Nonlinear Vibration Model Research Articles

Multiple nonlinear vibration model of drilling string and mechanism of drilling speed increase in ultra-HPHT oil & gas wells

Multiple nonlinear vibration model of drilling string and mechanism of drilling speed increase in ultra-HPHT oil & gas wells

Nonlinear forced vibration of the FGM piezoelectric microbeam with flexoelectric effect

In this paper, based on the extended dielectric theory, Euler beams theory and von Karman’s geometric nonlinearity, a nonlinear FGM piezoelectric microbeam model is established with flexoelectric effect. The governing equations, initial conditions and boundary conditions are obtained by applying Hamilton’s principle and then solved by combining the differential quadrature method (DQM) and iteration method. The innovation of this paper is to construct a nonlinear forced vibration model of piezoelectric microbeams. The coupling response between the inverse flexoelectric effect and the inverse piezoelectric effect is investigated. Various effects are examined, including the functional gradient index m and transverse distributed load q affecting the distribution of electric potential. Results indicated that the functional gradient index m, beam thickness h, and span-length ratio L/h have a significant impact on the dimensionless deflection of the FGM microbeam. The influence of the flexoelectric effect on dimensionless deflection increases with the decrease of scale. In addition, transverse load q and the functional gradient index m also have a significant impact on the distribution of electric potential. This paper will provide useful theoretical guidance for the design of micro-sensors and micro-actuators.

Dynamic Modeling and Theoretical Analysis of Planar Nonlinear Vibrations of an Articulated Hoop Truss Antenna

The dynamic behaviors of large-scale hoop truss antennas (HTAs) are significantly influenced by the nonlinear torque transmission properties of flexible hinges. Due to the nonlinearity of the hinge, the HTA can be easily excited to exhibit internal resonances that result in the energy exchange between the adjacent two or three modes. This paper focuses on the 3:1 internal resonant responses of an articulated HTA induced by the nonlinearity of the hinge. The analytical modes are obtained by using the global mode method (GMM) and are validated by finite element method (FEM). Then the partial differential equations (PDEs) of planar motions are discretized into ordinary differential equations (ODEs) by Galerkin’s technique. The multiple time scale method is employed to obtain the four-dimensional modulation equations of the HTA with primary and 3:1 internal resonance. By using the Newton–Raphson iteration and the pseudo arclength continuation, frequency–response and force–response curves are obtained to investigate the theoretical steady-state vibrations of the HTA. Moreover, the influence of the cubic spring stiffness, damping, and external excitation amplitude on the system’s nonlinear dynamic behaviors are investigated. The numerical simulation reveals that the articulated multi-beam hoop structure exhibits typical nonlinear phenomena such as hardening-spring type characteristic, jumps, bi-stability, and double peaks. The research findings contribute to the reliable structural design and vibration control of the articulated hoop truss antenna.

A nonlinear vibration model of fiber metal laminated thin plate treated with constrained layer damping patches

This paper explores the nonlinear vibration characteristics of fiber metal laminated thin plate(FMLTP) treated with constrained layer damping(CLD) patches. Additionally, based on the Jones-Nelson nonlinear material theory and the von Kármán geometric nonlinear theory, a theoretical model for FMLTP with CLD patches is established, considering material nonlinearity. Using the energy method, an energy equation is formulated and the nonlinear natural frequencies and vibration responses of the studied structure are iteratively determined through the Newton-Raphson method under various excitation amplitudes. Additionally, the modal damping ratios are calculated utilizing the complex modulus method. To validate the effectiveness of the model, a corresponding testing system is constructed, and tests are conducted on a 5-layer TA2/TC500 FMLTP treated with two pieces of CLD patches. The test results show excellent agreement with the theoretical results. Finally, the influences of different CLD patches treatment methods on the vibration characteristics of the FMLTP are discussed.

