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Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

About 30 years ago, I was among several students mentored by Professor Yang at Stony Brook to enter the field of particle accelerator physics. Since then, I have been fortunate to work on several major accelerator projects in USA and in China, guided and at times directly supported by Professor Yang. The field of accelerator physics is flourishing worldwide both providing indispensable tools for fundamental physics research and covering an increasingly wide spectrum of applications beneficial to our society.

Complex vibrations of an Euler–Bernoulli beam with different types of nonlinearities are considered. An arbitrary beam clamping is considered, and deflection constraints (point barriers) are introduced in some beam points along its length. The influence of a constraint, as well as of the amplitude and frequency of excitation on the vibrations is analyzed. Scenarios of transition to chaos owing to the introduced nonlinearities are reported.

In this paper, the theory of nonlinear interaction of two-layered beams and plates taking into account design, geometric and physical nonlinearities is developed. The theory is mainly developed relying on the first approximation of the Euler–Bernoulli hypothesis. Winkler type relation between clamping and contact pressure is applied allowing the contact pressure to be removed from the quantities being sought. Strongly nonlinear partial differential equations are solved using the finite difference method regarding space and time coordinates. On each time step the iteration procedure, which improves the contact area between the beams is applied and also the method of changeable stiffness parameters is used. A computational example regarding dynamic interaction of two beams depending on a gap between the beams is given. Each beam is subjected to transversal sign-changeable load, and the upper beam is hinged, whereas the bottom beam is clamped. It has been shown that for some fixed system parameters and with an increase of the external load amplitude, synchronization between two beams occurs with the upper beam vibration frequency. Qualitative analysis of the interaction of two noncoupled beams is also extended to the study of noncoupled plates. Charts of beam vibration types versus control parameters {q₀, ω_p}, i.e. the frequency and amplitude of excitation are constructed. Similar and previously described competitions have been reported in the case of two-layered plates.

We consider a thin multidomain of ℝ^N, N ≥ 2, consisting (e.g. in a 3D setting) of a vertical rod upon a horizontal disk. In this thin multidomain we introduce a bulk energy density of the kind W(D²U), where W is a convex function with growth p ∈ ]1,+∞[, and D²U denotes the Hessian tensor of a scalar (or vector-valued) function U. By assuming that the two volumes tend to zero with the same rate, under suitable boundary conditions, we prove that the limit model is well-posed in the union of the limit domains, with dimensions, respectively, 1 and N - 1. Moreover, we show that the limit problem is uncoupled if , "partially" coupled if , and coupled if N - 1 < p. The main result is applied in order to derive the equilibrium configuration of two joint beams, T-shaped, clamped at the three endpoints and subject to transverse loads. The main result is also applied in order to describe the equilibrium configuration of a wire upon a thin film with contact at the origin, when the thin structure is filled with a martensitic material.

Crack identification in thick beams has improved increasing considerations from the scientific and building areas since the unpredicted structural failure may cause disastrous, catastrophic and life trouble. The goal of the present examination is to predict the unknown crack location and its depth in thick beams from the information of frequency data obtained from experimental examination. The effectiveness of the proposed strategy is approved by numerical simulations in view of experimental data for a cantilever beam, free-free beam and simply supported beam. With the improvements in delicate figuring, optimization strategies are acknowledged to be an extremely proficient instrument to offer an answer for crack identification issue. In the simulation modeling, the parameters, for example, shift; modal assurance criterion (MAC) and stiffness, are predicted by utilizing optimized deep learning neural network (ODNN) approach in view of crack location and size. To improve the weight in DLNN, the opposition-based ant lion (OAL) is used by minimizing the mean square error (MSE) rate. The result shows that the proposed model achieves the optimal performance compared with existing techniques.

This paper presents an effective way in damage detection of beam structures using the wavelet analysis along with the general beam solution. Two case studies are considered: (1) a clamped beam with a damage point of zero bending moment; and (2) a simply supported beam with a transverse open crack. The proposed method is capable of revealing the precise damage locations which is generally difficult to be identified using the standard eigenvalue analysis.

The initial stresses due to dead loads have an influence on the natural frequencies of bridges. In this paper, a dynamic stiffness-based method is proposed for determining the natural frequencies of uniform elastic beams with allowance for the dead load effect. Firstly, the governing differential equation including the effect of dead loads is derived. Next, the analytical dynamic stiffness matrix is obtained by applying the displacements and forces boundary conditions at the ends of the beam. In order to solve analytically the governing differential equation, the modified dynamic stiffness matrix is defined by converting the governing quasi-static boundary value problem into an equivalent set of initial value problems. Finally, the Wittrick–Williams algorithm is implemented to extract the natural frequencies from the modified dynamic stiffness matrix. Numerical examples are presented and corresponding parameter studies have been performed to illustrate the applicability and reliability of the proposed method. It is demonstrated that the proposed dynamic stiffness matrix-based method is effective even though the beam is considered as a single element without adding additional nodes.

