Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads

The vibration of an Euler-Bernoulli beam, resting on a nonlinear Kelvin-Voight viscoelastic foundation, traversed by a moving load is studied in the frequency domain. The objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response. The parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient. The solution is obtained by using the Galerkin method in conjunction with the multiple scales method (MSM). The governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations. Subsequently, the solution is calculated for different harmonics by using the MSM as one of the powerful perturbation techniques. The steady-state responses of the main harmonic as well as its two super-harmonics are then obtained. As a case study, a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics. Moreover, a thorough stability analysis of the system is carried out.     

A substructure approach tailored to the dynamic analysis of multi-span continuous beams under moving loads

The paper deals with the dynamic analysis of multiply supported continuous beams subjected to moving loads, which in turn can be modelled either as moving forces or moving masses. A dedicated variant of the component mode synthesis (CMS) method is proposed in which the classical primary-secondary substructure approach (SA) is tailored to cope with slender (i.e. Euler-Bernoulli) continuous beams with arbitrary geometry. To do this, the whole structure is ideally decomposed in primary and secondary spans with convenient restraints, whose exact eigenfunctions are used as assumed local modes; the representation of the internal forces is improved with the help of two additional assumed modes for each primary span, while primary-secondary influence functions allow satisfying the kinematical compatibility between adjacent spans; the continuous beam is then re-assembled, and the Lagrange's equations of motion are derived in a compact block-matrix setting for both moving force and moving mass model. Numerical examples demonstrate accuracy and efficiency of the proposed procedure. An application with a platoon of high-speed moving masses confirms that the inertial effects neglected in the moving force model may have a significant impact in the structural response.     

Nonlinear response of shear deformable beams on tensionless nonlinear viscoelastic foundation under moving loads

In this paper, a boundary element method is developed for the geometrically nonlinear response of shear deformable beams of simply or multiply connected constant cross-section, traversed by moving loads, resting on tensionless nonlinear three-parameter viscoelastic foundation, undergoing moderate large deflections under general boundary conditions. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse moving loading as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the transverse displacement, to the axial displacement and to a stress functions and solved using the Analog Equation Method, a Boundary Element based method. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which is solved using an efficient time discretization scheme, from which the transverse and axial displacements are computed. The evaluation of the shear deformation coefficient is accomplished from the aforementioned stress function using only boundary integration. Analyses are performed to illustrate, wherever possible, the accuracy of the developed method, to investigate the effects of various parameters, such as the load velocity, load frequency, shear deformation, foundation nonlinearity, damping, on the beam displacements and stress resultants and to examine how the consideration of shear and axial compression affects the response of the system.     

Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load

The present paper investigates the convergence of the Galerkin method for the dynamic response of an elastic beam resting on a nonlinear foundation with viscous damping subjected to a moving concentrated load. It also studies the effect of different boundary conditions and span length on the convergence and dynamic response. A train–track or vehicle–pavement system is modeled as a force moving along a finite length Euler–Bernoulli beam on a nonlinear foundation. Nonlinear foundation is assumed to be cubic. The Galerkin method is utilized in order to discretize the nonlinear partial differential governing equation of the forced vibration. The dynamic response of the beam is obtained via the fourth-order Runge–Kutta method. Three types of the conventional boundary conditions are investigated. The railway tracks on stiff soil foundation running the train and the asphalt pavement on soft soil foundation moving the vehicle are treated as examples. The dependence of the convergence of the Galerkin method on boundary conditions, span length and other system parameters are studied.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Effects of surface irregularities upon the dynamic response of bridges under suspended moving loads

The effects of surface irregularities upon the dynamic response of bridges under suspended moving loads is analyzed by means of a theoretical study. The paper deals with geometrical imperfections of two types, global and local ones. The former may represent irregularities due to permanent loads, creep process or deflections produced by prestressing forces acting on young concrete. The local ones may represent the initial joint or local defects. The vehicle dynamics model is a simple degree of freedom oscillator and the method of analysis used is a Rayleigh-Ritz one. In some cases the amplification factors are found to be much greater than those given by current international design codes and the dynamic effects can no longer be neglected.     

A finite element for the vibration analysis of Timoshenko beams

A Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation. The element properties are based on a coupled displacement field; the lateral deflection is interpolated as a quintic polynomial function and the cross-sectional rotation is linked to the deflection by specifying satisfaction of the governing differential equation of moment equilibrium in the absence of the rotary inertia term. Numerical results confirm that this procedure does not preclude convergence to true Timoshenko theory solutions since rotary inertia is included in lumped form at element ends. The new Timoshenko beam element has good convergence characteristics and where comparison can be made in numerical studies it is shown to be generally more efficient than previous elements.     