Nonlinear combined harmonic resonances of composite cylindrical shells operating in hygro-thermo-electro-magneto-mechanical fields

By considering the effects of the hygro-thermo-electro-magnetic environments, von Karman nonlinear terms, and multi-harmonic excitations, a coupled nonlinear vibration modeling of composite cylindrical shells comprising a carbon nanotube-reinforced composite (CNTRC) core and two piezoelectric/magnetic composite (PEMC) skins is developed, and the nonlinear dynamic behaviors of such cylindrical shells under primary, super/sub-harmonic, and combined resonance states are investigated. In the theoretical modeling process, the effective mechanical properties of the CNTRC core are first determined using the mixing and Schapery laws, and the hygro-thermo-electro-magneto-mechanical constitutive relations of the PEMC skins are then formulated. Within the framework of Reissner-Mindlin shell theory, the Lagrangian of the system containing Green-Lagrange and von Karman nonlinear terms is derived, and solution techniques based on the multiscale method are provided to obtain nonlinear frequencies, dynamic responses, and phases of CNTRC-PEMC cylindrical shells under multi-physics fields. Consequently, comparison studies are conducted to validate the correctness of the proposed model from different aspects. Based on this, various resonance and chaos behaviors of such structures subjected to multiple load components are revealed and compared, and the influences of environmental factors and structure composition on the amplitude-frequency curves, time-history responses, and phase planes are explored, with several recommendations and findings being drawn.

Dynamic behaviors of delaminated nanofilms partly bonded on substrates with sub-nanoscale van der Waals dynamic boundaries

Abstract Dynamic buckling-induced delamination of nanofilms on substrates is a universal and essential phenomenon in nanoelectromechanical systems (NEMS). Van der Waals (vdWs) interactions play an important role in the dynamic buckling-induced delamination of nanofilms on substrates due to the interaction distances at nanoscale or even sub-nanoscale in NEMS. Therefore, it is interesting yet challenging to reveal the effect of intermolecular vdWs interactions on dynamic buckling-induced delamination of nanofilms on substrates. By considering sub-nanoscale dynamic boundary effects induced by intermolecular vdWs interactions, a parametric excitation nonlinear vibration model for dynamic buckling-induced delamination of nanofilms partly bonded on substrates is established. Effects of sub-nanoscale vdWs dynamic boundaries on transient and steady-state responses of dynamically delaminated nanofilms on substrates are analyzed. The sub-nanoscale vdWs dynamic boundaries lead the dynamic responses of delaminated-nanofilm/substrate systems very sensitive to initial conditions. The bending and shifting frequency response results demonstrated that the system nonlinearities can be greatly amplified by the sub-nanoscale vdWs dynamic boundary effect. Moreover, the spontaneous symmetry breaking and violent interfacial tearing/healing phenomena can be also triggered in the systems. Based on spontaneous symmetry breaking, a trans-scale relationship between nanofilm equilibrium positions and intermolecular vdWs interactions is established, which can provide a promising route for trans-scale measurements of molecular scale interfacial interactions. The work can also be helpful for the dynamic design of resonant NEMS devices based on nanofilm/substrate systems.

Nonlinear Coupled Vibration Behavior of BFRP Cables on Long-Span Cable-Stayed Bridges under Parametric Excitation

Based on the cable-stayed beam model, this paper studies the nonlinear coupled vibration behavior of basalt fiber-reinforced polymer (BFRP) cables on long-span cable-stayed bridges under parametric excitation. Considering the sag, damping of BFRP cables, and the coupled interactions between stayed cables and the main girder, the nonlinear coupling vibration model of the BFRP cable–beam composite structure has been established. Taking the longest cable of Sutong Bridge as a case study, the nonlinear coupled vibration behavior of BFRP cables under parametric excitation has been numerically analyzed using the finite difference method. The analysis results indicate that (1) under parametric excitation, the large amplitude nonlinear vibration of the BFRP cable will be induced with an evident “beat” phenomenon. (2) Under the same parametric excitation, the nonlinear coupling vibration response and the beta frequency of the BFRP cable were both smaller than that of the traditional steel cable. (3) The nonlinear coupling vibration response of the BFRP cable increased with an increment in excitation amplitude and a decrement in cable force. With the increase in the excitation frequency, weight per unit length, and axial stiffness, the nonlinear vibration response of the BFRP cable increased first and then decreased. Meanwhile, the damping ratio of the BFRP cable had no significant influence on the nonlinear coupling vibration.

Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System

The internal beveloid gear has good scientific research value and a broad application field. However, there is a lack of research on the nonlinear vibration characteristics of internal beveloid gears. The present study establishes a nonlinear vibration model that takes into account axial vibration, gear torsional vibration, and radial support vibration. The effects of various excitation parameters on the vibration characteristics of the gear transmission system are analyzed and computed using the Runge–Kutta method. After analyzing the dynamic model of the gear pair using numerical methods, the harmonic balance method is employed to obtain the theoretical analytical solution for a single−stage gear. The obtained results are compared with the velocity and acceleration time−domain response curves obtained from the numerical method, and they show excellent agreement.

Vertical nonlinear vibration analysis of cold rolling mill considering structural gap and dynamic rolling force

To reveal the impact of vertical nonlinear vibration on the stabilization of the precision cold rolling system, the vertical nonlinear vibration model with consideration of structural gap and dynamic rolling force has been established. Utilizing the Magnus series method with high accuracy and long-term numerical stabilization, we calculate the vibration response of the cold rolling system and conduct numerical simulations to determine the bifurcation characteristics, maximum Lyapunov exponent, and two-parameter dynamical behavior. The results reveal the parameter regions of bifurcation and chaos that cause the vibration instability of the cold rolling system. It has also been found that the system has the nonlinear phenomenon of transition from period-3 motion to chaotic motion. More importantly, two-parameter matching regions for the cold rolling system performing periodic and chaotic motions are determined. Two-parameter matching solutions for improving and deteriorating the stabilization of the cold rolling system are clarified, and the preferred matching regions for the parameters coupling with the structural gap and dynamic rolling force are revealed. The results reveal the impact of system parameters upon the vertical nonlinear vibration of the cold rolling mill, which can serve as an academic reference for reducing the vibration and enhancing the stabilization of the cold rolling system.

Reduced Order Model of Nonlinear Structures for Turbomachinery Aeroelasticity

AbstractThis work concerns the numerical modeling of geometric nonlinear vibrations of slender structures in rotation using an original reduced order model based on the use of dual modes along with the implicit condensation method. This approach is an improvement of the classical ICE method in the sense that the membrane stretching effect is taken into account in the dynamic resolution. The dynamics equations are first presented and the construction of the reduced order model (ROM) is then proposed. The second part of the paper deals with numerical applications using the finite element method, first for a three-dimensional cantilever beam, then for an Ultra-High Bypass Ratio (UHBR) fan blade subject to aerodynamic loads. In the applications considered, the proposed method predicts more accurately the geometrically nonlinear behavior than the ICE method.

Axial-lateral-torsional coupling nonlinear vibration of riserless drill-string considering both seawater and stratum parts

Riserless drilling has the obvious advantages of small platform load bearing and short well construction period because it discards the bulky conventional riser. However, because the drill-string is directly exposed to seawater, its dynamic behavior is extremely complex, which has an important impact on the safety and efficiency of drilling operations. Using the energy method and microelement method, an axial-lateral-torsional (ALT) coupling nonlinear vibration model of deepwater riserless drill-string considering both seawater and stratum sections is established. The model takes into account the platform heaving and slow drift motion, wind and ocean current, tidal current, wave load, wellbore contact, bit-rock interaction and seawater formation interaction. A bi-directional coupling scheme is proposed to obtain a good model solving efficiency. Using torsional vibration model results and field test data, the effectiveness of the model is presented in detail from the qualitative and quantitative aspects. The developed model and calculation program are used to analyze the dynamic behavior of a typical deep-water riserless drillings. The work in this paper can provide an effective analysis method and theoretical guidance for riserless drill-string design.

