Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance

Abstract In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω1), detuning parameter (σ1), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.

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Analytical and Experimental Analyses of Nonlinear Vibrations in a Rotary Inverted Pendulum

10.21203/rs.3.rs-273757/v1 ◽

2021 ◽

Author(s):

Mohammadjavad Rahimi dolatabad ◽

Abdolreza Pasharavesh ◽

Amir Ali Akbar Khayyat

Keyword(s):

Analytical Solution ◽

Numerical Solution ◽

Closed Loop ◽

Inverted Pendulum ◽

Multiple Scales ◽

Nonlinear Oscillations ◽

Wide Frequency Range ◽

Perturbative Method ◽

Full State Feedback ◽

Rotary Inverted Pendulum

Abstract Gaining insight into possible vibratory responses of dynamical systems around their stable equilibria is an essential step, which must be taken before their design and application. The results of such a study can significantly help prevent instability in closed-loop stabilized systems through avoiding the excitation of the system in the neighborhood of its resonance. This paper investigates nonlinear oscillations of a Rotary Inverted Pendulum (RIP) with a full-state feedback controller. Lagrange’s equations are employed to derive an accurate 2-DoF mathematical model, whose parameter values are extracted by both the measurement and 3D modeling of the real system components. Although the governing equations of a 2-DoF nonlinear system are difficult to solve, performing an analytical solution is of great importance, mostly because, compared to the numerical solution, the analytical solution can function as an accurate pattern. Additionally, the analytical solution is generally more appealing to engineers because their computational costs are less than those of the numerical solution. In this study, the perturbative method of multiple scales is used to obtain an analytical solution to the coupled nonlinear motion equations of the closed-loop system. Moreover, the parameters of the controller are determined, using the results of this solution. The findings reveal the existence of hardening- and softening-type resonances at the first and second vibrational modes, respectively. This led to a wide frequency range with moderately large-amplitude vibrations, which must be avoided when adjusting a time-varying set-point for the system. The analytical results of the nonlinear vibration of the RIP are verified by experimental measurements, and a very good agreement is observed between the results of both approaches.

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Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations

Fractal and Fractional ◽

10.3390/fractalfract2030021 ◽

2018 ◽

Vol 2 (3) ◽

pp. 21 ◽

Cited By ~ 8

Author(s):

Guy Eyebe ◽

Gambo Betchewe ◽

Alidou Mohamadou ◽

Timoleon Kofane

Keyword(s):

Analytical Solution ◽

Fractional Derivative ◽

Nonlinear Vibration ◽

Fractional Order ◽

Nonlocal Elasticity ◽

Multiple Scales ◽

Approximate Analytical Solution ◽

Nonlocal Elasticity Theory ◽

Pasternak Foundation ◽

D’Alembert Principle

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler–Pasternak foundation is studied using nonlocal elasticity theory. The D’Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

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A General Correlation for Pool Film Boiling Heat Transfer From a Horizontal Cylinder to Subcooled Liquid: Part 1—A Theoretical Pool Film Boiling Heat Transfer Model Including Radiation Contributions and Its Analytical Solution

Journal of Heat Transfer ◽

10.1115/1.2910396 ◽

1990 ◽

Vol 112 (2) ◽

pp. 430-440 ◽

Cited By ~ 49

Author(s):

A. Sakurai ◽

M. Shiotsu ◽

K. Hata

Keyword(s):

Heat Transfer ◽

Heat Transfer Coefficient ◽

Theoretical Model ◽

Analytical Solution ◽

Transfer Coefficient ◽

Numerical Solution ◽

Boiling Heat Transfer ◽

Approximate Analytical Solution ◽

Horizontal Cylinder ◽

Film Boiling

A rigorous numerical solution of a theoretical model based on laminar boundary layer theory for pool film boiling heat transfer from a horizontal cylinder including the contributions of liquid subcooling and radiation from the cylinder was obtained. The numerical solution predicted accurately the experimental results of pool film boiling heat transfer from a horizontal cylinder in water with high radiation emissivity for a wide range of liquid subcooling in the range of nondimensional cylinder diameters around 1.3, where the numerical solution was applicable to the pool film boiling heat transfer from a cylinder with negligible radiation emissivity. An approximate analytical solution for the theoretical model was also derived. It was given by the sum of the pool film boiling heat transfer coefficient if there were no radiation and the radiation heat transfer coefficient for parallel plates multiplied by a nondimensional radiation parameter similar to the expression for saturated pool film boiling given by Bromley. The approximate analytical solution agreed well with the rigorous numerical solution for various liquids of widely different physical properties under wide ranges of conditions.

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Analytical solution for monovalent-divalent ion exchange transport in groundwater

Canadian Geotechnical Journal ◽

10.1139/t99-064 ◽

1999 ◽

Vol 36 (6) ◽

pp. 1197-1201 ◽

Cited By ~ 2

Author(s):

Yee-Chung Jin ◽

Songlin Ye

Keyword(s):

Ion Exchange ◽

Analytical Solution ◽

Groundwater Flow ◽

Coordinate System ◽

Solute Transport ◽

Numerical Solution ◽

Approximate Analytical Solution ◽

Initial Conditions ◽

Direct Integration ◽

Advection Velocity

An approximate analytical solution is derived for solute transport with monovalent-divalent ion exchange in saturated, steady, groundwater flow. The analytical solution is obtained by simplifying the complicated heterovalent ion exchange. Using a coordinate system that moves at the average solute advection velocity, a solution is obtained by direct integration with given boundary and initial conditions. The results agree well with a numerical solution using the original isotherm. The analytical solution presented is similar to that of monovalent-monovalent exchange, which suggests that a simple ion exchange model may be assumed for approximation.

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Nonlinear vibration of a nonlocal nanobeam resting on fractional order viscoelastic pasternak foundations

10.20944/preprints201806.0259.v1 ◽

2018 ◽

Author(s):

Guy joseph Eyebe ◽

Gambo Betchewe ◽

Mohamadou Alidou ◽

Timoleon crepin Kofane

Keyword(s):

Analytical Solution ◽

Fractional Derivative ◽

Nonlinear Vibration ◽

Fractional Order ◽

Parametric Study ◽

Nonlocal Elasticity ◽

Multiple Scales ◽

Approximate Analytical Solution ◽

Nonlocal Elasticity Theory ◽

D’Alembert Principle

In the present study, nonlinear vibration of a nanobeam resting on fractional order viscoelastic Winkler-Pasternak foundaion is studied using nonlocal elasticity theory. D'Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. Detailled parametric study is conducted, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations

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Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

International Journal of Recent Technology and Engineering - 2 ◽

10.35940/ijrte.c4861.098319 ◽

2019 ◽

Vol 8 (3) ◽

pp. 5795-5802 ◽

Cited By ~ 1

Keyword(s):

Porous Medium ◽

Steady State ◽

Numerical Solution ◽

Viscous Dissipation ◽

Flow Problem ◽

Numerical Study ◽

Steady State Flow ◽

Nanofluid Flow ◽

Shooting Technique ◽

Order Four

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

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Numerical Approximations to Solution of Biochemical Reaction Model

International Journal of Chemical Reactor Engineering ◽

10.2202/1542-6580.2606 ◽

2011 ◽

Vol 9 (1) ◽

Author(s):

Ahmet Yildirim ◽

Ahmet Gökdogan ◽

Mehmet Merdan

Keyword(s):

Analytical Solution ◽

Approximate Analytical Solution ◽

Transformation Method ◽

Reaction Model ◽

Biochemical Reaction ◽

Graphical Form ◽

Kutta Method ◽

Transform Method ◽

Runge Kutta Method ◽

Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

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