Nonlinear Vibrations of Shallow Spherical Shells

1969 ◽  
Vol 36 (3) ◽  
pp. 451-458 ◽  
Author(s):  
P. L. Grossman ◽  
B. Koplik ◽  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equation Of Motion ◽  
Dynamic Equations ◽  
Forced Oscillations ◽  
Finite Deflection ◽  
Spherical Shells ◽  
Nonlinear Dynamic Equations ◽  
Edge Conditions ◽  
Spherical Caps

Based on a system of nonlinear dynamic equations and the associated variational equation of motion derived for elastic spherical shells (deep or shallow), an investigation of the axisymmetric vibrations of spherical caps with various edge conditions is made by carrying out a consistent sequence of approximations with respect to space and time. Numerical results are obtained for both free and forced oscillations involving finite deflection. The effect of curvature is examined, with particular emphasis on the transition from a flat plate to a curved shell. In fact, in such a transition, the nonlinearity of the hardening type gradually reverses into one of softening.

Axisymmetric Vibrations of Homogeneous and Sandwich Spherical Caps

1967 ◽  
Vol 34 (3) ◽  
pp. 667-673 ◽  
Author(s):  
B. Koplik ◽  
Yi-Yuan Yu
Keyword(s):  
Natural Frequencies ◽  
Complete System ◽  
Variational Equation ◽  
Circular Disk ◽  
Equation Of Motion ◽  
Spherical Shells ◽  
Spherical Cap ◽  
Spherical Caps ◽  
Thickness Shear ◽  
Clamped Edges

From the generalized variational equation of motion is derived a complete system of equations for sandwich as well as homogeneous spherical shells including the effect of thickness-shear deformation. Based on these equations the exact solution is obtained for the natural frequencies of axisymmetric vibrations of homogeneous and sandwich spherical caps with clamped edges. Emphasis in our investigation is on the transition from the vibrations of a circular disk to those of a spherical cap, for both homogeneous and sandwich cases, with numerical results showing the effects of thickness-shear and of curvature.

Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

1963 ◽  
Vol 30 (1) ◽  
pp. 79-86 ◽  
Author(s):  
Yi-Yuan Yu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Variational Equation ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Free Vibrations ◽  
Layered Plates ◽  
Variational Approximations ◽  
Elastic Systems ◽  
Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

Recognition of Neurons Impulses with the Use of Nonlinear Dynamic Equations

2001 ◽  
Vol 33 (5-8) ◽  
pp. 10
Author(s):  
Tatyana I. Aksenova ◽  
Igor V. Tetko ◽  
Olga K. Chibirova ◽  
Alexandro Villa
Keyword(s):  
Nonlinear Dynamic ◽  
Dynamic Equations ◽  
Nonlinear Dynamic Equations

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

Stability of Forced Oscillations With Nonlinear Second-Order Terms

1959 ◽  
Vol 26 (4) ◽  
pp. 499-502
Author(s):  
Chi-Neng Shen
Keyword(s):  
Continued Fraction ◽  
Variational Equation ◽  
Second Order ◽  
Iteration Procedure ◽  
Forced Oscillations ◽  
The Stability ◽  
Boundary Of Stability

Abstract A solution is obtained for forced oscillations with nonlinear second-order terms. The stability of this solution is given by its variational equation. The boundary of stability is analyzed by both the perturbation and continued fraction methods. The amplitude of osclllation with damping terms is also determined by the iteration procedure.

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