Multi-field coupling multiple nonlinear vibration model and fatigue failure mechanism of deep-ocean mining hydraulic lifting pipe

Multi-field coupling multiple nonlinear vibration model and fatigue failure mechanism of deep-ocean mining hydraulic lifting pipe

Coupled chaotic vibration during pressure detection of micro-resonant pressure sensor

In measuring gas pressure, the stiffness of the resonator changes in the micro-resonant pressure sensor. If the design parameters are not chosen correctly, significant variations in vibration, even chaotic vibrations, will occur in the process of gas pressure detection, seriously degrading the detection accuracy and the stability of the sensor. The micro-resonant pressure sensor is in an environment of deep coupling and mutual influence of multiple physical fields when measuring the external environmental pressure. In this work, a multi-field coupling nonlinear vibration model of the resonant sensor used in pressure detection is established, and the multi-field coupling bifurcation, chaos, and other complex vibration characteristics of the sensor system are explored. The influence of the initial gap, the length of the resonator, and the excitation voltage during pressure detection are analyzed, and the stable vibration range of each influencing parameter is determined. The results show that when the sensor measures the pressure, the initial gap of the sensor, the length of the resonator, and the stability of the excitation voltage all decrease, causing unstable vibration of the sensor. In order to obtain stable sensor detection performance, it is necessary to determine the stability range of sensor-related parameters correctly.

Bifurcations and Stability of Nonlinear Vibrations of a Three-Layer Composite Shell with Moderate Amplitudes

The authors derived a mathematical model of geometrically nonlinear vibrations of three-layer shells, which describes the vibrations of the structure with amplitudes comparable to its thickness. The high-order shear theory is used in the derivation of this model. Rotational inertia is also taken into account. At the same time, the middle layer is a honeycomb structure made thanks to additive FDM technologies. In addition, each shell layer is described by five variables (three displacement projections and two rotation angles of the normal to the middle surface). The total number of unknown variables is fifteen. To obtain a model of nonlinear vibrations of the structure, the method of given forms is used. The potential energy, which takes into account the quadratic, cubic, and fourth powers of the generalized displacements of the structure, is derived. All generalized displacements are decomposed by generalized coordinates and eigenforms, which are recognized as basic functions. It is proved that the mathematical model of shell vibrations is a system of nonlinear non-autonomous ordinary differential equations. A numerical procedure is used to study nonlinear periodic vibrations and their bifurcations, which is a combination of the continuation method and the shooting method. The shooting method takes into account periodicity conditions expressed by a system of nonlinear algebraic equations with respect to the initial conditions of periodic vibrations. These equations are solved using Newton's method. The properties of nonlinear periodic vibrations and their bifurcations in the area of subharmonic resonances are numerically studied. Stable subharmonic vibrations of the second order, which undergo a saddle-node bifurcation, are revealed. An infinite sequence of bifurcations leading to chaotic vibrations is not detected.

Nonlinear free vibration modeling of anisogrid lattice sandwich plates based on a weak formulation analysis

This research examines the geometrically nonlinear free vibrations of a type of sandwich plate. This plate consists of an anisogrid lattice core and two nanocomposite face sheets. The lattice core is made up of a set of straight and oblique ribs with rectangular cross-sections that meet at nodes that are not all in the same place (anisogrid nodes). The nanocomposite skins are reinforced with carbon nanotubes (CNTs) based on the various functionally graded models. The displacement field of the sandwich plate is estimated based on the third-order shear deformation plate theory. Also, the nonlinear von-Kármán strain–displacement relations are used. Based on the Hooke and Hamilton laws, the motion equations of the system for nonlinear free vibrations are derived in a weak form. Then, the nodal weak formulation method based on the Lagrange interpolation functions and Gauss–Lobatto–Chebyshev node distribution, for the first time, is employed to calculate the time-dependent equations. The residuals of the obtained equations are minimized by the Galerkin method. Next, a displacement control iterative technique is applied to compute the nonlinear frequency of the sandwich plate. The effect of anisogrid lattice core characteristics and nanocomposite face sheets properties on the linear and nonlinear frequencies and also mode shapes are examined in the results.

Mechanical Rectifier for Broadband Piezoelectric Vibration Energy Harvesting and Self-Adapting Synchronous Electric Charge Extraction

The synchronized switch interface circuit is believed to boost the piezoelectric power generation. The threshold voltage of electronic components causes rectification loss and delay of peak detection. This work reports a novel design of a mechanical rectification piezoelectric vibration energy harvester (MR-PVEH). Compact and low loss mechanical rectification is achieved by double switch with 5 electrodes contact at the cantilever tip. The switch as a mechanical stopper also contributes to broadband performance. The saturation voltage and response frequency range can be adjusted in modeling and simulation of nonlinear electromechanical coupling vibration. Self-adapting Mechanical rectification optimized synchronous electric charge extraction (M-OSECE) circuit is also proposed with less electronic components in this paper and shows an extra 60% increase in generated power than the traditional standard energy harvesting circuit (SEH) in test. This novel design gives a promising solution for self-adapting synchronous switching in broadband piezoelectric vibration energy harvesting.

