Vibration Of Timoshenko Beam Research Articles

An Efficient Numerical Method for Free and Forced Vibrations of Timoshenko Beams with Variable Cross-Section

An Efficient Numerical Method for Free and Forced Vibrations of Timoshenko Beams with Variable Cross-Section

Thermoelastic vibration of Timoshenko beam under the modified couple stress theory and the Moore–Gibson–Thompson heat conduction model

In modeling of micro/nanodevices, numerous approaches have been developed that take into account material length scale parameters to capture size effects. In this work, the variational formulation of the microstructure-dependent Timoshenko beam model is developed considering Hamilton’s principle and modified couple stress theory (MCST) which involves the material length scale parameter. The Moore–Gibson–Thompson (MGT) thermoelastic model is employed to present the governing equations of motion and boundary conditions. The numerical outcomes illustrated by classical and non-classical approaches reveal the behavior of both the deflection and thermal moment of the beam. In addition, the dissipation in the deflection and the thermal moment is visually depicted in standardized curves. Several peaks in graphical plots are then detected as a function of the dimensionless time, which bring to light some interesting features of MGT model in the prediction of thermoelastic vibration of the microstructure model of Timoshenko beam.

Minimum stiffness and optimal position of intermediate elastic support to maximize the fundamental frequency of a vibrating Timoshenko beam

Minimum stiffness and optimal position of intermediate elastic support to maximize the fundamental frequency of a vibrating Timoshenko beam

Modal analysis-based initial axial force recognition method for jacket platform

Jacket platform is the marine structure that anchor piles to the sea floor, among which the leg is one of the most important components of fixed marine structure. Most of the existing stress and strain monitoring methods for offshore platform ignore the existing forces in the structures caused by deck load and installation errors before the monitoring, which cannot be ignored in the subsequent monitoring and evaluation. Based on this, a method for axial force identification of marine structures is proposed in this paper. The partial differential equation of free vibration of Timoshenko beams with axial force under the influence of damping is derived. This method can accurately identify the axial force of beams without knowing the boundary conditions. The simulation results of the single beam show that this method can be used to identify the axial forces of beams with various cross sections and sizes. Then, the method is applied to the axial force identification of the legs of a four-legged jacket platform. The results show that the method can be applied to the axial force identification of the legs with the relative error (RE) of the identification results is less than 6.5%.

Green's functions based on the Timoshenko beam model for the longitudinal seismic dynamic response of tunnels crossing soft and hard rock strata

Weathered rocks typically exist in the portal section of mountain tunnels where the tunnels in the portal section often cross soft and hard rock strata. The interface between the soft and hard rocks is the control point in the seismic design of tunnels. However, there is currently no seismic design method based on dynamic methods that consider tunnels crossing soft and hard rock strata along their longitudinal direction. In this study, two elastic foundation beams in different rock strata were used to model the behaviour of a tunnel crossing soft and hard rock strata. Based on the Timoshenko beam vibration response theory, the higher-order differentials of the beam dynamic governing equation are transformed into algebraic equations using the Laplace transform. The closed-form solution of the dynamic response of the entire beam is derived by combining the boundary and deformation coordination conditions of two elastic foundation beams. Subsequently, the dynamic response of any point along the longitudinal direction of the elastic foundation beam (tunnel) crossing the soft and hard rock strata was obtained using Green's function. The validity of the method was verified by comparing it with a finite difference model. Finally, a parametric analysis was performed based on the analytical method derived in this study to investigate the influences of parameters such as the liner elastic modulus, liner thickness, modulus of elasticity ratio of hard rock to soft rock, and axial forces, on the seismic response of tunnels crossing soft and hard rock strata. The results suggest that with an increase in lining thickness, the bending moment and shear force of the tunnel become increasingly significant, but the transverse displacement response of the tunnel was slightly affected. With an increase in the elastic modulus of the lining, the difference in the peak values of the bending moment and shear force increases, whereas the transverse displacement response is slightly diminished. With an increase in the elastic modulus ratio of hard rock to soft rock, the difference in the peak values of the bending moment, shear force, and transverse displacement of the tunnel, varied more along the tunnel axis. With the increase in the axial force, the seismic responses from the bending moment and shear force on the tunnel cross-section were significantly weakened.

