Nonlinear Amplitude-frequency Research Articles

Nonlinear dynamic analysis of Sigmoid functionally graded circular cylindrical shells on elastic foundations using the third order shear deformation theory in thermal environments

This paper presents an analytical approach to investigate the nonlinear dynamic response and vibration of imperfect functionally graded materials (S-FGM) thick circular cylindrical shells surrounded on elastic foundation using the third order shear deformation shell theory. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a Sigmoid power law distribution (S-FGM) in terms of the volume fractions of constituents with metal–ceramic–metal layers. The S-FGM shells are subjected to mechanical, damping and thermal loads. The Galerkin method and fourth-order Runge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation and dynamic response of the shells. In numerical results, the effects of geometrical parameters, the material properties, imperfections, the elastic foundations and mechanical loads on the nonlinear dynamic response and nonlinear vibration of the shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.

Nonlinear LCO “amplitude–frequency” characteristics for plates fluttering at supersonic speeds

The nonlinear aeroelastic behavior of isotropic rectangular plates in supersonic gas flow is examined. Quadratic and cubic aerodynamic nonlinearities as well as cubic geometrical nonlinearities are considered in this study. While the aerodynamic nonlinearities are the results of the expansion of the nonlinear piston-theory aerodynamics loading up to the third-order, the geometrical nonlinearities are due to stiffening effects from the panel out-of-plane deformation consistent with the von Karman’s nonlinear plate theory. While in vacuum the typical nonlinear hardening frequency vs. oscillation amplitude, one characterized by monotonically increasing amplitudes at increasing frequencies, exists, in the presence of a high-speed flow, qualitative and quantitative changes of the nonlinear relationship are expected. This paper shows how the thin-plate behavior is influenced by the high-speed flows providing the “amplitude–frequency” dependency, which describes the nonlinear oscillations of the considered aeroelastic system.

Nonlinear vibrations of fluid-filled functionally graded cylindrical shell considering a time-dependent lateral load and static preload

In the recent years, functionally gradient materials (FGMs) have gained considerable attention with possible applications in several engineering fields, especially in a high-temperature or hazardous environment. In this work, the nonlinear vibrations of a simply supported fluid-filled functionally graded cylindrical shell subjected to a lateral time-dependent load and axial static preload are analyzed. To model the shell, the Donnell nonlinear shell theory is used. The fluid is assumed to be incompressible, nonviscous, and irrotational. A new function to describe the variation in the volume fraction of the constituent material through the shell thickness is proposed, extending the concept of sandwich structures to a functionally graded material. Material properties are graded along the shell thickness according to the proposed volume fraction power law distribution. A consistent reduced order model derived from a perturbation technique is used to describe the displacements of the shell and, the Galerkin method is applied to derive a set of coupled nonlinear ordinary differential equations of motion. Results show the influence of the variation of the two constituent materials along the shell thickness, internal fluid, static preload, and shell geometry on the natural frequencies, nonlinear frequency–amplitude relation, resonance curves, and bifurcation scenario of the FG cylindrical shell.

Nonlinear Characteristics of Floating Piles Under Rotating Machine Induced Vertical Vibration

This study presents a finite element based approach to evaluate the linear and nonlinear frequency–amplitude response of floating piles subject to rotating machine induced vertical vibrations. A Matlab program is developed to compute the response of a single pile with floating tip condition for a linear and two nonlinear soil models. The variation of complex soil stiffness parameters with frequency has also been presented using different boundary zone parameters (shear modulus reduction ratio, thickness ratio and damping ratio). A detailed investigation of soil–pile system stiffness and damping parameters has been done considering different values of soil–pile separation lengths and boundary zone parameters. To verify the effectiveness of this proposed approach vertical vibration tests were conducted in the field on a single pile of diameter 0.114 m and length of 2.85 m by constructing floating tip condition. The frequency–amplitude response obtained from field vibration test has been compared with the theoretical results for all the soil models. From the comparison results it is observed that the proposed theory can predict the nonlinear response of floating piles very efficiently with proper inclusion of boundary zone parameters and soil–pile separation lengths.

