Solution Of Nonlinear Dynamic Equations Research Articles

An efficient algorithm based on Broyden's and generalized-α method for large scale flexible multi-body systems

This paper designs a novel and efficient numerical strategy based on generalized-α algorithm and Broyden's method to solve dynamic equations of large scale flexible multi-body system. The dynamic models are established in view of Absolute Nodal Coordinate Formulation(ANCF) and the clearance joint model is described by nonlinear contact force model and modified Coulomb's friction model. The proposed strategy only needs a given initial Jacobian matrix and then the numerical solutions can be obtained by using Sherman-Morrison-Woodburg theorem, that can greatly increase the computational efficiency as a result of avoiding solving of the Jacobian matrix in each iteration. Newton-Raphson iteration and Trust-Region-Dogleg algorithm, as two typical numerical solutions for nonlinear dynamic equations of large scale flexible multi-body systems, are selected to be compared the effectiveness and efficiency with the proposed numerical method. Numerical results show that the proposed method is accurate and robust, and can improve the computational efficiency by several or even a dozen times than the other two common methods for large scale flexible multi-body systems including flexible double pendulum model, rigid and rigid-flexible coupling solar array systems.

Affine-periodic solutions for nonlinear dynamic equations on time scales

The existence of affine-periodic solutions for dynamic equations on time scales is studied. Mainly, via the topological degree theory, a general existence theorem is proved, which provides an effective method in the qualitative theory for nonlinear dynamic equations on time scales.

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

Numerical Modelling of Nonlinear Vibration Isolation System Free Oscillations

The article deals with numerical analysis of nonlinear vibration isolation system free oscillations. According to the latest recommendations [1], vibration isolation of precision equipment should be effective over the 1 – 100 Hz range [2]. This fact sets a challenge to develop a sub-herz vibration isolator. Such vibration isolators with 0.5 Hz natural frequency have been developed in latest works [3,4]. For mathematical representation of such a dynamical system, elastic characteristic should be approximated with a polynomial. In this paper, we consider the approximation of elastic characteristic by means of 5th and 7th order polynomials. This leads to numerical solution of nonlinear dynamic equations. We construct the solution of such equation on the phase plane and determine equilibrium points. The stability of the equilibrium points is investigated using linearization technique. The law of the isolated mass oscillations is derived from numerical solution of the dynamic equation by means of Dorman – Prince 4(5) and 8(7) solver and the system’s natural frequencies are calculated using the FFT technique.

Nonlinear oscillations of a three-layer and multi-layer plates and shells during periodic impacts (survey)

The analysis of the current state of research on the free and forced nonlinear oscillations of three-layer and multilayer thin elastic plates and shells under periodic influences was made. There were discussed some various approaches to the solution of nonlinear dynamic equations of laminated plates and shells, which are applied to this class of problems. There was also analyzed the impact of physical, mechanical, geometrical parameters, the packet structure of layers, forms of plates, boundary conditions on the nature of non-linear oscillations and form of amplitude-frequency characteristics of laminated plates and shells, as well as the available experimental results.

Dynamic Analysis of Multi-Supported Nonlinear Vibration Isolation System Mounted on Flexible Foundation

In view of its prototype in engineering application, a theoretical model of multi-supported nonlinear vibration isolation system installed on flexible foundation is studied, including derivation of system dynamic equations and analysis of system dynamic characteristics. For effectiveness evaluation of nonlinear vibration isolation systems, a generalized time-averaged power is proposed as an extension of classical theory of vibratory power flow, and a numerical solution method of time-averaged power is probed accompanying with the numerical solution of nonlinear dynamic equations. In a further concrete calculation example, an air spring vibration isolation system of a small UAV engine is numerically simulated based on Runge-Kutta method, and dynamic behavior and power flow transmission characteristics influenced by system parameters are investigated.

Soliton microdynamics of heat conduction in plutonium and uranium in the temperature range of martensitic phase transitions

The microdynamics of large-amplitude nonlinear lattice vibrations of plutonium and uranium has been investigated at high reactor temperatures in the ranges of martensitic phase transitions. Solutions of nonlinear dynamic equations have been obtained using the Lennard-Jones interatomic potential for the soliton generation and the energy transfer by solitons between the crystal boundaries in the shells. The synchronism of the soliton trajectories and peaks of energy fluxes demonstrates an analog of the shot effect. The temperature dependences of thermal conductivity coefficients are consistent with the experiment and exhibit local maxima in the ranges of phase transitions. A spectral analysis has revealed that the dominant heat transfer is provided by rarefaction solitons. The martensitic phase transitions are accompanied by a reorganization of the spectral density in the phase plane with a sharp increase in the high-frequency range. The spectral density has maxima of the quasi-biphonon type. The obtained data in dimensionless form, apart from plutonium and uranium, can be used for other monatomic crystals. The specific features of the thermal conductivity and microdynamics of the formation of vacancies and pores in crystals without shells have been discussed.

Existence of Nontrivial Solutions and Sign-Changing Solutions for Nonlinear Dynamic Equations on Time Scales

By using the topological degree theory and the fixed point index theory, the existence of nontrivial solutions and sign-changing solutions for a class of boundary value problem on time scales is obtained. We should point out that the integral operator corresponding to above boundary value problem is not assumed to be a cone mapping.

Nonlinear Damped Vibrations of a Laminated Composite Plate Subjected to Blast Load

This research deals with the derivation and solution of nonlinear dynamic equations of a clamped laminated plate exposed to blast load including structural damping effects. Dynamic equations of the plate are derived by the use of the virtual work principle. The geometric nonlinearity effects are taken into account with the von Karman large deflection theory of thin plates. An appropriate approximate solution is assumed for the space domain considering the finite element analysis results for the large deformation static analysis of the laminated plate. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. These nonlinear differential equations include structural damping effects. The finite difference method is applied to solve the system of coupled nonlinear equations. The results of the theoretical analysis are compared with the experimental results. Good agreement is found for the peak values and frequencies. Effects of aspect ratio and damping on the dynamic response of the plate are examined. The damping coefficient greatly affects the nonlinear dynamic response.

Approximating crossed symmetric solutions of nonlinear dynamic equations via quasilinearization

Crossed symmetric solutions of nonlinear boundary value dynamic problems play an important role in many applications, in particular in adaptive algorithm designs. This article is devoted to the continuation of our investigation on second-order nonlinear companion dynamic boundary value problems on time scales. Monotonically convergent upper and lower solutions of the problems and their quasilinear approximations are investigated. It is shown that, under proper smoothness constraints, the iterative sequences constructed not only converge to the analytic solutions of the desired companion problems monotonically, but also preserve important crossed symmetry properties. The quasilinearization offers an efficient way in the solution approximation. Computational examples are given to illustrate our results.

Periodic waveform relaxation solutions of nonlinear dynamic equations

We present a simple theorem to safeguard the convergence of waveform relaxation (WR) solutions of a dynamic system described by nonlinear ordinary differential equations (ODEs) with a periodic constraint. Namely, if a basic expression of certain constants issued from the system is less than one, the proposed WR algorithm is convergent to the exact solution. It is the first time that WR is used to treat periodic solutions of nonlinear dynamic systems. A numerical example is provided to confirm the theoretic work of the paper.

Nonlinear dynamic response of reinforced concrete under impulsive loading: Research status and needs

Current research status is presented for the multidimensional nonlinear analysis of reinforced concrete subjected to impulsive load conditions. Strategy for the solution of nonlinear dynamic equations is discussed. The description of the development of models for material behavior is given. Further research needs and interests for the development of improved analysis capabilities are indicated.