Nonlinear Vibration Equation Research Articles

Probabilistic analysis on parametric random vibration of a marine riser excited by correlated Gaussian white noises

A path integration method is used to discuss the stochastic response of a marine riser excited parametrically and externally by correlated Gaussian white noises. On the basis of the nonlinear vibration equation of a marine riser, the first-order vibration mode is studied and its governing equation is formulated. The response probability density function (PDF) is further developed by a path integration method. During the solution procedure, the Gaussian closure method and the short-time Gaussian approximation method are used to develop the transition probability density function. The results of different time intervals are compared with the results of Monte-Carlo simulation (MCS) to determine the adequate time interval. After that, the path integration method with a Gauss–Legendre scheme is used to obtain the stationary PDF solution of displacement and velocity. The path integration solutions (PIS) are compared with those of the equivalent linearization method (EQL) and MCS. Five demonstrative cases are analyzed to evaluate the effectiveness of the path integration method. The influences of nonlinearity and excitations are considered. It shows that the results of PIS coincide better with the results of MCS than those of EQL. The analysis shows that parametric excitation causes the softening behavior of the PDF distribution compared with those of EQL. The increase of excitation correlation results in the nonsymmetrical PDF distribution of response. The increase of nonlinearity in displacement can lead the PDF of displacement to showing a hardening behavior.

Nonlinear forced vibration analysis of composite beam combined with DQFEM and IHB

In this paper, a forced vibration model of composite beams under the action of periodic excitation force considering geometric nonlinearity is proposed. For the strain–displacement relationship, Timoshenko beam theory is used, and the element and system matrices are developed using the differential quadrature finite element method. Each node has 3 degrees of freedom. The incremental harmonic balance method is used to solve the nonlinear forced vibration equation. In order to prove the validity of the proposed model, the solution of the Duffing equation is calculated using the analytical method and the proposed method. Next, linear forced vibration analysis of the beam made of isotropic material is performed and compared with the result of ABAQUS. The results are very close. Based on these comparisons, nonlinear vibration phenomena of composite beams are studied under the action of periodic forces.

Nonlinear vibration of piezoelectric laminated nanobeams at higher modes based on nonlocal piezoelectric theory

The presented paper investigates the nonlinear vibration of a nanobeam with a piezoelectric layer bounded to its top surface considering the nonlocal piezoelectricity theory. To do this, Hamilton’s principle is implemented to derive the governing nonlinear vibration equations of the nanobeam by assumption of nonlocal piezoelectricity and a nonlinear strain–displacement relation. Then, the Galerkin separation method is applied to transform and simplify the partial differential equation of the nonlinear oscillation to an ordinary one with quadratic and cubic nonlinearities in the time domain. By implementing the multiple-scale perturbation method, an analytical relation for the nonlinear natural frequencies is obtained as a function of the oscillation amplitude and the nonlocal size scale parameter. Then, the nonlinear vibration characteristics of the nanobeam are investigated at higher modes of vibration and the size scale effects are reviewed comprehensively. It is observed that the nonlocal parameter decreases the nonlinear natural frequencies and becomes noticeable at higher modes of vibration. Moreover, by increasing the amplitude ratio, the nonlocal effects are decreased and the nonlocal nonlinear frequency approaches the local one. Also, the amplitude ratio has increasing effects on the nonlinear frequencies.

Nonlinear Vibration of the Flexoelectric Nanoplate with Surface Elastic Electrodes Under Active Electric Loading

This paper mainly studies the effects of surface elastic electrodes and electric loading on the nonlinear vibration of flexoelectric nanoplate. The governing equations are derived based on the von Karman type strain–displacement relations and the flexoelectricity theory. The nonlinear vibration equation is solved by an approximate method. Numerical results reveal that the surface elastic electrodes and the length–width ratio of flexoelectric nanoplate have a more significant effect on the nonlinear resonant frequency of simply-supported nanoplate than that of the all-edge-clamped nanoplate. Meanwhile, the effect of the graphene electrode on the nonlinear resonant frequency is more notable than that of the Al electrode. Additionally, the applied electric loading can significantly affect the nonlinear resonant frequency of the flexoelectric nanoplate. For the nanoplates with different boundary conditions, the applied electric loading has different effects on the nonlinear resonant frequency. This study has certain research significance in the structure design and examination of the flexoelectric nanoplate with electrodes.

