Accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness

2007 ◽  
Vol 307 (3-5) ◽  
pp. 720-736 ◽  
Author(s):  
S.K. Lai ◽  
C.W. Lim
Keyword(s):  
Free Vibration ◽  
Analytical Solutions ◽  
Nonlinear Stiffness ◽  
Nonlinear Free Vibration ◽  
Approximate Analytical Solutions ◽  
Linear And Nonlinear

Analytical solutions of linear and nonlinear Klein-Fock-Gordon equation

2015 ◽  
Vol 4 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Sajida Rasheed
Keyword(s):  
Analytical Solutions ◽  
Gordon Equation ◽  
Series Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Convergence Region ◽  
Relativistic Version ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Linear And Nonlinear

AbstractIn this paper, we deal with some linear and nonlinear Klein-Fock-Gordon (KFG) equations, which is a relativistic version of the Schrödinger equation. The approximate analytical solutions are obtained by using the homotopy analysis method (HAM). The efficiency of the HAM is that it provides a practical way to control the convergence region of series solutions by introducing an auxiliary parameter }. Analytical results presented are in agreement with the existing results in open literature, which confirm the effectiveness of this method.

Nonlinear free vibration analysis of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes

2019 ◽  
Vol 25 (14) ◽  
pp. 2063-2078 ◽  
Author(s):  
Mahsa Heidari ◽  
Hadi Arvin
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Direct Method ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Timoshenko Beams ◽  
Nonlinear Free Vibration ◽  
Reinforced Composite ◽  
Linear And Nonlinear

In this paper, the linear and nonlinear free vibrations of functionally graded rotating composite Timoshenko beams reinforced by carbon nanotubes are presented. The formulation is based on the assumptions of Timoshenko beam theory in addition to consideration of the nonlinear von Karman strain–displacement relationship. The effective material properties of carbon nanotube reinforced composites are determined employing the Mori–Tanaka micromechanics model and the extended mixture rule. For the carbon nanotube reinforced composite beams, uniform distribution and four types of functionally graded distribution patterns of single-walled carbon nanotube reinforcements are considered. A differential transform method is applied on the nondimensionalized equations of motion to release the flapping modeshapes and the associated natural frequencies. The direct method of multiple scales is implemented to derive the effective nonlinearity and the corresponding nonlinear natural frequency. The accuracy of the present outcomes is validated by the comparison with the results given in the literature. The numerical results are presented in both tabular and graphical forms to investigate the effects of nanotube volume fractions, distribution types of the carbon nanotubes, and rotation speed on linear and nonlinear free vibration characteristics of carbon nanotube reinforced composite beam. The results demonstrate the important role of carbon nanotube distribution profile on linear and nonlinear free vibration features.

scholarly journals LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE

2021 ◽  
Vol 12 (1) ◽  
pp. 15-25
Author(s):  
Edward Adah ◽  
David Onwuka ◽  
Owus Ibearugbulem ◽  
Chinenye Okere
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Large Deflection ◽  
Closed Form Solution ◽  
Middle Surface ◽  
Rectangular Plates ◽  
Nonlinear Free Vibration ◽  
Surface Displacements ◽  
Linear And Nonlinear

The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

scholarly journals Theoretical Analysis on the Nonlinear Free Vibration of a Tri-Cross String

2017 ◽  
Vol 2017 ◽  
pp. 1-17
Author(s):  
Bo Pan ◽  
Jingda Tang ◽  
Ryuichi Tarumi ◽  
Fulin Shang ◽  
Yanbo Wang ◽  
...  
Keyword(s):  
Constitutive Equation ◽  
Theoretical Analysis ◽  
Free Vibration ◽  
Natural Frequency ◽  
Analytical Solutions ◽  
Vibration Amplitude ◽  
Natural Frequencies ◽  
Geometrical Nonlinearity ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Here we present a theoretical analysis on the nonlinear free vibration of a tri-cross string system, which is an element of space net-antennas. We derived the governing equations from Hamilton’s principle and obtained a linearized solution by the standard perturbation method. The semi-analytical solutions of the governing equations have not been provided referring to the solution of plate vibrating problem. This analysis revealed that natural frequencies of the tri-cross string depend on the vibration amplitude due to the geometrical nonlinearity in the constitutive equation. The geometric parameters, such as the diameters and the lengths of the constituent strings, also affect the frequency through the nonlinearity of the tri-cross string. The nonlinear natural frequency shows coupled characteristic; that is, the natural frequency of the tri-cross string varies with that of the constituent strings, but the contribution of each constituent string to the natural frequency is in different proportions.

Analytical Solutions for Nonlinear Free Vibration of Micro-Scale Timoshenko Beams

2016 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Free Vibration ◽  
Analytical Solutions ◽  
Equations Of Motion ◽  
Variational Iteration Method ◽  
Strain Gradient Theory ◽  
Physical Parameters ◽  
Gradient Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Nonlinear free vibration of micro-beams based on the Timoshenko beam model is studied. The governing equations of motion using the strain gradient theory, Von Kármán strain tensors and Hamiltonian principle, are developed. The Galerkin method is applied to the governing equations and the coupled nonlinear ordinary differential equations of system are obtained. The variational iteration method is utilized to determine the time responses of the micro-beam and also a close form expression for the frequency-amplitude is found. The analytical solutions obtained for different values of parameters are compared with those found from different numerical methods. The effects of geometrical and physical parameters on the dynamics of micro-beam are also examined. Moreover, the analytical formulation for frequency ratio, i.e., the ratio of nonlinear natural frequency to the linear one is obtained and the sensitivity of this ratio to the variations of various parameters is evaluated. It is proved that the proposed solution methods and the results obtained are accurate and reliable when dynamics of such micro structures are studied.

Nonlinear Free Vibration of a Symmetrically Conservative Two-Mass System With Cubic Nonlinearity

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
T. Pirbodaghi ◽  
S. Hoseini
Keyword(s):  
Free Vibration ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Maximum Relative Error ◽  
Nonlinear Free Vibration ◽  
Mass System ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
First Order Approximation ◽  
The Mathematical Model

In this study, the nonlinear free vibration of conservative two degrees of freedom systems is analyzed using the homotopy analysis method (HAM). The mathematical model of such systems is described by two second-order coupled differential equations with cubic nonlinearities. First, novel approximate analytical solutions for displacements and frequencies are established using HAM. Then, the homotopy Padé technique is applied to accelerate the convergence rate of the solutions. Comparison between the obtained results and those available in the literature shows that the first-order approximation of homotopy Padé technique leads to accurate solutions with a maximum relative error less than 0.068 percent for all the considered cases.

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