Nonlinear Oscillations of Hyperelastic Annular Membranes With Varying Density

2017 ◽  
Author(s):  
Renata M. Soares ◽  
Paulo B. Gonçalves
Keyword(s):  
Hypergeometric Functions ◽  
Nonlinear Oscillations ◽  
Outer Boundary ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Element Formulation ◽  
The Galerkin Method ◽  
Stretching Ratio ◽  
Annular Membrane ◽  
Membrane Density

This research presents the mathematical modeling for the nonlinear oscillations analysis of a pre-stretched hyperelastic annular membrane with varying density under finite deformations. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean and the variation of the membrane density in the radial direction is investigated. The membrane is first subjected to a uniform radial traction along its outer circumference and the stretched membrane is fixed along the outer boundary. Then the equations of motion of the pre-stretched membrane are derived. From the linearized equations of motion, the natural frequencies and mode shapes of the membrane are obtained analytically. The vibration modes are described by hypergeometric functions, which are used to approximate the nonlinear deformation field using the Galerkin method. The results are compared with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the influence of the stretching ratio and varying density on the linear and nonlinear oscillations of the membrane.

Nonlinear Vibrations of Rectangular Mooney-Rivlin Membrane Resting on a Nonlinear Elastic Foundation

2014 ◽  
Author(s):  
Renata M. Soares ◽  
Paulo B. Gonçalves
Keyword(s):  
Elastic Foundation ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Incompressible Material ◽  
Mode Shapes ◽  
Element Formulation ◽  
Nonlinear Elastic Foundation ◽  
The Galerkin Method ◽  
Nonlinear Elastic ◽  
Foundation Parameters

The aim of the present work is to investigate the nonlinear vibration response of a pre-stretched rectangular hyperelastic membrane resting on a nonlinear elastic foundation. The membrane is composed of an isotropic, homogeneous and hyperelastic material, which is modeled as a Mooney-Rivlin incompressible material. The elastic foundation is described by a Winkler type nonlinear model with cubic nonlinearity. First the exact solution of the membrane under a biaxial stretch is obtained. Then the equations of motion of the pre-stretched membrane resting on the nonlinear foundation are derived. From the linearized equations, the natural frequencies and mode shapes of the membrane are obtained analytically. Then the natural modes are used to approximate the nonlinear deformation field using the Galerkin method. The results compare well with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the strong influence of the initial stretching ratio and foundation parameters on the linear and nonlinear oscillations and stability of the membrane.

Free Vibration Analysis of a Carbon Nanotube Reinforced Composite Beam

2014 ◽  
Vol 592-594 ◽  
pp. 2041-2045 ◽  
Author(s):  
B. Naresh ◽  
A. Ananda Babu ◽  
P. Edwin Sudhagar ◽  
A. Anisa Thaslim ◽  
R. Vasudevan
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Element Formulation ◽  
Reinforced Composite ◽  
Vibration Responses

In this study, free vibration responses of a carbon nanotube reinforced composite beam are investigated. The governing differential equations of motion of a carbon nanotube (CNT) reinforced composite beam are presented in finite element formulation. The validity of the developed formulation is demonstrated by comparing the natural frequencies evaluated using present FEM with those of available literature. Various parametric studies are also performed to investigate the effect of aspect ratio and percentage of CNT content and boundary conditions on natural frequencies and mode shapes of a carbon nanotube reinforced composite beam. It is shown that the addition of carbon nanotube in fiber reinforced composite beam increases the stiffness of the structure and consequently increases the natural frequencies and alter the mode shapes.

Nonplanar Dynamics of Fixed-Free Beams With Low Torsional Stiffness

2012 ◽  
Author(s):  
Eulher Chaves Carvalho ◽  
Paulo Batista Gonçalves ◽  
Zenon J. G. N. del Prado
Keyword(s):  
Nonlinear Dynamics ◽  
Internal Resonance ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Torsional Stiffness ◽  
Complex Dynamic ◽  
Runge Kutta Method ◽  
The Galerkin Method

The three-dimensional motions of a clamped-free, inextensible beam subject to lateral harmonic excitation are investigated in this paper. Special attention is given to the nonlinear oscillations of beams with low torsional stiffness and its influence on the bifurcations and instabilities of the structure, a problem not tackled in the previous literature on this subject. For this, the nonlinear integro-differential equations describing the flexural-flexural-torsional couplings of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by numerical integration using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. By varying the beam stiffness parameters, and using several tools of nonlinear dynamics, a complex dynamic behavior of the beam is observed near the region where a 1:1:1 internal resonance occurs. In this region several bifurcations leading to multiple coexisting solutions, including planar and nonplanar motions are obtained. Finally, the paper shows how the tools of nonlinear dynamics can help in the understanding of the global integrity of the model, thus leading to a safe design.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

scholarly journals Parametric vibrations of graphene sheets based on the double mode model and the nonlocal elasticity theory