This paper presents an investigation on the influence of stress gradient on the elastic critical stress of distortional buckling of cold-formed steel sections supporting wall sheeting or roof cladding in buildings. The critical stress of distortional buckling of cold-formed steel section beams subjected to a uniformly distributed transverse loading is calculated using the model proposed recently by Li and Chen. The sections investigated in the paper include channel, zed, and sigma sections. Numerical examples are provided that highlight the influence of stress gradient, section dimensions, and sheeting lateral restraints on the critical stress of distortional buckling.

Results from finite element modeling (FEM) of large-scale steel-concrete composite beams strengthened in flexure with prestressed carbon fiber-reinforced polymer (CFRP) plate were validated with experimental results and presented in this paper. The effect of varying the level of prestressing as percentage of the ultimate tensile strength of the CFRP plate was investigated. Comparison was carried out in terms of overall load-deflection behavior, strain profile along the length of the CFRP plate, and strain distribution across the depth of the beam at mid-span section. Very good agreement was observed between the finite element (FE) and the experimental results. The validated FE models were used to perform a comprehensive parametric study to investigate the changes in the behavior through wider range of prestressing levels and then, determine the optimum prestressing level that maintain the unstrengthened beams' original ductility (or energy absorption). An iterative analytical model was also developed, validated with both the FE model and the experimental results, and showed good agreement. A parametric study was carried out to investigate the effect of changing the yield strength of the steel and the concrete compressive strength on the moment of resistance of the section and the strain in the CFRP plate at ultimate.

Fundamental principles controlling the deflection behavior of a simply supported beam responding to the impact action of a solid object is revealed in this paper. The significant mitigating effects that the mass of the beam have upon its impact resistant behavior have been illustrated with examples. It is a myth that the static resistance of the beam is indicative of its impact resistance. The important effects of "cushioning" and the higher modes phenomenon have also been identified by the analytical study presented herein. Hand calculations and computer analysis methods are introduced and evaluated by comparison with results obtained from finite element analyses using LS-DYNA.

Cracks in structural members lead to local changes in their stiffness and consequently their static, dynamic and stability behavior is altered. The influence of cracks on dynamic characteristics like free vibration, buckling and parametric resonance characteristics of a cracked beam with a transverse crack using finite element method (FEM) is investigated in the present work. Modal testing of beams with transverse open crack is conducted using FFT analyzer to verify the frequencies of vibration of beams. The crack is assumed to be open type and the analysis is linear based on small deformation theory neglecting damping. The loading on the beam is considered as axial with a simple harmonic fluctuation with respect to time. A two-noded Timoshenko beam element with provision of crack is used in this study. The equation of motion represents a system of second-order differential equations with periodic coefficients of the Mathieu–Hill type. The development of the regions of instability arises from Floquet's theory and the periodic solution is obtained by Bolotin's approach using FEM. It is observed that the frequencies of vibration and buckling load of the beam are influenced significantly by location and depth of cracks. It is observed that, for a given location of crack, the onset of instability occurs earlier with increase in depth of crack. As the location of crack moves from the fixed end to the free end the excitation frequency increases. The instability occurs later and the width of the instability regions reduces. When the damage is near to the free end, the instability region almost coincides with the instability region for the undamaged beam. This means that the damage near the fixed end is more severe on the dynamic instability behavior than that of the crack located at other positions. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance and safety.

The free vibration of functionally graded (FG) beams with various boundary conditions resting on a two-parameter elastic foundation in the thermal environment is studied using the third-order shear deformation beam theory. The material properties are temperature-dependent and vary continuously through the thickness direction of the beam, based on a power-law distribution in terms of the volume fraction of the material constituents. In order to discretize the governing equations, the differential quadrature method (DQM) in conjunction with the Hamilton’s principle is adopted. The convergence of the method is demonstrated. In order to validate the results, comparisons are made with solutions available for the isotropic and FG beams. Through a comprehensive parametric study, the effect of various parameters involved on the FG beam was studied. It is concluded that the uniform temperature rise has more significant effect on the frequency parameters than the nonuniform case.

Beams and columns subjected to the axial pressure are studied. Critical buckling loads are established for stepped beams clamped at one end and elastically fixed at the other end. The beams under consideration are of piecewise constant thickness and are weakened by cracks emanating from re-entrant corners of steps. The cracks are assumed to be stable part-through surface cracks. The influence of a crack on the stability of the beam is modeled by the method of distributed line spring known in the elastic fracture mechanics. Numerical results are presented for beams with a single step making use of different stress correction functions.

