Nonlinear forced vibrations analysis of imperfect stiffened FG doubly curved shallow shell in thermal environment using multiple scales method

2021 ◽  
Vol 256 ◽  
pp. 113090
Author(s):  
Habib Ahmadi ◽  
Aliakbar Bayat ◽  
Nguyen Dinh Duc
Keyword(s):  
Shallow Shell ◽  
Multiple Scales ◽  
Thermal Environment ◽  
Forced Vibrations ◽  
Multiple Scales Method ◽  
Doubly Curved

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

2011 ◽  
Vol 46 (1) ◽  
pp. 170-179 ◽  
Author(s):  
F. Alijani ◽  
M. Amabili ◽  
F. Bakhtiari-Nejad
Keyword(s):  
Multiple Scales ◽  
Multiple Scales Method ◽  
Shallow Shells ◽  
Non Linear ◽  
Linear Vibrations ◽  
Doubly Curved

Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels

2014 ◽  
Vol 85 ◽  
pp. 341-349 ◽  
Author(s):  
Vijay K. Singh ◽  
Subrata K. Panda
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Shallow Shell ◽  
Free Vibration Analysis ◽  
Nonlinear Free Vibration ◽  
Doubly Curved

scholarly journals Perturbations of nonlinear autonomous oscillators

1994 ◽  
Vol 35 (4) ◽  
pp. 445-468 ◽  
Author(s):  
P. B. Chapman
Keyword(s):  
General Theory ◽  
Fourier Coefficients ◽  
Multiple Scales ◽  
Second Order ◽  
Multiple Scales Method ◽  
Suitable Parameter ◽  
Non Linear

AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

The Multiple Scales Method

2011 ◽  
pp. 359-360
Author(s):  
Michael Paidoussis ◽  
Stuart Price ◽  
Emmanuel de Langre
Keyword(s):  
Multiple Scales ◽  
Multiple Scales Method

scholarly journals Nonlinear Free and Forced Vibrations of a Hyperelastic Micro/Nanobeam Considering Strain Stiffening Effect

Nanomaterials ◽  
2021 ◽  
Vol 11 (11) ◽  
pp. 3066
Author(s):  
Amin Alibakhshi ◽  
Shahriar Dastjerdi ◽  
Mohammad Malikan ◽  
Victor A. Eremeyev
Keyword(s):  
Size Effect ◽  
Boundary Conditions ◽  
Frequency Response ◽  
Multiple Scales ◽  
Forced Vibrations ◽  
Response Force ◽  
Nonlinear Frequency ◽  
Large Rotation ◽  
Free And Forced Vibrations ◽  
Stiffening Effect

In recent years, the static and dynamic response of micro/nanobeams made of hyperelasticity materials received great attention. In the majority of studies in this area, the strain-stiffing effect that plays a major role in many hyperelastic materials has not been investigated deeply. Moreover, the influence of the size effect and large rotation for such a beam that is important for the large deformation was not addressed. This paper attempts to explore the free and forced vibrations of a micro/nanobeam made of a hyperelastic material incorporating strain-stiffening, size effect, and moderate rotation. The beam is modelled based on the Euler–Bernoulli beam theory, and strains are obtained via an extended von Kármán theory. Boundary conditions and governing equations are derived by way of Hamilton’s principle. The multiple scales method is applied to obtain the frequency response equation, and Hamilton’s technique is utilized to obtain the free undamped nonlinear frequency. The influence of important system parameters such as the stiffening parameter, damping coefficient, length of the beam, length-scale parameter, and forcing amplitude on the frequency response, force response, and nonlinear frequency is analyzed. Results show that the hyperelastic microbeam shows a nonlinear hardening behavior, which this type of nonlinearity gets stronger by increasing the strain-stiffening effect. Conversely, as the strain-stiffening effect is decreased, the nonlinear frequency is decreased accordingly. The evidence from this study suggests that incorporating strain-stiffening in hyperelastic beams could improve their vibrational performance. The model proposed in this paper is mathematically simple and can be utilized for other kinds of micro/nanobeams with different boundary conditions.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

2021 ◽  
Vol 16 (3) ◽  
Author(s):  
Yuanbin Wang ◽  
Weidong Zhu
Keyword(s):  
Governing Equation ◽  
Transverse Vibration ◽  
Harmonic Balance ◽  
Fourier Transforms ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Frequency Responses ◽  
Runge Kutta ◽  
Nonlinear Transverse Vibration

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam

2019 ◽  
Vol 24 (11) ◽  
pp. 3514-3536
Author(s):  
Mohsen Tajik ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Nonlinear Vibration ◽  
Bifurcation Analysis ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Twist Angle ◽  
Equations Of Motion ◽  
Multiple Scales Method ◽  
Vibration Stability ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, an Euler–Bernoulli model has been used for nonlinear vibration, stability, and bifurcation analysis of spinning twisted beams with linear twist angle, and with large transverse deflections, near the primary and parametric resonances. The equations of motion, in the case of pure single mode motion are analyzed by two methods: directly applying multiple scales method and using multiple scales method after discretization by Galerkin’s procedure. It is observed that the same final relations are obtained in the two methods. Effects of twist angle, damping ratio, longitudinal to transverse stiffness ratio, and eccentricity on the frequency responses are investigated. Then, the results are compared with the results obtained from Runge–Kutta numerical method on ODEs in a steady state, and confirmed with some previous research. Finally, the results show a good correlation, and it shows that with increasing the twist angle from 0 to 90°, the natural frequencies increase in the first two modes.

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