Size-dependent nonlinear dynamic modeling and vibration analysis of piezo-laminated nanomechanical resonators using perturbation method

This article investigates size-dependent nonlinear dynamic modeling and vibration analysis of nanomechanical resonators equipped with piezoelectric layers employing perturbation methods. For nonlinear dynamic modeling, the nanomechanical beam resonator is modeled based on the nonlocal elasticity theory and von-karman type of nonlinear formulation. Then, the Hamilton’s principle is employed to derive the nonlinear vibration equation of the piezoelectric laminated beam resonator. The Galerkin separation approach is employed to develop the governing equation as a time-dependent nonlinear differential one which is solved by the multiple scales perturbation method. Obtaining an analytical formulation, a detailed study is conducted to investigate the size effects on nonlinear free vibration of the beam at several modes of vibration. The obtained results indicate that the nonlocal parameter causes a reduction effect on the nonlinear frequencies particularly for its higher values and higher modes of vibration. Also, it is observed that the nonlocal effect on the natural frequencies diminishes with an increase in the vibration amplitude.

Effect of the Time-Varying Damping on the Vibration Isolation of a Quasi-Zero-Stiffness Vibration Isolator

This study focuses on the effect of damping changes on the vibration isolation of a quasi-zero-stiffness vibration isolator. A nonlinear-vibration equation for the quasi-zero-stiffness vibration isolator is found and solved using the multiscale method. Then, the vibration characteristics before, in the process of and after the damping change, are also examined. The results show that time-varying damping can be equivalent to the addition of a stiffness term to the vibration system, which leads to a change of the vibration amplitude frequency response, leakage of power spectrum, and corresponding linear spectrum features being weakened. When the damping changes rapidly, the vibration system tends to be divergent rather than stable. After the change, the number of stable focuses of the proposed quasi-zero-stiffness vibration isolator increases from one to two, and the system will see decline in its vibration stability.

Parametric Vibration Analysis of Submerged Floating Tunnel Tension Legs

According to the characteristics of submerged floating tunnel anchored by tension legs, simplifying the tube as point mass and assuming that the tension leg is a nonlinear beam model hinged at both ends, the nonlinear vibration equation of the tension leg is derived. The equation is solved by the Galerkin method and Runge-Kutta method. Subsequently, numerical analysis of typical submerged floating tunnel tension leg is carried out. It is shown that, the parametric vibration response of the submerged floating tunnel tension leg is related to the amplitude and frequency of the end excitation. Without considering axial resonance and transverse resonance, it is reasonable that higher order modes are abandoned and only the first three modes are considered. The axial resonance amplitude of the second or third order mode is equivalent to the first order mode axial resonance amplitude, which should not be ignored.

Анализ динамики нелинейных моделей конструкций ракетно-космической техники при нестационарных случайных нагрузках

In aerospace engineering, it is customary to employ stochastic analysis methods at the design stage to investigate how the mechanical system responds to random external forces. This is relevant due to high reliability requirements for spacecraft. We developed a method for probabilistic estimation of the dynamic properties of a structure subjected simultaneously to external (additive) and parametric (multiplicative) vibrations. An ordinary non-linear vector differential equation describes the vibrations in the elastic structure. Non-linear position and velocity properties of kinematic pairs may have cusps and discontinuities. We assume that the probabilistic dispersions of respective phase coordinates are close to the normal distribution of probability density. The initial non-linear vibration equations are statistically linearised. The system of differential equations is not rewritten in the canonical form, which means that it is possible to carry out the probabilistic analysis of the system for any external non-steady-state effect. The fundamental matrix of the linearised system is used to find the expected value vector and the correlation function matrix of the phase coordinate vector. The solution consists of a matrix integro-power series containing linear and quadratic terms. Using the method makes it possible to assess the contribution of each external force component to the total result. We consider an example of a non-linear system responding to a stepwise non-steady-state external influence.

Effect of multiphase internal flows considering hydrate phase transitions on the streamwise oscillation of marine risers

Marine risers are the main transportation channels of offshore oil and gas from the wellheads to the floating platforms. In mixed oil and gas transportation, natural gas hydrates are produced at low temperatures and high pressures, which makes the multiphase internal flow quite complex. In order to explore the effects of multiphase internal flows considering hydrate phase transitions on the streamwise oscillation of marine risers, a numerical multiphase internal flow model considering hydrate phase transitions is established, where the flow parameters are solved with the finite difference method. Subsequently, based on the virtual work principle, a non-linear vibration equation for marine risers is deduced. The parameters of the multiphase internal flow are introduced and solved numerically with the finite element method. The results show that the gas pressure and density decrease, and the gas velocity and acceleration increase, from the riser inlet to the outlet due to gas expansion and the hydrate phase transitions. The natural frequencies of the riser increase with an increase of gas intake, and the changing trends at higher modes are more obvious. Under current excitation, the amplitude envelope of the riser decreases with an increase of gas intake. Under wave excitation, the gas intake has little effect on the vibration frequency of the riser. However, when gas intake changes, the natural frequency of the riser changes, which leads to a change of the resonance amplification coefficient, thus affecting the vibration amplitude of the riser.

