Multi-span Beam Research Articles

Free vibrations of a multi-span Timoshenko beam carrying multiple spring-mass systems

Structural elements supporting motors or engines are frequently seen in technological applications. The operation of a machine may introduce additional 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 on Timoshenko multi-span beams carrying multiple spring-mass systems is fewer. This paper aims at determining the natural frequencies and mode shapes of a Timoshenko multi-span beam. The model allows to analyse the influence of the shear effect and spring-mass systems on the dynamic behaviour 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 natural frequencies of Timoshenko multi-span beam calculated by using secant method for non-trivial solution are compared with the natural frequencies of multi-span beam calculated by using Bernoulli-Euler Beam Theory (EBT) in literature; the mode shapes are presented in graphs.

On the natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of concentrated elements

This paper adopts the numerical assembly method (NAM) to determine the exact solutions of natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and springmass systems. First, the coefficient matrix for an intermediate station with various concentrated elements, cross-section change and/or pinned support and the ones for the left-end and right-end supports of a beam are derived. Next, the overall coefficient matrix for the entire beam is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact solutions for the natural frequencies of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and the associated mode shapes are obtained by substituting the corresponding values of integration constants into the associated eigenfunctions.

On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring–mass systems. First, the coefficient matrices for an intermediate pinned support, an intermediate concentrated element, left- and right-end support of a Timoshenko beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.

Orthogonality condition for a multi-span beam, and its application to transient vibration of a two-span beam

This paper treats the orthogonality condition for a multi-span beam, and its application to forced (transient) vibration of a two-span beam. The beam is modeled as a Bernoulli–Euler beam. The boundary conditions for the particular case of two-span beam are clamped–pinned–pinned. An exact closed-form solution is obtained for this problem. Even though there has been an enormous amount of work on beam vibration, most of the studies are conducted on a single-span beam. There are some studies on the multi-span beam vibration. However, their treatment is rather specialized in terms of the applied loading and the initial conditions. None of the studies in the past treats an exact solution for a forced (transient) vibration of a general two-span beam with arbitrary initial conditions and arbitrary forcing functions. Therefore, the solution obtained in this paper is new. The key development in the solution is the orthogonality condition for a multi-span beam. The method of solution developed in this paper establishes a general methodology for the forced (transient) vibration of a multi-span beam. The closed-form solution obtained in this paper can be used as a benchmark solution for the transient vibration of a two-span beam.

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.

LOGICAL MATRIX FOR THE LONGITUDINAL AND TORSIONAL VIBRATIONS OF CONTINUOUS BEAMS

The formal procedure for calculations of longitudinal and torsion vibrations of the multispan beams with continuous mass distribution on the basis of the method of start parameters, associated matrices and elements of mathematical logic is proposed.

Free vibration analysis of circularly curved multi-span Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the free vibration analysis of circularly curved multi-span Timoshenko beams. Each section of the beam has its own basis functions, and the continuity conditions at the intermediate supports as well as the boundary condition are treated as the constraints of the basis functions so that the number of degrees of freedom matches the number of the pseudospectral expansion coefficients. The computed natural frequencies are compared with those of existing literature, where it is shown that they are in good agreement. Numerical examples are provided for pinned-pinned, clamped-clamped and free-pinned boundary conditions for different numbers of sections and for different thickness-to-length ratios.

A Study on the Effects of Damage Models and Wavelet Bases for Damage Identification and Calibration in Beams

Abstract: Damage detection and calibration in beams by wavelet analysis involve some key factors such as the damage model, the choice of the wavelet function, the effects of windowing, and the effects of masking due to the presence of noise during measurement. A numerical study has been performed in this article addressing these issues for single and multispan beams with an open crack. The first natural modeshapes of single and multispan beams with an open crack have been simulated considering damage models of different levels of complexity and analyzed for different crack depth ratios and crack positions. Gaussian white noise has been synthetically introduced to the simulated modeshape and the effects of varying signal‐to‐noise ratio have been studied. A wavelet‐based damage identification technique has been found to be simple, efficient, and independent of damage models and wavelet basis functions, once certain conditions regarding the modeshape and the wavelet bases are satisfied. The wavelet‐based damage calibration is found to be dependent on a number of factors including damage models and the basis function used in the analysis. A curvature‐based calibration is more sensitive than a modeshape‐based calibration of the extent of damage.

