Explore the vast range of titles available.

The internal resonances between the longitudinal and transversal oscillations of a forced Timoshenko beam with an axial end spring are studied in depth. In the linear regime, the loci of occurrence of 1 :  ir , i r ∈ N , internal resonances in the parameters space are identified. Then, by means of the multiple time scales method, the 1 : 2 case is investigated in the nonlinear regime, and the frequency response functions and backbone curves are obtained analytically, and investigated thoroughly. They are also compared with finite element numerical simulations, to prove their reliability. Attention is paid to the system response obtained by varying the stiffness of the end spring, and it is shown that the nonlinear behaviour instantaneously jumps from hardening to softening by crossing the exact internal resonance value, in contrast to the singular (i.e. tending to infinity) behaviour of the nonlinear correction coefficient previously observed (without properly taking the internal resonance into account).

This study is aimed at analysing damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems, taking into account both external (non-rotating) and internal (rotating) damping distributions. As case-study giving rise to a set of coupled differential Mathieu–Hill equations with both damping and gyroscopic terms, a balanced shaft is considered, modelled as a spinning Timoshenko beam loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs or flywheels. After discretization of the equations of motion into a set of coupled ordinary differential Mathieu–Hill equations, stability is studied via eigenproblem formulation, obtained by applying the harmonic balance method. The occurrence of simple and combination parametric resonances is analysed introducing the notion of characteristic circle on the complex plane and deriving analytical expressions for critical solutions, including combination parametric resonances, valid for a large class of rotors. A numerical algorithm is then developed for computing global stability thresholds in the presence of both damping and gyroscopic terms, also valid when closed-form expressions of critical solutions do not exist. The influence on stability of damping distributions and gyroscopic actions is then analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince–Strutt diagrams.

Vibration mitigation has been an important research interest in the past decades. In this paper, the enhancement of vibration suppression of thick beams is investigated. The Timoshenko beam is considered, and finite element method is used to discretize governing equations for the beam consisting of axial load. The stability of the system is studied both numerically by using Floquet theory, and analytically by employing averaging perturbation method. Effects of the thickness change, also boundary conditions are provided. The results demonstrate that, by adding extra boundary condition, the stability of the beam increases under the same circumstances. It means that, boundary condition can play important role in mitigating the vibration. Moreover, considering the thick beam reveals that the equivalent damping of the beam enhances. In this case, the excitation amplitude as well as the excitation frequency will increase. Therefore, under the same condition, the thicker the beam is, the more stable it will be.

In this paper, a FE formulation for nonlinear vibration of fully geometrically exact Timoshenko beams is derived. Firstly, the strong form of the governing equation of motion is obtained without any approximation. Next, the weak relations are used to derive the FE formulation. To obtain the vibration response, a direct integration method is employed to solve highly nonlinear formulation for geometrically exact beams. Finally, some examples are investigated, and very good results with analytical solutions available in the literature are achieved. The results show that the formulation presented in this paper can predict the vibration response of the Timoshenko beam with high accuracy. Moreover, investigating the effect of parameters shows that increasing the length parameter leads to increasing natural frequencies. Besides, the frequency value increases when the amplitude of vibration increases. Furthermore, adding an axial spring can lead to asymptotic behavior in vibration response of the fully exact Timoshenko beams.

Purpose In engineering design, attention is usually paid to avoiding the low-order modal resonance of elastomers, but it is difficult to avoid high-order modal resonance. Although the amplitude of high-order modal resonance is generally not large, due to the higher frequency, it will also lead to vibration fatigue of the structure. To accurately analyze the higher order modes of a one-dimensional elastomer, the vibration reduction efficiency of a nonlinear energy sink (NES) cell element at higher order excitation frequencies is explored. Methods First, a Timoshenko beam dynamic model of transverse nonlinear vibration of elastic body is established. The natural frequencies and the modes of the first eight orders of the elastic beam are analyzed and compared with the numerical results of ANSYS software. The correctness of modal analysis of elastic beam is verified. To suppress the resonance of the higher order modes of the elastomer, the centralized and distributed control strategies of the NES cell are introduced. The dynamic model of forced vibration of Timoshenko beam coupled with NES cell is established. The harmonic balance method (HBM) and the pseudo arc-length method are combined to solve the partial differential–integral equation of Timoshenko beam coupled with multiple NES cell nonlinear differential equations. The amplitude–frequency response curve of the system is obtained. The approximate analytical results are numerically verified by the Runge–Kutta method (RKM). The resonant steady-state response of the nonlinear vibration of Timoshenko beam without NES cell control is compared. Results and Conclusion For both different vibration reduction strategies, the NES cell element has good vibration reduction efficiency. As the number of NES cell elements increases, the vibration reduction efficiency also increases. However, the vibration reduction efficiency does not keep increasing, but tends to a stable value. For both vibration reduction strategies, the NES cell elements have highly similar vibration reduction efficiencies at higher order resonant frequencies.

