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The nonlinear mechanical oscillator with X-shaped-spring suspension has attracted relevant attention in the past couple of years due to its ability to tailor the corresponding stiffness characteristic on purpose and fit diverse engineering needs. This paper deeply investigates the conditions for such a configuration to exhibit a quasi-zero stiffness behaviour with both the linear and cubic elastic term of the force-displacement curve equal to zero. The resulting quintic characteristic leads to a vibration isolator with a much larger isolation frequency band respect to the classical isolator with non-zero cubic elastic term only. A systematic comparison between the quintic and cubic characteristics is performed, emphasizing the asymptotic trends, and thus showing the fundamental behaviours. It is found that the critical damping for an unbounded response of the quintic isolator is proportional to the square of the normalized excitation amplitude, while that of the cubic isolator is directly proportional to the normalized displacement amplitude. This makes the critical damping of the quintic isolator much lower, and thus more favourable in practical engineering applications, where the values of the normalized excitation amplitude are less than one. [ABSTRACT FROM AUTHOR]

This article focuses on the nonlinear dynamics of transverse vibration of the axially moving yarn system. In this article, the motion of yarn in actual production is analyzed, and the transverse nonlinear vibration equation of the viscoelastic axially moving yarn system with Kelvin model is established by using Newton's second law. In the modeling process, the viscoelasticity of yarn material and the periodic fluctuation of tension in the process of movement are considered. The partial differential control equations of the system are transformed into ordinary differential equations by the Galerkin method. In the first three modes, when the speed is low, the related trial functions occupy the main part, and with the increase of speed, the trial functions of other orders occupy more proportion in the modal functions. Through numerical analysis, it is found that when the moving yarn is subjected to external excitation, the lateral vibration displacement has an increasing trend, and it is found that the nonlinear dynamic phenomenon alternates with the change of system parameters, such as single periodic motion, double periodic motion, and multiple periodic motion. Finally, an experimental platform for lateral vibration of yarn transmission was built and its working principle introduced. The lateral vibration of axially moving yarn was studied from the experimental point of view. It was proved that there are abundant phenomena of periodic motion, period doubling motion, and chaotic motion in the axially moving yarn system, which provides an experimental reference for theoretical research of lateral nonlinear vibration of axially moving yarn. [ABSTRACT FROM AUTHOR]

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally. [ABSTRACT FROM AUTHOR]

According to the nonlocal theory, the dynamic instability and vibration of functionally graded (FG) porous sandwich nanobeams resting on a visco-elastic foundation under axial harmonic load are studied. In this paper the Timoshenko and nonlocal continuum theory are employed to take shear deformation, rotational bending and small-scale effects into account. The governing equations of motion are extracted using the nonlinear Von-Karman, and Hamilton approaches. Then, to turn the partial differential equations (PDEs) into ordinary differential equations (ODEs) in the case of equations of motion, the method of Galerkin is employed, followed by the use of the multiple time scale method to solve the resulting equations. According to the results of this study, it can be claimed that the porosity ratio is more effective than porosity distribution and nonlocal parameters on the amplitude response and dynamic instability regions. Moreover, to examine thermal increments, foundation characteristics, slenderness, and thickness ratios, the bifurcation diagrams are drawn and discussed. It is found that foundation and geometric characteristics are of the highest significance in the nonlinear response of the Timoshenko nanobeams. The main intention of this study is to present a holistic idea about the porous materials used in sandwich structures which can be used in many composite structures in aircraft and automobiles. [ABSTRACT FROM AUTHOR]

To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations' residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system's forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system. [ABSTRACT FROM AUTHOR]

In this paper, a curve-shaped based doubly clamped piezoelectric energy harvester (CD-PEH) is explored for improving the energy harvesting performance. The harvester consists of a composed beam constructed with two arc-shaped structures and a flat beam, as well as two proof masses. A method based on chained beam constraint model theory (CBCM) is first applied to build the nonlinear restoring force model of the CD-PEH, the developed analytical model is validated by the finite element analysis (FEA). Then the electromechanically coupled model for the CD-PEH is built to investigate the effect of excitation amplitudes, geometric parameters and load resistance on the output characteristics. Due to the geometric nonlinearity caused by the arc-shaped configuration, the CD-PEH orderly exhibits quasi-linear, softening nonlinear and mixed hardening & softening nonlinearity behavior with the increasing of excitation level, which could effectively extend the frequency bandwidth of the system. For the excitation of A = 8 m/s2, the effective working bandwidth of the CD-PEH is increased by 633% compared with the effective bandwidth in the case of A = 2 m/s2. Moreover, comparison experiments demonstrate that the output voltage and the effective bandwidth are increased by 225 and 450%, respectively, compared with the typical doubly-clamped piezoelectric energy harvester (T-PEH) under the same excitation amplitude. Overall, this study provides a new way and theoretical framework for the design of high-efficiency doubly clamped piezoelectric energy harvester. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical insight on the performance of a vibration isolator consisting of a combination of linear mechanical springs arranged to achieve a specific form of geometric nonlinearity. In particular, positive and negative stiffness nonlinearities are combined to achieve a sigmoidal shape of the force-deflection curve, which is proven to be beneficial for vibration isolation purposes when the amplitude of vibration is relatively large. Such behaviour is fundamentally different from that of the classical quasi-zero-stiffness isolator, which presents a low dynamic stiffness at the equilibrium configuration and is thus effective for relatively low amplitude of vibration. The analytical findings are validated by numerical simulations, providing useful guidelines for the design of such isolators. [ABSTRACT FROM AUTHOR]