Numerical modeling of nonlinear viscoelastic vibrations of vehicles

Nonlinear viscoelastic oscillations of a vehicle in a vertical direction excited by static imbalance are considered. When taking into account the rheological properties of the suspension material and spring, the Boltzmann-Volterra principle was used. Mathematical models of the problem under consideration are obtained, which are described by systems of nonlinear integro-differential equations. A solution method based on quadrature formulas has been developed, and a computer program has been compiled on its basis. The results obtained are reflected in graphs. The influence of the material's rheological properties and the nonlinear characteristics of the suspension and spring on the amplitude and frequency of vertical vibrations of vehicles was investigated.

Modeling of nonlinear torsional vibrations of a truncated conical rod

Разработана нелинейная математическая модель нестационарных крутильных колебаний усеченного конического стержня из упругого материала с учетом нелинейной связи между напряжениями и деформациями. Выведено нелинейное уравнение для крутильных колебаний усеченного конического стержня относительно главной части крутильного перемещения оси симметрии стержня. Показано, что полученное уравнение нелинейных крутильных колебаний усеченного конического упругого стрежня в частных случаях совпадает с известными уравнениями, полученными другими авторами. С помощью полученного уравнения можно однозначно определить напряженно-деформированное состояние произвольного сечения конического стержня по пространственной координате и времени. На основе построенной модели численно решена задача о нестационарных крутильных колебаниях усеченного конического стержня при действии торцевой и поверхностной динамических нагрузок в условиях, когда широкий конец стержня жестко заделан, а узкий является свободным.

A hybrid averaging and harmonic balance method for weakly nonlinear asymmetric resonators

Models for nonlinear vibrations commonly employ polynomial terms that arise from series expansions about an equilibrium point. The analysis of symmetric systems with cubic stiffness terms is very common, and the inclusion of asymmetric quadratic terms is known to modify the effective cubic nonlinearity in weakly nonlinear systems. When using low (second, in this case)-order perturbation methods, the net effect in these cases is found to be a monotonic dependence of the free vibration frequency on the amplitude squared, with a single term that depends on the coefficients of the quadratic and cubic terms. However, in many applications, such a monotonic dependence is not observed, necessitating the use of techniques for strongly nonlinear systems, or the inclusion of higher-order terms and perturbation methods in weakly nonlinear formulations. In either case, the analysis involves very tedious and/or numerical approaches for determining the system response. In the present work, we propose a method that is a hybrid of the methods of averaging and harmonic balance, which provides, with relatively straightforward calculations, good approximations for the free and forced vibration response of weakly nonlinear asymmetric systems. For free vibration, it captures the correct amplitude–frequency dependence, including cases of non-monoticity. The method can also be used to determine the steady-state response of damped, harmonically driven vibrations, including information about stability. The method is described, and general results are obtained for an asymmetric system with up to quintic nonlinear terms. The results are applied to a numerical example and validated using simulations. This approach will be useful for analyzing a variety of system models with polynomial nonlinearities.

Nonlinear vibration model of sandwich beam with a shear thickening fluid core

Dynamic performance of sandwich beam with shear thickening fluid (STF) core subjected to a periodic excitation is investigated. The addition of STF enables the sandwich beam to have a tuning stiffness and damping capacity under dynamic deformation. Here, the constitutive model of STF is described by a complex shear modulus related to the shear rate. The transverse moving beam is modeled based on Newtonian law and derived nonlinear equations are solved by finite difference method. It is found that the resonant frequency of the beam increases with the amplitude of the external excitation by a power law. In addition, the initial damping ratio decreases first and then increases with the increase of the amplitude of the initial excitation for the STF with shear thinning zone. The minimum value of the initial damping ratio should be avoided, which is caused by initial shear thinning. This work can be used to guide the designing of structures with enhanced damping and the preparation of STF materials for application to such structures.