On applications of generalized functions to the analysis of Euler–Bernoulli beam–columns with jump discontinuities

In this article some applications of the distribution theory of Schwarz to the analysis of beam–columns with various jump discontinuities are offered. The governing differential equation of an Euler–Bernoulli beam–column with jump discontinuities in flexural stiffness, displacement, and rotation, and under an axial force at the point of discontinuities, is obtained in the space of generalized functions. The auxiliary beam–column method is introduced. Using this method, instead of solving the differential equation of the beam–column in the space of generalized functions, another differential equation can be solved in the space of classical functions. Some examples of beam–columns and columns with various jump discontinuities are solved. Deflections of beam–columns and buckling loads for columns with jump discontinuities are calculated using the Laplace transform method in the space of generalized functions.     

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.     

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.     

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.     

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.     

Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading

In this paper, the function for describing the distribution of flexural stiffness K(x) of a non-uniform column is arbitrary, and the distribution of axial distributed loading N(x) acting on the column is expressed as a function of K(x) and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of variable transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 14 cases. The analytical buckling solutions of one-step non-uniform columns are thus found. Then the obtained analytical solutions are used to derive the eigenvalue equation for buckling of a multi-step non-uniform column for several boundary supports by using the transfer matrix method. A numerical example shows that the proposed procedure is an efficient method for buckling analysis of multi-step non-uniform columns.     

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.     

Free vibration of SDOF systems with arbitrary time-varying coefficients

A new analytical approach for determining the exact solutions for free vibration of single-degree-of-freedom (SDOF) systems with non-periodically time-varying coefficients (mass and stiffness) is presented herein. In this paper, the function for describing the variation of mass of a SDOF system with time is an arbitrary one, and the variation of the stiffness is expressed as a functional relation with the mass function and vice versa. Using appropriate functional transform, the governing differential equation for the title problem is reduced to a Bessel’s equation or other analytically solvable equations. Exact solutions for free vibration of SDOF systems with non-periodically varying coefficients are obtained for six important cases. In order to simplify the free vibration analysis of a SDOF system with multi-step time-varying coefficients, the fundamental solutions that satisfy the normalization conditions are constructed based on the exact solutions derived. It is more convenient to determine the displacement response of the SDOF system by using the fundamental solutions and a recurrence formula developed in this paper. Numerical example shows that the proposed procedure is a simple, efficient and exact method.     

Analysis of dynamic properties of boring bars concerning different clamping conditions

Boring bars are frequently used in the manufacturing industry to turn deep cavities in workpieces and are usually associated with vibration problems. This paper focuses on the clamping properties’ influence on the dynamic properties of clamped boring bars. A standard clamping housing of the variety commonly used in industry today has been used. Both a standard boring bar and a modified boring bar have been considered. Two methods have been used: Euler–Bernoulli beam modeling and experimental modal analysis. It is demonstrated that the number of clamping screws, the clamping screw diameter sizes, the screw tightening torques, the order the screws are tightened has a significant influence on a clamped boring bars eigenfrequencies and its mode shapes orientation in the cutting speed—cutting depth plane. Also, the damping of the modes is influenced. The results indicate that multi-span Euler–Bernoulli beam models with pinned boundary condition or elastic boundary condition modeling the clamping are preferable as compared to a fixed-free Euler–Bernoulli beam for modeling dynamic properties of a clamped boring bar. It is also demonstrated that a standard clamping housing clamping a boring bar with clamping screws imposes non-linear dynamic boring bar behavior.     

Development of a Wittrick–Williams algorithm for the spectral element model of elastic–piezoelectric two-layer active beams

In this paper, a Wittrick–Williams algorithm is developed for the elastic–piezoelectric two-layer active beams. The exact dynamic stiffness matrix (or spectral element matrix) is used for the development. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the sign change of the determinant of spectral element matrix via infinity instead of via zero. The uniform and partially patched active beams are considered as the illustrative examples to confirm the present algorithm.     

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.     

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.     

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.     

Effect of number of blades and distribution of cracks on vibration localization in a cracked pre-twisted blade system

A number of several blades can be grouped at their tips to increase their stiffness. This work examines how the effect of number of grouped blades and distribution of cracks affect the mode localization of a mistuned blade system. The pre-twisted blade and the effect of twist angle on localization are also considered in this article. Dynamic characteristics of blades in a blade system are focused to study. Periodically coupled pre-twisted beams were used to approximate shrouded blades. The Euler–Bernoulli beam model was employed to characterize the tapered pre-twisted blade. The mode localization equations associated with the local blade crack defects in the rotating grouped blade system were formulated using Hamilton's principle. The Galerkin method was used to discretize the localization equations of the mistuned system. The numerical results herein reveal that the number of grouped blades and the distribution of multi-disorders in a rotating blade system may markedly affect the localization phenomenon.     

Transverse vibrations of an Euler–Bernoulli uniform beam carrying two particles in-span

Vibrations of beams carrying different combinations of particles, heavy bodies and spring-mass systems which are located on or off resilient supports have been tackled by several researchers. Most of the approaches were based mainly on various approximate methods. In this paper an analytical solution based on the classical beam eigenvalue technique is presented for the vibrations of a beam carrying two particles. For purpose of analysis, the beam was divided into a portion from one end to the first particle, a portion between particles and a portion from the second particle to the other end. The frequency equation is expressed in closed form as a 2nd order determinant equated to zero. Schemes are presented to compute the 4 elements of the determinant 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 are tabulated for 16 combinations of the classical boundary conditions and several combinations of the location and mass of the particles. The beam mode shape is the juxtaposition of the mode shapes of the three portions of the beam. Some examples of normalised beam mode shapes and location/s of node/s are also presented. The results may be used to judge the accuracy of values obtained by approximate methods.     

New rational interpolation functions for finite element analysis of rotating beams

A rotating beam finite element in which the interpolating shape functions are obtained by satisfying the governing static homogenous differential equation of Euler–Bernoulli rotating beams is developed in this work. The shape functions turn out to be rational functions which also depend on rotation speed and element position along the beam and account for the centrifugal stiffening effect. These rational functions yield the Hermite cubic when rotation speed becomes zero. The new element is applied for static and dynamic analysis of rotating beams. In the static case, a cantilever beam having a tip load is considered, with a radially varying axial force. It is found that this new element gives a very good approximation of the tip deflection to the analytical series solution value, as compared to the classical finite element given by the Hermite cubic shape functions. In the dynamic analysis, the new element is applied for uniform, and tapered rotating beams with cantilever and hinged boundary conditions to determine the natural frequencies, and the results compare very well with the published results given in the literature.     

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.     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

Variations of instability in a rotating spindle system with various bearings

To drive the speed of spindle faster and faster, especially for micro-via-drilling, the gas bearing–spindle is a must. However, most investigations of the dynamic characteristics of the spindle system are limited to the ball bearing type of spindle. This work examines the dynamic instability of a rotating spindle system with various bearings to elucidate the difference between the ball and gas bearing–spindle systems. A round Euler–Bernoulli beam is used to approximate the spindle. The Hamilton principle is applied to derive the equation of motion for the spindle system, and the multiple scales perturbation method is employed to solve the instability solution of the system. The effects of bearing types and speeds of rotation on the dynamic characteristics and instability of a rotating spindle system are further studied.