To improve the machining quality of the splicing area of the hardened steel mold, this paper proposes a bifurcation and chaos analysis method for the milling vibration of the splicing area of the mold under the coupled thermal-mechanical effect. First, based on the instantaneous milling force model at the splice, the milling dynamics equations of a two-degree-of-freedom rectangular spliced mold part are established. By applying the Galerkin method, the resulting nonlinear partial differential equation is reduced to a single-degree-of-freedom nonlinear system. Meanwhile, by using the Melnikov function method, the criteria for chaos motion are obtained by the Smale horseshoe mapping. Then, the fourth-order Runge-Kutta algorithm is applied to solve the bifurcation diagram, largest Lyapunov exponent diagram, displacement waveform diagram, and phase plane diagram of the nonlinear dynamic system in the mold splicing area. Subsequently, the influence of spindle speed and cutting depth on the nonlinear vibration system in the mold splice area is quantitatively analyzed. Finally, the effect of spindle speed and cutting depth on the vibration system of the workpiece is investigated through experiments. The obtained results indicate that the milling vibration amplitude in the rectangular splicing mold area can be controlled more easily and effectively by changing the cutting depth than by changing the spindle speed. [ABSTRACT FROM AUTHOR]

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system. [ABSTRACT FROM AUTHOR]

This paper deals with the nonlinear vibration behavior of sandwich panels composed of Graphene platelet reinforced composite (GPLRC) skins and auxetic core embedded with piezoelectric layers subjected to the thermo-magneto-electric loading. The nonlinear equations are obtained by applying Hamilton's principles in the framework of higher-order shear deformation theory (HSDT) and von Karman's nonlinear theory. Linear and nonlinear equations of motion are solved by applying the differential quadrature method (DQM) and homotopy perturbation technique, respectively. In the numerical illustration, at first validation of the present formulation is carried out by comparing the numerical results with those available in the open literature. Then the effects of several parameters such as geometric parameters of auxetic core, applied voltage, magnetic potential, GPL volume fraction, different boundary conditions, and temperature rise on the nonlinear frequency response of sandwich panel are studied. From numerical results, it was concluded that the sandwich panel incorporating auxetic core and piezoelectric layers could increase the magneto-electrical power output. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear energy sink (NES) and a giant magnetostrictive-piezoelectric (GMP) energy harvester (NES-GMP) are investigated to suppress the nonlinear aeroelastic responses and to absorb the mechanical energy of an embedded plate interacting with external subsonic airflow. The analytical model of the embedded plate attaching the NES-GMP is established by using the Hamilton's principle based on the Kirchhoff plate theory and incompressible subsonic aerodynamic model. The natural frequencies of the plate with the NES-GMP and with the NES are analyzed by solving the generalized eigenvalue problems. The global amplitude-frequency responses of the pure plate, the plate with NES, and the plate with NES-GMP are compared to show the vibration suppression effects of the proposed method. Based on the energy analysis, the energy transfer mechanisms of the plate with the NES-GMP are studied. The results reveal that the input energy of the NES-GMP converts into the strain energy by the Terfenol-D layer and then transforms into the electric energy by the piezoelectric layer. Furthermore, numerical simulations also indicate that the maximum harvested energy is obtained when the airflow velocity approaches the critical divergence flow velocity of the plate, and the most effective harvesting installation position is in a specific annular region near the edge of the plate. [ABSTRACT FROM AUTHOR]

Recently, dynamic analysis of a beam structure with nonlinear energy sink (NES) and various supports is attracting great attention. Most of the existing studies are about the beam structure with NES or nonlinear boundary supports with zero rotational restraint, respectively. However, there is little research accounting for such two types of complex factors simultaneously. In this work, the dynamic behavior of an axially loaded beam with both NES and general boundary supports is modeled and studied. The Galerkin truncated method (GTM) is employed to make the prediction of dynamic behavior of such a beam system, in which the mode functions of axially loaded Euler–Bernoulli beam with linear elastic boundary conditions are selected as the trail and weight functions. Then, the Galerkin condition is used to discretize the nonlinear governing equation of the beam system and establish the residual equations. The Runge–Kutta method is used to solve the residual matrix which consists of residual equations directly, and the harmonic balance method is also used to verify the results from the GTM. The influence of NES on vibration suppression and dynamic behavior of the beam structure is investigated and discussed. Results show that the vibration states of the beam structure can be transformed effectively through the change of NES parameters. On the other hand, the NES with suitable parameters has a beneficial effect on the vibration suppression at both ends of the beam structure. [ABSTRACT FROM AUTHOR]

In this study, a new type of vibration isolator based on fluidic actuators and a composite slab was tested experimentally with an unbalanced disturbance. Quasi-zero stiffness vibration isolation techniques are advanced and provide effective isolation performance for non-nominal loads. The isolation performance of the proposed isolator was compared to that of a nonlinear vibration isolator equipped with fluidic actuators and a mechanical coil spring (NLVIFA). The NLVIFA system is better suited to non-nominal loads; however, the mechanical spring axial deflection leads to limited amplitude reduction in the system. To address this issue, a cross buckled slab was developed to replace a mechanical coil spring for absorbing vertical deflection by transverse bending, which is made of a specially developed composite material of Basalt fiber reinforced with epoxy resin and enhanced with graphene nano pellets. This current study was concerned with the theoretical analysis and experimental investigations of the proposed nonlinear vibration isolator with fluidic actuators and composite material (NLVIFA-CM), which performs under quasi-zero stiffness characteristics. Because of its reduced axial deflection, the theoretical and experimental results show that the NLVIFA-CM system outperforms the NLVIFA system and other linear type vibration isolators in terms of isolation performance. Furthermore, the proposed vibration isolator makes a significant contribution to low-frequency vibration. [ABSTRACT FROM AUTHOR]

