中心刚体-外Timoshenko梁系统的建模与分岔特性研究

对于中心刚体固结悬臂梁系统,当不考虑梁剪应力(即Euler-Bernoulli梁)影响时,匀速转动梁的平凡解是稳定的。而对于深梁,有必要考虑剪应力(即Timoshenko梁)的影响,此时其匀速转动平凡解将出现拉伸屈曲。为此采用广义Hamilton变分原理建立了中心刚体固结Timoshenko梁这类刚-柔耦合系统的非线性动力学模型,应用数值方法研究了匀速转动Timoshenko梁非线性系统的分岔特性,以及失稳的临界转速。     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Piezoelectric beams under small strains but large displacements and rotations

Small strains are consistently incorporated into a model to describe the behavior of piezoelectric beams subjected to large displacements and rotations. While the displacement is assumed to vary in accordance with the Timoshenko assumption, the electric potential has linear variation through each piezoelectric layer thickness. The strong geometric nonlinear effect on the beam electroelastic response is illustrated by static problems of cantilever beams with tip loads and distributed sensing or actuation. The present work seems to be the first to obtain analytical solutions for beams under large displacements and rotations with piezoelectric sensors. One believes that the quality of such solutions can be valuable to validate results predicted by approximate methods.     

Dynamic analysis of rotating double-tapered cantilever Timoshenko nano-beam using the nonlocal strain gradient theory

Based on the nonlocal strain gradient theory, the coupling nonlinear dynamic equations of a rotating double-tapered cantilever Timoshenko nano-beam are derived using the Hamilton principle. The equation of motion is discretized via the differential quadrature method. The effects of the angular velocity, nonlocal parameter, slenderness ratio, cross-section parameter, and taper ratios are examined and discussed. It is shown that taper ratios and cross-section parameter play a significant role in the vibration response of a rotating cantilever nano-beam. Further as rotational angular velocity increases, the taper ratios and cross-section parameter effect on the frequency response are increased for first modes of vibration.     

Modelling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring

Using Hamilton variation principle, a nonlinear dynamic model of the system with a finite deforming Rayleigh beam clamped radially to the interior of a rotating rigid ring, under the assumption that the constitutive relation of the beam is linearly elastic, is discussed. The bifurcation behavior of the simple system with the Euler-Bernoulli beam is also discussed. It is revealed that these two models have no influence on the critical bifurcation value and buckling solution in the steady state. Then we use the assumption model method to analyse the bifurcation behavior of the steadily rotating Euler-Bernoulli beam and get two different types of bifurcation behavior which physically exist. Finite element method and shooting method are used to verify the analytical results. The numerical results confirm our research conclusion. Project supported by the National Natural Science Foundation of China (Grant No. 19332022) and Space High Technology Foundation of China.     

Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring

The free and forced vibrations of a Kelvin-Voigt viscoelastic beam, supported by a nonlinear spring are analytically investigated in this paper. The governing equations of motion along with the compatibility conditions are obtained employing Newton’s second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model. The method of multiple timescales, a perturbation technique, is employed which ultimately leads to approximate analytical expressions for vibration response, and provides better insight into how the system parameters influence the vibration response. Finally, the effect of system parameters on the linear and nonlinear natural frequencies, vibration responses and frequency-response curves of the system is characterized.     

Analytical approximate solutions to oscillation of a current-carrying wire in a magnetic field

This paper is concerned with analytical approximate solutions to dynamic oscillation of a current-carrying wire in a magnetic field generated by a fixed current-carrying conductor parallel to the wire. The wire is restrained to a fixed wall by linear elastic springs. The periodic oscillation solutions are obtained by generalizing the Newton-harmonic balance method. The procedure yields rapid convergence with respect to the “exact” solution obtained by numerical integration. In general, the results are valid for small as well as large oscillation amplitude. The method presented in this paper can be applied to other strongly nonlinear oscillators with more general restoring forces of rational form.     

Study on asymptotic analytical solutions using HAM for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass

Presented herein is to establish the asymptotic analytical solutions for the fifth-order Duffing type temporal problem having strongly inertial and static nonlinearities. Such a problem corresponds to the strongly nonlinear vibrations of an elastically restrained beam with a lumped mass. Taking into consideration of the inextensibility condition and using an assumed single mode Lagrangian method, the single-degree-of-freedom ordinary differential equation can be derived from the governing equations of the beam model. Various parameters of the nonlinear unimodal temporal equation stand for different vibration modes of inextensible cantilever beam. By imposing the homotopy analysis method (HAM), we establish the asymptotic analytical approximations for solving the fifth-order nonlinear unimodal temporal problem. Within this research framework, both the frequencies and periodic solutions of the nonlinear unimodal temporal equation can be explicitly and analytically formulated. For verification, numerical comparisons are conducted between the results obtained by the homotopy analysis and numerical integration methods. Illustrative examples are selected to demonstrate the accuracy and correctness of this approach. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.     

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

On modeling beam-rigid-body microgyroscopes

A comprehensive mathematical model for the bending–bending vibration of a rotating cantilever beam carrying an end rigid body at its free-end is derived using extended Hamilton’s principle. The beam rotates about its longitudinal axis, excited in two orthogonal directions along the end rigid body. The model is compared to the existing simplified model of the beam-mass gyroscope. The discretized model is obtained using the method of assumed mode. Through the stationary, the eigenvalue, and the dynamic analyses of the system response, the model is evaluated.     