Active control of wave propagation in multi-span beams using distributed piezoelectricactuators and sensors

This paper presents an integrated model of a one dimensional periodic structure withdistributed piezoelectric actuators and sensors for complete active or passive wavepropagation control. The periodic structures exhibit unique dynamic characteristics thatmake them act as mechanical filters for wave propagation. As a result, waves can propagatealong the periodic structures only within specific frequency bands called ‘pass bands’ andwave propagation is completely blocked within other frequency bands called ‘stop bands’.This basic property of periodic structures is enhanced by the application of distributedpiezoelectric actuators and sensors, to actively control the wave propagation over thefrequency range of interest.The finite element model, based on the transfer matrix approach, is developed to study theflexural wave propagation of the uniform beams resting on periodically spaced, rigid simplesupports with distributed piezoelectric actuators and sensors. The model is used to predictpass and stop frequency bands for different proportional and derivative control gains. Theresults indicate that the location and width of the stop bands as well as the attenuationcharacteristics in the beam can be modified by proper choice of the control gains. Thenumerical predictions demonstrate that the attenuation characteristics can be maximizedwithin different frequency bands by proper tuning of the control gains. Also, the finiteelement method is used to find the response of the active periodic beams to a convectedharmonic pressure field. Computed results show that the response amplitudes atcoincidence frequency can be actively controlled by proper selection of control gains.The tunable characteristics of the piezoelectric inserts are also used to introduceirregularities in the periodic structure. The source of disorder is the variance in the controlgains of the inserts. Disorder in the periodicity typically extends the stop bands intoadjacent propagation zones. More importantly, it produces localization of the vibrationenergy near the excitation source. The results obtained demonstrate the localizationphenomenon and its control through appropriate tuning of the level of disorder in thecontrol gains.

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.

Free vibration analysis of a uniform multi-span beam carrying multiple spring–mass systems

The literature regarding the free vibration analysis of single-span beams carrying a number of spring–mass systems is plenty, but that of multi-span beams carrying multiple spring–mass systems is fewer. Thus, this paper aims at determining the “exact” solutions for the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple spring–mass systems. Firstly, the coefficient matrices for an intermediate pinned support, an intermediate spring–mass system, left-end support and right-end support of a uniform beam are derived. Next, the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. In this paper, the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made, thus, the last solutions are the exact ones. The effects of attached spring–mass systems on the free vibration characteristics of the 1–4-span beams are studied.

Analytical models for assessment of the safety of multi-span steel beams based on average strains from long gage optic sensors

In structural health monitoring, the safety of a multi-span steel beam is assessed by a comparison between the maximum stress measured during monitoring and the allowable stress of the beam calculated by a design code. The locations and magnitudes of the maximum stress to be monitored keep changing since multi-span beams in buildings and infrastructures are actually subjected to physical loads that have variable positions as well as variable magnitudes. However, most fiber optic sensors measuring the strain only at a local or specified point of a beam cannot consider the variations in the positions of applied loads. Long gage fiber optic sensors that measure average or integrated strain over a relatively long length can consider the variations in magnitudes and positions. On the other hand, the maximum strain in a beam cannot be measured with long gage optic sensors. Therefore, in this paper, analytical models are presented for estimation of the maximum values of strains or stresses in multi-span beams based on the average strains measured by long gage optic sensors. The model was tested in an experiment by comparing the maximum strains directly obtained from electrical gages and the estimated maximum strains from long gage optic sensors.

