Nonlinear Amplitude-frequency Research Articles

Nonlinear vibration characteristics of virtual mass systems for wind turbine blade fatigue testing

Abstract. The biaxial fatigue test of wind turbine blades is helpful to shorten the test time and is more suitable for the actual operating conditions. Adding tuning masses to the blade is a common method for blade uniaxial testing at present, and its purpose is to adjust the load distribution in one direction of the blade. However, the tuning masses on the blade will affect the load distribution in the direction of the blade flap-wise and edge-wise at the same time in the biaxial test, so the concept of “virtual masses” is proposed to realize the decoupling of the load distribution in the biaxial test. Due to the limitation of the size of the virtual mass mechanism and the complex motion trajectory of the blade, the actual inertial effect provided by the virtual masses is different from the ideal situation, which will affect the resonance characteristics of the test system and the load distribution of the blade. Therefore, in order to evaluate the effect of the nonlinear effect introduced by the virtual masses on the resonance characteristics of the test system and the blade load distribution, the equivalent dynamic model of the bladed virtual mass test system was established by using the Lagrange method. Then, the nonlinear effects of blade amplitude and virtual mass installation parameters on the test system are obtained by a numerical method. Then, based on the nonlinear vibration theory, the approximate nonlinear amplitude–frequency characteristics of the test system are obtained, that is, the resonance frequency of the test system will decrease with the increase in the blade amplitude. Through the simulation analysis of two blades over 80 m in length, the applicability of the theoretical method is verified. It can be seen from the simulation results of the simulated uniaxial test that larger amplitudes of the blade and shorter connection rods will reduce the resonance frequency of the test system. When the vibration amplitude at the excitation point is the same, a lower resonance frequency results in a smaller load distribution level, that is, the area which is actually fully tested will be reduced. In the biaxial simulation test, the resonance frequency of the test system will be further reduced because the virtual masses will be affected by the coupled motion in both directions at the same time. Furthermore, the introduction of an external mechanism of the virtual mass will also cause deformation of the envelope of the blade biaxial trajectory, which will further affect the load distribution of the blade. This work explores the nonlinear influence of virtual masses on the actual fatigue test. The theoretical analysis is helpful to provide the basis and reference for the preliminary preparation work of the test organization, including adjusting the tuning mass scheme, adjusting the load distribution and selecting the appropriate excitation equipment.

Nonlinear Forced Vibration of Functionally Graded Graphene-Reinforced Composite (FG-GRC) Laminated Cylindrical Shells under Different Boundary Conditions with Thermal Repercussions

The key aim of this research is to develop an analytical model for the forced vibration of graphene-reinforced composite (GRC) cylindrical shell with viscous damping and thermal environment effects under several boundary conditions. The prescribed mechanical and thermal characteristics of the GRC layer are evaluated utilizing a micromechanical extended Halpin–Tsai technique. Further, the governing equations are determined to be grounded on the shear deformation theory (SDT) with the von Kármán-form in terms of the geometric nonlinearity. The basic nonlinear partial differential governing equation is transformed into a nonlinear ordinary differential equation with the Galerkin-based method. The resulting ordinary governing differential equations are systematically solved based on the multiple scales scheme in order to obtain the nonlinear forced vibration frequency response of the laminated GRC cylindrical shell in the presence of damping impact and subjected to multiple boundary conditions. The validation of the obtained expressions is achieved by comparing them with the available literature data where the comparison revealed a clear consistency between them. Moreover, a parametric investigation is provided to illustrate the impacts of the distribution of graphene layers, elastic foundation coefficients, damping ratios, temperature effects and dimensionless radial excitation amplitude on the forced nonlinear amplitude-frequency ratio responses for a functionally-graded graphene-reinforced composite (FG-GRC)-layered cylindrical shells. The analytical outcomes indicate that the different distribution types of graphene, elastic foundation coefficients, damping ratios, temperature, and radial excitation parameters have a noteworthy impact on the frequency–amplitude behaviors of an FG-GRC-layered cylindrical shell. In addition, the new insights gained from this research might contribute to a deeper understanding of the nonlinear forced vibration responses in subsequent analysis and design techniques for giving appropriate benchmark findings.

Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

An exact analytical solution for the frequency response of a system consisting of a mass grounded by linear and nonlinear springs in series was derived. The system is applicable in the study of beams carrying intermediate rigid mass. The exact solution was derived by naturally transforming the integral of the governing nonlinear ODE into a form that can be expressed in terms of elliptic integrals. Hence, the exact frequency–amplitude solution was derived in terms of the complete elliptic integral of the first and third kinds. Periodic solutions were found to exist for all real values of [Formula: see text] and for [Formula: see text]. On the other hand, linear or weakly nonlinear frequency–amplitude responses were found to occur during small-amplitude vibrations ([Formula: see text]), very large-amplitude vibrations with strong hardening nonlinearity ([Formula: see text] and [Formula: see text]), and for all amplitudes when [Formula: see text]. Simulations showed that the system’s periodic response is significantly influenced by the nonlinearity parameter ([Formula: see text]), linearity parameter ([Formula: see text]), and amplitude of vibration ([Formula: see text]). Besides, the existence of bifurcation points at [Formula: see text] and different values of [Formula: see text] was confirmed. Lastly, an approximate frequency solution obtained using the He’s frequency–amplitude formulation was found to produce errors less than 1.0% for a wide range of input parameters. In conclusion, the present study provides a benchmark solution for verification of other approximate solutions, and the He’s frequency–amplitude formulation can be used to obtain fast and accurate solutions for complex nonlinear systems.

Nonlinear vibration and stability of sandwich functionally graded porous plates reinforced with graphene platelets in subsonic flow on elastic foundation

This paper investigates the nonlinear vibration and stability of a functionally graded porous sandwich plate reinforced with graphene platelets (GPLR-SFGP) interacting with subsonic airflow on elastic foundation. The plate comprises of a functionally graded porous core with graphene platelet reinforcement and two metal face layers. Utilizing Hamilton's principle, the nonlinear equation of the plate is exported and discretized into ordinary equations using the assumed modes method. The influence of porosity, GPL weight fraction, surface thickness ratio and Winkler Pasternak elastic foundation arguments on the critical divergence velocity of the plate under subsonic flow is revealed by calculating the system characteristic values. The Matcont toolbox is occupied to generate nonlinear amplitude frequency resonance curves, allowing for a comprehensive examination of the influence of these parameters on the nonlinear resonance behavior of the system. The GPLR-SFGP plate exhibits outstanding characteristics, including superior stiffness and a reduced mass, rendering it a suitable choice for exterior applications in airplanes, automobiles, and high-speed railways. The findings in this study can provide valuable insight into the key design parameters that significantly affect the performance of GPLR-SFGP plates, enabling future design efforts aimed at enhancing their efficacy and robustness in real-world applications.

Incorporating Geometric Nonlinearity in Theoretical Modeling of Muscle-Powered Soft Robotic Bio-Actuators

Abstract Biohybrid actuators aim to leverage the various advantages of biological cells over artificial components to build novel compliant machines with high performance and autonomy. Significant advances have been made in bio-fabrication technologies, enabling the realization of muscle-powered bio-actuators. However, the mechanics of muscle-scaffold coupling has been relatively understudied, limiting the development of bio-actuators to intuitive or biomimetic designs. Here, we consider the case of implementing muscle-based actuation for soft robotic swimmers operating at low Reynolds numbers. We develop an analytical model to describe the elasto-hydrodynamic problem and identify key design parameters. Muscle contraction dynamics is characterized experimentally and the implications of nonlinear amplitude-frequency relationship of muscle-based actuation are discussed. We show that a novel bio-actuator with high performance can be developed by introducing compliant flexural mechanisms undergoing large deflection. Geometric nonlinearities are accounted for in the analysis of the force-deflection relationship for the flexural mechanism. Our results show that for expected muscle contraction forces, this novel bio-actuator can outperform previous muscle-powered swimmers by up to two orders of magnitude in swimming speed.

Multi-harmonic resonance of pipes conveying fluid with pulsating flow

The parametric excitation analysis of pipes conveying pulsating fluid usually focuses on the principal parametric resonance. In this paper, the principal parametric resonance and the combination resonance under parametric excitation are compared for the first time. A dynamic model of a fluid-conveying pipe with elastic torsional constraints at both ends is established. The nonlinear partial differential-integral equation is derived for governing the transverse vibration. The natural frequencies and modes of pipe are obtained. Moreover, the analytical solutions of stable region and nonlinear amplitude-frequency responses for three kinds of parametric resonances are derived. Then, the analytical results are verified numerically. The results show that there is no differential-type combination parametric resonance for pipes supported at both ends. Increasing material damping or torsional stiffness and decreasing mean velocity can increase the stable region of the pipe. However, it is interesting to note that when combination parametric resonance occurs between symmetric and antisymmetric modes, the effect of mean velocity on stability is reduced. In addition, more than one form of resonance occurs in some regions. The sum-type combination resonance is easier to be excited than the principal parametric resonance. Among them, the combination of first-fourth mode is the most sensitive to the torsional boundary, and its response is the largest. In summary, this paper demonstrates that the combination parametric resonance, which has not been researched deeply yet, can be more destructive than the principal parametric resonance.

