Axisymmetric vibrations of closed spherical shells: equations of motion and bifurcation analysis

2006 ◽  
Vol 13 (1) ◽  
pp. 388-416 ◽  
Author(s):  
A.H. Nayfeh ◽  
H.N. Arafat
Keyword(s):  
Bifurcation Analysis ◽  
Equations Of Motion ◽  
Spherical Shells

Bifurcation Analysis of Ship Steering in Canals

2003 ◽  
Vol 46 (02) ◽  
pp. 92-100
Author(s):  
Fotis A. Papoulias ◽  
Panos E. Kapasakis
Keyword(s):  
Bifurcation Analysis ◽  
Complete System ◽  
Equations Of Motion ◽  
Open Loop ◽  
System Stability ◽  
Control Law ◽  
Time Lags ◽  
Essential Dynamics ◽  
Ship Steering ◽  
Loop Dynamics

The problem of ship steering in canals and confined waters is analyzed with emphasis on stability and bifurcation analysis. The classical maneuvering equations of motion augmented with a model for ship-canal interaction are used to model open-loop dynamics. Coupling of a control law and a guidance scheme with appropriate time lags is employed to model the essential dynamics of a helmsman. The complete system is analyzed using both linear and nonlinear techniques in order to assess its stability under finite disturbances. The results indicate that for certain regions of parameters, limit cycle oscillations may develop that could compromise system stability and safety of operations.

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.

scholarly journals A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mariame Nassit ◽  
Abderrahmane El Harif ◽  
Hassan Berbia ◽  
Mourad Taha Janan
Keyword(s):  
Analytical Method ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Current Method ◽  
Free Edge ◽  
Free Vibrations ◽  
Thin Shells ◽  
Element Calculation ◽  
Spherical Shells ◽  
Clamped Edge

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

Extensional Vibrations of Elastic Orthotropic Spherical Shells

1961 ◽  
Vol 28 (2) ◽  
pp. 229-237 ◽  
Author(s):  
W. H. Hoppmann ◽  
W. E. Baker
Keyword(s):  
Orthotropic Material ◽  
Orthotropic Shell ◽  
Equations Of Motion ◽  
Special Consideration ◽  
Uniform Thickness ◽  
Spherical Shells ◽  
Isotropic Shell ◽  
Principal Directions ◽  
Limiting Case ◽  
Orthotropic Shells

The extensional vibrations (momentless) of spherical shells of elastic orthotropic material have been studied theoretically. Equations of motion have been derived and solved. The principal directions of the elastic compliances are assumed to be along parallels of latitude and along meridians. In addition to the case of orthotropic shells of uniform thickness, the analysis may be applied in the case of shells with stiffeners attached. Special consideration is given to the isotropic shell as a limiting case of the orthotropic shell.

Fractional Derivative Viscoelastic Model for the Analysis of Two Colliding Spherical Shells

2012 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Wave Fronts ◽  
Spherical Shells ◽  
Reflected Waves ◽  
Standard Linear Solid ◽  
Standard Linear ◽  
The Moment ◽  
Contact Domain

The collision of two isotropic spherical shells is investigated for the case when the viscoelastic features of the shells represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives. Thus, the problem concerns the shock interaction of two shells, wherein the generalized fractional-derivative standard linear law instead of the Hertz contact law is employed as a low of interaction. The pans of the shells beyond the contact domain are assumed to be elastic, and their behavior is described by the equations of motion which take rotary inertia and shear deformations into account. The model developed here suggests that after the moment of impact quasi-longitudinal and quasi-transverse shock waves are generated, which then propagate along the spherical shells. Due to the short duration of contact interaction, the reflected waves are not taken into account. The solution behind the wave fronts is constructed with the help of the theory of discontinuities. To determine the desired values behind the wave fronts, one-term ray expansions are used, as well as the equations of motion of the contact domains for the both spherical shells.

Nonlinear Dynamics of Closed Spherical Shells

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Internal Resonances ◽  
Spherical Shells ◽  
Linear Eigenvalue Problem ◽  
Nonlinear Responses ◽  
Nonlinear Equations Of Motion ◽  
Primary Resonance Excitation

We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.

Rigid-Body Approximations to Turbulent Motion in a Liquid-Filled, Precessing, Spherical Cavity

1972 ◽  
Vol 39 (1) ◽  
pp. 18-24 ◽  
Author(s):  
J. P. Vanyo ◽  
P. W. Likins
Keyword(s):  
Rigid Body ◽  
Viscous Liquid ◽  
Spherical Cavity ◽  
State Energy ◽  
Layer Structure ◽  
Equations Of Motion ◽  
Solid Sphere ◽  
Turbulent Motion ◽  
Rotating Fluid ◽  
Spherical Shells

Rigid-body approximations for turbulent motion in a liquid-filled, spinning and precessing, spherical cavity are presented. The first model assumes the turbulent liquid to spin and precess as a rigid solid sphere coupled to the cavity wall by a thin layer of massless viscous liquid. The second model replaces the layer of massless viscous liquid by a series of n concentric rigid spherical shells. The number and thickness of the shells can be varied so that the interior sphere varies from a negligible diameter to nearly the diameter of the cavity. Although these models do not provide solutions of the fluid equations of motion, they yield steady-state energy dissipation rates that compare favorably with existing experimental data associated with turbulent flow in such a cavity. The models also duplicate several other important features of rotating fluid flow theory. In particular, the motions of the concentric shells exhibit characteristics associated with a classic Ekman layer structure.

Numerical Continuation Methods and Bifurcation Analysis Applied to Multibody System Dynamics

2003 ◽  
Author(s):  
J. P. Meijaard
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Multibody System ◽  
Equations Of Motion ◽  
Algebraic Manipulation ◽  
Analysis Procedure ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Double Pendulum ◽  
Variational Equations

It is shown how a standard public-domain program for the numerical continuation of stationary and periodic solutions of dynamical systems, and of their bifurcations, can be used to analyse the behaviour of solutions of the equations of motion for a multibody system. The equations of motion are derived with the aid of a symbolic multibody program. From these, the variational equations and the derivatives with respect to parameters can be easily obtained with the underlying algebraic manipulation routines. The analysis procedure is illustrated in the example of a double pendulum, where some results can be checked against analytically derived results.

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