This research introduces a procedure for signal denoising based on linear combinations of intrinsic mode functions (IMFs) extracted using empirical mode decomposition (EMD). The method, termed component-scaled signal reconstruction, employs the standard EMD algorithm, with no enhancements to decompose the signal into a set of IMFs. The problem of mode mixing is leveraged for noise removal by constructing an optimal linear combination of the potentially mixed IMFs. The optimal linear combination is determined using an optimization routine with an objective function that maximizes and minimizes the information and noise, respectively, in the denoised signal. The method is demonstrated by applying it to a computer-generated voice sample and the displacement response of a cantilever beam with local stiffness nonlinearity. In the first application, the noise is introduced into the sample manually by adding a Gaussian white-noise signal to the signal. In the second application, the response of the entire beam is filmed using two 1-megapixel cameras, and the three-dimensional displacement field is extracted using digital image correlation. The noise in this application arises entirely from the images captured. The proposed method is compared to existing EMD, ensemble EMD, and LMD based denoising approaches and is found to perform better. [ABSTRACT FROM AUTHOR]

In this study, harvesting energy from a nonlinear functionally graded (FG) piezoelectric cantilever beam under harmonic excitation is investigated. The material properties of the piezoelectric are assumed to be a combination of piezo-ceramic and piezo-polymer materials for high performance and flexibility. The neutral axis is obtained in order to eliminate bending–stretching coupling. The geometrical nonlinearity and electromechanical coupling are incorporated in the coupled nonlinear equations that are derived using the generalized Hamilton's principle and solved using the combination of modal reduction and finite element methods. The shooting method is employed to obtain steady-state periodic response of an FG nonlinear harvester with appropriate initial conditions. Also it is shown that at least two-mode approximation is required for accurate estimation of nonlinear response and harvested power. Using the method of nonlinear modal reduction, the unstable branches for frequency domain solution are estimated and the computational time is reduced considerably compared to full finite element method. A case study is also accomplished in detail to analyze the effects of base amplitude values and material distribution on harvested power and bandwidth. [ABSTRACT FROM AUTHOR]

In this article, the vibrational behavior of conical panels in the nonlinear regime made of functionally graded graphene platelet–reinforced composite having a hole with various shapes is investigated in the context of higher-order shear deformation theory. To achieve this aim, a numerical approach is used based on the variational differential quadrature and finite element methods. The geometrical nonlinearity is captured using the von Karman hypothesis. Also, the modified Halpin–Tsai model and rule of mixture are applied to calculate the material properties of graphene platelet–reinforced composite for various functionally graded distribution patterns of graphene platelets. The governing equations are derived by a variational approach and represented in matrix-vector form for the computational purposes. Moreover, attributable to using higher-order shear deformation theory, a mixed formulation approach is presented to consider the continuity of first-order derivatives on the common boundaries of elements. In the numerical results, the nonlinear free vibration behaviors of functionally graded graphene platelet–reinforced composite conical panels including square/circular/elliptical hole and with crack are studied. The effects of boundary conditions, graphene platelet reinforcement, and other important parameters on the vibrational response of panels are comprehensively analyzed. [ABSTRACT FROM AUTHOR]

In this study, nonlinear vibration of a conical shell with a piezoelectric sensor patch and a piezoelectric actuator patch is evaluated. The strain displacement relation is considered to be nonlinear and based on that the equations of motion and piezoelectric sensor output signal are extracted. For the first time, nonlinear electromechanical equations of motion for the conical shell with a piezoelectric sensor layer and a piezoelectric actuator layer are extracted in detail and presented. The producer of extraction of conical shell electromechanical equations of motions is presented for further clarifications. The output signal of the piezoelectric sensor patch considering the nonlinear strain displacement equation of motion is derived. For controlling the vibration actively, a proportional controller is defined by which the applied actuator signal is determined. The nonlinear electromechanical equations of motion are expanded based on conical shell strains. By applying the Galerkin method, the nonlinear time-domain equations are extracted. The complicated attained equations are solved using the harmonic balance method. Conical shell nonlinear frequency response in uncontrolled condition and controlled condition is computed and compared with each other. The frequency response shows that the active vibration control of the conical shell increases the nonlinearity of the system and reduces the vibration amplitude. In addition, free nonlinear vibration of the proposed system is calculated and presented. The results show higher impacts of piezoelectric patches on the conical shell free vibration response and they can reduce the vibration amplitude dramatically. [ABSTRACT FROM AUTHOR]

The research on the nonlinear dynamic characteristics of the central bevel gear transmission system was carried out for the requirement of the vibration control and fault diagnosis of the central bevel gear transmission system in an aero engine. This paper proposes the dynamic model for the first time in order to investigate nonlinear dynamic behavior of the central bevel gear transmission system, and the internal and external excitations are taken into account by using lumped parameter modeling. Time-varying meshing stiffness, gear backlash, and gear transmission error is included in the internal excitations, and the input power, the speed fluctuation of the driven shaft and the high-pressure rotor unbalance excitation are included in the external excitations. The effects of the external excitations on the complex nonlinear dynamic behavior of the system are studied. The results show that reasonable control of input power, speed fluctuations, and high-pressure rotor mass imbalance can improve engine vibration. [ABSTRACT FROM AUTHOR]

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined. [ABSTRACT FROM AUTHOR]

