Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems

1981 ◽  
Vol 48 (4) ◽  
pp. 959-964 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung
Keyword(s):  
Variational Principle ◽  
Nonlinear Vibrations ◽  
Elliptic Integral ◽  
Nonlinear Problems ◽  
Free Vibrations ◽  
Linear Solution ◽  
Starting Point ◽  
Highly Nonlinear ◽  
Elastic Systems ◽  
Incremental Step

The incremental method has been widely used in various types of nonlinear analysis, however, so far it has received little attention in the analysis of periodic nonlinear vibrations. In this paper, an amplitude incremental variational principle for nonlinear vibrations of elastic systems is derived. Based on this principle various approximate procedures can be adapted to the incremental formulation. The linear solution for the system is used as the starting point of the solution procedure and the amplitude is then increased incrementally. Within each incremental step, only a set of linear equations has to be solved to obtain the data for the next stage. To show the effectiveness of the present method, some typical examples of nonlinear free vibrations of plates and shallow shells are computed. Comparison with analytical results calculated by using elliptic integral confirms that excellent accuracy can be achieved. The technique is applicable to highly nonlinear problems as well as problems with only weak nonlinearity.

Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

1963 ◽  
Vol 30 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Layered Plates ◽  
Variational Approximations ◽  
Elastic Systems ◽  
Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

A Variable Parameter Incrementation Method for Dynamic Instability of Linear and Nonlinear Elastic Systems

1982 ◽  
Vol 49 (4) ◽  
pp. 849-853 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung ◽  
S. Y. Wu
Keyword(s):  
Nonlinear Vibrations ◽  
Dynamic Instability ◽  
Parametric Instability ◽  
Variable Parameter ◽  
Nonlinear Problems ◽  
Geometrically Nonlinear ◽  
Weak Nonlinearity ◽  
Elastic Systems ◽  
Nonlinear Elastic

A variable parameter incrementation method is proposed and then applied to the determination of parametric instability boundary of columns. Attention is particularly paid to the geometrically nonlinear problems including the instability of nonlinear vibrations. Although only beam and column problems are treated at present, the approach is believed to be general in methodology. This method is not subjected to the limitations of small exciting parameters and weak nonlinearity.

A Comprehensive Investigation of Ensembles of Gaussian-Based and Gradient-Based Transformed Reliability Analyses: When and How to Use Them

2016 ◽  
Author(s):  
Po Ting Lin ◽  
Wei-Hao Lu ◽  
Shu-Ping Lin
Keyword(s):  
Design Optimization ◽  
Design Space ◽  
Optimization Problems ◽  
Nonlinear Problems ◽  
Kernel Functions ◽  
Sensitivity Analyses ◽  
Normal Space ◽  
Highly Nonlinear ◽  
Standard Normal ◽  
Gradient Based

In the past few years, researchers have begun to investigate the existence of arbitrary uncertainties in the design optimization problems. Most traditional reliability-based design optimization (RBDO) methods transform the design space to the standard normal space for reliability analysis but may not work well when the random variables are arbitrarily distributed. It is because that the transformation to the standard normal space cannot be determined or the distribution type is unknown. The methods of Ensemble of Gaussian-based Reliability Analyses (EoGRA) and Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) have been developed to estimate the joint probability density function using the ensemble of kernel functions. EoGRA performs a series of Gaussian-based kernel reliability analyses and merged them together to compute the reliability of the design point. EGTRA transforms the design space to the single-variate design space toward the constraint gradient, where the kernel reliability analyses become much less costly. In this paper, a series of comprehensive investigations were performed to study the similarities and differences between EoGRA and EGTRA. The results showed that EGTRA performs accurate and effective reliability analyses for both linear and nonlinear problems. When the constraints are highly nonlinear, EGTRA may have little problem but still can be effective in terms of starting from deterministic optimal points. On the other hands, the sensitivity analyses of EoGRA may be ineffective when the random distribution is completely inside the feasible space or infeasible space. However, EoGRA can find acceptable design points when starting from deterministic optimal points. Moreover, EoGRA is capable of delivering estimated failure probability of each constraint during the optimization processes, which may be convenient for some applications.

Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amin Gholami ◽  
Davood D. Ganji ◽  
Hadi Rezazadeh ◽  
Waleed Adel ◽  
Ahmet Bekir
Keyword(s):  
Approximate Solution ◽  
Nonlinear Vibrations ◽  
Nonlinear Problems ◽  
Iterative Approach ◽  
Mathematical Pendulum ◽  
Science And Engineering ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
Strong Candidate ◽  
Iteration Technique

Abstract The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering.