Analytical analysis for large-amplitude oscillation of a rotational pendulum system

This paper deals with large amplitude oscillation of a nonlinear pendulum attached to a rotating structure. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-quintic Duffing equation. The resulting Duffing type temporal problem is solved by an analytic iteration approach. Two approximate formulas for the frequency (period) and the periodic solution are established for small as well as large amplitudes of motion. Illustrative examples are selected and compared to those analytical and exact solutions to substantiate the accuracy and correctness of the approximate analytical approach.     

Analytical approximate solutions for the generalized nonlinear oscillator

He's energy balance method (HEBM) is employed in this article to obtain the analytical approximate solution of the generalized nonlinear oscillator. Existence of periodic solutions is analytically verified and consequently the relationship between the natural frequency and the initial amplitude is obtained in an analytical form. A number of numerical simulations are carried out and accuracy of the HEBM is then examined within an error analysis. The exact values of the natural frequency numerically obtained via the elliptic integrals are taken into account as the references bases and the relative error is then evaluated for a range of oscillation amplitudes. Excellent correlation of the approximate frequencies with the exact ones demonstrates that the approximate solutions are quite consistent even for large amplitudes of oscillation.     

Nonlinear analysis of buckling,free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment

Employing Euler–Bernoulli beam theory and the physical neutral surface concept, the nonlinear governing equation for the functionally graded material beam with two clamped ends and surface-bonded piezoelectric actuators is derived by the Hamilton’s principle. The thermo-piezoelectric buckling, nonlinear free vibration and dynamic stability for the piezoelectric functionally graded beams, subjected to one-dimensional steady heat conduction in the thickness direction, are studied. The critical buckling loads for the beam are obtained by the existing methods in the analysis of thermo-piezoelectric buckling. The Galerkin’s procedure and elliptic function are adopted to obtain the analytical solution of the nonlinear free vibration, and the incremental harmonic balance method is applied to obtain the principle unstable regions of the piezoelectric functionally graded beam. In the numerical examples, the good agreements between the present results and existing solutions verify the validity and accuracy of the present analysis and solving method. Simultaneously, validation of the results achieved by rule of mixture against those obtained via the Mori–Tanaka scheme is carried out, and excellent agreements are reported. The effects of the thermal load, electric load, and thermal properties of the constituent materials on the thermo-piezoelectric buckling, nonlinear free vibration, and dynamic stability of the piezoelectric functionally graded beam are discussed, and some meaningful conclusions have been drawn.     

Prediction of the dynamic response of a mini-cantilever beam partially submerged in viscous media using finite element method

In this work, the use of mini cantilever beams for characterization of rheological properties of viscous materials is demonstrated. The dynamic response of a mini cantilever beam partially submerged in air and water is measured experimentally using a duel channel PolyTec scanning vibrometer. The changes in dynamic response of the beam such as resonant frequency, and frequency amplitude are compared as functions of the rheological properties (density and viscosity) of fluid media. Next, finite element analysis (FEA) method is adopted to predict the dynamic response of the same cantilever beam. The numerical prediction is then compared with experimental results already performed to validate the FEA modeling scheme. Once the model is validated, further numerical analysis was conducted to investigate the variation in vibration response with changing fluid properties. Results obtained from this parametric study can be used to measure the rheological properties of any unknown viscous fluid.     

Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators

This paper presents a new approach for solving accurate approximate analytical higher-order solutions for strong nonlinear Duffing oscillators with cubic–quintic nonlinear restoring force. The system is conservative and with odd nonlinearity. The new approach couples Newton’s method with harmonic balancing. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton’s method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations without analytical solution. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. These approximate solutions are valid for small as well as large amplitudes of oscillation. In addition, it is not restricted to the presence of a small parameter such as in the classical perturbation method. Illustrative examples are presented to verify accuracy and explicitness of the approximate solutions. The effect of strong quintic nonlinearity on accuracy as compared to cubic nonlinearity is also discussed.     

Analytical and approximate solutions to autonomous,nonlinear, third-order ordinary differential equations

Analytical solutions to autonomous, nonlinear, third-order nonlinear ordinary differential equations invariant under time and space reversals are first provided and illustrated graphically as functions of the coefficients that multiply the term linearly proportional to the velocity and nonlinear terms. These solutions are obtained by means of transformations and include periodic as well as non-periodic behavior. Then, five approximation methods are employed to determine approximate solutions to a nonlinear jerk equation which has an analytical periodic solution. Three of these approximate methods introduce a linear term proportional to the velocity and a book-keeping parameter and employ a Linstedt–Poincaré technique; one of these techniques provides accurate frequencies of oscillation for all the values of the initial velocity, another one only for large initial velocities, and the last one only for initial velocities close to unity. The fourth and fifth techniques are based on the Galerkin procedure and the well-known two-level Picard’s iterative procedure applied in a global manner, respectively, and provide iterative/sequential approximations to both the solution and the frequency of oscillation.     

Brief note: viscoelastic modelling of cantilever vibration absorbers

Internal damping is considered in a cantilever beam supported on a single degree-of-freedom vibrating system. The internal damping mechanism is modelled by a Voigt viscoelastic coefficient in the stress-strain law. The governing equations reduce to a partial differential equation with time-dependent boundary conditions. The solution yields expressions for the amplitude ratio in terms of the tuning and frequency ratios, as well as the viscoelastic damping coefficient. These results are simulated numerically on the computer to obtain three different plots of the frequency response. The first plot has parametric values of the number of cantilevers. In the second plot, three parametric values of the internal damping coefficient are portrayed; the third plot shows the frequency response with various values of the natural frequency of the main system. These results show advantages in using cantilever beams with internal damping as dynamic vibration absorbers.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.