Mathematical methods for the study of nonlinear vibration problems

1974 ◽  
Vol 55 (S1) ◽  
pp. S1-S1
Author(s):  
P. R. Sethna
Keyword(s):  
Nonlinear Vibration ◽  
Mathematical Methods ◽  
Vibration Problems

MODELING OF NONLINEAR VIBRATION PROBLEMS WITH FLUID FLOWS THROUGH PIPELINES

2018 ◽  
pp. 44-47
Author(s):  
F.J. Тurayev
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Viscoelastic Properties ◽  
Construction Material ◽  
Fluid Flows ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Flowing Liquid ◽  
Vibration Problems

In this paper, mathematical model of nonlinear vibration problems with fluid flows through pipelines have been developed. Using the Bubnov–Galerkin method for the boundary conditions, the resulting nonlinear integro-differential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integro-differential equations with both constant and variable coefficients as functions of time.A system of algebraic equations is obtained according to numerical method for the unknowns. The influence of the singularity of heredity kernels on the vibrations of structures possessing viscoelastic properties is numerically investigated.It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

scholarly journals Extension of method of moment equation to nonlinear deterministic vibrations

2002 ◽  
Vol 24 (3) ◽  
pp. 133-141
Author(s):  
Nguyen Dong Anh ◽  
Ninh Quang Hai
Keyword(s):  
Nonlinear Vibration ◽  
Moment Equation ◽  
Numerical Results ◽  
Equation Approach ◽  
Method Of Moment ◽  
Vibration Problems

The paper present the so-called "an extended averaged equation approach" to the investigation of nonlinear vibration problems. The numerical results in analysing the vibration systems with weak, middle and strong non-linearity show the advantages of the method.

Assessment of Solutions to Vibration Problems Involving Piecewise Constant Exertions

2000 ◽  
Author(s):  
L. Dai
Keyword(s):  
Nonlinear Vibration ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Piecewise Constant ◽  
Truncation Errors ◽  
Piecewise Constant Argument ◽  
Vibration Problems ◽  
Linear And Nonlinear ◽  
Constant Argument

Abstract Direct analytical and numerical solutions are constructed for linear and nonlinear vibration problems involving piecewise constant exertions. Existence and uniqueness of the solutions and the truncation errors of the numerical calculations are also analysed. With the employment of a piecewise constant argument, vibration systems with piecewise constant exertions are connected with the corresponding systems with continuous exertions.

Amplitude Incremental Generalized Potential Energy Principles with Two Kinds of Variables for Nonlinear Vibration

2012 ◽  
Vol 157-158 ◽  
pp. 1130-1134
Author(s):  
Zong Min Liu ◽  
Bai Tao Sun ◽  
Gui Xin Zhang ◽  
Yuan Yuan Zhang
Keyword(s):  
Potential Energy ◽  
Nonlinear Vibration ◽  
Quantitative Method ◽  
Theoretical Foundation ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Generalized Potential ◽  
Vibration Problems ◽  
Energy Principles ◽  
Ihb Method

For nonlinear vibration problems, IHB method is an effective definite quantitative method. The theoretical foundation of IHB method is amplitude incremental variational principle. In this paper, amplitude incremental generalized potential energy principle and generalized quasi-potential energy principle with two variables are established, thus improving theoretical foundation of IHB method.

scholarly journals Applying the method of normal forms to second-order nonlinear vibration problems

2010 ◽  
Vol 467 (2128) ◽  
pp. 1141-1163 ◽  
Author(s):  
Simon A. Neild ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Normal Forms ◽  
Equations Of Motion ◽  
Second Order ◽  
Form Analysis ◽  
First Order ◽  
Invariance Properties ◽  
Vibration Problems ◽  
Second Order Equations

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

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