Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Solutions of some classes of second order non-linear ordinary differential equations

A second order non-linear ordinary differential equation satisfied by a homogeneous function of u and v where u is a solution of the linear equation ÿ + p(t)ÿ + r(t)y = 0 and v = ωu, ω being an arbitrary function of t, is obtained. Defining ω suitably in two specific cases, solutions are obtained for a non-linear equation of the form ÿ + p(t)ÿ + q(t)y = μÿ²y^{−1 +} f(t)yⁿ where μ ≠ 1, n≠ 1. Applying our results, some classes of equations of the above type possessing solutions involving two or one or no arbitrary constants are derived. Some illustrative examples are also discussed.     

Accurate Higher-Order Analytical Approximate Solutions to Large-Amplitude Oscillating Systems with a General Non-Rational Restoring Force

A new approximate analytical approach for accurate higher-order nonlinear solutions of oscillations with large amplitude is presented in this paper. The oscillatory system is subjected to a non-rational restoring force. This approach is built upon linearization of the governing dynamic equation associated with the method of harmonic balance. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. This approach also explores large parameter regions beyond the classical perturbation methods which in principle are confined to problems with small parameters. It has significant contribution as there exist many nonlinear problems without small parameters. Through some examples in this paper, we establish the general approximate analytical formulas for the exact period and periodic solution which are valid for small as well as large amplitudes of oscillation.     

Generalization of the method of full approximation and its applications

This paper presents a generalized form of the method of full approximation.By usingthe concept of asymptotic linearization and making the coordinate transformationsincluding the nonlinear functionals of dependent variables,the original nonlinear problemsare linearized and their higher-order solutions are given in terms of the first-termasymptotic solutions and corresponding transformations.The analysis of a model equationand some problems of weakly nonlinear oscillations and waves with the generalized methodshows that it is effective and straightforward.     

ON LIAPOUNOFF’S METHOD IN THE THEORY OF STABILITY OF LAMINAR FLUID FLOWS

In[1]Zhou extended some Liapounoff‘s theorems of the theory of stability in the case of plane laminar fluid flows.In[2]Zhou and Li investigated the eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation,and obtained some new results.In this paper,based on the results of[1]and[2]we shall prove:(1)For the linearized problem the definition of stability according to the eigenvalues of Orr-Sommerfeld equation and that according to the perturbation.energy are equivalent;(2)The method of linearization is admissible for the stability pro-blem of plane laminar fluid flows for sufficiently small initial disturbance.     

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

     

无额外自由度广义有限元非线性分析

将无额外自由度的广义有限元法由线弹性分析扩展到弹塑性大变形分析.局部强化函数的构建依赖于已有节点,不引入额外自由度,避免了线性相关性问题.在更新拉格朗日框架下,通过控制方程弱形式的线性化推导得到了节点内力的率形式,并分为材料和几何两部分.考虑超弹性和亚弹-塑性两种材料模型,采用Newton-Raphson迭代求解,给出...     

Nonlinear vibration analysis by an extended averaged equation approach

The paper presents an extended averaged equation approach to the investigation of nonlinear vibration problems. The proposed method is applied to some free and self-excited oscillators, the Duffing's forced oscillators including main resonance, subharmonic resonance and super harmonic resonance. The results in analyzing the vibration systems with arbitrary non-linearity show advantages of the method.     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

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A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.     

Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only     

高维非线性振动系统参数识别

将增量谐波平衡非线性识别推广到高维振动系统, 推导了基于增量谐波平衡的多自由度非线性系统的识别方程. 针对一个两自由度系统进行了数值模拟计算, 讨论了系统在单周期、倍周期和混沌运动状态下的参数识别, 以及噪声对识别结果的影响, 验证了增量谐波平衡非线性识别在多自由度系统中的有效性. 结果表明, 该方法具有较高的计算效率和识别精度, 以及良好的抗噪能力.      

A PREDICT-CORRECT NUMERICAL INTEGRATION SCHEME FOR SOLVING NONLINEAR DYNAMIC EQUATIONS

A new numerical integration scheme incorporating a predict-correct algorithm forsolving the nonlinear dynamic systems was proposed in this paper. A nonlinear dynamic systemgoverned by the equation v=F(v,t) was transformed into the form as v=Hv f(v,t). Thenonlinear part f(v,t) was then expanded by Taylor series and only the first-order term retained inthe polynomial. Utilizing the theory of linear differential equation and the precise time-integrationmethod, an exact solution for linearizing equation was obtained. In order to find the solution of theoriginal system, a third-order interpolation polynomial of v was used and an equivalent nonlinearordinary differential equation was regenerated. With a predicted solution as an initial value andan iteration scheme, a corrected result was achieved. Since the error caused by linearization couldbe eliminated in the correction process, the accuracy of calculation was improved greatly. Threeengineering scenarios were used to assess the accuracy and reliability of the proposed method andthe results were satisfactory.     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Homoclinic orbits for some （2＋1）-dimensional nonlinear Schrodinger-like equations

Chaos is closely associated with homoclinic orbits in deterministic nonlinear dynamics. In this paper, analytic expressions of homoclinic orbits for some （2＋1）- dimensional nonlinear Schrodinger-like equations are constructed based on Hirota＇s bilinear method, including long wave-short wave resonance interaction equation, generalization of the Zakharov equation, Mel＇nikov equation, and g-Schrodinger equation are constructed based on Hirota＇s bilinear method.     

Classical Solvability of the Coupled System Modelling a Heat-Convergent Poiseuille-Type Flow

We consider the coupled system of two nonlinear scalar parabolic equations modelling a simple uni-directional Poiseuille-type flow of a homogeneous incompressible Newtonian fluid whose viscosity is a temperature-dependent function. The energy balance equation of this system takes into account the phenomena of the viscous energy dissipation. We prove existence of a classical solution to this system on an arbitrary interval of time. The smooth solution turns out to be unique in a wider class of weak solutions.     

On the theory of three-dimensional multihump solitons in active-dissipative media

Stable localized nonlinear coherent structures, i.e. solitons, play a key role in the stochastization of the processes occurring in active-dissipative media. In this study, three-dimensional multi-hump solitons are investigated for a model equation which qualitatively describes the wave processes in some physical systems. The existence of 3D multihump solitons is demonstrated numerically and the soliton behavior is studied. The results are generalized to describe multihump solitons in descending viscous-fluid layers [1]. An unusual physical phenomenon observed in experiments [1], namely, stable two-hump coherent structures on the surface of a downflowing viscous-fluid layer, is explained qualitatively.     

Theory of long-term microdamage of physically nonlinear homogeneous materials

A theory of long-term damage of physically nonlinear homogeneous materials is proposed. Damage is modeled by randomly dispersed micropores. The failure criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle fracture on the difference between the equivalent stress and its limit, which is the ultimate strength, according to the Huber–Mises criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in a physically nonlinear material at an arbitrary time is formulated. Algorithms of calculating the time dependence of microdamage and macrostresses are developed and the corresponding curves are plotted. The effect of the nonlinearity of the material on its macrodeformation and damage is analyzed