Large amplitude forced vibrations of Timoshenko beams using coupled displacement field method

Forced vibration behaviour of a Timoshenko beam, with a concentrated mass at the centre, subjected to a concentrated harmonic force is investigated using the coupled displacement field method. The expressions for the transverse displacement were deduced from the assumed admissible function having a single term undetermined coefficient for a total rotation of the beam using the coupled displacement field method. The equation of motion for the forced vibration condition is derived using the principle of conservation of total energy. The Newmark method is employed to solve the equation of motion to obtain the dynamic response of the Timoshenko beam. The dynamic forced vibration response for the two boundary conditions, simply supported and clamped-clamped, is computed by applying the mode superposition technique for the obtained coupled displacement field equations. The authors also carried out the finite element analysis using ANSYS 21 software to validate the results obtained by the coupled displacement field method, and the results are also validated with the existing literature wherever possible. The obtained results are very closely matching with the FEM and other published results.

Effect of phase-lag on thermoelastic vibration of Timoshenko beam

An explicit formula of coupled three-phase-lag (TPL) thermoelasticity theory under the Timoshenko beam is constructed for microbeam resonators. The constructed mathematical model is based on the modified couple stress theory which implies a prediction of size-dependent effects in microbeam resonators. By using Hamilton’s principle, governing equations for motion and boundary conditions are derived. The thermal moment and thermal deflection of microbeam resonators are studied analytically and numerically. A comparison of the results between modified coupled stress theory and classical theory is executed for TPL, GN-II, and Lord–Shulman (LS) models. Also, a comparison of the results between TPL, GN-II, and LS models for modified coupled stress theory is done. Besides, the result is presented for silicon microbeam for different aspect ratios and phase-lags. It demonstrated the result corresponding to the behavior of thermoelastic frequencies of microbeam resonators.

Direct Determination of Dynamic Elastic Modulus and Poisson’s Ratio of Rectangular Timoshenko Prisms

AbstractIn this paper, the exact solution of the Timoshenko beam vibration frequency equation under free-free boundary conditions is determined with an accurate shear shape factor. The exact soluti...

Nonlinear Modeling of Bolted Lap Jointed Structure with Large Amplitude Vibration of Timoshenko Beams

This paper aims at investigating the nonlinear behavior of a system which is consisting of two free-free beams which are connected by a nonlinear joint. The nonlinear system is modelled as an in-extensional beam with Timoshenko beam theory. In addition, large amplitude vibration assumption is taken into account in order to obtain exact results. The nonlinear assumption in the system necessities existence of the curvature-related and inertia-related nonlinearities. The nonlinear partial differential equations of motion for the longitudinal, transverse, and rotation are derived using the Hamilton’s principle. A set of coupled nonlinear ordinary differential equations are further obtained with the aid of Galerkin method. The frequency-response curves are presented in the section of numerical results to demonstrate the effect of the different dimensionless parameters. It is shown that the nonlinear bolted-lap joint structure exhibits a hardening-type behavior. Furthermore, it is found that by adding a nonlinear spring the system exhibits a stronger hardening-type behavior. In addition, it is found that the system shows nonlinear behavior even in the absence of the nonlinear spring due to the nonlocal nonlinearity assumption. Moreover, it is shown that considering different engineering beam theories lead to different results and it is found that the Euler-Bernoulli beam theory over-predict the resonance frequency of the structure by ignoring rotary inertia and shear deformation.

Spectral element analysis of free - vibration of Timoshenko beam

Spectral element analysis of free - vibration of Timoshenko beam

Study of non-linear dynamic behavior of open cracked functionally graded Timoshenko beam under forced excitation using harmonic balance method in conjunction with an iterative technique