Nonlinear Vibration Analysis of Submerged Floating Tunnel Tether

To study the nonlinear vibration characteristics of submerged floating tunnel tether, the nonlinear vibration partial differential equation of tether is set up, in which the inclination, sag, initial tension force, and length of tether are taken into account. According to the linear vibration mode of tether, the partial differential equation is converted into ordinary differential equations by Galerkin method and the nonlinear vibration equation of tether for the in‐plane first four modes and the out‐of‐plane first four modes are obtained. The results show that the change of inclination results is in the change of sag. The initial configuration of sag is symmetric. Thus, the sag only affects the damping ratio of the in‐plane symmetric mode, namely, that of the first and the third mode. The sag has no effect on the second and forth order modes which are antisymmetric modes. Therefore, the sag is only existed in plane and has no effect on the out‐of‐plane mode. The first and the third in‐plane modal damping ratios of tether are in direct proportion to their inclination, whereas in inverse proportion to their sag. The first modal damping ratio of tether (both in‐plane and out‐of‐plane) is in direct proportion to its length, whereas in inverse proportion to its initial tension.

Study on Vibration Characteristics of Cables Excited by Lateral Displacement of Ends

Considering the axial vibration, the partial differential nonlinear vibration equations of the cable excited by the lateral displacement of the end is established. The Galerkin method is used to perform the modal truncation and transform the equations into the ordinary differential nonlinear equations. The typical examples are selected and the vibration characteristics are studied by numerical simulation. The conclusions of the study are as follows: when the frequency ratio is close to 2, the resonance phenomenon occurs. When the frequency ratio is greater than 3, the nonlinear characteristic is obvious. The amplitude of the in-plane vibration of the cable increases with the increase of the excitation amplitude, and small displacement excitation at the end causes large amplitude vibrations in the cable surface. The initial disturbance outside the plane is introduced. When the excitation frequency is close to 2.1 times the natural frequency of the in-plane vibration, the resonance phenomenon occurs and the amplitude is large.

A method to derive thermoelastic free vibration in a simply supported annulus elliptic plate

The present article investigates the thermoelastic free vibration for a simply supported elliptic plate subjected to a thermal load. The realistic solution involving the Mathieu functions and also their derivatives for the heat conduction differential equation subjected to sinusoidal sectional heating on the upper face with the lower face and the curved inner surface is kept at zero temperature, and the outer curve is kept thermally insulated and is derived using the classical method. The strain energy due to bending of the middle surface of the plate undergoing large deflection was well-thought-out by neglecting the second strain invariant terms for the analysis of large amplitude (nonlinear)-free vibrations of a simply supported plate. Furthermore, nonlinear free vibration equation of elliptic structure is developed with the aid of Berger assumption and Hamilton’s principle and obtained its solution using a new integral transform involving Mathieu and modified Mathieu functions. A closed-form bending stress function obtained has been equated with those obtained by Berger’s methodology. The thermal stress components are obtained in terms of resultant bending moments and resultant forces for numerical analysis. The free-vibration mode of the corresponding nonlinear problem, the Jacobi elliptic function, is obtained from the exact resolution of the natural frequency of the simply supported elliptic plate. For the particular case by applying limiting conditions, the elliptic region can degenerate into the problem of the circular zone, and some numerical results have also been plotted in a few instances.

Nonlinear Vibration of Truncated Conical Shells: Donnell, Sanders and Nemeth Theories

AbstractNonlinear free vibration of truncated conical shells has been investigated for three different shell theories; Donnell, Sanders and Nemeth to investigate the effect of their simplifying assumptions. The displacement field of a finite element model that was obtained from the exact solution of equilibrium equations of Sander’s improved first-approximation theory is used to define the nonlinear strain energy of conical shells. Employing generalized coordinates method the equations of motion are derived and subsequently the amplitude equation of nonlinear vibration of conical shells was developed. The amplitude equation is solved for multiple cases of isotropic materials. Linear and nonlinear free vibration results are validated against the existing studies in scientific literature and demonstrate good accordance. The validated model is used to investigate effects of different parameters including circumferential mode number, cone-half angle, length to radius ratio, thickness to radius ratio and boundary conditions for the nonlinear vibration of conical shells.