Optimization of the Closure-Weld Region of Cylindrical Containers for Long-Term Corrosion Resistance Using the Successive Heuristic Quadratic Approximation Technique*

2003 ◽  
Vol 125 (3) ◽  
pp. 533-539 ◽  
Author(s):  
Zekai Ceylan ◽  
Mohamed B. Trabia
Keyword(s):  
Finite Element ◽  
Random Search ◽  
Nonlinear Problems ◽  
Search Space ◽  
Induction Coil ◽  
Quadratic Approximation ◽  
Approximation Technique ◽  
General Corrosion ◽  
Highly Nonlinear

Welded cylindrical containers are susceptible to stress corrosion cracking (SCC) in the closure-weld area. An induction coil heating technique may be used to relieve the residual stresses in the closure-weld. This technique involves localized heating of the material by the surrounding coils. The material is then cooled to room temperature by quenching. A two-dimensional axisymmetric finite element model is developed to study the effects of induction coil heating and subsequent quenching. The finite element results are validated through an experimental test. The container design is tuned to maximize the compressive stress from the outer surface to a depth that is equal to the long-term general corrosion rate of the container material multiplied by the desired container lifetime. The problem is subject to several geometrical and stress constraints. Two different solution methods are implemented for this purpose. First, an off-the-shelf optimization software is used. The results however were unsatisfactory because of the highly nonlinear nature of the problem. The paper proposes a novel alternative: the Successive Heuristic Quadratic Approximation (SHQA) technique. This algorithm combines successive quadratic approximation with an adaptive random search within varying search space. SHQA promises to be a suitable search method for computationally intensive, highly nonlinear problems.

Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand’s Equation

2013 ◽  
Vol 6 (2) ◽  
pp. 76-82 ◽  
Author(s):  
U. Filobello- ◽  
H. Vazquez-Le ◽  
K. Boubaker ◽  
Y. Khan ◽  
A. Perez-Sesm ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Nonlinear Problems ◽  
Highly Nonlinear

scholarly journals Contraction Integral Equation for Three-Dimensional Electromagnetic Inverse Scattering Problems

2019 ◽  
Vol 5 (2) ◽  
pp. 27 ◽  
Author(s):  
Yu Zhong ◽  
Kuiwen Xu
Keyword(s):  
Integral Equation ◽  
Inverse Scattering ◽  
Contraction Mapping ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Optimization Method ◽  
Scattering Problems ◽  
Physical Reason ◽  
Inverse Scattering Problems ◽  
Highly Nonlinear

Inverse scattering problems (ISPs) stand at the center of many important imaging applications, such as geophysical explorations, industrial non-destructive testing, bio-medical imaging, etc. Recently, a new type of contraction integral equation for inversion (CIE-I) has been proposed to tackle the two-dimensional electromagnetic ISPs, in which the usually employed Lippmann–Schwinger integral equation (LSIE) is transformed into a new form with a modified medium contrast via a contraction mapping. With the CIE-I, the multiple scattering effects, i.e., the physical reason for the nonlinearity in the ISPs, is substantially suppressed in estimating the modified contrast, without compromising physical modeling. In this paper, we firstly propose to implement this new CIE-I for the three-dimensional ISPs. With the help of the FFT type twofold subspace-based optimization method (TSOM), when handling the highly nonlinear problems with strong scatterers, those with higher contrast and/or larger dimensions (in terms of wavelengths), the performance of the inversions with CIE-I is much better than the ones with the LSIE, wherein inversions usually converge to local minima that may be far away from the solution. In addition, when handling the moderate scatterers (those the LSIE modeling can still handle), the convergence speed of the proposed method with CIE-I is much faster than the one with the LSIE. Secondly, we propose to relax the contraction mapping condition, i.e., different contraction mappings are used in updating contrast sources and contrast, and we find that the convergence can be further accelerated. Several numerical tests illustrate the aforementioned interests.

scholarly journals A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

2019 ◽  
Vol 24 (1) ◽  
pp. 311-331 ◽  
Author(s):  
Prashant Kumar ◽  
Carmen Rodrigo ◽  
Francisco J. Gaspar ◽  
Cornelis W. Oosterlee
Keyword(s):  
Monte Carlo ◽  
Variance Reduction ◽  
Nonlinear Problems ◽  
Picard Iteration ◽  
Richards Equation ◽  
Porous Media Flow ◽  
Soil Parameters ◽  
Multilevel Monte Carlo ◽  
Highly Nonlinear ◽  
Non Gaussian

AbstractWe present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator.

HAM Solutions for Vortex-Induced Vibration of a Circular Cylinder

2014 ◽  
Author(s):  
Zhiliang Lin ◽  
Longbin Tao
Keyword(s):  
Circular Cylinder ◽  
Practical Importance ◽  
Nonlinear Problems ◽  
Van Der Pol Oscillator ◽  
Fluid Structure ◽  
Vortex Induced Vibration ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Simple Linear Equation ◽  
Wake Oscillation

The vortex-induced vibration (VIV) phenomenon is result of fluid-structure interaction which occurs in many engineering fields. The study of VIV of a circular cylinder is of practical importance (such as in marine cables and flexible risers in petroleum production). In this paper, one classical phenomenological VIV model — the motion of the cylinder is modeled by a simple linear equation, and the fluctuating nature of the vortex wake oscillation is modeled by a van der Pol oscillator, is analyzed. Firstly, the homotopy analysis method (HAM), a powerful technique for highly nonlinear problems, is developed to solve the coupled fluid-structure dynamical system with the convergence of the homotopy series solutions being demonstrated. Based on the HAM solutions, some properties of the fully nonlinear classical coupled VIV model are presented. All the results proved that the proposed HAM scheme has potential to be an effective analytic technique to study the VIV problems.

Sign in / Sign up

Export Citation Format

Share Document