Finite element methods for second order differential equations with significant first derivatives

Galerkin finite element methods based on symmetric pyramid basis functions give poor accuracy when applied to second order elliptic equations with large coefficients of the first order terms. This is particularly so when the mesh size is such that oscillations are present in the numerical solution. In the present note asymmetric linear and quadratic basis functions are introduced and shown to overcome this difficulty in an appropriate two point boundary value problem. In particular symmetric quadratic basis functions are oscillation free and highly accurate for a working range of mesh sizes.     

Van der Pol oscillator with time variable parameters

In this paper, the analytical solving procedure of the oscillator with slow time variable mass is developed. The solution is based on the Jacobi elliptic function whose properties: frequency, amplitude and modulus are obtained according to the requirements given for the amplitude of the displacement and the amplitude of the velocity of vibration and also period of vibration. The suggested procedure is applied for the solution of the time variable Van der Pol oscillator. The limit value of the initial mass of the oscillator is determined which separates the case when the limit cycle motion occurs, and the case when the amplitude of vibration tends to zero independently of the initial displacement. A numerical example is considered. The analytical solution is compared with the numerically obtained one and it is concluded that they are in good agreement.     

Modelling vortex-induced fluid-structure interaction

The principal goal of this research is developing physics-based, reduced-order, analytical models of nonlinear fluid-structure interactions associated with offshore structures. Our primary focus is to generalize the Hamilton's variational framework so that systems of flow-oscillator equations can be derived from first principles. This is an extension of earlier work that led to a single energy equation describing the fluid-structure interaction. It is demonstrated here that flow-oscillator models are a subclass of the general, physical-based framework. A flow-oscillator model is a reduced-order mechanical model, generally comprising two mechanical oscillators, one modelling the structural oscillation and the other a nonlinear oscillator representing the fluid behaviour coupled to the structural motion.Reduced-order analytical model development continues to be carried out using a Hamilton's principle-based variational approach. This provides flexibility in the long run for generalizing the modelling paradigm to complex, three-dimensional problems with multiple degrees of freedom, although such extension is very difficult. As both experimental and analytical capabilities advance, the critical research path to developing and implementing fluid-structure interaction models entails-formulating generalized equations of motion, as a superset of the flow-oscillator models; and-developing experimentally derived, semi-analytical functions to describe key terms in the governing equations of motion.The developed variational approach yields a system of governing equations. This will allow modelling of multiple d.f. systems. The extensions derived generalize the Hamilton's variational formulation for such problems. The Navier-Stokes equations are derived and coupled to the structural oscillator. This general model has been shown to be a superset of the flow-oscillator model. Based on different assumptions, one can derive a variety of flow-oscillator models.     

Direct finite element computation of non-linear modal coupling coefficients for reduced-order shell models

We propose a direct method for computing modal coupling coefficients—due to geometrically nonlinear effects—for thin shells vibrating at large amplitude and discretized by a finite element (FE) procedure. These coupling coefficients arise when considering a discrete expansion of the unknown displacement onto the eigenmodes of the linear operator. The evolution problem is thus projected onto the eigenmodes basis and expressed as an assembly of oscillators with quadratic and cubic nonlinearities. The nonlinear coupling coefficients are directly derived from the FE formulation, with specificities pertaining to the shell elements considered, namely, here elements of the “Mixed Interpolation of Tensorial Components” family. Therefore, the computation of coupling coefficients, combined with an adequate selection of the significant eigenmodes, allows the derivation of effective reduced-order models for computing—with a continuation procedure —the stable and unstable vibratory states of any vibrating shell, up to large amplitudes. The procedure is illustrated on a hyperbolic paraboloid panel. Bifurcation diagrams in free and forced vibrations are obtained. Comparisons with direct time simulations of the full FE model are given. Finally, the computed coefficients are used for a maximal reduction based on asymptotic nonlinear normal modes, and we find that the most important part of the dynamics can be predicted with a single oscillator equation.     

Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.     