Application of Iteration Perturbation Method in studying dynamic pull-in instability of micro-beams

In the present study, dynamic pull-in instability of electrostatically-actuated micro-beams is investigated through proposing the nonlinear frequency amplitude relationship. An approximate analytical expression of the fundamental natural frequency is presented by modern asymptotic approach namely Iteration Perturbation Method (IPM). Influences of vibrational amplitude as well as different parameters on dynamic pull-in voltage are investigated. It is demonstrated that two terms in series expansions is sufficient to produce an acceptable solution of the mentioned microstructure. The simulations from numerical methods verify the validity of the analytical procedure.

Nonlinear free vibrations of quintic inextensional beams lying on Winkler elastic substrate based on three-mode assumptions

The vibrational behavior of quintic nonlinear inextensional beam on two-parameter elastic substrate based on the three-mode assumptions is investigated in this paper. Using the Galerkin decomposition principle by considering three-mode assumptions, the nonlinear governing equations are derived. An asymptotic approach called parameter expansion method is employed to obtain the approximate expressions of nonlinear frequency–amplitude relationship for the first, second, and third modes of vibrations. Afterwards, it is clearly shown that the first term in series expansions gives excellent results in contrast with numerical calculations. Finally, the effect of the initial configuration of the beam as well as system parameters on the natural frequencies of quintic nonlinear beams is studied.

Research on Nonlinear Dynamics of Drive Mode inZ-Axis Silicon Microgyroscope

A novel drive method is proposed to increase the vibration amplitude of drive mode of silicon vibratory gyroscopes based on the nonlinear dynamics research made in this paper. A nonlinear dynamic equation, considering the nonlinear mechanical-elastic and electrostatic force, was firstly deduced according to the simplified structure schematic of drive mode. Then, experiments were made to characterize the nonlinear amplitude-frequency response characteristic as well as its dependence on drive voltage. Besides, by solving the nondimensional nonlinear dynamics using Matlab and Simulink, the influence of initial status of drive mode, that is, the displacement and the velocity, on the stable vibration amplitude was presented. The amplitude-frequency characteristic was also analyzed by simulation. The result of simulation shows good agreement with that of experiments. When nonlinearity is present in the amplitude-frequency characteristic, there is a wide flat region with larger amplitude than the peak amplitude obtained in the linear characteristic. Finally, a novel drive method was presented to excite the nonlinear vibration of large amplitude in the downwards-sweeping characteristic curve. It is numerically proved that this merit can be utilized to enlarge the drive mode’s vibration amplitude so as to enhance silicon gyroscopes’ performance.

Nonlinear vibrations and multiple resonances of fluid filled arbitrary laminated circular cylindrical shells

Nonlinear forced vibrations of water-filled, laminated circular cylindrical shells are studied by using the Amabili–Reddy nonlinear higher-order shear deformation theory and energy approach in the Lagrangian description. The fluid is modeled by potential flow. It is assumed that the shell is subjected to a steady harmonic concentrated force acting in the radial direction. Pseudo arc-length continuation and collocation technique is used to carry out bifurcation analysis and to obtain nonlinear frequency–amplitude responses. Direct time integration of the equations of motion has also been performed by using Gear’s backward differentiation formula (BDF) to obtain time histories, phase space diagrams and Poincaré maps. The effect of internal fluid and lamination angle on the frequency–amplitude response in the neighborhood of the resonance frequency, traveling wave solution and internal resonances of simply supported shells are investigated. It is shown that water-filled composite shells may exhibit complex nonlinear dynamics including a rare and intricate 1:1:1:1 internal resonance.

The effect of the temperature-dependent nonlinearities on the temperature stability of micromechanical resonators

Micromechanical resonators show a discrepancy between the frequency-temperature (f-T) characteristics they have in open-loop and closed-loop measurements, and this discrepancy adversely affects resonator's temperature stability performance. We explain the discrepancy with a model that combines the temperature-dependent quality factor (Q) with the nonlinear amplitude-frequency (A-f) effect; we then experimentally verify the model using two types of double-ended tuning fork resonators. In addition, we present an improved closed-loop system that removes the discrepancy, thus improving the temperature stability.

