Transverse vibrations of an Euler–Bernoulli uniform beam carrying several particles

The Euler–Bernoulli uniform beam considered in this paper consist of n portions and carry (n+1) particles, two of which are at the beam ends. For the classical beam eigen-value technique developed here, n co-ordinate systems are chosen with origins at the particle locations. The mode shape of the jth portion of the beam is expressed in the form Y_j(X_j)=AU_j(X_j)+BV_j(X_j) in which U_j(X_j) and V_j(X_j) are ‘modified’ mode shape functions applicable to that portion but the constants A and B are common to all the portions. From the boundary conditions at the right end, the frequency equation was expressed in closed form as a second-order determinant equated to zero. Schemes are presented to compute the four elements of the determinant (from a recurrence relationship) and to evaluate the roots of the frequency equation. Computational difficulties were not encountered in the implementation of the schemes. The first three natural frequency parameters of 16 combinations of the classical boundary conditions are tabulated for beams with three and up to nine portions for selected particle location and mass parameters. Frequency parameters of beams with one and up to 500 equi-spaced, equi-mass systems are also tabulated. The approaches in previous publications include those based on various approximate methods like finite element, Rayleigh–Ritz, Galerkin, transfer matrix, etc. The results in the present paper may be used to judge the accuracy of values obtained by approximate methods.     

On the methods to derive frequency equations of beams carrying multiple masses

Two commonly used methods to generate the eigenfrequency equation of a beam carrying multiple concentrated masses at an arbitrary location are of main concern in this paper. Both the methods of frequency determinant and the method of Laplace transform are considered and compared here for the effectiveness of each derivation for the frequency equation of the same system. The dimensionless frequency equation for a clamped–clamped beam carrying two different masses is obtained for comparison. The computational times to generate the same expression by using the two methods are noted. It is found that the processing and computational time by using the Laplace transform is longer although the resultant system equations are often claimed to be more compact than those by the method of frequency determinant.     

Comparisons of experimental and numerical frequencies for classical beams carrying a mass in-span

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for ten combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. The resulting transcendental frequency equations are numerically solved. A parametric study on the effects of the mass and its location for each respective case is presented. To verify the validity of the transcendental equations, the results for the fixed-fixed cases are compared with those obtained experimentally. On the other hand, approximate results are given, using the Rayleigh’s method with two static deflection shape functions. The effects of the position and magnitude of the mass, as well as comparisons of the different results obtained analytically, are investigated and discussed. The comparisons clearly show that the eigenfrequencies of the beam–mass system can be accurately predicted by solving the transcendental equation, whereas the closed-form Rayleigh’s expression is suggested for a quick estimation of fundamental frequency.     

A comparative study of the eigenvalue solutions for mass-loaded beams under classical boundary conditions

A comparative study of the eigenfrequency analysis for an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented in this short note. The dimensionless frequency equation for different combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. Two formulation methods have been commonly used for the boundary-value problem. One is to adopt a single frame originated from the beam's left-end, while another is by dual frames associated with the concentrated mass. It is found that the forms derived by dual frames are more compact than the corresponding expressions by using the single frame. Nevertheless, the comparison for all the cases shows that the dual-frame expressions need more time to obtain the same set of eigenvalues if compared with the time by using the single-frame expressions.     

A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions associated to the masses. The resulting transcendental equations are numerically solved for the eigenvalue. On the other hand, the eigenvalue can be predicted merely from the individual beam system carrying a single mass, by virtue of the Dunkerley's formula. A parametric study on the effects of the two masses and their locations is presented for the beam with different boundary conditions. It is found that the Dunkerley's expression can generally yield good approximation if compared with the result associated with the original characteristic equation. The computation time saved owing to the modified Dunkerley method is also illustrated in a comparison. The Dunkerley's method is recommended for the beam carrying more than two masses at different positions, owing to its good approximation and the saving in computational time.     