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

Nonlinear functions are crucial points and terms in engineering problems and the solutions of many important physical problems are centered on finding accurate solutions to these functions. In this paper, a new method called max-min method has been presented for deriving accurate/approximate analytical solution to strong nonlinear oscillators. Furthermore, it is shown that a large class of linear or nonlinear differential equations can be solved without the tangible restriction of sensitivity to the degree of the nonlinear term, adding that the method is quite convenient due to reduction in size of calculations. Results obtained by max-min are compared with Homotopy Analysis Method (HAM), energy balance and numerical solution and it is shown that, simply one term is enough to obtain a highly accurate result in contrast to HAM with just one term in series solution. Finally, the phase plane to show the stability of systems is plotted and discussed.     

Thermocapillary flow in a liquid layer at minimum in surface tension

Summary In the present paper a class of similarity solutions for the two-dimensional Navier-Stokes and energy equations describing thermocapillary flows in a liquid layer of constant width and infinite extent is presented. The layer is bounded by a horizontal rigid plate from one side and opened to the ambient gas from the other one. The physical properties of the liquid are assumed to be constant except the surface tension which varies as a quadratic function with temperature. It is supposed that a constant temperature gradient exists along either the liquid free surface (case I) or the rigid boundary (case II).In both cases, by means of a similarity transformation, the equations of motion and energy are reduced to a system of three ordinary differential equations, one for the velocity and two for the temperature. The equation for the velocity can be solved separately from the other equations and its solution, found numerically, exists only for the Marangoni number less than a certain finite value. The solution of the whole system depends also on the Prandtl number. The solution of one of the temperature equations is presented in an analytical form and the other equation is solved numerically. Asymptotic formulas of the functions are also obtained for small and large Marangoni numbers. Flow pattern and temperature fields are presented. One convective roll exists in every semi-infinite layer. Fluid velocities at different points of the free surface are evaluated for an aqueous solution of n-heptanol and compared with those measured in the experiments.     

Elastic structural response of anisotropic sandwich plates with a first-order compressible core impacted by a Friedlander-type shock loading

The foundation of the non-linear theory of asymmetric anisotropic sandwich plates with a first order compressible weak orthotropic core under a Friedlander-type explosive blast is presented. The equations of motion are developed by means of Hamilton’s Principle. Within the theory, the face sheets are asymmetric while adopting the Love-Kirchoff model. In addition, the core layer is assumed to be compressible (extensible) in the transverse direction thereby capturing any wrinkling or global instabilities. The theory is then simplified and applied for the case of sandwich plates with symmetric unidirectional fiber reinforced laminated composite facings with the axes of orthotropy not necessarily coincident with the geometrical axes. The governing solution is developed using the Extended-Galerkin method resulting in two coupled non-linear second order ordinary differential equations which are then solved using the 4th-order Runge–Kutta method for a system of differential equations.     

Model-size reduction for the analysis of symmetric structures with asymmetric boundary conditions

A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (a) restructuring the governing finite element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (b) successive application of the finite element method and the classical Rayleigh–Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh–Ritz technique. A tracing parameter is introduced which identifies all the contributions to the antisymmetric response. The global approximation vectors are selected to be the solution corresponding to a zero value of the tracing parameter and the various-order derivatives of the solution with respect to this parameter, evaluated at zero value of the parameter. The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving non-linear problems is discussed.     

Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM

This paper presents a finite element model based on the first order shear deformation theory to investigate the dynamic behavior of laminated composite plates traversed by a moving oscillator. The oscillator model is assumed to be consisting of two nodal masses that are connected by means of a spring-damper unit. The governing equations of motion of two sub-systems are separately integrated by applying the Newmark’s time integration procedure. Then, the obtained equations are coupled and the responses of system components are calculated in each time step. The accuracy of algorithm is verified by comparing the numerical results of static, free vibration and simplified moving force problems analysis with the available exact solutions and other numerical results in the literature. Also, the effects of mass ratio, damping ratio of system components, stiffness of suspension system, velocity and eccentricity of moving oscillator on dynamic responses is parametrically studied. This algorithm can be applied to various boundary conditions, lamination schemes and fiber angels.     