Based on modified strain gradient theory, nonlinear vibration analysis of a functionally graded piezoelectric doubly curved microshell in thermal environment has been performed in this research. Three scale parameters have been included in the modeling of thin doubly curved microshell in order to capture micro-size effects. Graded material properties between the top and bottom surfaces of functionally graded piezoelectric doubly curved microshell have been considered via incorporating power-law model. It is also assumed that the microshell is exposed to a temperature field of uniform type and the material properties are temperature-dependent. By analytically solving the governing equations based on the harmonic balance method, the closed form of nonlinear vibration frequency has been achieved. Obtained results indicate the relevance of calculated frequencies to three scale parameters, material gradation, electrical voltage, curvature radius, and temperature changes. [ABSTRACT FROM AUTHOR]

In this article, the coupling effects of the unbalanced magnetic pull and ball bearing on nonlinear vibration of the three-phase asynchronous motor are investigated with the experimental and numerical methods. A test rig of a motor whose rotor supported by ball bearings is used and a 2 degrees of freedom magnetic solid coupling dynamic model of the motor rotor system is presented. The nonlinear dynamic response and spectrum are obtained from experiments and numerical analysis. The numerical results are in good agreement with test data, thus validating the presented model. It is found that the unbalanced magnetic pull and ball bearing forces possess the significantly interactional and nonlinear influences on the rotor dynamic characteristics. Small magnetic pull could impact the nonlinear bearing-rotor system, resulting in remarkable changes in the dynamic characteristics of the system. The effects of rotational speed and the rotor mass eccentricity on dynamic behaviors of the motor are discussed, and the results show that the magnetic pull gradually increases the amplitude of the ball bearing-rotor system, and its effect decreases with the increment of the rotational speed and mass eccentricity. [ABSTRACT FROM AUTHOR]

Numerous applications have been found in various microelectromechanical systems for shape memory alloys. This study analyzes the nonlinear free vibrations of a sandwiched microbeam, consisting of a shape memory alloy core and two elastic layers on either side. The behavior of the shape memory layer is modeled with the one-dimensional Brinson model. The equation of motion is derived from Hamilton's principle and the modified couple stress theory for the Euler–Bernoulli beam and then truncated into a reduced-order model through Galerkin's technique. An approximate analytical solution is obtained using the Lindstedt–Poincare method for different temperatures. The effects of temperature, material length scale parameter, aspect ratio, and maximum vibration amplitude on nonlinear normalized frequency have been investigated. The results have been compared with those of some previous studies and can provide a better understanding of the design of sandwiched microbeams, which have a shape memory alloy layer. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear vibration responses of laminated composite conical shells surrounded by elastic foundations under S-S and C-C boundary conditions via an approximate approach. The laminated composite conical shells are modeled based on classical shell theory of Love employing von Karman nonlinear theory. Nonlinear vibration equation of the conical shells is extracted by handling Lagrange method. The linear and nonlinear vibration responses are obtained via an approximate method which combines Lindstedt-Poincare method with modal analysis. The validation of this study is carried out through the comparison of the results of this study with results of published literature. The effects of several parameters including the constants of elastic foundations, boundary conditions, total thickness, length, large edge radius and semi-vertex angle on the values of fundamental linear frequency and curves of amplitude parameter versus nonlinear frequency ratio for laminated composite conical shells with both S-S and C-C boundary conditions are investigated. [ABSTRACT FROM AUTHOR]

This paper presents a study of dynamics analysis and optimization of a nonlinear cantilevered beam with the combination of the acoustic black hole (ABH) and the nonlinear energy sink (NES). We establish the nonlinear equations of a cantilevered beam with both the ABH and a NES. Then, we conduct sensitivity analysis of the parameters of the NES for a further understanding. To discover the potential of the NES efficiency, we define a multi-objective optimization problem and solve it by using cell mapping method. The numerical results show that the nonlinear ABH beam with the optimal NES can dissipate 97.7% of the input energy due to the energy concentration capability of the ABH and the irreversible energy transfer behavior of the NES. [ABSTRACT FROM AUTHOR]

In this paper, energy harvesting from nonlinear vibration of a functionally graded beam covered by a piezoelectric patch under multi-moving oscillators is studied. The material of both the substructure and the piezoelectric patch is assumed to be functionally graded in the thickness direction. A coupled system of equations considering Euler-Bernoulli beam theory and von-Karman nonlinearity as well as electromechanical coupling are derived using the generalized Hamilton's principle. Finite element method as well as Newmark time integration scheme are used to solve the coupled nonlinear time dependent problem. The effects of different parameters including material distribution, velocity of the moving oscillators, piezoelectric patch thickness and load resistance on the output voltage and harvested power are investigated. Moreover, the effects of oscillator characteristics such as damping ratio and stiffness on the nonlinear behavior of the beam and harvested power are also studied. Results indicate that the aforementioned parameters have considerable effects on the harvested power. It is also shown that ignoring nonlinear effects may lead to erroneous and unacceptable results. To the best of authors' knowledge, there is no study about energy harvesting from nonlinear vibration of beams under moving oscillators. [ABSTRACT FROM AUTHOR]

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff's theory and then incorporate von Kármán nonlinearity and employ Hamilton's principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamic behavior of a geometrically nonlinear nanobeam resting on the fractional visco-Pasternak foundation and subjected to dynamic axial and transverse loads. The fractional-order governing equation of the system is derived and then discretized by using the single-mode Galerkin discretization. Corresponding forced Mathieu-Duffing equation is solved by using the perturbation multiple time scales method for the weak nonlinearity and by the semi-numerical incremental harmonic balance method for the strongly nonlinear case. A comparison of the results from two methods is performed in the validation study for the weakly nonlinear case and a fine agreement is achieved. A parametric study is performed and the advantages and deficiencies of each method are discussed for order two and three superharmonic resonance conditions. The results demonstrate a significant influence of the fractional-order damping of the visco-Pasternak foundation as well as the nonlocal parameter and external excitation load on the frequency response of the system. The proposed methodology can be used in pre-design procedures of novel energy harvesting and sensor devices at small scales exhibiting nonlinear dynamic behavior. [ABSTRACT FROM AUTHOR]