A generalized mechanical model with high-flexibility of viscoelastic damping materials and devices considering frequency and amplitude dependence effects

Owing to the complex mechanical properties influenced by excitation frequency and amplitude, the modeling of viscoelastic damping materials and devices faces the challenge of poor coordination among simplicity, accuracy, generality and flexibility. To address this issue, a nonlinear frequency–amplitude model was established by combining the proposed generalized complex stiffness model with the Berg friction model. The generalized complex stiffness model was proposed to characterize the frequency dependence, and the Berg friction model was employed to study the amplitude dependence. A dynamic analysis method was proposed based on a state space approach to realize a complete and accurate representation of the proposed mechanical model. Property tests of a viscoelastic damper and viscoelastic isolator were then conducted to study the dynamic mechanical performance under different excitation frequencies and amplitudes. Finally, the accuracy of the nonlinear frequency–amplitude model was validated using experimental data.

Mathematical modeling and solution of nonlinear vibration problem of laminated plates with CNT originating layers interacting with two-parameter elastic foundation

Generalizing the first-order shear deformation plate theory (FOPT) proposed by Ambartsumyan (Theory of anisotropic plates, Nauka, Moscow, 1967 (in Russian)) to the heterogeneous laminated nanocomposite plates and the nonlinear vibration problem is analytically solved taking into account an elastic medium in this study for the first time. The Pasternak-type elastic foundation model (PT-EF) is used as the elastic medium model. After creating the mathematical models of laminated rectangular plates with CNT originating layers on the PT-EF, the large amplitude stress–strain relationships and motion equations are derived in the form of nonlinear partial differential equations (PDEs) within FOPT. Then, by applying Galerkin's method to the derived equations, it is reduced to a nonlinear ordinary differential equation (NL-ODE) containing the second- and third-order nonlinear terms of the deflection function for laminated rectangular plates composed of nanocomposite layers. The NL-ODE is solved by the semi-inverse method, and the nonlinear frequency–amplitude relationship for the laminated plates consisting of CNT originating layers resting on the PT-EF is established within FOPT for the first time. From these relations, similar relations can be obtained particularly for the unconstrained laminated and monolayer CNT patterns plates. After comparing the accuracy of the obtained formulas with the reliable results in the literature, comprehensive numerical analyses are performed.

A Piezoelectrically Excited ZnO Nanowire Mass Sensor with Closed-Loop Detection at Room Temperature.

One-dimensional nanobeam mass sensors offer an unprecedented ability to measure tiny masses or even the mass of individual molecules or atoms, enabling many interesting applications in the fields of mass spectrometry and atomic physics. However, current nano-beam mass sensors suffer from poor real-time test performance and high environment requirements. This paper proposes a piezoelectrically excited ZnO nanowire (NW) mass sensor with closed-loop detection at room temperature to break this limitation. It is detected that the designed piezo-excited ZnO NW could operate at room temperature with a resonant frequency of 417.35 MHz, a quality factor of 3010, a mass sensitivity of -8.1 Hz/zg, and a resolution of 192 zg. The multi-field coupling dynamic model of ZnO NW mass sensor under piezoelectric excitation was established and solved. The nonlinear amplitude-frequency characteristic formula, frequency formula, modal function, sensitivity curve, and linear operating interval were obtained. The ZnO NW mass sensor was fabricated by a top-down method and its response to ethanol gas molecules was tested at room temperature. Experiments show that the sensor has high sensitivity, good closed-loop tracking performance, and high linearity, which provides great potential for the detection of biochemical reaction process of biological particles based on mechanics.