Oscillator with non-integer order nonlinearity and time variable parameters

In this paper, the vibrations of the oscillator with nonlinearity of integer or non-integer order and with mass variable parameters are considered. New appreciative analytical procedures are developed: first, based on the generating solution that is the exact analytic solution of the system with constant parameters and the second, based on the approximate solution in the form of a trigonometric function with exact period of vibration of the system with constant parameters. For the both methods, the assumed trial solutions represent the perturbed versions of the solutions of the equations with constant parameters, where the amplitude and phase of vibration are supposed to be time variable. The amplitude and phase functions are determined using the averaging procedure over the period of vibration. The obtained approximate analytic solutions are compared with numerical ones. It is shown that the developed methods are accurate for the monotone slow time variable systems. The example of mass variable oscillator is considered. The influence of mass variation, small linear viscous damping and of the reactive force is investigated, too.     

基于正交展开方法的Duffing振子随机最优控制

基于多维Hermite多项式的经典均相混沌展开,考察了Duffing振子随机最优多项式控制的正交展开方法,阐明了多项式系数演化与振子系统反应、最优控制力概率特性之间的联系.系统输入采用Karhunen-Loève展开表现的随机地震动.为降低混求解规模,引入位移-速度范数准则,发展了自适应混沌多项式展开策略.同时,基于Lyapunov稳定条件设计控制器的控制增益参数.数值算例分析表明,受控后系统位移和加速度的均方特征得到改善、振子系统的非线性程度减小,基于混沌多项式展开的最优控制方法能明显降低系统的随机涨落和显著改善系统的非线性反应性态.     

Probabilistic solution of non-linear vibration energy harvesters driven by Poisson impulses

This paper develops a new solution procedure for obtaining the joint probability density function (PDF) of non-linear energy harvesters under Poisson impulses using the state-space-split method and the exponential-polynomial closure method. In the beginning, a generic electromechanical energy harvester is introduced and its governing equations are transformed into non-dimensional ones. According to the non-dimensional governing equations, a Duffing-type oscillator with piezoelectric conversion mechanism is further considered in this study. The proposed solution procedure includes three steps. First the joint PDF of this system is governed by the generalized Fokker-Plank-Kolmogorov (FPK) equation. The state-space-split method is used to reduce this generalized FPK equation to a low-dimensional one only about displacement and velocity. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation. Finally, the joint PDF of displacement, velocity and voltage can be approximated by the product of the obtained PDF and the conditional Gaussian PDF of voltage. Four cases are investigated considering the effects of non-linearity coefficient, impulse arrival rate and a bistable Duffing oscillator. The PDFs of these state variables and the harvested power are compared with the simulation results, respectively. The good agreement between the obtained PDFs and the simulated results is achieved. Compared with the results of the equivalent linearization method, the strong non-linearity in displacement and lower impulse rate lead the tail PDF of the harvested power to exhibiting hardening and softening behaviors, respectively. For the bistable Duffing oscillator, the PDF of the harvested power differs significantly from the result of the equivalent linearization method in the tail region.     

A vorticity-only formulation and a low-order asymptotic expansion solution near Hopf bifurcation

A vorticity-only formulation is used in order to study the behavior of the solutions of the Navier-Stokes equations for two-dimensional incompressible flows as the Reynolds number is increased. This approach allows one to limit the numerical-solution domain to the vortical region of the flow, thereby reducing the number of the state variables of the system. The vorticity-only formulation is obtained from a vorticity-stream function formulation by inverting the Poisson equation relating the vorticity to the stream function and substituting the expression for the velocity in the vorticity-transport equation. The vorticity at the solid boundary is determined from the boundary conditions. The resulting dynamical system consists of a set of first-order ordinary differential equations having only quadratic nonlinearities. This system is then used to address the behavior of the solution beyond the stability boundary, within the context of the theory of dynamical systems. This part of the paper is general and is based on the use of a singular-perturbation technique, known as the method of multiple time scales; formulae for the nonlinear analysis near a Hopf bifurcation are given explicitly in terms of the coefficients of the dynamical system. Preliminary numerical results for the validation of the formulation are presented for the limited case of two-dimensional flows around a circular cylinder. These results include the steady-state solution with varying Reynolds number, the eigenvalues and the eigenvectors of the related stability matrix, and the characteristics of the corresponding limit cycle.     

