Analysis of Nonlinear Forced Vibrations of Shallow Shells With Cut-Outs by Using the R-Function Method

2011 ◽  
Author(s):  
Galyna Pilgun ◽  
Marco Amabili
Keyword(s):  
Equations Of Motion ◽  
Continuation Method ◽  
Forced Vibrations ◽  
Second Step ◽  
Geometrically Nonlinear ◽  
Shallow Shells ◽  
Natural Modes ◽  
Mesh Free ◽  
Nonlinear Equations Of Motion ◽  
Admissible Functions

Geometrically nonlinear forced vibrations of shells based on the domains with cut-outs are investigated. Classical nonlinear shallow-shell theories retaining in-plane inertia is used to calculate the strain energy; the shear deformation is neglected. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allowed for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. Shell displacements are expanded by using Chebyshev orthogonal polynomials. A two-step approach is implemented to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using Lagrange approach on both steps. The convergence of the solution is studied by using different multimodal expansions. The pseudo-arclength continuation method and bifurcation analysis are used to study the nonlinear equations of motion. Numerical responses are obtained in the spectral neighbourhood of the lowest natural frequency. When possible, obtained results are compared to those available in the literature.

Multi-Modal Nonlinear Forced Vibrations of Circular Cylindrical Shells of Arbitrary Shapes

2011 ◽  
Author(s):  
Galyna Pilgun ◽  
Marco Amabili
Keyword(s):  
Strain Energy ◽  
Cylindrical Panel ◽  
Forced Vibrations ◽  
Second Step ◽  
Natural Modes ◽  
Nonlinear Responses ◽  
Mesh Free ◽  
Circular Cylindrical Shells ◽  
Spectral Neighborhood ◽  
Admissible Functions

Large-amplitude nonlinear forced vibrations of a circular cylindrical panel with a complex base, clamped at the edges are investigated. The Sanders-Koiter and the Donnell nonlinear shell theories are used to calculate the strain energy; in-plane inertia is retained. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. The problem is solved in two steps: a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using the Lagrange approach on both steps. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency. The convergence of nonlinear responses is investigated.

A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

2015 ◽  
Vol 07 (05) ◽  
pp. 1550076 ◽  
Author(s):  
Reza Ansari ◽  
Mostafa Faghih Shojaei ◽  
Vahid Mohammadi ◽  
Raheb Gholami ◽  
Mohammad Ali Darabi
Keyword(s):  
Boundary Conditions ◽  
Surface Stress ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Free Vibrations ◽  
Model Parameters ◽  
Geometrically Nonlinear ◽  
Residual Surface Stress ◽  
Various Boundary Conditions ◽  
Shear Deformable

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.

Multiharmonic Forced Response Analysis of a Turbine Blading Coupled by Nonlinear Contact Forces

2009 ◽  
Author(s):  
Christian Siewert ◽  
Lars Panning ◽  
Jo¨rg Wallaschek ◽  
Christoph Richter
Keyword(s):  
Frequency Domain ◽  
Steam Turbine ◽  
Equations Of Motion ◽  
Forced Vibrations ◽  
Forced Response ◽  
Contact Forces ◽  
Dynamic Loads ◽  
Turbine Stage ◽  
Nonlinear Equations Of Motion ◽  
Turbine Blading

The rotor blades of a low pressure (LP) steam turbine stage are subjected to high static and dynamic loads during operation. The static loads are mainly due to the centrifugal force and thermal strains, whereas the dynamic loads are caused by fluctuating gas forces resulting in forced vibrations of the blades. The forced vibrations can lead to high cycle fatigue (HCF) failures causing substantial damage and high maintenance effort. Thus, one of the main tasks in the design of LP steam turbine blading is the vibration amplitude reduction in order to avoid high dynamic stresses that could damage the blading. The vibration amplitudes of the blades in a LP steam turbine stage can be reduced significantly to a reasonable amount if adjacent blades are coupled by shroud contacts that reinforce the blading, see Fig. 1. Furthermore, in the case of blade vibrations, relative displacements between neighboring blades occur in the contacts and friction forces are generated that provide additional damping to the structure due to the energy dissipation caused by micro- and macroslip effects. Therefore, the coupling of the blades increases the overall mechanical damping. A three-dimensional structural dynamics model including an appropriate spatial contact model is necessary to predict the contact forces generated by the shroud contacts and to describe the vibrational behavior of the blading with sufficient accuracy. To compute the nonlinear forced vibrations of the coupled blading, the nonlinear equations of motion are solved in the frequency domain owing to the high computational efficiency of this approach. The transformation of the nonlinear equations of motion into the frequency domain can be carried out by representing the steady-state displacement in terms of its harmonic components. After that transformation, the nonlinear forced response is computed as a function of the excitation frequency in the frequency domain.

scholarly journals Limit Cycle Predictions of a Nonlinear Journal-Bearing System

1990 ◽  
Vol 112 (2) ◽  
pp. 168-171 ◽  
Author(s):  
M. T. M. Crooijmans ◽  
H. J. H. Brouwers ◽  
D. H. van Campen ◽  
A. de Kraker
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Discretization ◽  
Linear Stability Theory ◽  
Algebraic Equations ◽  
System Parameters ◽  
Nonlinear Algebraic Equations ◽  
System Equations ◽  
Nonlinear Equations Of Motion

An analysis is presented of the self-excited vibrations of a journal carried in a cylindrical fluid film bearing. Using linear stability theory, the values of the system parameters at the point of loss of stability are determined. These values agree well with those of previous investigators. Solutions of the nonlinear system equations are obtained by time discretization and by an arc-continuation method for solving the obtained nonlinear algebraic equations. In this way periodic solutions of the nonlinear equations of motion are calculated as a function of the system parameters. The behavior of the journal can be explained by the results of these calculations.