The nonlinear dynamic and vibration behaviors of a cantilevered carbon nanotube-reinforced composite trapezoidal plate with two surface-bonded piezoelectric layers as an actuator in micro air vehicles are considered in this article. The plate is reinforced by single-walled carbon nanotubes and is exposed to subsonic airflow under combined parametric and external excitations. The large deflection von Karman plate assumptions and the classical laminated plate theory are applied to derive the governing equations of the motion of the piezoelectric nanocomposite laminated trapezoidal plate by using Hamilton's principle. The geometry of the trapezoidal plate is mapped into a rectangular computational domain. The Galerkin's approach is used for transforming the nonlinear partial differential equations of motion into nonlinear two-degrees-of-freedom ordinary differential equations of cubic nonlinearities. The case of 1:3 internal resonance and primary resonance is considered, and the multiple scales method is employed. The aerodynamic pressure distribution formula is modeled by linear potential flow theory. The frequency and time history responses and phase portrait in free forced vibrations are obtained to analyze the nonlinear dynamic behavior of the plate. The effects of different parameters such as the plate geometry, volume fraction of carbon nanotubes, and different excitations on the nonlinear vibration of the thin laminated plate are also discussed. A complex softening nonlinearity with two peaks in the higher mode is observed in frequency response curves. The influence of electrical excitation with several amplitudes and frequencies on dynamic stability is investigated using time response curves. [ABSTRACT FROM AUTHOR]

The aim of this embodiment is to present an analytical analysis of a functionally graded piezoelectric energy harvester consisting of a flexible functionally graded piezoelectric layers carrying magnetic mass at the free end. The magnetic tip mass is in interaction with a permanent magnet which is located at a distance from the top of the tip mass. The oscillation of the harvester happens via excitation of the base. Using Rayleigh's beam theory and Hamilton's principle and considering geometric nonlinearity, the coupled electromechanical governing equations have been developed. The nonlinear frequency response of the piezoelectric energy harvester beam has also been studied under base excitation. A parametric study has been carried out to investigate the effect of grading index and magnetic force on responses of both free vibration and induced excitation cases. The results were compared with those obtained using three-dimensional finite element model developed in COMSOL Multiphysics 5.5 commercial software and good agreement has been observed. The results from both the analytical method and simulation confirm that tuning the design parameters of grading index and magnetic gap to the optimal value results in a considerable change in the performance of the energy harvester. [ABSTRACT FROM AUTHOR]

This article proposes a semianalytical approach for the nonlinear free and forced asymmetric vibration of corrugated sandwich functionally graded cylindrical shells containing fluid under harmonic radial load. The corrugation is considered with two types: trapezoidal corrugation and round corrugation. By using the Donnell shell theory, taking into account the geometrical nonlinearity of von Kármán and an equivalent model for corrugated structures, the governing equations of shell are established. The nonlinear motion equations are reduced to nonlinear differential equation in terms of time by applying the Galerkin procedure. The numerical investigations for the nonlinear dynamic response of cylindrical shells are obtained by using the fourth-order Runge–Kutta method. Numerical results show the very large effects of corrugation and fluid on the natural frequency and nonlinear vibration behavior of shells. [ABSTRACT FROM AUTHOR]

It has been proved that using pure nonlocal elasticity, especially in differential form, leads to inconsistent and unreliable results. Therefore, to obviate these weaknesses, Eringen's two-phase local/nonlocal elasticity has been recently used by researchers to consider the nonlocal size dependency of nanostructures. Given this, for the first time, the size-dependent nonlinear free vibration of nanobeams is investigated in this article within the framework of two-phase elasticity by using the Galerkin method. Contrary to differential nonlocal elasticity, the size dependency of the axial tension force, due to von Karman nonlinearity, is considered, and its effect on the nonlinear vibration is examined. The correct procedure of using the Galerkin method for studying the nonlinear vibration of two-phase nanobeams is introduced. It is shown that, although it is possible to extract the linear mode shapes of two-phase nanobeams using its equal differential equation, integral form of two-phase elasticity should be considered in the Galerkin method. Furthermore, it is observed that in two-phase elasticity, applying classic mode shapes in the Galerkin method leads to significant errors, especially in higher nonlocal parameters and lower values of local phase fraction and amplitude ratios. [ABSTRACT FROM AUTHOR]

This article presents a nonlinear vibration signature study of high-speed defective cylindrical roller bearings under unbalance rotor conditions. Qualitative analysis is conducted considering a spall defect of a specific size on major elements such as outer race, inner race, and rollers. A spring-mass model with nonlinear stiffness and damping is formulated to study the dynamic behavior of the rotor-bearing model. The set of nonlinear differential equations are solved using the fourth-order Runge–Kutta method to predict the characteristics of the discrete spectra and analyze the stability of the system. The results show that higher impulsive forces are generated because of outer race defects than defects in the inner race and roller. This can be explained as every time the roller passes through the defect in the outer race during rotation, the energy is released. However, in the case of both the roller and inner race defects, the impulsive force generated in the load zone is averaged because of the force generated in the unloading zone. The route to chaos from periodic to quasiperiodic response has been observed and analyzed that vibration signature is very much sensitive not only to the defects of bearing components but also to the rotor speed. [ABSTRACT FROM AUTHOR]

