Forced Vibration of Delaminated Timoshenko Beams under the Action of Moving Oscillatory Mass

This paper presents the dynamic response of a delaminated composite beam under the action of a moving oscillating mass. In this analysis the Poisson's effect is considered for the first time. Moreover, the effects of rotary inertia and shear deformation are incorporated. In our modeling linear springs are used between delaminated surfaces to simulate the dynamic interaction between sub-beams. To solve the governing differential equations of motion using modal expansion series, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes necessary for forced vibration analysis. The obtained results for the free and forced vibrations of beams are verified against reported similar results in the literatures. Moreover, the maximum dynamic response of such beam is compared with an intact beam. The effects of different parameters such as the velocity of oscillating mass, different ply configuration and the delamination length, its depth and spanwise location on the dynamic response of the beam are studied. In addition, the effects of delamination parameters on the oscillator critical speed are investigated. Furthermore, different conditions under which the detachment of moving oscillator from the beam will initiate are investigated

Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity

This study focuses on the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a moving force under the influence of three types of motions, including accelerating, decelerating and constant velocity types of motion, respectively. The beam’s nonlinear governing coupled partial differential equations (PDEs) of motion for the bending rotation of warped cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. To obtain the dynamic response of the beam under the action of a moving force, the derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic response is obtained using mode summation technique. Furthermore, the calculated results are verified with those obtained by finite element method (F.E.M.) analysis. In the next step a parametric study on the response of the beam is conducted by changing the magnitude of the traveling concentrated force, its velocity and beam’s boundary conditions and likewise their sensitivity on the beam’s dynamic response are studied, respectively. It is observed that the existence of quadratic-cubic nonlinearity in the governing coupled PDEs of motion renders hardening/softening behavior on the dynamic response of the beam. Moreover, it is noticed that any restriction on the beam mid-plane stretching will introduce nonlinear behavior in the beam’s PDEs of motion.Вивчено нелiнiйну динамiчну реакцiю нахиленої балки Тимошенка з рiзними граничними умовами пiд дiєю рухомої сили, включаючи вплив рухiв трьох типiв, зокрема руху з прискоренням, уповiльненням та сталою швидкiстю. З допомогою принципу Гамiльтона отримано нелiнiйнi зв’язанi рiвняння з частинними похiдними для вигину обертання деформованого перетину, поздовжнього та поперечного зсувiв. Для встановлення динамiчної реакцiї балки пiд дiєю рухомої сили було розв’язано отриманi нелiнiйнi зв’язанi рiвняння з використанням методу Гальоркiна. Далi динамiчну реакцiю балки було одержано з використанням технiки модального пiдсумовування. Встановленi результати було перевiрено за допомогою методу скiнченних елементiв. На наступному кроцi було проведено параметричний аналiз реакцiї балки при змiнi величини рухомої концентрованої сили, її швидкостi та граничних умов, а також чутливостi реакцiї балки на цi параметри. Було помiчено, що наявнiсть квадратичної або кубiчної нелiнiйностi у зв’язаних рiвняннях з частиними похiдними робить динамiчну реакцiю балки бiльш твердою або м’якою, а будь-якi обмеження на розтягування середньої площини вводять нелiнiйнiсть у рiвняння руху балки

Nonlinear vibration and comfort analysis of high-speed trains moving over railway bridges

The ride comfort of high-speed trains passing over railway bridges is studied in this paper. The effects of some nonlinear parameters in a carriage-track-bridge system are investigated such as the load-stiffening characteristics of the rail-pad and the ballast, rubber elements in the primary and secondary suspensions systems. The influence of the track irregularity and train speed on two comfort indicators, namely Sperling's comfort index and the maximum acceleration level, are also studied. Timoshenko beam theory is used for modelling the rail and bridge and two layers of parallel damped springs in conjunction with a layer of mass are used to model the rail-pads, sleepers and ballast. A randomly irregular vertical track profile is modelled, characterised by a power spectral density (PSD). The 'roughness' is generated for three classes of tracks. Nonlinear Hertz theory is used for modelling the wheel-rail contact

Response of beams on nonlinear viscoelastic foundations to harmonic moving loads

The response of infinite beams supported by nonlinear viscoelastic foundations subjected to harmonic moving loads is studied. A straightforward solution technique applicable in the frequency domain is presented in this paper. The governing equations are solved using a perturbation method in conjunction with complex Fourier transformation. A closed-formed solution is presented in an integral form based on the presented Green’s function and the theorem of residues is used for the calculation of integrals. The solution is directed to compute the deflection and bending moment distribution along the length of the beam. A parametric study is carried out and influences of the load speed and frequency on the beam responses are investigated. It is found that for an excitation frequency of Ω there exist superharmonics of 3Ω O(ε), 5Ω O(ε2), …, (2n-1) × Ω O(εn-1), n = 1, 2,

Ride comfort of high-speed trains travelling over railway bridges

The ride comfort of high-speed trains passing over railway bridges is studied in this paper. A parametric study is carried out using a time domain model. The effects of some design parameters are investigated such as damping and stiffness of the suspension system and also ballast stiffness. The influence of the track irregularity and train speed on two comfort indicators, namely Sperling's comfort index and the maximum acceleration level are also studied. Two types of railway bridges, a simple girder and an elastically supported bridge are considered. Timoshenko beam theory is used for modelling the rail and bridge and two layers of parallel damped springs in conjunction with a layer of mass are used to model the rail-pads, sleepers and ballast. A randomly irregular vertical track profile is modelled, characterized by its power spectral density (PSD). The ‘roughness' is generated for three classes of tracks. Nonlinear Hertz theory is used for modelling the wheel-rail contact. The influences of some nonlinear parameters in a carriage-track-bridge system, such as the load-stiffening characteristics of the rail-pad and the ballast and that of rubber elements in the primary and secondary suspension systems, on the comfort indicators are also studied. Based on Galerkin's method of solution, a new analytical approach is developed for the combination between the rigid and flexural mode shapes, which could be used not only for elastically supported bridges but also other beam-type structures

Parametrically excited vibration of a timoshenko beam on random viscoelastic foundation subjected to a harmonic moving load

The vibration response of a Timoshenko beam supported by a viscoelastic foundation with randomly distributed parameters along the beam length and subjected to a harmonic moving load, is studied. By means of the first-order two-dimensional regular perturbation method and employing appropriate Green's functions, the dynamic response of the beam consisting of the mean and variance of the deflection and of the bending moment are obtained analytically in integral forms. Results of a field measurement for a test track are utilized to model the uncertainty of the foundation parameters. A frequency analysis is carried out and the effect of the load speed on the response is studied. It is found that the covariance functions of the stiffness and the loss factor both have the shape of an exponential function multiplied by a cosine function. Furthermore, it is shown that in each frequency response there is a peak value for the frequency, which changes inversely with the load speed. It is also found that the peak value of the mean and also standard deviation of the deflection and bending moment can be a decreasing or increasing function of the load speed depending on its frequency