Study on nonlinear vibration of simplified solid rocket motor model

The unstable combustion of the large aspect-ration solid rocket in the end phase flight is of great interest and importance to the solid rocket design.The large aspect-ratio solid rocket motor (SRM) tends to produce unstable combustion in flight. As shown in published literatures, it is believed that the main reason for this phenomenon is the internal flow field. However, in terms of the intense pressure oscillation, the contribution of the motor casing should not be ignored. In this paper, the nonlinear natural frequencies of different motor shells in flight are mainly concerned. The SRM model in the working state is simplified to a double-layer time-varying axially moving free-free beam. The governing equation of nonlinear lateral vibration is derived according to the Hamilton principle. The Galerkin method and multi-scale method are adopted to solve the governing equations. Firstly, to verify the accuracy of the presented method, the validating experiments base on the finite difference method are carried out. According to the comparison, the results obtained through the Galerkin method coupled with multi-scale method coincides well with the results obtained by the finite difference method as well as the results in the public literatures. The influence of structural change and axial velocity on the nonlinear coefficients and natural frequencies of the SRM shell is investigated by numerical experiments. The results show that the natural frequency of the SRM shell change during the whole flight and the increase of aspect ratio has a significant influence on the natural frequency. Under the initial excitation, the nonlinear natural frequency will change abruptly. Under some particular excitations, it is possible that the resonance coupling between the shell vibration and internal flow field occurs and it contributes to the amplification of the unstable combustion further.

Calculation of Natural Frequencies on Transverse Vibration for drill strings in Non-Uniform Temperature Field

It is necessary to consider the effect of temperature which changes the mechanical properties of materials when drill strings work in a non-uniform temperature field. In this paper, the natural frequencies of transverse vibration are studied under axial compression in the nonuniform temperature field. Using an asymptotic solution method, first-order approximation formula of the nonlinear vibration equation is obtained. The effects of temperature changes and axial pressure on the natural frequencies are discussed by an example. The results show that the influence of temperature on the critical load is small; change rate of natural frequency increases with the increase of axial pressure; while the axial pressures become larger, the influence of temperature on the first-order frequency is greater than on the second-order’s.

Numerical Solution of Nonlinear Vibration by Euler-Cromer Method

Nonlinear equations are able to present many behaviours of physical systems better than linear equations. Analytical solutions to nonlinear vibration equations have intractability characteristics, while limitations in computational software resources make it difficult to study systematically the phenomena in many systems. In this paper, the Euler-Cromer method is used to solve numerically the vibration equation nonlinear. The nonlinear vibrations of the equation of harmonic, Van der Pol, and Duffing oscillator motion are used as the physical case models to solve. Simulation results show that the Euler-Cromer method provides a numerical solution that is easy to implement and accurate.

Kinematic Aspects in Modeling Large-Amplitude Vibration of Axially Moving Beams

This paper clarified kinematic aspects of motion of axially moving beams undergoing large-amplitude vibration. The kinematics was formulated in the mixed Eulerian–Lagrangian framework. Based on the kinematic analysis, the governing equations of nonlinear vibration were derived from the extended Hamilton principle and the higher-order shear beam theory. The derivation considered the effects of material parameters on the beam deformation. The proposed governing equations were compared with a few previous governing equations. The comparisons show that proposed equations are with higher precision. Besides, the proposed equations can be viewed as the asymptotic governing equations of Lagrange’s equations of motion for large displacement. Finally, the corresponding boundary conditions and the comparison between the presented model equation and classical model equation were provided.

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.

Effects of van der Waals force on nonlinear vibration of electromechanical integrated electrostatic harmonic actuator

An electromechanical integrated electrostatic harmonic actuator is promising for the miniaturization of electromechanical devices. As the dimensions of the actuator decrease, the effects of the van der Waals force become obvious. In this study, by considering the nonlinearity of electrostatic and van der Waals forces, nonlinear vibration equations of the flexible ring of an electrostatic harmonic actuator are deduced. Using these equations, the nonlinear free vibration and nonlinear forced response of the actuator are investigated. The effects of the van der Waals force on the nonlinear vibration of the flexible ring are analyzed. Results show that these effects of the van der Waals force are relatively obvious under some conditions and should be considered.

Magneto-elastic internal resonance of an axially moving conductive beam in the magnetic field

The Hamiltonian principle is applied to the nonlinear vibration equation of an axially moving conductive beam in the magnetic field with consideration of the axial velocity, axial tension, electromagnetic coupling effect and complex boundary conditions. Nonlinear vibration characteristics of the free vibrating beam under 1:3 internal resonances are studied based on our approach. For beams with one end fixed and the other simply supported, the nonlinear vibration equation is dispersed by the Galerkin method, and the vibration equations are solved by the multiple-scales method. As a result, the coupled relations between the first-order and second-order vibration modes are obtained in the internal resonance system. Firstly, the influence of initial conditions, axial velocity and the external magnetic field strength on the vibration modes is analysed in detail. Secondly, direct numerical calculation on the vibration equations is carried out in order to evaluate the accuracy of the perturbation approach. It is found that through numerical calculations, in the undamped system, the vibration modes are more sensitive to the initial value of vibration amplitude. The amplitude changes of the first-order and second-order modes resulting from the increase of the initial amplitude value of the vibration modes respectively are very special, and present a “reversal behaviour”. Lastly, in the damped system, the vibration modes exhibit a trend of coupling attenuation with time. Its decay rate increases when the applied magnetic field strength becomes stronger.