A multimode perturbation method for frequency response analysis of nonlinearly vibrational beams with periodic distribution parameters is proposed. The partial differential equation with spatial varying parameters for nonlinear vibration of beams with periodic parameters under harmonic excitations is derived. The procedure of the multimode perturbation method includes three main steps: first, the nonlinear partial differential equation is transformed into linear partial differential equations with varying parameters by applying perturbation method; second, the linear partial differential equations are transformed into ordinary differential equations with multimode coupling by applying Galerkin method, where multiple vibration modes of the beams are used and the equations are suitable to nonlinear vibration of periodic structures with high parameter-varying wave in wide frequency band; third, the ordinary differential equations are solved by applying harmonic balance method to obtain vibration response of the nonlinear periodic beam, which is used for characteristics analysis of frequency response and spatial mode. Furthermore, the stability problem of nonlinear harmonic vibration as multidegree-of-freedom system with periodic time-varying parameters is solved by applying direct eigenvalue analysis approach. The proposed method can incorporate multiple vibration modes into response analysis of nonlinear periodic structures and consider mode-coupling effects due to structural nonlinearity and parametric periodicity. Finally, a nonlinear beam with periodic supports under harmonic excitations is studied. Numerical results on frequency response of the beam are given to illustrate an application of the proposed method, new frequency response characteristics, and influences of periodic parameters on structural response. The results have potential application to nonlinear structural vibration control and support damage detection of nonlinear structures with periodic supports. [ABSTRACT FROM AUTHOR]

In passive vibration isolation, a bistable composite plate can be used to generate high-static low-dynamic stiffness, and in vibratory energy harvesting, when used with piezoelectric film flexing, the bistable composite plate can enhance the performance via snap-through motion. In the present work, we designed a bistable piezo-composite plate for both vibration isolation and energy harvesting. The analytical model for performance prediction is derived from the virtual work principle and the composite elasticity. Displacement transmissibility and output voltage generation were analytically determined for different mechanical and electrical parameters. The results illustrate that the bistable piezo-composite plate improves the feasibility and effectiveness of the integration design. Hardening and softening nonlinear phenomena coexist in both the displacement transmissibility and the output voltage plot. Approximate analytical solutions and numerical simulations are in good agreement. Both analytical and numerical results demonstrate that, at a certain frequency bands, enhanced energy harvesting is accompanied by a vibration transmissibility reduction. Compared with a linear system with the bistable piezo-composite plate removed, it achieved a reduction in the displacement transmissibility of about 13 dB at 100 Hz, simultaneously, and produced a considerable output voltage of about 0.05 V. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to comprehensively study the nonlinear frequency response of viscoelastic piezoelectric nanoplates exposed to dual harmonic external excitation and thermo-electro-mechanical loads. To achieve this goal, firstly, a piezoelectric nanoplate resting on a viscoelastic foundation is modeled. Secondly, using the nonlocal piezoelectricity theory, Kelvin-Voigt model, von Karman nonlinear relations and Hamilton's principle, the nonlinear governing differential equation of motion is derived. In the next step, employing the Galerkin technique and multiple time scales method, the partial differential equation is transformed to an ordinary one and solved. Finally, the modulation equation of viscoelastic piezoelectric nanoplates for combinational excitation is obtained. Emphasizing the effect of dual harmonic excitation and thermo-electro-mechanical loads on nonlinear frequency response of the system, jump and resonance phenomena are discussed. A detailed parametric study is conducted to examine the effect of nonlinearity, damping coefficient, nonlocal parameter, combinational excitation, electric voltage, initial stress and thermal environment. [ABSTRACT FROM AUTHOR]

Investigation of flexoelectric effect on nonlinear forced vibration of piezoelectric/functionally graded porous nanocomposite is the objective of this study. The nanocomposite is exposed to electric voltage and external parametric excitation. First, a functionally graded porous core nanoplate is modeled and then two piezoelectric layers are glued with core. It is also rested on a visco-Pasternak foundation. Second, to derive governing equation of motion, two theories including Mindlin and Kirchhoff plate theories and Hamilton's principle are utilized. In the next step, to obtain and solve ordinary differential equation, Galerkin technique and multiple time scales method are used, respectively. At the end, modulation equation of piezoelectric/functionally graded porous nanocomposite for both primary and secondary resonances is obtained and discussed. Emphasizing the effect of piezoelectric and flexoelectric, von Karman nonlinear deformation and parametric external excitation are simultaneously taken into account. It is found that electric voltage has no effect on the performance of piezoelectricity and flexoelectricity of the material on vibration behavior. The results of this study can be useful as benchmark for the next investigations in field of energy harvesting systems and piezoelectric structures. [ABSTRACT FROM AUTHOR]

A dynamic analysis of a hemispherical and cylindrical contact, material properties of which are graded elastically and plastically along the radius, is presented. The static force–displacement behavior of a hemisphere and a semi-cylinder in contact with a rigid flat is obtained using finite element software. The force–displacement is used in a further dynamic analysis for undamped-free as well as for forced-damped vibration of the contact interface. For the undamped free vibration, variation of natural frequency w.r.t. initial displacement is furnished for different values of elastic and plastic gradation parameter. In addition, variation of maximum initial displacement for contact loss is also demonstrated. The forced-damped vibration characteristics of the spherical and cylindrical contact interfaces are presented in the form of frequency response curves with jump up and jump down frequencies. Spherical and cylindrical contact interfaces are found to exhibit softening and hardening type nonlinearity, respectively. [ABSTRACT FROM AUTHOR]

