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This paper presents an analysis of the non-linear vibrations of beams, which play a crucial role in various industrial and construction structures. Understanding the transverse vibrations of beams and accurately determining their frequency response is essential for achieving optimal design and structural performance. The novelty of this study lies in conducting a transverse non-linear vibration analysis of a three-dimensional beam while considering the effect of mid-plane elongation. By incorporating this aspect into the analysis, the study aims to provide deeper insights into the dynamic behavior of beams subjected to non-linear effects. A multiple-time scale approach has been adopted to conduct this research. To verify the accuracy of the method as well as the accuracy of the outcomes gained from this method, a contrast has been made with the 4th-order Runge-Kutta technique, which indicates that the results obtained are acceptable. The frequency response of the beam indicates the presence of a phenomenon of splitting into two non-linear branches during the three-dimensional vibrations of the beam, as well as a hardening state in the frequency response as a result of stretching the middle plane of the beam. Furthermore, a parametric study was conducted in which different parameters were examined to determine the starting point of non-linear bifurcation. As a result, the damping coefficient and resonance deviation parameter are two factors that affect the preference for critical bifurcation over safe bifurcation. Furthermore, the stretching of the middle plane results in a higher non-linear term coefficient in the vibration equations of the beam, which increases the oscillation frequency of the beam.

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.

This paper presents a state-dependent switching law to achieve asymptotic stability of a fourth-order switched system, in which each subsystem has two pairs of conjugate characteristic roots with positive real parts. Firstly, a linear vibration system of two-degrees-of-freedom is introduced to match the standard fourth-order linear subsystem. An energy function of each subsystem is then defined as its kinetic and potential energies. Secondly, an invertible transformation is introduced to convert a general fourth-order subsystem into a standard one, thereby constructing its energy function with definite physical meaning. Thirdly, in order to reasonably define an energy ratio function between two subsystems, an intermediate subsystem is introduced so that an identical invertible transformation can be used to convert the subsystem and the intermediate subsystem into standard fourth-order ones simultaneously. Based on the obtained energy ratio function, a state-dependent switching law is constructed to maximize the energy loss in a switching loop, thus stabilizing the fourth-order switched system at a fast speed. In the end, the effectiveness of the proposed method is illustrated by four numerical simulations.

Summary This paper presents an improved crack model incorporating non‐linearity of flexural damage in concrete to reproduce changes in vibration properties of cracked reinforced concrete beams. A reinforced concrete beam model with multiple‐distributed flexural cracks is developed, in which the cracked regions are modelled using the fictitious crack approach and the undamaged parts are treated in a linear‐elastic manner. The model is subject to incremental static four‐point bending, and its dynamic behaviour is examined using different sinusoidal excitations including swept sine and harmonic signals. From the swept sine excitations, the model simulates changes in resonant frequency with increasing damage. The harmonic excitations are utilised to investigate changes in modal stiffness extracted from the restoring force surfaces, and changes in the level of non‐linearity are deduced from the appearance of super‐harmonics in the frequency domain. The simulation results are compared with experimental data of reinforced concrete beams subject to incremental static four‐point bending. The comparisons revealed that the proposed crack model is able to quantitatively predict changes in vibration characteristics of cracked reinforced concrete beams. Changes are sensitive to support stiffness, where the sensitivity increases with stiffer support conditions. Changes in the level of non‐linearity with damage are not suitable for damage detection in reinforced concrete structures because they do not follow a monotonic trend. Copyright © 2014 John Wiley & Sons, Ltd.

Funicular shells are thin, doubly curved shallow shells that are in compression under dead weight due to their shape. In this study, an analytical approach is employed to consider the forced linear vibration of concrete funicular shells with a rectangular base under impulse loads based on shallow shells theory. Two boundary conditions, simply supported and clamped, are considered. The solution is obtained by Lagrangian approach. The accuracy of the results was considered by comparing the results with those of finite element method. The results indicated that tensile stresses in addition to compressive stresses formed in funicular shells under impulse loads.

Purpose The purpose of this study is to facilitate the engineering technical personnel to easily choose the appropriate vibration analysis model in the design of symmetry composite sandwich structure with composite face layers and viscoelastic core. Therefore, the applicable condition of two linear vibration models was obtained by considering the vibration analysis of three-layered sandwich structures. Design/methodology/approach Equilibrium equations of three kinds of vibration models were deduced using Hamilton’s principle and were solved using the closed-form Navier solution. Then, numerical results were compared with those reported in the literature to verify the accuracy of the deduced calculating formulas. Findings The applicable conditions of two linear vibration models are obtained by examining the influence of the ratio of length to total thickness (q) and that of viscoelastic thickness to total thickness (c) on the solutions. Two cases are considered about the applicable conditions of two linear vibration models based on the error analysis. Originality/value The paper obtains the application conditions of two linear vibration models by using error analysis for the first time and provides the reference for engineering staff to easily choose a suitable vibration model to design a symmetry composite sandwich plate.

Dynamical and structural systems are susceptible to sudden excitations and loadings such as wind gusts, blasts, earthquakes, and others which may cause destructive vibration amplitudes and lead to catastrophic impact on human lives and economy. Therefore, various vibration absorbers of linear and nonlinear coupling dynamics have been widely studied in plenty of publications where some have been applied in real-world practical applications. Firstly, the tuned-mass-damper (TMD), the first well-known linear vibration absorber that has been well-studied in the literature and applied with various structural and dynamical systems, is discussed. The linear vibration absorbers such as TMDs are widely used in real-life small- and large-scale structures due to their robust performance in vibration suppression of the low natural frequency structural modes. However, the TMD performs efficiently at narrowband frequency range where its performance is deteriorated by any changes in the frequency content in the structure and the TMD itself. Therefore, the targeted-energy-transfer mechanism which is found to be achieved by nonlinear energy sinks (NESs) has ignited the interest in passive nonlinear vibration suppression. Unlike TMDs, the NESs are dynamical vibration absorbers that achieve vibration suppression for wide range of frequency-energy levels. Given the very rapid growth in this field and the extensive research studies supporting the robustness of the NESs, this paper presents the different types of NESs and their applications with main emphasis on the rotary-based and impact-based NESs since they are of high impact in the literature due to their strong nonlinear dynamical behavior and robust targeted energy transfer.

This study proposes a novel magnetic circuit design to reduce the size of horizontal linear vibration motors (HLVMs) used in vehicle touchscreens. The HLVM prototype uses two thick permanent magnets to create a magnetic circuit below the voice coil; however, the novel design places four thin permanent magnets above and below the voice coil. Moreover, the coil position has been changed to a yoke center to create an effective magnetic circuit with short magnets. Compared with the vertical linear vibration motor, the force calculation method of the HLVM is significantly different. In this study, a new force calculation method is used to analyze the electromagnetic–mechanical coupling of the HLVM. As a prototype, the novel design is small in size (−28.85%) but possesses similar acceleration. The experimental results verify the analysis results of the HLVM in the displacement and acceleration on a dummy jig.

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.

To improve the calibration accuracy of quadratic error coefficients in the static error model of gyroscopes, a whole-period vibration calibration method on a linear vibration table is proposed. By considering parasitic rotation and attitude error of the vibration table, angular vibration during testing, and gyroscope installation error, a calibration model for the gyroscope is established. A multi-position calibration method is adopted, where the quadratic error coefficients are calibrated by combining the stationary and whole-period vibration states of the vibration table. Error analysis demonstrates that the proposed method accurately identifies the quadratic error model coefficients of the gyroscope, achieving a calibration accuracy of 10 −4 (°/h/g 2 ).