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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.

The nonlinear behavior and in-plane stability of parabolic shallow arches with elastic rotational supports are investigated. A central concentrated load is applied to create the compression in the supports. Nonlinear buckling analysis based on the virtual work formulation is carried out to obtain the critical load for both symmetric snap-through buckling and anti-symmetric bifurcation buckling. It is found that the effect of rotational stiffness of the elastic supports on the critical loads is significant. The critical load increases as either the initial stiffness coefficient α or the stiffening rate β increases. The effect of the stiffening rate β on the critical loads decreases, as the initial stiffness coefficient α increases. In addition, the influence of rotational stiffness on the bifurcation buckling is larger than that on the snap-through buckling. The limit of the geometric parameter λ between the bifurcation and snap-through buckling modes also increases with the increase in the stiffening rate β. In addition, the snap-through buckling mode governs the buckling of the arch with larger stiffening rates.

This paper deals with the nonlinear dynamic response of elastically supported stiffened plates with initial stresses under impact loads. A stiffened plate is assumed to be composed of a plate with some stiffeners, which are treated separately. The plate is modeled by the thin plate theory, whereas the stiffeners are considered as geometrically nonlinear Euler–Bernoulli beams. First, the equations of both the kinetic energies and strain energies of the plate and stiffeners are established. Then, the dynamic equilibrium equations for the stiffened plate are derived as the Lagrange’s equation of the functional. A parametric analysis is performed to evaluate how initial stresses, initial geometric imperfections, elastic supports, impact loads and configuration of stiffeners affect the time-history responses of the stiffened plates. Some useful nonlinear dynamic properties are obtained, which serve as references for engineering design and application.

This paper investigates the vibration suppression of an elastically supported nonlinear cantilever beam attached to an inertial nonlinear energy sink (NES). The nonlinear terms introduced by the NES are transferred as the external excitations acting on the beam. The governing equations of the nonlinear beam with an inertial NES are derived according to the Lagrange equations and the assumed mode method. The linear and nonlinear frequencies of the beam are numerically obtained by the Rayleigh–Ritz method and the direct iteration method, respectively. The frequencies are verified by the results of the finite element analysis and literature. The responses of the beam under shock excitations and harmonic excitations are numerically solved by the fourth-order Runge–Kutta method. The suppression effect of the inertial NES on the transverse vibration of the beam is evaluated through the values of amplitude reduction and energy dissipation. In addition, a parametric analysis of the inertial NES is conducted to improve the vibration reduction effect of the NES on the beam.

The vibrations of rotating joined conical–conical shells with classical supported conditions have been studied extensively. As a matter of fact, in some cases, these classical boundary conditions cannot exactly model actual situations. Moreover, theoretical frameworks on them are still limited. This research aims to investigate the fundamental frequencies and dynamic mode shapes of the traveling wave of the rotating porous metal material joined conical–conical thin shells (PJCS) with elastic supports. By utilizing artificial spring technology, arbitrary elastic supported boundary conditions and classical boundary conditions are achieved efficiently. A new dynamic model has been formulated with the help of the first-order shear deformation theory (FSDT) and Hamilton’s principle. By employing the generalized differential quadrature (GDQ) method along with stress boundary conditions and generalized eigenvalues, various factors such as porosity, semi-vertex angles and stiffness are analyzed for their impact on the fundamental frequencies of forward wave (FW), backward wave (BW) and mode shapes. The presented results are validated through the convergence and comparison studies from literatures. The interesting and novel results indicate that the in-plane displacement constraints have the most significant impact on the critical speed, while the lateral displacement constraint has the least effect. The vibrations are more easily excited for the part with a larger half vertex angle. Rotating PJCS with Type 1 has the biggest critical rotating speed.