Softening-spring nonlinearity in large amplitude vibration of unsymmetric double-layer lattice truss core sandwich beams

In recent years, many novel sandwich structures with multilayer or graded lattice truss cores that exhibited superior structural performance have been proposed. However, only limited works have studied the nonlinear behavior of sandwich structures with unsymmetric lattice truss cores. This paper aims to provide an analytical study on the nonlinear vibration of the unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). The double-layer LTCSB is designed to be unsymmetric and possesses varying material property and structural geometry in each layer. In this study, six unsymmetric cases of LTCSB classified as two categories according to the midplane locations are considered. Subsequently, an analytical model for the unsymmetric double-layer LTCSB is developed based on the Allen’s model and von Kármán nonlinear theory. The axial displacement of the midplane of LTCSB is considered in the analytical model, therefore the proposed model is more generalized compared with previous models for the symmetric double-layer LTCSB. The Ritz method with a direct iterative procedure is applied to solve the nonlinear governing equations and determine the nonlinear frequencies for the unsymmetric double-layer LTCSB. Finally, the effects of six unsymmetric cases of LTCSBs on the nonlinear frequency ratio versus amplitude curve under three different types of boundary conditions are discussed detailly. An interesting phenomenon of softening-spring nonlinearity is found for hinged–hinged and clamped–hinged sandwich beams with large bending–extension coupling.

Multiple internal resonances of rotating composite cylindrical shells under varying temperature fields

Composite cylindrical shells, as key components, are widely employed in large rotating machines. However, due to the frequency bifurcations and dense frequency spectra caused by rotation, the nonlinear vibration usually has the behavior of complex multiple internal resonances. In addition, the varying temperature fields make the responses of the system further difficult to obtain. Therefore, the multiple internal resonances of composite cylindrical shells with porosities induced by rotation with varying temperature fields are studied in this paper. Three different types of the temperature fields, the Coriolis forces, and the centrifugal force are considered here. The Hamilton principle and the modified Donnell nonlinear shell theory are used to obtain the equilibrium equations of the system, which are transformed into the ordinary differential equations (ODEs) by the multi-mode Galerkin technique. Thereafter, the pseudo-arclength continuation method, which can identify the regions of instability, is introduced to obtain the numerical results. The detailed parametric analysis of the rotating composite shells is performed. Multiple internal resonances caused by the interaction between backward and forward wave modes and the energy transfer phenomenon are detected. Besides, the nonlinear amplitude-frequency response curves are different under different temperature fields.

Nonlinear Modes in Shear Horizontal Wave Propagation–Analytical and Numerical Analysis

Nonlinear shear horizontal guided wavefield is investigated analytically in plates. The major focus is on the effect of nonlinear material parameters, excitation frequency, wave number and wave amplitude on nonlinear mode generation. The analytical solution is based on the multiple-scale perturbation method and modal decomposition approach. The approximate analytical solution is validated using the Local Interaction Simulation Approach, implemented for nonlinear shear wave propagation. The results show that the analysed shear horizontal wavefield is distorted by material nonlinearity. The resulting displacement field can be interpreted in a form of nonlinear modes.

Homotopy perturbation method for strongly nonlinear oscillators

This paper reveals the effectiveness of the homotopy perturbation method for strongly nonlinear oscillators. A generalized Duffing oscillator is adopted to elucidate the solving process step by step, and a nonlinear frequency–amplitude relationship is obtained with a relative error of 0.91% when the amplitude tends to infinity, the solution morphology is also discussed, and the zero-th approximate solution is enough for conservative nonlinear oscillators, while the accuracy of the frequency can be improved if the iteration continues.

Tuning the nonlinearity of graphene mechanical resonators by Joule heating

As an inherent property of the device itself, nonlinearity in micro-/nano- electromechanical resonators is difficult to eliminate, and it has shown a wide range of applications in basic research, sensing and other fields. While many application scenarios require tunability of the nonlinearity, inherent nonlinearity of a mechanical resonator is difficult to be changed. Here, we report the experimental observation of a Joule heating induced tuning effect on the nonlinearity of graphene mechanical resonators. We fabricated multiple graphene mechanical resonators and detected their resonant properties by an optical interference method. The mechanical vibration of the resonators will enter from the linear to the nonlinear intervals if we enhance the external driving power to a certain value. We found that at a fixed drive power, the nonlinearity of a mechanical resonator can be tuned by applying a dc bias current on the resonator itself. The tuning mechanism could be explained by the nonlinear amplitude-frequency dependence theory. Our results may provide a research platform for the study of mechanical nonlinearity by using atomic-thin layer materials.

