A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.     

Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators

In this paper we consider the problem which can appear at the determination of the dynamical stability of the responses of oscillators with discontinuous or steep derivative of the restoring characteristic obtained in the frequency domain. For that purpose, a simple one degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Confirmation of the results obtained in the frequency domain is done by comparing with the results obtained in the time domain by the method of piecing the exact solutions. Determination of the dynamical stability can be made more reliable by using the proposed plot of maximum modulus of the eigenvalues of the monodromy matrix in dependence of non-dimensional frequency and the number of harmonics included in the supposed approximate solution.     

Fault Detection and Diagnosis of Components of the Vehicle Vertical Dynamics

In this paper, model and signal-based methods for supervision ofsuspension elements of a car are presented. A model-based approach usingparameter estimation is used to determine the current parameters of avehicle suspension. Exploiting the analytical redundancy, different faults,like weak dampers or sensor faults, can be distinguished. For faultdetection the use of parity equations is shown. In addition, a signal-basedapproach is presented which makes a detection of tire pressure losspossible. All presented results were drawn from a test rig or from a drivingcar.     

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.     

A Method for Obtaining Approximate Analytic Periods for a Class of Nonlinear Oscillators

This paper deals with nonlinear oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. By combining the linearization of the governing equation with the method of harmonic balance, we establish two approximate analytic formulas for the period. These two formulas are valid for small as well as large amplitudes of oscillation. Three examples are used to illustrate that the proposed formulas can give very satisfactory approximate results. Sommario. Questo lavoro tratta il problema delle oscillazioni nonlineari di sistemi conservativi ad un grado di libertà con nonlinearità simmetriche. Combinando opportunamente la tecnica di linearizzazione dellequazione del moto con il metodo del bilancio armonico si perviene a due formule analitiche approssimate per il periodo. Le formule ottenute sono valide sia per piccole che per grandi ampiezze di oscillazione. Si utilizzano tre esempi classici di oscillatori nonlineari per illustrate lefficacia del metodo nel produrre risultati approssimati soddisfacenti.     

Visco-elastic systems under both deterministic harmonic and random excitation

IntroductionForlinearviscoelasticsystemsunderbothadditiveandmultiplicativebroad_bandexcitationexcitations,Ariaratnam[1]studiedthestochasticstabilityofthesystembyusingthemethodofstochasticaveragingprocedure .Itwasshownthatthevisco_elasticforcecontributedtowarddamping ,hence ,stabilityofthesystem .However,thestiffnesseffectofthevisco_elasticcomponentwasnotfullyaccountedfor.FurthermoreAriaratnam[2 ]studiedthestochasticstabilityofthesystembutthemodelislinear.Inthetheoryofnonlinearrandomvibration…     

The inlet effect on the drag factor of a sphere in a tube

The creeping motion Ground a sphere situated axisymmetrically near the entrance of asemi-infinite circular cylindrical tube is analyzed using infinite series solutions for thevelocity components. pressure and the stream function. Truncating the infinite series. thecorresponding coefficients in the series are determined by a collocation technique. The dragfactor and the stress distribution on the surface of the sphere are calculated for the sphere inmotion in quiescent fluid and for the flow with uniform velocity at the entrance past a rigidlyheld sphere. The results indicate that a sphere near the entrance which has a uniformentrance velocity profile will suffer larger drag than that in infinite tube.Theconvergence of the collocation technique is tested by numerical calculation. It is shown thatthe technique has good convergence properties.     

Non-linear vibration of bimaterial magneto-elastic cantilever beam with thermal loading

Formulas and numerical results are studied for the transient vibration and dynamic instability of a bimaterial magneto-elastic cantilever beam which is subjected to alternating magnetic field and thermal loading. Materials are assumed isotropic, and the physical properties are assumed to have unique values in each layer. The governing equation of motion is derived by the extended Hamilton's principle, in which the damping factor, the electromagnetic force, the electromagnetic torque, and the thermal load are considered. The solution of thermal effect is obtained by superposing certain fundamental linear elastic stress states which are compatible with the Euler–Bernoulli beam theory. The axial stresses results are found to be in good agreement with some known numerical solutions. Using Galerkin's method, the equation of motion is reduced to a time-dependent Mathieu equation. The numerical results of the regions of dynamic instability are determined by the incremental harmonic balance (IHB) method, and the transient vibratory behaviors are presented by the fourth-order Runge–Kutta method. The results show that the responses of the transient vibration and dynamic instability of the system are influenced by the magnetic field, the thickness ratio, the excitation frequency, but not by the temperature increase in this study.     

Engineering Models for Numerical Analysis of Composite Bending Members∗

Abstract

ABSTRACT The two-step numerical analysis of a composite beam structure is presented in this paper. The first step, based on the idea of dividing the cross section into laminas, leads to the estimation of the moment-curvature relation for different types of cross sections used in composite beams. The second step adopts this constitutive relation, which is expressed in the space of generalized stresses and strains, into finite element nonlinear code. Some numerical examples are given, to show the agreement of numerical calculations with results of the authors' experiments, when the shrinkage of a concrete encasement and stresses due to welding processes in steel beams are considered. In addition, the numerical concept presented here seems to reduce the sensitivity of the final results obtained to finite element discretization error.     