On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli-Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. Using the normal mode method, a second approximate frequency equation is established which is then used for the derivation of a sensitivity formula for the eigenfrequencies. The frequency equations of some simpler systems are obtained from the general equation as special cases. These frequency equations are then numerically solved for various combinations of physical parameters. The comparison of the numerical results with those from exact frequency equations indicate clearly that the eigenfrequencies of the combined system described above can be accurately determined by the present method.     

Vibration and stability of an Euler–Bernoulli beam with up to three-step changes in cross-section and in axial force

The paper treats the vibration of beams with up to three-step changes in cross-section and in which the axial force in each portion is constant but different. The system parameters are the step positions, the flexural rigidity, the mass per unit length and the axial force in the beam portions—all of which were normalized. The frequency equation for 16 combinations of classical boundary conditions are expressed as fourth-order determinants equated to zero. The first three frequency parameters are tabulated for sets of system parameters (arbitrarily chosen and which includes a stepped beams under tensile or compressive axial end force). Critical compressive end force which causes a stepped beam to buckle are tabulated. Buckling under a system of axial forces, one of which is critical is discussed and several critical combinations of the system parameters are tabulated. Beams of constant depth and step change in breadth, of constant breadth and step change in depth and shafts with step change in diameter are considered. It is shown that stepped shafts are inferior machine elements if dynamic properties are the prime consideration.     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

Crack detection in beams using experimental modal data and finite element model

In this paper, an analytical, as well as experimental approach to the crack detection in cantilever beams by vibration analysis is established. An experimental setup is designed in which a cracked cantilever beam is excited by a hammer and the response is obtained using an accelerometer attached to the beam. To avoid non-linearity, it is assumed that the crack is always open. To identify the crack, contours of the normalized frequency in terms of the normalized crack depth and location are plotted. The intersection of contours with the constant modal natural frequency planes is used to relate the crack location and depth. A minimization approach is employed for identifying the cracked element within the cantilever beam. The proposed method is based on measured frequencies and mode shapes of the beam.     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。     

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.     

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.     

利用传递矩阵法对阶梯悬臂梁的裂纹参数识别

基于Bernoulli-Euler梁振动理论,以等效弹簧模拟裂纹引起的局部软化效应,利用传递矩阵法推导阶梯悬臂梁振动频率的特征方程,对于含多个裂纹以及复杂边界条件的阶梯梁,仅需求解4×4的行列式即可获得相应的频率特征方程。直接利用该特征方程,提出两种有效估计裂纹参数的方法———等值线法和目标函数最小化法,并应用两段阶梯悬臂梁的数值算例说明方法的有效性。算例结果表明,只需结构前三阶频率即可识别裂纹位置和深度。应用“零设置”可减小计算频率与理论频率不相等对识别结果的影响。等值线法可以直观给出裂纹位置和裂纹深度参数,目标函数最小化法可给出最优的裂纹参数结果,并且该方法可推广应用到含多个裂纹复杂梁(如非完全固支、弹性支撑等)结构的裂纹参数识别中。     

Vibration of an Euler–Bernoulli beam on elastic end supports and with up to three step changes in cross-section

Compared to the bibliography on the transverse vibration of Euler–Bernoulli beams with one step change in cross-section, publications on beams with more than one step changes is not extensive. In this paper an analytical method is proposed to calculate the frequencies of beams with up to three step changes in cross-section. Combinations of the classical clamped, pinned, sliding, free, ‘general’ and ‘degenerate’ types of elastic end supports are considered. The frequency equations of stepped beams were expressed as fourth order determinant equated to zero. A scheme to calculate the elements of the determinant and a scheme to evaluate the roots of the determinant are presented. Special consideration is given to three types of stepped beams frequently encountered in engineering applications. The first three frequency parameters of beams with two and three step changes in cross-section are tabulated for selected sets of system parameters and 45 types of end supports. Computational difficulties were not encountered. The method proposed may be extended to tackle beams with any number of step changes in cross-section. The tabulated results may be used to judge the frequencies calculated by numerical methods.     

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.     