In this article, the nonlinear hygrothermally induced vibrational behavior of bidirectional functionally graded porous beams is studied through a numerical approach. Two-dimensional material and temperature distributions, even and uneven porosity distributions, temperature-dependent nature of material properties, and hygroscopic effects are all taken into account in studying beam’s lateral deflection. All material properties are assumed to vary along both thickness and axial directions of beam following a modified power-law distribution in terms of volume fractions of the material constituents, which are considered temperature dependent using Touloukian experiments. Beam's upper surface is subjected to a sudden temperature rise, while its lower surface is kept at reference temperature or is thermally adiabatic; meanwhile, left and right boundaries are thermally insulated. Two-dimensional transient heat conduction equation is solved using generalized differential quadrature (GDQ) method for discretizing spatial derivatives, while time derivatives are approximated using Newmark-beta integration method. Nonlinear sinusoidal moisture concentration is assumed through the thickness direction. Governing equations of motion are derived based on Timoshenko beam theory (TBT) and with the assumption of Von-Kármán geometrical nonlinearity, which is solved afterward using an iterative scheme in conjunction with GDQ and Newmark's method. Finally, the effects of porosity volume fractions, porosity cases, thermal boundary conditions, moisture concentration, FG indexes, slenderness ratio, and temperature rise on maximum non-dimensional lateral deflection are investigated considering various boundary conditions.

In this study, an attempt is made to model and investigate the dynamic behavior of the spinning Timoshenko beam-disk with nonlinear elastic boundaries, in which an unbalanced concentrated mass and axial loads are considered. In order to satisfy the elastic boundary conditions of the spinning beam-disk containing translational and rotational stiffnesses as well as nonlinear stiffnesses, an improved version of Fourier series is employed for the admission function construction. Nonlinear dynamic behavior of the spinning beam-disk and its boundary supporting system are described based on energy principle, and the system governing equations of spinning beam-disk are formulated by the Lagrange equation of the second type. The time-domain response is then obtained by solving the system dynamic equations through Runge–Kutta method, while the reliability of the current model is verified through the comparison with those predicted by harmonic balance method. Then, the effect of sweep direction and nonlinear elastic boundary parameters on system dynamic behavior of the spinning Timoshenko beam-disk is investigated and addressed. The results show that the dynamic responses of the spinning beam-disk with nonlinear elastic boundary are sensitive to the initial values of calculation, and the nonlinear elastic boundary parameters make the spinning beam-disk exhibit complex dynamic behavior. Analysis of Poincare points in the phase diagram can better determine the dynamic behavior of spinning beam-disk, and a set of suitable nonlinear elastic boundary parameters can suppress the complex dynamic response of the spinning beam-disk.

Natural frequencies and mode shapes of a Timoshenko beam with arbitrary non-uniformities, discontinuities, discrete spring/mass constraints and boundary conditions are computed by developing a new method based on the state transition matrix for spatially varying state equations. Algorithms to treat discontinuities in material and geometrical properties and discontinuities due to discrete spring constraints are clearly presented. Equations for natural frequencies and mode shapes are derived for following boundary conditions: clamped-free, attached springs/masses at both ends and pinned–pinned. Numerical results are presented and compared to those in the existing literature.

PurposeThis study aims to perform dynamic response analysis of damaged rigid-frame bridges under multiple moving loads using analytical based transfer matrix method (TMM). The effects of crack depth, moving load velocity and damping on the dynamic response of the model are discussed. The dynamic amplifications are investigated for various damage scenarios in addition to displacement time-histories.Design/methodology/approachTimoshenko beam theory (TBT) and Rayleigh-Love bar theory (RLBT) are used for bending and axial vibrations, respectively. The cracks are modeled using rotational and extensional springs. The structure is simplified into an equivalent single degree of freedom (SDOF) system using exact mode shapes to perform forced vibration analysis according to moving load convoy.FindingsThe results are compared to experimental data from literature for different damaged beam under moving load scenarios where a good agreement is observed. The proposed approach is also verified using the results from previous studies for free vibration analysis of cracked frames as well as dynamic response of cracked beams subjected to moving load. The importance of using TBT and RLBT instead of Euler–Bernoulli beam theory (EBT) and classical bar theory (CBT) is revealed. The results show that peak dynamic response at mid-span of the beam is more sensitive to crack length when compared to moving load velocity and damping properties.Originality/valueThe combination of TMM and modal superposition is presented for dynamic response analysis of damaged rigid-frame bridges subjected to moving convoy loading. The effectiveness of transfer matrix formulations for the free vibration analysis of this model shows that proposed approach may be extended to free and forced vibration analysis of more complicated structures such as rigid-frame bridges supported by piles and having multiple cracks.

This study is aimed at investigating the effects of anisotropic supports on the stability of slender rotors parametrically excited by external loads. An axisymmetric shaft described by scaling a spinning Timoshenko beam on anisotropic supports is studied, loaded by oscillating axial end thrust and twisting moment, with the possibility of carrying additional inertial elements like discs, which represents a model including all the general features of slender rotors which are relevant for this kind of stability analysis, gyroscopic effects comprised. Stability is studied after discretization of the equations of motion into a set of coupled ordinary differential Mathieu-Hill equations. The influence on stability of angular speed combined with anisotropy in the supports (including principal stiffness, principal damping and cross-elements) is analysed with respect to frequency and amplitude of the external loads on stability charts in the form of Ince-Strutt diagrams. The occurrence of different kinds of critical solutions, simple and combination, is investigated, highlighting their dependency on both the degree of anisotropy in the supports and angular speed.