This study investigates the nonlinear vibration of magnetoelectroelastic composite cylindrical nanoshells embedded in elastic media for the first time. The small-size effect and thermoelectromagnetic loadings are considered. Based on the nonlocal elasticity theory and Donnell's nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived using Hamilton's principle. Then, the Galerkin method is utilized to transform the governing equations into a nonlinear ordinary differential equation and subsequently the method of multiple scales is employed to obtain an approximate analytical solution to nonlinear frequency response. The present results are verified by the comparison with the published ones in the literature. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external magnetic potential, the external electric potential, the temperature change, and the elastic media on the nonlinear vibration characteristics of magnetoelectroelastic composite nanoshells. [ABSTRACT FROM AUTHOR]

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled graphene-reinforced composite (GRC) laminated plates resting on elastic foundations. The piecewise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the plate. The anisotropic and temperature-dependent material properties of the FG-GRC layers are estimated through the extended Halpin–Tsai micromechanical model. Based on the Reddy's higher order shear deformation plate theory and the von Kármán strain–displacement relationships, the motion equations of the plates are derived. The foundation support, the thermal effect, and the initial deflection caused by thermal postbuckling are also included in the derivation. A two-step perturbation approach is applied to determine the thermal postbuckling equilibrium paths as well as the nonlinear vibration solutions for the FG-GRC laminated plates. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled FG-GRC laminated plates under a uniform temperature field. The effects of graphene reinforcement distributions and foundation stiffnesses on the vibration responses of FG-GRC laminated plates are examined in detail. [ABSTRACT FROM AUTHOR]

Landing gear shimmy remains a challenge in aircraft design despite abundant advances in aircraft engineering in the past few decades. Accurate shimmy prediction is closely tied to availability of dynamic models with all relevant types of motions and key nonlinear elements, a matter which has been accomplished in the present study through including rotational, lateral, longitudinal, and axial degrees of freedom and tire, shock absorber, and Coulomb friction nonlinearities. Using multi-body dynamic simulations, stability of the nose landing gear is studied as a function of key system parameters. Influences of nonlinearities are investigated in isolation, with a more in-depth look at the Coulomb friction effect, which is modeled as a function of the shock absorber stroke rate and rotational shimmy speed. It is found that Coulomb friction is a key factor in determining the onset and type of shimmy. The effect of friction parameters is then studied using nonlinear sensitivity analyses, and witnessed trends are utilized to draw design recommendations. [ABSTRACT FROM AUTHOR]

Theoretical and experimental studies are conducted to control nonlinear vibration of a two-link flexible manipulator via internal resonance. A vibration control method is proposed and an effective vibration absorber is implemented based on a servomotor to establish a 2:1 internal resonance relationship with the flexible manipulator. By way of perturbation analysis, it is proven that internal resonance can be successfully established for the flexible manipulator undergoing rigid motion. In the presence of damping, the vibration energy of the flexible manipulator can be transferred to and dissipated by the vibration absorber via internal resonance. Numerical simulations and experimental investigation have verified the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Frequency up-converting vibration energy harvester can bridge the gap between high-frequency response and low-frequency input, greatly increasing the efficiency of energy conversion. This article proposed a novel frequency up-converting energy harvester based on 1:3 internal resonance in 2 degree-of-freedom cubic nonlinear systems. The harvester consists of two asymmetric cantilevers corresponding to two vibration degrees-of-freedom. The ratio of cantilevers’ first-order resonances is (or close to) 1:3. When excited frequency matches the resonant frequency of the first assisting cantilever, 1:3 internal resonance of the harvester system occurs, leading to drastic vibration of the second generating cantilever at its resonance. The generated voltage frequency is then three times increased. Finally, simulated and experimental results clearly proved this frequency up-converting principle. In addition, the resonant frequency tuning and wideband behaviors of the harvester were also investigated, which increased the viability of the proposed harvester under the practical environment vibrations. [ABSTRACT FROM AUTHOR]

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined. [ABSTRACT FROM AUTHOR]

This paper investigates nonlinear vibration of a simply supported rectangular membrane based on large deflection theory. Dynamic stress caused by transverse displacement of the membrane is considered in modeling the membrane. The assumed mode method and the nonlinear finite element method (FEM) are both used as discretization methods for the membrane. In the assumed mode method, an approximate analytical formula of the natural frequency is derived. In the nonlinear FEM, a three-node triangular membrane element is proposed. The difference between the membrane’s dynamical characteristics obtained by these two discretization methods is revealed. Simulation results indicate that natural frequency of the membrane will rise along with the increasing of the vibration amplitude of the membrane, and the natural frequency obtained by the nonlinear FEM is larger than that obtained by the assumed mode method. When the membrane vibration is small, the assumed mode method may achieve a reasonable result, but it may lead to a big error when the membrane vibration is large. [ABSTRACT FROM AUTHOR]

Rods are one of significant engineering’s structures and vibration analysis of a rod because of extended application of it in engineering is very important. Due to large amplitude or to excitations at frequencies close to their resonance frequencies, these structural elements can experience large amplitude, hence nonlinear vibration. Therefore, understanding of longitudinal nonlinear vibration of rods with different boundary conditions and large amplitude is very useful to reach an appropriate designing. In this paper, vibration of a rod with different boundary conditions undergoing finite strain, without simplification in strain–displacement equation, is investigated. For obtaining governing equation, Green–Lagrange strain, structural damping and Hamilton principle are used and then Galerkin method is employed to convert nonlinear partial differential equation to nonlinear ordinary differential equation. In spite of many papers that only use of cubic term for nonlinearity, the governing equation has quadratic and cubic terms. At the first, free vibration equations are solved with multiple time scales method and for verifying the accuracy of this method, the results are compared with results of Runge–Kutta numerical method, which have good accuracy. Then, forced vibration is investigated in the cases of primary resonance and hard excitation including sub-harmonic and super-harmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including nonlinear coefficients, damping coefficient and external excitation are discussed. [ABSTRACT FROM AUTHOR]