Free Vibration Analysis of a Cross-Ply Laminated Composite Beam on Pasternak Foundation

In this study, free vibration analysis of a cross-ply laminated composite beam (LCB) on Pasternak foundation was investigated. Natural frequencies of beam on Pasternak foundation are computed using finite element method (FEM) on the basis of Timoshenko beam theory. Effect of both shear deformation and rotary inertia are implemented in the modeling of stiffness and mass matrices. The model was designed in such a way that it can be used for single-stepped cross-section, stepped foundation and multi-span beams. Results of few examples are compared with finding in literature and good agreements were achieved. Natural frequencies of LCBs with different layers arrangements (symmetric and non-symmetric) are compared. For multi-span beam, variation of frequency with respect to number of spans was also studied.

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.

VIBRATIONS OF CONTINUOUS BEAMS WITH INTERMEDIATE SUPPORTS

Results of investigations, based upon logical models application, are used for account of intermediate supports of continuous multispan beams. The matrices obtained take into account all possible combinations of elastic fastenings of both the end and intermediate sections of continuous beams. In this, any additional solutions for the bar parts placed in the whereabouts of localized masses, elastic supports or joints are excluded.

On the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple point masses

Multi-span beams carrying multiple point masses are widely used in engineering applications, but the literature for free vibration analysis of such structural systems is much less than that of single-span beams. The complexity of analytical expressions should be one of the main reasons for the last phenomenon. The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of a multi-span uniform beam carrying multiple point masses. First, the coefficient matrices for an intermediate pinned support, an intermediate point mass, left-end support and right-end support of a uniform beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the related eigenfunctions respectively. The effects of in-span pinned supports and point masses on the free vibration characteristics of the beam are also studied.

Design sensitivity analysis of multi-span non-linear elastic beam–columns with elastic supports

The incremental problem arising from the design perturbation of a materially non-linear elastic beam–column with intermediate supports and elastic end supports is formulated via a second variation of the potential energy functional, augmented with constraint conditions. Design variables are distributed parameters (bending stiffness and transverse load), discrete (compression force, end points dead loads and elasticity of supports) and domain (length and intermediate support points). The state variables perturbations are the solution of an incremental BVP (equilibrium differential equation, static, kinematic and jump conditions) derived from an incremental virtual work equation. A sensitivity formula is obtained by the adjoint method.

FREEVIBRATIONOFSUSPENDEDPIPELINECROSSINGS

The article is devoted to the question of defining own frequencies of trailing pipeline passages with the use of calculation model as a multi-span regular beam (pipe) on elastic supports (suspension brackets together with a cable). The example of calculation is given in comparison with calculations by other methods.

A general approach to formulating the frequency equation for a beam carrying miscellaneous attachments

Beams carrying various lumped attachments have been studied extensively over the years. In this paper, a simple approach is proposed that can be used to readily determine the eigenvalues of an arbitrarily supported single-span or multi-span beam carrying any combination of lumped mass, rotary inertia, grounded translational or torsional spring, grounded translational or torsional viscous damper, an undamped or damped oscillator with or without a rigid body degree of freedom. Rather than solving a generalized eigenvalue problem to obtain the eigenvalues of the system, a frequency equation is formulated instead whose solution can be easily solved either numerically or graphically. The proposed scheme is easy to code, and can be easily modified to accommodate beams with arbitrary supports and carrying any number of miscellaneous lumped attachments.

Free vibration analysis of multi-span beams with intermediate flexible constraints

This paper deals with the free vibration analysis of a multi-span beam with an arbitrary number of flexible constraints. Each span of the continuous beam is assumed to obey Timoshenko beam theory. Considering the compatibility requirements on each constraint point, the relationships between two adjacent spans can be obtained. By using a transfer matrix method, eigensolutions of the entire system can be determined. Some numerical results are shown to present the effects of support stiffness and locations.