2021 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
Olga Mazur
Keyword(s):  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Nonlinear Oscillations ◽  
Scale Effects ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Small Scale ◽  
Largest Lyapunov Exponent ◽  
Graphene Sheets ◽  
Parametric Vibrations

AbstractParametric vibrations of the single-layered graphene sheet (SLGS) are studied in the presented work. The equations of motion govern geometrically nonlinear oscillations. The appearance of small effects is analysed due to the application of the nonlocal elasticity theory. The approach is developed for rectangular simply supported small-scale plate and it employs the Bubnov–Galerkin method with a double mode model, which reduces the problem to investigation of the system of the second-order ordinary differential equations (ODEs). The dynamic behaviour of the micro/nanoplate with varying excitation parameter is analysed to determine the chaotic regimes. As well the influence of small-scale effects to change the nature of vibrations is studied. The bifurcation diagrams, phase plots, Poincaré sections and the largest Lyapunov exponent are constructed and analysed. It is established that the use of nonlocal equations in the dynamic analysis of graphene sheets leads to a significant alteration in the character of oscillations, including the appearance of chaotic attractors.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

A Method for Fatigue Analysis of Pipelines on Topsides of FPSO Systems

2005 ◽  
Author(s):  
Pol Spanos ◽  
Alba Sofi ◽  
Juan Wang ◽  
Berry Peng
Keyword(s):  
Fatigue Life ◽  
Random Vibration ◽  
Equations Of Motion ◽  
Plug Flow ◽  
Fatigue Curve ◽  
Sea Waves ◽  
Random Loading ◽  
Euler Beam ◽  
Stress Spectrum ◽  
The Galerkin Method

Pipelines located on the decks of FPSO systems are exposed to damage due to sea waves induced random loading. In this context, a methodology for estimating the fatigue life of conveying-fluid pipelines is presented. The pipeline is subjected to a random support motion which simulates the effect of the FPSO heaving. The equation of motion of the fluid-carrying pipeline is derived by assuming small amplitude displacements, modeling the empty pipeline as a Bernoulli-Euler beam, and adopting the so-called “plug-flow” approximation for the fluid (Pai¨doussis, 1998). Random vibration analysis is carried out by the Galerkin method selecting as basis functions the natural modes of a beam with the same boundary conditions as the pipeline. The discretized equations of motion are used in conjunction with linear random vibration theory to compute the stress spectrum for a generic section of the pipeline. For this purpose, the power spectrum of the acceleration at the deck level is determined by using the Response Amplitude Operator of the FPSO hull. Finally, the computed stress spectrum is used to estimate the pipeline fatigue life employing an appropriate S-N fatigue curve of the material. An illustrative example concerning a pipeline simply-supported at both ends is included in the paper.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

scholarly journals A Simulation Model for Computing the Loads Generated at Landing Site During Helicopter Take-Off or Landing Operation

2019 ◽  
Vol 2019 (2) ◽  
pp. 59-75
Author(s):  
Jarosław Stanisławski
Keyword(s):  
Equations Of Motion ◽  
Landing Site ◽  
Simulation Method ◽  
Rigid Bodies ◽  
Landing Gear ◽  
Rotor Blades ◽  
Wind Gust ◽  
The Galerkin Method ◽  
Lumped Masses ◽  
And Control

Summary The paper presents simulation method and results of calculations determining behavior of helicopter and landing site loads which are generated during phase of the helicopter take-off and landing. For helicopter with whirling rotor standing on ground or touching it, the loads of landing gear depend on the parameters of helicopter movement, occurrence of wind gusts and control of pitch angle of the rotor blades. The considered model of helicopter consists of the fuselage and main transmission treated as rigid bodies connected with elastic elements. The fuselage is supported by landing gear modeled by units of spring and damping elements. The rotor blades are modeled as elastic axes with sets of lumped masses of blade segments distributed along them. The Runge-Kutta method was used to solve the equations of motion of the helicopter model. According to the Galerkin method, it was assumed that the parameters of the elastic blade motion can be treated as a combination of its bending and torsion eigen modes. For calculations, data of a hypothetical light helicopter were applied. Simulation results were presented for the cases of landing helicopter touching ground with different vertical speed and for phase of take-off including influence of rotor speed changes, wind gust and control of blade pitch. The simulation method may help to define the limits of helicopter safe operation on the landing surfaces.

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