含立方刚度和滞回的隔振系统动力学特性分析 1)

包含立方刚度和Bouc-Wen 型滞回的隔振系统具有复杂的非线性动力学特性。系统无阻尼响应模型可基于无滞回恢复力建立,利用谐波平衡法和泰勒展开求得近似解析解。系统有阻尼响应模型可利用解析/数值联合方法求解,该方法基于谐波平衡法和Levenberg-Marquardt 迭代算法,对于滞回产生的多值非光滑函数项,先计算时域响应再通过快速傅里叶变换求解谐波项系数。上述方法在含水平绞制梁的非线性隔振系统分析中得到有效应用。分析表明,在Bouc-Wen 型滞回和立方刚度的综合影响下,隔振系统呈现渐软–渐硬特性,滞回阻尼和线性阻尼都可以有效抑制共振,但前者高频隔振效果优于后者。     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度,将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中,然后对传统打靶法进行改造,将周期也作为一个参数一起参与打靶法的迭代过程,迭代过程包含对周期轨道和周期的求解,迭代过程中的增量通过优化方法选择,从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式,理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统,求出了在某些参数下转子的周期解及其近似解析表达式,通过与四阶Runge-Kutta数值积分结果比较,验证了方法的有效性,计算结果对于转子系统运动的定量控制有重要理论指导意义。     

悬索桥的非线性分析

为进行悬索桥的几何非线性分析，建立了梁单元的刚度矩阵及有关算法，通过算例验证了方法是可行的。     

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.     

Localized Basis Function Method for Computing Limit Cycle Oscillations

An alternate approach to the standard harmonic balance method (based on Fourier transforms) is proposed. The proposed method begins with an idea similar to the harmonic balance method, i.e. to transform the initial set of differential equations of the dynamics to a set of discrete algebraic equations. However, as distinct from previous harmonic balance techniques, the proposed method uses a set of basis functions which are localized in time and are not necessarily sinusoidal. Also as distinct from previous harmonic balance methods, the algebraic equations obtained after the transformation of the differential equations of the dynamics are solved in the time domain rather than the frequency domain. Numerical examples are provided to demonstrate the performance of the method for autonomous and forced dynamics of a Van der Pol oscillator.     

A nonlinear oscillatory system subjected to driving forces of elliptic type

The aim of this paper is to show how Jacobi elliptic functions in combination with the averaging and the harmonic balance methods can be applied to obtain the approximate solution of two coupled, ordinary differential equations having a spring with cubic nonlinearity and subjected to driving forces of elliptic type. By an appropriate choice of the system parameter values, it is possible to show that our derived solution represents the exact steady-state solution of the undamped Duffing equation with driving force of elliptic type. At the end of this work, we also demonstrate the validity of our derived solution by comparing the amplitude–time response curves with those of the numerical integration solutions.     

Nonlinear Vibration of Plane Structures by Finite Element and Incremental Harmonic Balance Method

A nonlinear steady state vibration analysis of a wide class of planestructures is analyzed. Both the finite element method and incrementalharmonic balance method are used. The usual beam element is adopted inwhich the nonlinear effect arising from longitudinal stretching has beentaken into account. Based on the geometric nonlinear finite elementanalysis, the nonlinear dynamic equations including quadratic and cubicnonlinearities are derived. These equations are solved by theincremental harmonic balance (IHB) method. To show the effectiveness andversatility of this method, some typical examples for a wide variety ofvibration problems including fundamental resonance, super- andsub-harmonic resonance, and combination resonance of plane structuressuch as beams, shallow arches and frames are computed. Most of theseexamples have not been studied by other researchers before. Comparisonwith previous results are also made.     

一种单元谐波平衡法

基于有限元离散,对于工程中的非线性响应问题,提出一种单元谐波平衡法．与常规的谐波平衡法不同,本文将谐波平衡方程建立在有限元素上,从而兼顾了有限元素法和常规谐波平衡法两大优势．有限元技术的应用能使得求解问题的范围扩大到复杂工程结构,而谐波平衡概念的使用将使得含有复杂变形和复杂本构关系的动力学响应问题得到有效解决．所提方法能适用于工程结构中具有复杂非线性关系的动力学响应问题．由于谐波平衡法的实施依赖于谐波系数方程及其切线刚度矩阵的解析推导,尽管已经局限到有限元素上,但对于较为复杂一些的本构关系,推导仍非易事．为解决这些问题,放弃通常对于变形梯度和应变张量所作的向量假设,而是从连续介质力学中基本的几何关系入手,提出一种矩阵分解形式．通过利用张量的内蕴导数定义以及关于迹函数的有关性质,给出应力增量的一种新的表现形式．当它与变形梯度的矩阵分解相结合时,使得切线刚度矩阵的导出变得十分简单,而且所得计算形式也比通常紧凑和方便许多．     

Elliptic balance solution of two-degree-of-freedom, undamped, forced systems with cubic nonlinearity

The solution of a system of two coupled, nonhomogeneous undamped, ordinary differential equations with cubic nonlinearity and sinusoidal driving force is obtained by the use of Jacobian elliptic functions and the elliptic balance method. To assess the accuracy of our proposed solution, we consider an example that arises in the study of the finite amplitude, nonlinear vibration of a simple shear suspension system. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters.