Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

Preface
Contents
Part I A Short Introduction to the Optimal Auxiliary Functions Method
1 Introduction
	References
2 The Optimal Auxiliary Functions Method
	References
Part II The Optimal Auxiliary Functions Method in Engineering Applications
3 The First Alternative of the Optimal Auxiliary Functions Method
	3.1 Dynamics of an Angular Misaligned Multirotor System
		3.1.1 The Governing Equations
		3.1.2 Optimal Auxiliary Functions Method for Nonlinear Vibration of Misaligned Multirotor System
		3.1.3 Numerical Examples
	References
4 Oscillations of a Pendulum Wrapping on Two Cylinders
	4.1 Equation of Motion
	4.2 Application of OAFM to a Pendulum Wrapping on Two Cylinders
	4.3 Numerical Examples
	References
5 Free Oscillations of Euler–Bernoulli Beams on Nonlinear Winkler-Pasternak Foundation
	5.1 Nonlinear Euler–Bernoulli Beam Model
	5.2 OAFM for Free Oscillations of Euler–Bernoulli Beam
	5.3 Numerical Example
	References
6 Nonlinear Vibrations of Doubly Clamped Nanobeam Incorporating the Casimir Force
	6.1 Nonlinear Equation for Nanobeam
	6.2 Galerkin Formulation
	6.3 Application of OAFM to Eqs. (6.11) and (6.12)
	6.4 Numerical Example
	References
7 Transversal Oscillations of a Beam with Quintic Nonlinearities
	7.1 The Governing Equation
	7.2 OAFM for Nonlinear Differential Eq. (7.11)
	7.3 Numerical Example
	References
8 Approximate Analytical Solutions to Jerk Equations
	8.1 OAFM for Jerk Equations
	8.2 Numerical Examples
		8.2.1 Case 1
		8.2.2 Case 2
	References
9 Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection
	9.1 Equation of Motion
	9.2 Application of OAFM to Vibration of Nonlinear Nonlocal Elastic Column with Initial Imperfection
	9.3 Numerical Example
	References
10 Nonlinear Vibration of Bernoulli–Euler Beam on a Winkler Elastic Foundation
	10.1 System Description
	10.2 Discretization and Free Vibration of the Beam Under Study
	10.3 Numerical Example
	References
11 The Nonlinear Thermomechanical Vibration of a Functionally Graded Beam (FGB) on Winkler-Pasternak Foundation
	11.1 The Governing Equations
	11.2 Application of OAFM to Eqs. (11.33) and  (11.36)
	11.3 Numerical Examples
	References
12 Nonlinear Free Vibration of Microtubes
	12.1 Problem Formulation
	12.2 Free Vibration of the Microtube
	12.3 OAFM for Eqs. (12.16) and  (12.18)
	12.4 Numerical Example
	References
13 Nonlinear Free Vibration of Elastically Actuated Microtubes
	13.1 Problem Formulations
	13.2 Free Vibration of the Microtube
	13.3 Application of OAFM to Elastically Actuated Microtube
	13.4 Numerical Examples
	References
14 Analytical Investigation to Duffing Harmonic Oscillator
	14.1 OAFM for Duffing Harmonic Oscillator
	14.2 Numerical Examples
	References
15 Free Vibration of Tapered Beams
	15.1 OAFM for Free Vibration of Tapered Beams
	15.2 Numerical Examples
	References
16 Dynamic Analysis of a Rotating Electrical Machine Rotor-Bearing System
	16.1 Application of OAFM to the Investigation of Nonlinear Vibration of the Considered Electrical Machine
	16.2 Numerical Example
	References
17 Investigation of a Permanent Magnet Synchronous Generator
	17.1 Governing Equations of PMSG
	17.2 Approximate Solution of Eqs. (17.11) and (17.10)
	References
18 Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust
	18.1 Approximate Solution of the Dynamic Model of the Wind-Power System
	References
19 Axisymmetric Flow and Heat Transfer on a Moving Cylinder
	19.1 Equations of Motion
	19.2 Optimal Auxiliary Functions Method for Solving the System (3.17.9)–( 3.17.18)
	19.3 Numerical Results
	References
20 Blasius Problem
	20.1 The Governing Equation
	20.2 Approximate Solution of the Blasius Problem
	20.3 Discussion
	References
21 Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder
	21.1 Governing Equations of Thin Film Flow of a Fourth Grade Fluid Down a Vertical Cylinder
	21.2 Approximate Solution of the Eqs. (21.8) and (21.9)
	21.3 Numerical Example for the First Alternative
	21.4 Numerical Results by OAFM (The Second Alternative)
	References
22 Viscous Flow Due to a Stretching Surface with Partial Slip
	22.1 The Governing Equations
	22.2 Application of OAFM to Viscous Fluid Given by Eqs. (22.8), (22.10), (22.11) and (22.12)
	22.3 Numerical Examples
	References
23 Axisymmetric MHD Flow and Heat Transfer to Modified Second Grade Fluid
	23.1 The Governing Equations
	23.2 OAFM for Solving the System (23.18), (23.19), (23.21), (23.22)
	23.3 Numerical Examples
		23.3.1 Example 1
		23.3.2 Example 2
		23.3.3 Example 3
	References
24 Thin Film Flow of an Eyring Powel Fluid on a Vertical Moving Belt
	24.1 The Governing Equation of Motion
	24.2 Numerical Examples
		24.2.1 Case 1
		24.2.2 Case 2
		24.2.3 Case 3
	References
25 The Steady Flow of a Fourth Grade Fluid in a Porous Medium
	25.1 The Governing Equations
	25.2 OAFM for the Steady Flow of a Fourth Grade Fluid in a Porous Medium
		25.2.1 Case 1
		25.2.2 Case 2
		25.2.3 Case 3
		25.2.4 Case 4
		25.2.5 Case 5
		25.2.6 Case 6
		25.2.7 Case 7
		25.2.8 Case 8
		25.2.9 Case 9
		25.2.10 Case 10
		25.2.11 Case 11
		25.2.12 Case 12
		25.2.13 Case 13
		25.2.14 Case 14
	References
26 Thin Film Flow of an Oldroyd Six-Constant Fluid Over a Moving Belt
	26.1 Governing Equations
	26.2 OAFM for Eqs. (26.15) and  (26.16)
	References
27 Cylindrical Liouville-Bratu-Gelfand Problem
	27.1 OAFM for Cylindrical Liouville-Bratu-Gelfand Problem
		27.1.1 Case A
		27.1.2 Case B
	27.2 Numerical Examples
		27.2.1 Case A
		27.2.2 Case B
		27.2.3 Case C
	References
28 The Polytrophic Spheres of the Nonlinear Lane—Emden—Type Equation Arising in Astrophysics
	28.1 The Nonlinear Lane—Emden Equation
	28.2 OAFM for the Polytrophic Spheres of the Lane—Emden Equation
		28.2.1 Case 1
		28.2.2 Case 2
	References
Part III Some Variants and Modifications of the Basic Optimal Auxiliary Functions Method
29 The Second Alternative to the Optimal Auxiliary Functions Method
	29.1 Dynamic Response of a Permanent Magnet Synchronous Generator to a Wind Gust
		29.1.1 Application of an Alternative of OAFM to the Considered Problem (29.16) and (29.17)
		29.1.2 Numerical Examples
	29.2 Lambert W Function with Application in Electronics and Seismic Waves
		29.2.1 Evaluation of the Lambert W Function by OAFM
		29.2.2 Application of the Lambert Function in Electronics and Seismic Waves
	29.3 Nonlinear Blasius and Sakiadis Flows
		29.3.1 Approximate Solutions for the Blasius and Sakiadis Problems Using the Alternative of the OAFM
	29.4 Poisson–Boltzman (P.B) Equations
		29.4.1 P.B Equation for a Charged Rod in Absence of Added Salt
		29.4.2 OAFM for P.B given by Eqs. (29.134) and (29.135)
		29.4.3 Numerical Examples
	References
30 Piecewise Optimal Auxiliary Functions Method
	30.1 The Lane-Emden Equation of the Second Kind
		30.1.1 The Nonlinear Lane-Emden Equation of the Second Kind
		30.1.2 POAFM for the Lane-Emden Equation of Second Kind
	References
31 Some Exact Solutions for Nonlinear Dynamical Systems by Means of the Optimal Auxiliary Functions Method
	31.1 Some Exact Solutions for MHD Flow and Heat Transfer to Modified Second Grade Fluid with Variable Thermal Conductivity in the Presence of Thermal Radiation and Heat Generation/Absorption
		31.1.1 Some Exact Solutions for Eqs. (31.6)–(31.9) Using OAFM
	31.2 Exact Solutions of Nonlinear Dynamical Systems Arising in Fluid Dynamics
		31.2.1 Case 1. The Flow of a Fourth Grade Fluid Past a Porous Plate
		31.2.2 Case 2. The Flow of a Second Grade Fluid Over a Stretching Sheet with Suction/Injection
		31.2.3 Case 3. Thin Film of an Oldroyd 6-Constant Fluid Over a Moving Belt
		31.2.4 Case 4. Viscous Flow Due to a Stretching Surface with Partial Slip
		31.2.5 Case 5. Thermal Radiation on MHD Flow Over a Stretching Porous Sheet
		31.2.6 Case 6. Upper-Convected Maxwell Fluid Over a Porous Stretching Plate
		31.2.7 Case 7. Unsteady Viscous Flow Over a Shrinking Cylinder
		31.2.8 Case 8. The Flow of a Viscous Incompressible Fluid Over a Porous Stretching Wall
	31.3 Exact Solutions to Oscillations of Some Nonlinear Dynamical Systems
		31.3.1 Oscillations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia
		31.3.2 Nonlinear Jerk Equations
	31.4 Exact Solutions to Duffing Equation
	31.5 Solutions of the Double-Well Duffing Equation
	References