A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches

This paper intends to promote the application of modern analytical approaches to the governing equation of transversely vibrating quintic nonlinear beams. Four new studied methods are Stiffness analytical approximation method, Homotopy Perturbation Method with an Auxiliary Term, Max–Min Approach (MMA) and Iteration Perturbation Method (IPM). The powerful analytical approaches are used to obtain the nonlinear frequency–amplitude relationship for dynamic behavior of vibrating beams with quintic nonlinearity. It is demonstrated that the first terms in series expansions of all methods are sufficient to obtain a highly accurate solution. Finally, a numerical example is conducted to verify the integrity of the asymptotic methods.

Large amplitude free vibration of size-dependent circular microplates based on the modified couple stress theory

The present investigation proposes a nonclassical mathematical model and an algorithm for the axisymmetrically nonlinear free vibration analysis of a circular microplate. Based on the modified couple stress theory and von Kármán geometrically nonlinear theory, the governing equations for microplates are established in variational form from Hamilton principle containing only one material length scale parameter which can capture the size-dependent behavior. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed simple harmonic function in time. The resulting nonlinear spatial boundary value problem is then solved numerically by shooting method, and the size-dependent characteristic relationships of nonlinear vibration frequency versus central amplitude of the microplates are obtained. The parametric studies are conducted for immovable clamped and simply supported edge conditions, some of the results in special cases are verified by comparing with those in the literature. The numerical results indicate that the microplates modeled by the modified couple stress theory cause more stiffness than modeled by the classical continuum plate theory, such that the differences between the results of these two theories are large when the thickness to material length scale ratio is small, whereas they are decreasing with the increase of the ratio.

Nonlinear free vibration of a microscale beam based on modified couple stress theory

This paper presents a nonlinear free vibration analysis of the microbeams based on the modified couple stress Euler–Bernoulli beam theory and von Kármán geometrically nonlinear theory. The governing differential equations are established in variational form from Hamilton principle, with a material length scale parameter to interpret the size effect. These partial differential equations are reduced to corresponding ordinary ones by eliminating the time variable with the Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the size-dependent characteristic relations of nonlinear vibration frequency vs. central amplitude of the microbeams are obtained successfully. The comparisons with available published results show that the current approach and algorithm are of good practicability. A parametric study is conducted involving the dependency of the frequency on the length scale parameter along with Poisson ratio, which shows that the nonlinear vibration frequency predicted by the current model is higher than that by the classical one.

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.

Nonlinear vibration of functionally graded circular cylindrical shells based on improved Donnell equations

In the present work, the study of the nonlinear vibration of a functionally graded cylindrical shell subjected to axial and transverse mechanical loads is presented. Material properties are graded in the thickness direction of the shell according to a simple power law distribution in terms of volume fractions of the material constituents. Governing equations are derived using improved Donnell shell theory ignoring the shallowness of cylindrical shells and kinematic nonlinearity is taken into consideration. One-term approximate solution is assumed to satisfy simply supported boundary conditions. The Galerkin method, the Volmir's assumption and fourth-order Runge–Kutta method are used for dynamical analysis of shells to give explicit expressions of natural frequencies, nonlinear frequency–amplitude relation and nonlinear dynamic responses. Numerical results show the effects of characteristics of functionally graded materials, pre-loaded axial compression and dimensional ratios on the dynamical behavior of shells. The proposed results are validated by comparing with those in the literature.

Trispectrum and correlation dimension analysis of magnetorheological damper in vibration screen

In order to improve the screening efficiency of vibrating screen and make vibration process smooth, a new type of magnetorheological (MR) damper was proposed. The signals of displacement in the vibration process during the test were collected. The trispectrum model of autoregressive (AR) time series was built and the correlation dimension was used to quantify the fractal characteristics during the vibration process. The result shows that, in different working conditions, trispectrum slices are applied to obtaining the information of non-Gaussian, nonlinear amplitude-frequency characteristics of the signal. Besides, there is correlation between the correlation dimension of vibration signal and trispectrum slices, which is very important to select the optimum working parameters of the MR damper and vibrating screen. And in the experimental conditions, it is found that when the working current of MR damper is 2 A and the rotation speed of vibration motor is 800 r/min, the vibration screen reaches its maximum screening efficiency.

Nonlinear damping and forced vibration analysis of laminated composite beams

The purpose of the present work it to study the damping and forced vibrations of three-layered, symmetric laminated composite beams. In the analytical formulation, both normal and shear deformations are considered in the core by using the higher-order zig-zag theories. The harmonic balance method is coupled with a one mode Galerkin procedure for a simply supported beam. The geometrically nonlinear coupling leads to a nonlinear frequency amplitude equation governed by several complex coefficients. In the first part of the paper, linear and nonlinear damping parameters of laminated composite beams are obtained. In the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. The frequency response curves are presented and discussed for various geometric and material properties.

Nonlinear Response of Cantilevered Carbon Nanotube in Mass Sensing

This study predicts the nonlinear response of a cantilevered carbon nanotube(CNT) for biomolecular mass sensing using simple mechanical model analysis. The electrostatically resonating CNT cantilever can measure the frequency shift due to the added mass such as virus, bacterium and various macromolecules on the CNT. The analysis model of the resonating CNT has geometric and inertial nonlinearities to understand the complex nonlinear phenomena as increasing the applied AC and DC voltages with attached mass on the CNT. The CNT cantilever as a nanomechanical biosensor shows nonlinear frequency amplitude and phase responses such as softening at primary resonance through saddle-node bifurcation, and subharmonic resonance through period-doubling bifurcation with different attached mass and Q-factors. Figure 1: Concept of electrostatically actuated CNT cantilever. (a) Without tip mass (, ) (b) With tip mass (, ) Figure 2: Frequency response of CNT cantilever on and (mass ratio ). Acknowledgment This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government(MEST). (NRF-2011-0005249).

Nonlinear Amplitude-Frequency Response of a Helmholtz Resonator

Nonlinear amplitude-frequency response of an acoustic Helmholtz resonator is investigated by incorporating linear damping, nonlinear damping, and nonlinear restoration. Method of multiple time scale is used for theoretical analysis, and a reasonable explanation of an experiment observation, i.e., the downward shift of resonance frequency, is given. We also discuss the response of two-frequency driving, and find that amplitude-frequency response depends on the phase difference of the driving. In addition, the response amplitude of two-frequency driving can increase approximately 10% as compared with the single frequency driving at a certain phase difference for the parameters we choose to simulate.

Complex modes based numerical analysis of viscoelastic sandwich plates vibrations

In this paper, a numerical method for linear and nonlinear vibrations analysis of viscoelastic sandwich beams and plates is developed with finite element based solution. This method couples the harmonic balance technique to complex mode Galerkin's procedure. This results in a scalar nonlinear complex amplitude-frequency relationship involving numerical computation of three coefficients. A general formulation taking into account the frequency dependence of the viscoelastic behaviour allowing to intoduce any viscoelastic law is given. Complex eigenmodes are numerically computed in a general procedure and used as Galerkin's basis. The free and steady-state vibrations analyses of viscoelastic sandwich beams and plates are investigated for constant and frequency dependent viscoelastic laws and for various boundary conditions. The equivalent frequencies and loss factors as well as forced harmonic response and phase curves are performed. The obtained results show the efficiency of the present approach to large amplitudes vibrations of viscoelastic sandwich structures with nonlinear frequency dependence.

Study on Trispectrum Characteristics of MR Damper in Vibration Processes

In order to find out a new type of MR damper for block making machine, the signals of displacement in the vibration process during the test are collected and the time series AR model of trispectrum for analyzing the dynamic characteristics of the MR damper is built. It turns out that , in different working conditions, slices of trispectrum are applied to obtain the signal’s non-Gaussian, nonlinear amplitude-frequency characteristics which are very important for us to select the optimum working parameters of the MR damper.