Parametric vibrations of a horizontal beam with a concentrated mass at one end

This investigation treats the steady state response of parametric vibration of a simply supported horizontal beam, carrying a concentrated mass at one end and subjected to a periodic axial displacement excitation at the other end under the influence of gravity. Non-linear terms arising from longitudinal inertia of a concentrated end mass and beam elements are included in the equation of motion. By using the one mode approximation and applying Galerkin's method, the governing equation of motion is reduced to a non-linear ordinary differential equation with periodic coefficient. The harmonic balance method is applied to solve the equation and the dynamic response is derived. Experimentally determined amplitude-frequency curves are presented, and are found to be in good agreement with the theory.     

An exact approach for free vibration analysis of rectangular plates with line-concentrated mass and elastic line-support

An exact approach for free vibration of an isotropic rectangular plate carrying a line-concentrated mass and with a line-translational spring support or carrying a line-spring-mass system is presented in this paper. The mode shape function of vibration of such a plate is expressed in terms of the four fundamental solutions derived in this paper. The main advantage of the proposed method is that the resulted frequency equation for such a rectangular plate can be conveniently obtained from a second-order determinant. The proposed method is thus computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures which usually led to an eighth-order determinant for solving the title problem. Two numerical examples are given to illustrate the efficiency of the proposed method and to investigate the effects of the location and the magnitude of a line-concentrated mass and elastic line-support as well as the influence of the aspect ratio on the natural frequencies of a rectangular plate.     

连续小波变换在梁结构损伤诊断中的应用研究

为了检测出梁中的裂缝或因刚度降低引起的损伤,对有损伤简支梁的振型曲线进行连续小波变换．从小波系数出现模极大值有效地识别损伤的存在以及裂缝位置和刚度下降段的位置。基本振型是用小波变换识别裂缝的最佳振型．用损伤位置处振幅较大的振型曲线来识别最清楚,对有噪声影响的振型曲线同样可以用本文方法进行识别。通过分析和计算获得满意结果．在梁结构损伤诊断中具有较高的应用价值。     

Non-uniform Euler-Bernoulli beams under a single moving oscillator: An approximate analytical solution in time domain

Dynamic responses of simply supported non-uniform beams traversed by a moving oscillator are analysed in this paper. An approximate analytical method based on Rayleigh-Ritz (R-R) formulation is developed. The fundamental approximate mode obtained from R-R method is used in the present formulation to determine the responses of the beam and the oscillator. Effects of surface irregularities on the displacement and acceleration responses of the beam and the vehicle are also analysed. The results are compared with those obtained using Finite element method (FEM). A numerical example is provided to illustrate the validity of the present method which shows that the proposed method is simple, computationally more efficient compared to FEM and gives fairly good results. Though the single-mode approach used in the present paper is a classical one and numerous studies on the responses of uniform beams under moving loads have been reported in the past, its application to non-uniform beams (for which there does not exist any closed form expression for mode shapes) under a moving load, especially a moving oscillator, is presented for the first time.     

DETERMINATION OF UNKNOWN IMPACT FORCE ACTING ON A SIMPLY SUPPORTED BEAM

This work presents a predictive model for determining the location and amplitude of an unknown impact force acting on a simply supported beam. Both time and frequency domain prediction methods are developed, respectively. The structural modal parameters can be first obtained by theoretical modal analysis (TMA) or by experimental modal analysis (EMA). The structural response at time and frequency domains due to an unknown impact force can then be measured and recorded. The predicted response can also be formulated and expressed as functions of amplitude and location of the impact force. The sum of square errors between the predicted and measured response is then defined as the objective function, while the amplitude and the location of the unknown impact force are defined as design variables. The optimisation problem is thereby constructed and can be solved for the amplitude of the impact force. The mode shape information associated to the location of the impact force can also be resolved and compared to the structural mode shapes to determine the location of the unknown impact force. Both numerical and experimental prediction results are presented. Results show that the predictive model is feasible and leads to the prediction of magnitude and location of the unknown impact force for arbitrary structures as well.