In this article, harvesting of electrical power from nonlinear vibration of an asymmetric bimorph piezoelectric plate is presented based on the classical plate theory with von Kármán strain–displacement nonlinear relations in the presence of temperature change effect. Two piezoelectric layers with different thicknesses cover top and bottom layers of a substructure. The structure has been excited under harmonic transverse forces in case of primary resonance. Two coupled ordinary differential equations for displacement and voltage have been derived and solved via multiple time scales method. The voltage equation has been defined by electric displacement and Gauss’s law. Analytic relations for voltage and harvested electrical power have been derived. The analytic relation for power is based on different parameters of the plate such as thicknesses of the layers, dimensions of plate, load resistance, frequency of harmonic excitation, and mechanical properties of structure. The parameters are optimized using genetic algorithm method with consideration of power relation as cost function. To harvest the maximum energy from the plate, the thicknesses of two piezoelectric layers, load resistance, and detuning parameter have been optimized. To illustrate the effectiveness of the optimization, results are depicted in three different simulations. [ABSTRACT FROM AUTHOR]

This study investigates the stiffening effect due to rotation on the nonlinear vibrational characteristics for cracked Timoshenko beam for the first time. Fixed end of the beam is attached to a rotating hub. Functionally graded material is taken into consideration, in which the properties vary as a continuous function along the depth of the beam. An elastic mass-less rotational spring is assumed in the place of crack, which splits the beam into two different parts. The point on the neutral axis at the fixed end is assumed to be the center of rotation of the beam. Centrifugal force is considered to act towards the spanwise direction and along the neutral axis. An additional displacement due to rotation of the beam along with the centrifugal force is incorporated with the energy formulation. Timoshenko beam theory and classical Ritz method is employed to derive the governing equations. In order to solve the nonlinear governing equations, direct substitution iterative technique is used. Effects of various parameters such as rotating speeds, radius of hub, depth of crack, location of crack, and different functionally graded material properties on linear and nonlinear vibration characteristics are studied. Validity of the present methodology is assured by comparing the results with some of the results from the existing literatures. [ABSTRACT FROM AUTHOR]

The identification of nonlinear squeeze-film damper (SFD) bearings, typically used in aero-engines, has so far focused on their forward model (i.e. displacement input/force output). The contributions of this paper are the non-parametric identification of the inverse model of the SFD bearing (force input/displacement output) from empirical data, and its application to a nonlinear inverse rotor-bearing problem. This work is motivated by the need for a reliable substitute for internal instrumentation, to enable the identification of rotor unbalance using vibration data from externally mounted sensors, in applications where the rotor is inaccessible under operating conditions and there is no adequate linear connection between rotor and casing. The identification of the inverse model is fundamentally different from that of the forward model due to the need to account for system memory. A suitably trained Recurrent Neural network (RNN) is shown to be capable of identifying the inverse model of an actual SFD through two validation studies. In the first study, the RNN model satisfactorily predicted the SFD journal’s displacement time histories for given periodic time histories of the Cartesian SFD forces, although it could not predict the user-applied static offset in the SFD since it was not trained to do so. This was no limitation for the second study where, for both centred and non-centred SFD conditions, the RNN proved to be a reliable substitute for actual instrumentation as part of the inverse problem solution process for identifying the amplitudes and phases of the external excitation forces on a simple test rig. [ABSTRACT FROM AUTHOR]

The aim of the present work is to study the vibration behavior of sandwich composites with shear damages on the foams. The effect of the densities of damage on the linear and nonlinear vibration parameters is studied. The sandwich materials used in this study is constructed with glass fiber laminates as skins and with PVC closed-cell foams with density 60 and 100 kg m−3 as core. After a serie of vibration tests, the change of natural frequencies and modal damping due to the damage and the foam densities. For an intact specimen, the resonance frequency was constant when we increase the excitation amplitude. For damaged specimens, the resonance frequencies decrease and the loss factors increase proportionally with the increasing excitation amplitude. The nonlinear parameters corresponding to the elastic modulus and damping were determined for each frequency modes and each damage and foam densities. The results showed that nonlinear dissipative parameters were more sensitive to damage than linear elastic and dissipative parameters. [ABSTRACT FROM AUTHOR]

A large amplitude vibration analysis is presented for nanocomposite doubly curved panels resting on elastic foundations in thermal environments. The doubly curved nanocomposite panels are studied with the consideration of different types of distributions of uniaxial aligned single-walled carbon nanotubes (SWCNTs). The material properties of the functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to be graded in the thickness direction according to linear distributions of the volume fraction of CNTs and are estimated through a micromechanical model. The motion equations are based on a higher order shear deformation theory and von Kárma'n strain-displacement relationships. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. The motion equations are solved by a two-step perturbation approach to determine the nonlinear frequencies of the CNTRC doubly curved panel. The numerical illustrations cover small- and large-amplitude vibration characteristics of CNTRC doubly curved panels resting on Pasternak elastic foundations. The present solutions also highlight the effects of CNT volume fraction, temperature variation, foundation stiffness, panel curvature ratio as well as in-plane boundary conditions on the nonlinear free vibration behaviors of CNTRC doubly curved panels. [ABSTRACT FROM AUTHOR]