The steady state response of geometrically non-linear shallow arches

The non-linear steady state response of structures with curvature is investigated through the expository example of a shallow circular arch. A consistent mass finite element formulation is employed to derive the governing non-linear differential equations. These equations are solved by assuming a single mode expansion reducing the governing equations to the single degree-of-freedom Duffing's equation with a quadratic term. The non-symmetric amplitude-frequency curve is derived and compared with results previously obtained by direct integration of the equations of motion.     

Brinkman-Forchheimer方程的加罚有限元方法

Brinkman-Forchheimer方程(BF方程)是具有强非线性项并满足无散度条件的流动控制方程,其中无散度条件的精确满足对控制方程的数值求解极其重要.为了放松无散度条件的限制,本文采用了加罚方法.为了得到加罚问题解的适定性,首先,利用加罚关系将压力项消去,证明了速度所满足的具有单调性的非线性椭圆变分问题等价于对应能量泛函的极小化问题,从而得到了速度的存在唯一性.进一步,利用LBB条件证明了BF方程加罚问题压力的存在唯一性.其次,证明了BF方程加罚问题的Galerkin变分问题的解关于加罚参数收敛到BF方程的Galerkin变分问题的解.最后,给出了BF方程加罚问题Galerkin变分问题的有限维逼近问题及其解的存在唯一性,并且得出了采用协调有限元离散的误差估计.数值算例表明加罚方法是有效的.     

Long-term frequency aging for unpowered space-class oscillators

Very little data exists regarding the long-term aging performance of space-rated oscillators in the non-operating mode. This paper provides empirical evidence that may be used to estimate the performance of unpowered oscillators for long-term space missions, as well as an aid in validating the worst-case analyses that are routinely performed on these oscillators. The proper operation of any space vehicle is dependent on the performance of the onboard master oscillator. For maximum reliability, the onboard clock is often provided as a dualor triple-redundant ensemble, with one active oscillator. The backup oscillator(s) are usually powered off and may be off for a significant percentage of the mission. A significant parameter of importance for oscillator performance is the aging rate. Papers have been presented on the aging performance of active oscillators in space, but very little data exists regarding the long-term aging performance of oscillators in the nonoperating mode. The aging rate data are extremely important for predicting the expected performance of these backup oscillators after 10 or 15 years in of non-operation in space. This paper presents performance data derived from temperature controlled crystal oscillators and oven controlled crystal oscillators that have been dormant for decades.     

定轴转动与基础激励下梁的非线性动力学

采用Kane方程，建立了含耦合的几何及惯性非线性项的定轴转动与轴向基础激励联合作用下柔性梁的非线性动力学控制方程组，该方程组不仅包含二次及三次非线性项，而且体现了参数激励与外激励的联合作用。运用多尺度法，研究了匀转速，顺臂安装下悬臂梁的一阶模态主参激共振与外激励1/2次亚谐共振同时作用时梁的一阶近似稳态响应。结果表明，梁的一阶模态幅频特性将受到转速，旋转半径和激励幅值等参数变化的显著影响。     

Survival probability of nonlinear oscillators endowed with fractional derivative element and subjected to evolutionary excitation: A stochastic averaging treatment with path integral concepts

An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.     

AMCXO and its test system

In this paper, we introduce a novel temperature compensated crystal oscillator--analog memory compensated crystal oscillator (AMCXO), a microcomputer compensated crystal oscillator (MCXO) based on an analogue approach, and we compare the performance between MCXO and AMCXO, and analyze the necessity of developing AMCXO. Both AMCXO and MCXO, combined with their test systems, use computers to generate the control signals used for the compensated crystal oscillators according to temperature variation. However, the execution devices are much different from each other in the practical temperature compensated crystal oscillators. A MCXO uses a set of digital units, including a microprocessor, to realize the compensation. For it, equations or tables are used to express the relationship between the control signal and temperature. An AMCXO uses an analogue memory, which has the same functions as the digital units of MCXO. For AMCXO, a curve or figure based on the equations or tables is used to express the same relationship. Their test systems have obvious distinctions in temperature experiment, data acquisition, and processing. A better performance will be obtained by using a more complicated AMCXO test system, and the cost can be reduced at the same time.