. Results show that deflection decreases and vibration frequencies increase exponentially with increasing connection spring stiffness. Moreover, both variables remain steady when connection spring stiffness reaches a considerable value. Lastly, an experimental study is conducted to investigate the vibration characteristics of a discontinuous beam with a curvic coupling, and the results exhibit a good match with the proposed model.

     

Static analysis of a Bickford beam by means of the DQEM

In this paper the recently proposed differential quadrature element method is employed in order to solve the equilibrium equations of a higher-order beam. A simple five-node element is introduced, in which the vertical displacement is approximated by a sixth-order polynomial, whereas the rotation is consistently approximated by a fourth-order polynomial, and the resulting weighting coefficient matrix is given. Moreover, a general procedure is outlined, for an N-node element, in which vertical displacements and rotations are given by polynomials of order N+1 and N-1, respectively. Numerical examples are aimed both at checking the convergence of the results for increasing values of the nodes, and at comparing the used cubic beam theory with the simpler, linear, Timoshenko theory.     

Influence of pre-buckling displacements on the elastic critical loads of thin walled bars of open cross section

Nonlinear expressions for the strains occurring in thin walled bars of open cross section, when subjected to axial, flexural and torsional displacements, are incorporated in a general instability analysis based on the vanishing of the second variation of the total potential energy. It is shown that the influence of the pre-buckling displacements is automatically included in the analysis. A closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending agrees exactly with a solution based on the governing differential equations. Solutions obtained using numerical methods are also presented. The significance of the second order axial strains induced by rotation about the shear centre, is investigated by considering the instability of an inverted T-beam subjected to uniform bending.     

The elastically supported Timoshenko beam subjected to an axial force and a moving load

A solution for the flexural vibration of an elastically supported Timoshenko beam which is subjected to an axial force and a moving transverse load is obtained. The influences of the axial force and the load velocity on the beam response are studied and the characteristics of the various resonances are examined. The results are also compared with those by the Euler beam theory.     

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.     

Three-dimensional vibration analysis of thick, tapered rods and beams with circular cross-section

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, tapered rods and beams with circular cross-section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four- digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3-D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler–Bernoulli beam theory.     

Stick-slip vibration of a moving oscillator on an axially flexible beam

This study investigates the stick-slip vibration between an axially flexible beam fixed at both ends and an oscillator moving on the beam. After deriving the equations of motion for the stick and slip states, the stick-slip vibrations between the oscillator and the beam are analyzed. In addition, to obtain the irregularly changed contact position due to the axial deformation of the beam and oscillator movement, a mathematical expression for the contact position is derived. It is found that the long-period stick-slip vibration is influenced mainly by the oscillator and the short-period vibration is influenced mainly by axial deformation of the beam. Furthermore, the dynamic responses show that even if a high damping ratio is applied to the oscillator, stick-slip vibration due to axial deformation of the beam can occur. Finally, the analysis shows that a kind of the internal resonance occurs between the oscillator and the beam when the harmonics of the natural frequency of the oscillator match the natural frequencies of the beam.

     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.