Vibration Characteristic Analysis of a Coupling System for Compound Biaxial and Inter-Shaft Bearings

In order to analyse the system spatial vibration of compound biaxial and inter-shaft bearings during unbalanced excitation, the theory of rotor dynamics and bearing contact deformation are combined to establish a coupling system model of compound biaxial and inter-shaft bearings. Considering the moment of inertia, gyro effect, and structural damping, a one-dimensional beam model is used to perform the MATLAB numerical calculation of the coupling system to obtain the modal characteristics of the system, and based on the ANSYS simulation analysis of the three-dimensional model, a modal test is used to obtain the system modal shape and test frequency response function to verify the rationality of the coupling system model. Based on the Runge-Kutta iterative method and Hamilton’s principle, the nonlinear amplitude-frequency response of the system is calculated while taking into account the influence of the compound biaxial eccentricity, as well as the stiffness and radial clearance of the inter-shaft bearing. The results show that the changes in the eccentricity of the compound biaxial and the structural parameters of the inter-shaft bearing cause the unbalanced excitation of the system, additional external load changes, and different coupled vibrations between the composite compound biaxial and inter-shaft bearings. Therefore, it is reasonable to match the structural parameters of the coupling system and improve the dynamic performance to avoid system resonance.

Nonlinear splitting of magnetoelastic resonance line in powerfully excited ferrite

The nonlinear splitting of magnetoelastic resonance line in powerfully excited ferrite is investigated. It is shown that the amplitude of splited resonance has the same order of value that the amplitude of basis resonance and its frequency is determined by the upper boundary of nonlinear amplitude-frequency characteristic of magnetic system. It is found the threshold character of splitting additional resonance from general. It is determined the possibility of operation of splited elastic resonance frequency by changing of exited magnetic field value which has practical importance. Keywords: ferromagnetic resonance, magnetoelastic interaction, nonlinear vibrations.\

Нелинейное расщепление линии магнитоупругого резонанса в сильно возбужденном феррите

The nonlinear splitting of magnetoelastic resonance line in powerfully excited ferrite is investigated. It is shown that the amplitude of splited resonance has the same order of value that the amplitude of basis resonance and its frequency is determined by the upper boundary of nonlinear amplitude-frequency characteristic of magnetic system. It is found the threshold character of splitting additional resonance from general. It is determined the possibility of operation of splited elastic resonance frequency by changing of exited magnetic field value which has practical importance.

Vibration and nonlinear dynamic response of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio

This paper reports a study on natural frequency, nonlinear dynamic response and frequency amplitude relation of temperature-dependent FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio. The nonlinear motion equations and the compatibility equation are retrieved from first order shear deformation theory including the effect of von Karman nonlinearity terms, elastic foundations and initial imperfection. The system of dynamic governing equations is obtained by utilizing the Airy stress function and Bubnov–Galerkin procedure. The nonlinear dynamic response is also obtained by applying the 4th-order Runge–Kutta method. The frequency–amplitude relations of the shell is determined by using the assumption of inertia caused by a rotation. Besides, the analytical method and the finite element method are proposed to calculate the vibrational frequencies. Five types of FG-CNTRC laminate shell, three types of the structures and different thermal environments with the shell’s material properties depending on the temperature are also considered in this study. The numerical results are presented, verified with available studies in the literature. Finally, the effects of elastic foundations, initial imperfection, and temperature-dependence, typing double curved shallow shell, angle of lamination against the shell x-axis on the natural frequency and nonlinear dynamic response of FG-CNTRC laminated double curved shallow shell with positive and negative Poisson’s ratio are discussed.

Nonlinear Forced Vibration of Spinning Pipes Conveying Fluid Under Lateral Harmonic Excitation

As a crucial offshore drilling equipment, the drill strings often encounter instability due to various external excitations. In order to deeply understand the dynamical mechanism of such issues, in this paper, the working drill strings are modeled as a spinning pipe conveying fluid, and the forced vibration characteristics of such system under an equivalent lateral excitation are studied analytically and numerically. The system considered is essentially under triple excitations: the fluid–structure interaction (FSI), spinning motion and external force. The viscoelastic material and geometrical nonlinearity due to extensible pipe axis are taken into account. The governing equation is derived by the Hamilton principle and then treated by the numerical simulation. The Lissajous trajectory and whirling motion shape of the pipe are obtained, in which the subcritical and supercritical cases are both discussed. It is revealed that the system conducts periodic or quasi-periodic motion at subcritical state, while presents chaotic motion at supercritical flow velocity. Perturbation analysis is also carried out to investigate the nonlinear frequency–amplitude features of the system. Comprehensive parametric studies are performed and the flow velocity, spinning speed, nonlinearity, excitation and damping are all found to have significant effects on the moving trajectory and frequency response of the pipe.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.