• An easier method based on harmonic balance with an iterative technique is proposed. • Influence of higher order harmonics on non-linear dynamic response is highlighted. • Effects of cracks on nonlinear dynamic response of C C and C F beams are obtained. • Effects of material indices on nonlinear forced vibration response are obtained. The nonlinear modeling and subsequent dynamic analysis of cracked Timoshenko beams with functionally graded material (FGM) properties is investigated for the first time using harmonic balance method followed by an iterative technique. Crack is assumed to be open throughout. During modeling, nonlinear strain–displacement relation is considered. Rotational spring model is adopted in order to model the open cracks. Energy formulations are established using Timoshenko beam theory. Nonlinear governing differential equations of motion are derived using Lagrange's equation. In order to incorporate the influence of higher order harmonics, harmonic balance method is employed. This reduces the governing differential equations into nonlinear set of algebraic equations. These equations are solved using two different iterative techniques. Methodology is computationally easier and efficient as well. This is observed that although assumption of simple harmonic motion (SHM) simplifies the problem, it yields to erroneous results at higher amplitude of motion. However, accuracy of the solution is improved considerably when the contribution of higher order harmonic terms are considered in the analysis. Results are compared with the available results, which confirm the validity of the methodology. Subsequent to that a parametric study on influence of forcing term, material indices and crack parameters on large amplitude vibration of Timoshenko beams is performed for two different boundary conditions.

Study of Natural Frequencies of Timoshenko Beam With Crack Using SERR Method

This paper presents vibration analysis of cracked Timoshenko beam. The compliance coefficients to account for local flexibility due to crack have been obtained by concept of strain energy release rate (SERR) and stress intensity factors (SIF). Governing equation for transverse vibration of Timoshenko beam is solved considering boundary condition for cantilever beam arrangement to obtain natural frequencies. Results are obtained for different crack locations and crack depths. It is verified that presence of crack decreases the natural frequency of beam.

Frequency equation using new set of fundamental solutions with application on the free vibration of Timoshenko beams with intermediate rigid or elastic span

Stepped beams are crucial power transmission components in many mechanical engineering systems. These beams may be dynamically analyzed using the stepped Timoshenko model or the rigid mass model. In this paper, a new set of fundamental solutions is derived in order to normalize the Timoshenko beam equation at the origin of the coordinates. This set of solutions is used to derive the frequency equation of both stepped and rigid mass models. The validity ranges of these models were investigated by comparing the modal frequency results of both models. In addition, selected cases were compared using mode shape analysis. Three different models with classical end conditions are considered through this work. These are free–free, pinned–pinned and clamped–free beam configurations. The numerical results of the current work show that increasing the intermediate diameter ratio and decreasing the length ratio of the rigid mass results in decreasing the percentage deviation between the rigid mass model results and the elastic model results.

Comparing Nonlinear Free Vibrations of Timoshenko Beams with Mechanical or Geometric Curvature Definition

The nonlinear free oscillations of a planar, initially straight Timoshenko beam are investigated by means of the asymptotic development method. Attention is focus on the difference in considering the “mechanical” vs the “geometric” curvature of the axis of the beam, which are different for extensible beams. A comparison of the results obtained by the two models is proposed, and it is shown when they are equivalent and when they give different nonlinear behaviours. A parametric analysis showing the effects of the slenderness (arbitrary, not necessarily large) and of the stiffness of the right-end axial spring is performed.

Coupled Flexural–Longitudinal Vibrations of Timoshenko Double-Beam Systems Induced by Mass Eccentricities

In many instances, marine vessel may suffer unexpected flexural–longitudinal vibrations as the consequences of the unevenly distribution of the inner structures and equipment. In order to study the dynamic behaviors of the marine vessel with the involvement of the mass unevenness, the whole vessel is modeled as a discretely connected double-beam system in this paper. In this model, the misalignment between geometric and gravitational center are considered as the simplification of the mass unevenness. The governing equations of coupled flexural–longitudinal vibrations of Timoshenko beam due to mass eccentricity are derived. Extending to the double-beam model, the natural frequencies, modal shapes and forced response of the system are obtained analytically by using the modified transfer matrix method. The comparison between the present method and the conventional finite element method shows a great match. The result indicates that the existence of mass eccentricity can cause significant coupled flexural–longitudinal vibrations, thus, even only vertical force is applied, the vibration in longitudinal direction can be excited and controlled by the natural frequencies of the flexural vibration.

Axial-transversal coupling in the free nonlinear vibrations of Timoshenko beams with arbitrary slenderness and axial boundary conditions.