Inelastic response of beams under sinusoidal and random loads

A new analytical model has been suggested for the hysteretic behaviour of beams. The model can be directly used in a response analysis without bothering to locate the precise point where the unloading commences. The model can efficiently simulate several types of realistic softening hysteretic loops. This is demonstrated by computing the response of cantilever beams under sinusoidal and random loadings. Results are presented in the form of graphs for maximum deflection, bending moment and shear.     

A sector finite element for dynamic analysis of thick plates

A 24 degree of freedom sector finite element is developed for the static and dynamic analysis of thick circular plates. The element formulation is based on Reissner's thick plate theory. The convergence characteristic of the elements is first studied in a static example of an unsymmetrically loaded annular plate. The obvious advantageous effect of including the twist derivatives of deflection as degrees of freedom is shown. The elements are then used to analyze the natural frequencies of an annular plate with various ratios of inner to outer radius. The results are in good agreement with an alternative solution in which thick plate theory is used. The versatility of this finite element is finally demonstrated by performing free vibration analysis of an example of clamped sector plates with various thicknesses and different sectorial angles.     

A time delay control for a nonlinear dynamic beam under moving load

The bifurcation resulted from moving force may lead to instability for the system. Based on time delay feedback controller, a nonlinear beam under moving load is discussed in the case of the primary resonance and the 1/3 subharmonic resonance. The bifurcation may be eliminated or the bifurcation point's position may be changed. The perturbation method is used to obtain the bifurcation equation of the nonlinear dynamic system. The result indicates time delay feedback controller may work well on this system, but the selection of a proper time delay and its coefficient may depend on the engineering condition. This paper presents some theoretical results.     

Steady state response of an infinite beam on a viscoelastic foundation with moving distributed mass and load

Compared with the moving concentrated load model, it is more realistic and proper to use the moving distributed mass and load model to simulate the dynamics of a train moving along a railway track. In the problem of a moving concentrated load, there is only one critical velocity, which divides the load moving velocity into two categories: subcritical and supercritical. The locus of a concentrated load demarcates the space into two parts: the waves in these two domains are called the front and rear waves,respectively. In comparison, in the problem of moving distributed mass and load, there are two critical velocities, which results in three categories of the distributed mass moving velocity. Due to the presence of the distributed mass and load, the space is divided into three domains, in which three different waves exist. Much richer and different variation patterns of wave shapes arise in the problem of the moving distributed mass and load. The mechanisms responsible for these variation patterns are systematically studied. A semi-analytical solution to the steady-state is also obtained, which recovers that of the classical problem of a moving concentrated load when the length of the distributed mass and load approaches zero.     

Studies of the performance of particle dampers under dynamic loads

This paper presents a systematic investigation of the performance of particle dampers (vertical and horizontal) attached to a primary system (single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF)) under different dynamic loads (free vibration, stationary random excitation as well as nonstationary random excitation, with single component or multi-component), and the optimum operating regions are all determined. The amount of dissipated energy due to impact and friction, and the concept of “Effective Momentum Exchange” are shown to be suitable “global” measures to interpret the physics involved in the behavior of particle dampers. Using the well-established discrete element method, the motion of vertical particle dampers can be analyzed and classified into three different regions, and the associated damping characteristics can be interpreted. The first mode of a MDOF primary system can be effectively controlled by a properly designed particle damper; however, the higher modes are more affected by other parameters. Consequently, extensive parametric studies are presented to evaluate the effects of various system parameters, such as: mass ratio, primary system damping, coefficient of restitution, container dimensions, excitation amplitude and components, input locations and damper locations.     

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.     

Reverberation-ray analysis of moving or distributive loads on a non-uniform elastic bar

The method of reverberation-ray matrix has been developed and successfully applied to analyze the wave propagation in a multibranched framed structure or in a layered medium. However, the formulation is confined to the case of external concentrated loads applied at the junctions. This paper aims to extend the formulation of reverberation-ray matrix to cases of continuously distributed loads and point moving loads. To this end, a non-uniform bar subjected to these new types of loads is considered for illustration. The difference lies largely in the exact solutions, which include the particular parts due to the loads considered in this paper. The compatibility between displacements in the dual coordinates for a single member is utilized to derive the phase relations. For several types of loadings, numerical results are given and compared with the exact solutions or those obtained by other available method. Exact agreement is observed, thereby validating the present approach. The commonly adopted method that transforms distributed load to equivalent nodal forces is also discussed.     

A conserved quantity and the stability of axially moving nonlinear beams

Free nonlinear transverse vibration is investigated for an axially moving beam modeled by an integro-partial-differential equation. Based on the equation, a conserved quantity is defined and confirmed for axially moving beams with pinned or clamped ends. The conserved quantity is applied to demonstrate the Lyapunov stability of the straight equilibrium configuration in transverse nonlinear of beam with a low axial speed.     

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation.In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.