Determination of the initial conditions by solving boundary value problem method for period-one walking of a passive biped walking robots

Robotica ◽  
2015 ◽  
Vol 35 (1) ◽  
pp. 166-188 ◽  
Author(s):  
Masoumeh Safartoobi ◽  
Morteza Dardel ◽  
Mohammad Hassan Ghasemi ◽  
Hamidreza Mohammadi Daniali
Keyword(s):  
Boundary Value Problem ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Basin Of Attraction ◽  
Initial Guess ◽  
Walking Robots ◽  
Nonlinear Equations Of Motion

SUMMARYWith regard to the small basin of attraction of the passive limit cycles, it is important to start from a proper initial condition for stable walking. The present study investigates the passive dynamic behaviors of two-dimensional bipedal walkers of a compass gait model with different foot shapes. In order to find proper initial conditions for stable and unstable period-one gait limit cycles, a method based on solving the nonlinear equations of motion is presented as a boundary value problem (BVP). An initial guess is required to solve the related BVP that is obtained by solving an initial value problem (IVP). For parametric analysis purposes, a continuation method is applied. Simulation results reveal two, period-one gait cycles and the effects of parameters variation for all models.

Ship Maneuverability Analysis Using Steady-State Techniques

1991 ◽  
Vol 28 (03) ◽  
pp. 163-180
Author(s):  
Volf Asinovsky ◽  
Kiang-Ning Huang ◽  
Mark C. Oakes
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Second Step ◽  
Hydrodynamic Characteristics ◽  
Kinematic Parameters ◽  
Step Method ◽  
Analytical Functions ◽  
Ship Maneuverability ◽  
Nonlinear Equations Of Motion ◽  
The Relationship

This paper discusses the application of steady-state techniques to maneuverability, position-keeping and track-keeping analyses in the ship design process. A two-step method of calculating the diagram of steering is described. The first step gives an approximate solution based on a preliminary analysis of the relationship between the drift angle and the angular velocity of ship in the steady turn. The second step solves the nonlinear equations of motion using an iterative process. In the second step of the solution, the hull's hydrodynamic characteristics are introduced into the equations of motion directly in numerical form without a preliminary approximation by analytical functions. Assumptions about the kinematic parameters of motion are not used. This results in increased accuracy of the calculations. The hydrodynamic characteristics of the hull, rudder and appendages, and their interaction are considered separately. Position and track-keeping analysis using an approach similar to that used for the calculation of the diagram of steering is discussed.

Nonlinear Vibrations of Circular Cylindrical Panels: Theory and Experiments

2003 ◽  
Author(s):  
M. Amabili ◽  
M. Pellegrini
Keyword(s):  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Elastic Strain Energy ◽  
Laminated Composite ◽  
Harmonic Excitation ◽  
Cylindrical Panels ◽  
Radial And Axial Displacements ◽  
Nonlinear Equations Of Motion ◽  
Spectral Neighborhood

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical panels subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances are investigated. The Donnell’s nonlinear thin-shell theory is used to calculate the elastic strain energy. The formulations is also valid for orthotropic and symmetric cross-ply laminated composite shells; geometric imperfections are taken into account. Comparison of calculations to numerical results available in the literature is also performed. The nonlinear equations of motion are studied by using a code based on arclength continuation method that allows bifurcation analysis. Vibration response of three thin circular cylindrical panels of different materials (stainless steel, copper and composite) to harmonic excitation in the neighborhood of the first three natural frequencies has been measured for different force levels. The experimental boundary conditions approximate (i) on the curved edges: zero radial, axial and circumferential displacements; all rotations were allowed; (ii) on the straight edges: zero radial and axial displacements; all rotations and circumferential displacements were allowed. The different levels of excitation permitted to reconstruct the relatively strong, softening type nonlinearity of the panels.

Nonlinear Vibrations of Cantilever Circular Cylindrical Shells

2010 ◽  
Author(s):  
M. Amabili ◽  
Ye. Kurylov
Keyword(s):  
Large Amplitude ◽  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Experimental Studies ◽  
Elastic Strain Energy ◽  
Displacement Fields ◽  
Variable Boundary ◽  
Nonlinear Equations Of Motion

Only experimental studies are available on large amplitude vibrations of cantilever shells. In the present paper, large-amplitude nonlinear vibrations of cantilever circular cylindrical shell are investigated. Shells with perfect and imperfect shape are studied. The Sanders-Koiter nonlinear shell theory, which includes shear deformation, is used to calculate the elastic strain energy. Shell’s displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series: harmonic functions for the circumferential variable; Chebyshev polynomials for the longitudinal variable. Boundary conditions are exactly satisfied. The Lagrangian approach is applied to obtain a system of nonlinear ordinary differential equations. The nonlinear equations of motion are studied by using arclength continuation method and bifurcation analysis. Numerical responses in the spectral neighborhood of the lowest natural frequency are obtained.

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