The nonlinear free oscillations of a straight planar Timoshenko beam are investigated analytically by means of the asymptotic development method. Attention is focused for the first time, to the best of our knowledge, on the nonlinear coupling between the axial and the transversal oscillations of the beam, which are decoupled in the linear regime. The existence of coupled and uncoupled motion is discussed. Furthermore, the softening versus hardening nature of the backbone curves is investigated in depth. The results are summarized by means of behaviour charts that illustrate the different possible classes of motion in the parameter space. New, and partially unexpected, phenomena, such as the changing of the nonlinear behaviour from softening to hardening by adding/removing the axial vibrations, are highlighted.

Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section

A novel method is proposed to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. For a wide class of Timoshenko beams, this method enables us to reduce the coupled governing differential equations with variable coefficients to a pair of uncoupled second-order differential equations of Sturm–Liouville type with respect to the rotation angle due to bending. The reduced equations contain two important parameters, one describing the variations of translational inertia and bending rigidity along the beam axis, and the other reflecting the comprehensive effect of rotatory inertia and shear deformation. A series of exact analytical solutions are derived from the reduced equations for the first time, and several examples are also provided as benchmarks.

Exact vibration of Timoshenko beam combined with multiple mass spring sub-systems

This paper deals with the analysis of the natural frequencies, mode shapes of an axially loaded beam system carrying ends consisting of non-concentrated tip masses and three spring-two mass subsystems. The influence of system design and sub-system parameters on the combined system characteristics is the major part of this investigation. The effect of material properties, rotary inertia and shear deformation of the beam system is included. The end masses are elastically supported against rotation and translation at an offset point from the point of attachment. Sub-systems are attached to center of gravity eccentric points out of the beam span. The boundary conditions of the ordinary differential equation governing the lateral deflections and slope due to bending of the beam system including developed shear force frequency dependent terms, due to the sub–system suspension, have been formulated. Exact formulae for the modal frequencies and the modal shapes have been derived. Based on these formulae, detailed parametric studies are carried out. The geometrical and mechanical parameters of the system under study have been presented in non-dimensional analysis. The applied mathematical model is presented to cover wide range of mechanical, naval and structural engineering applications.

A weak form quadrature element method for nonlinear free vibrations of Timoshenko beams

Purpose– The purpose of this paper is to highlight the implementation of a recently developed weak form quadrature element method for nonlinear free vibrations of Timoshenko beams subjected to three different boundary conditions.Design/methodology/approach– The design of the paper is based on considering the geometrically nonlinear effects of axial strain, bending curvature, and shear strain. Then the quadrature element formulation of the beam is introduced.Findings– The efficiency of the method is demonstrated by a convergence study. Ratios of the nonlinear fundamental frequencies to the corresponding linear frequencies are extracted. Their variations with the ratio of amplitude to radius of gyration and the slenderness ratio are examined. The effects of the nonlinearity on higher order frequencies and mode shapes are also investigated.Originality/value– The computed results show fast convergence and compare well with available literature results.

Large amplitude free vibrations of Timoshenko beams at higher modes using coupled displacement field method

A simple but accurate continuum solution for the shear flexible beam problem using the energy method involves in assuming suitable single term admissible functions for the lateral displacement and total rotation. This leads to two non-linear temporal differential equations in terms of the lateral displacement and the total rotation and are difficult, if not impossible, to solve to obtain the large amplitude fundamental frequencies of beams as a function of the amplitude and slenderness ratios of the vibrating beam. This situation can be avoided if one uses the concept of coupled displacement field where in the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation. In this paper the lateral displacement field is assumed and the field for the total rotation is evaluated through the coupling equation. This approach leads to only one undetermined coefficient which can easily be used in the principle of conservation of total energy of the vibrating beam at a given time, neglecting damping. Finally, through a number of algebraic manipulations, one gets a nonlinear equation of Duffing type which can be solved using any standard method. To demonstrate the simplicity of the method discussed above the problem of large amplitude free vibrations of a uniform shear flexible hinged beam at higher modes with ends immovable to move axially has been solved. The numerical results obtained from the present formulation are in very good agreement with those obtained through finite element and other continuum methods for the fundamental mode, thus demonstrating the efficacy of the proposed method. Also some interesting observations are made with variation of frequency Vs amplitude at different modes.