Evaluating the performance of beam-like structures in terms of their current boundary conditions, stiffness and modal properties can be challenging as the structures behave differently from their designed conditions due to aging. The purpose of the current study is to determine the flexural rigidity of beam-like structures when their support conditions are not fully understood. A novel optimization scheme is proposed for estimation of the flexural stiffness and the capacity of the beam-like structures under moving loads. The proposed method is applied to various profiles of the beams made of different materials with unknown boundary conditions, and the effects of damage, excitation and optimization algorithm are rigorously investigated. The results of the numerical and experimental studies showed that the proposed substructural bending rigidity identification (SBI) method can correctly assess the in-service flexural stiffness, fixity of the boundary condition and the load-carrying capacity. This technique can be considered as a cost-effective method for periodic monitoring, load rating and model updating of the beam-like structures.

This paper studies the nonlinear dynamic responses of graphene-reinforced composite (GRC) beams in a thermal environment. It is assumed that a laminated beam rests on a Pasternak foundation with viscosity and consists of GRC layers with various volume fractions of graphene reinforcement to construct a functionally graded (FG) pattern along the transverse direction of the beam. An extended Halpin–Tsai model which is calibrated against the results from molecular dynamics (MD) simulations is used to evaluate the material properties of GRC layers. The mechanical model of the beam is on the establishment of a third-order shear deformation beam theory and includes the von-Kármán nonlinearity effect. The model also considers the foundation support and the temperature variation. The two-step perturbation technique is first applied to solve the beam motion equations and to derive the nonlinear dynamic load–deflection equation of the beam. Then a Runge–Kutta numerical method is applied and the solutions for this nonlinear equation are obtained. The influence of FG patterns, visco-elastic foundation, ambient temperature and applied load on transient response behaviors of simply supported FG-GRC laminated beams is revealed and examined in detail.

This study uses the piezoelectric technology to collect vibration energy from the fixed-fixed nonlinear elastic beams attached with the piezo-patch between the two ends. Both single elastic steel sheet (SESS) and double elastic steel sheet (DESS) systems are investigated and correlated. To simulate the power generation of the vibration energy harvester (VEH) of both the SESS and the DESS in different engineering elements, the simple harmonic external force generated by a shaker at the location of the piezo-patch is used as the source. With this, more vibration converted electric energy is derived from the transverse deformation and flapping from the DESS than the SESS beam. The equation of a nonlinear Euler–Bernoulli beam is coupled with the electric energy equation of the piezo-patch to simulate the SESS VEH system. The flapping force from the DESS VEH system can be considered the concentrated external load applied on the SESS beam model. The method of multiple scales (MOMS) is employed to analyze this nonlinear problem. The fixed points plots and the numerical results confirm this theory presented for the two beam systems, which can be used for evaluating similar engineering systems. Experiments are also performed in this study. The Taguchi method is used to analyze the optimum locations of the shaker and piezo-patch, as well as the confidence level of the factors. The method of nonlinear analysis presented in this study demonstrates its accuracy compared with the linear case. The transverse DESS VEH model proposed is proved to be feasible and more effective than the SESS system.

In this work, transverse vibrations of the piston rod of a hydraulic cylinder, which is connected with a system characterized by its own high weight, e.g. a tank gate, were considered. The case is considered when the actuator is fully extended and the piston rod is not affected by external static axial forces (the actuator acts in a horizontal position, as a result of which the weight load at its end does not compress the piston rod). To analyze this issue, a beam model was developed taking into account the longitudinal inertia of the mass element associated with one of the ends of the system. The boundary problem of vibrations was formulated using the Bernoulli–Euler theory. Taking into account mass inertia (directed longitudinally) results in the appearance of nonlinear terms in equations describing the behavior of the system during vibrations. The small parameter method was used to finally formulate the problem of nonlinear vibrations. In the description thus adopted, the vibrations of the considered system strictly depend on the dead weight of the gate valve as well as the amplitude of the piston rod oscillations. The results of numerical simulations are presented taking into account the impact of piston rod stiffness and stiffness of mounting to the valve on its vibrations. In the considered range of masses, the effect of amplitude on the value of the natural frequency of the system is presented. Theoretical considerations have been confirmed to some extent by experimental studies.

In this paper, the internal and external cancellation phenomena for damped beams subjected to multi-moving loads are investigated in detail. To start, the theory for the vibration of a simply supported beam is revisited by including the effect of damping. For the first time, a simple expression is derived for the free vibration of the damped beam under multi-moving loads. Based on the concept of local minimum, two cancellation conditions are identified. One is the internal cancellation, which relates to the inherent property of the beam and is conventionally known. The other is the newly formulated external cancellation that relates to the number and spacing of moving loads. For comparison, both the resonant condition and the optimal criterion for span length of the bridge are also briefed. By comparing with the classical solution, the present simple expression for the free vibration of the beam is firstly validated. Then the factors affecting the cancellation are investigated against various load cases and damping levels. The results show that external cancellation occurs more frequently due to the increase in the number and spacing of the moving loads. The damping of the beam has a leaking effect on cancellation, in that nonzero vibration may occur, but it is also quickly damped out by damping itself.