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.

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.

Nonlinear vibration analysis of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation

2016 ◽  
Vol 231 (11) ◽  
pp. 2147-2158 ◽  
Author(s):  
M Zamanian ◽  
A Karimiyan ◽  
SAA Hosseini ◽  
H Tourajizadeh
Keyword(s):  
Nonlinear Vibration ◽  
Electrostatic Force ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Electrostatic Actuation ◽  
Concentrated Force ◽  
Assumed Mode Method ◽  
Lagrange’S Equation ◽  
Comparison Functions ◽  
Horizontal Part

This article studies the nonlinear vibration of a ┴-shaped mass attached to a clamped–clamped microbeam under electrostatic actuation considering the effect of stretching. The DC and AC electrostatic force is applied to the horizontal part of ┴-shaped mass. The dynamic solution is studied using two methods of modeling. In the first model, the ┴-shaped mass is considered as a rigid body between two flexible microbeams. Then, the discretized equation of motion is derived using Lagrange’s equation combined with assumed mode method. The vibration mode shape of linear system is used as the comparison functions. In the second model, the dynamical effect of ┴-shaped mass is modeled as a concentrated force and moment, and it is introduced in the equation of motion by the Dirac function. Afterwards, the equation of motion is discretized using Galerkin method. In both methods of modeling, the equations of motion are solved using two methods. The first method is approximate analytical perturbation and the other one is Runge–Kutta numerical method. The effect of geometrical dimension of ┴-shaped mass on the nonlinear shift of resonance frequency and dynamic pull-in voltage is studied. The efficiency and accuracy of the presented formulations is verified by comparing the obtained results by two methods of modeling and two methods of solution.

Analysis of Nonlinear Vibration for the Cracked Rotors With Unsymmetrical Viscoelastic Supported Conditions

2005 ◽  
Author(s):  
Changping Chen ◽  
Liming Dai ◽  
Yiming Fu
Keyword(s):  
Nonlinear Vibration ◽  
Nonlinear Vibrations ◽  
Equations Of Motion ◽  
Rotor System ◽  
Governing Equations ◽  
Response Curves ◽  
Disc Position ◽  
On Line ◽  
Cracked Rotor System ◽  
Line Crack

The nonlinear governing equations of motion for the cracked rotor system with unsymmetrical viscoelastic supported condition are derived and the nonlinear vibrations of the system are analyzed. The effects of the cracked depth, the cracked position and the disc position on the response curves of the rotational speed-the nonlinear vibration amplitude and the response curve of the nonlinear amplitude-frequency are discussed in detail. The results can be used for the on-line crack monitor of the rotor system.

Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems

1981 ◽  
Vol 48 (4) ◽  
pp. 959-964 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung
Keyword(s):  
Variational Principle ◽  
Nonlinear Vibrations ◽  
Elliptic Integral ◽  
Nonlinear Problems ◽  
Free Vibrations ◽  
Linear Solution ◽  
Starting Point ◽  
Highly Nonlinear ◽  
Elastic Systems ◽  
Incremental Step

The incremental method has been widely used in various types of nonlinear analysis, however, so far it has received little attention in the analysis of periodic nonlinear vibrations. In this paper, an amplitude incremental variational principle for nonlinear vibrations of elastic systems is derived. Based on this principle various approximate procedures can be adapted to the incremental formulation. The linear solution for the system is used as the starting point of the solution procedure and the amplitude is then increased incrementally. Within each incremental step, only a set of linear equations has to be solved to obtain the data for the next stage. To show the effectiveness of the present method, some typical examples of nonlinear free vibrations of plates and shallow shells are computed. Comparison with analytical results calculated by using elliptic integral confirms that excellent accuracy can be achieved. The technique is applicable to highly nonlinear problems as well as problems with only weak nonlinearity.

scholarly journals Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Haiming Yuan ◽  
Xian-Hui Ge
Keyword(s):  
Boundary Condition ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Boundary Theory ◽  
Hydrodynamic Analysis ◽  
Dimensionless Variables ◽  
Pass Through ◽  
Special Points

Abstract The “pole-skipping” phenomenon reflects that the retarded Green’s function is not unique at a pole-skipping point in momentum space (ω, k). We explore the universality of pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a more straightforward way to derive a pole-skipping point. We use this method in Lifshitz, AdS2 and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $$ \frac{\omega }{2\pi T} $$ ω 2 πT and $$ \frac{\left|k\right|}{2\pi T} $$ k 2 πT pass through pole-skipping points $$ \left(\frac{\omega_n}{2\pi T},\frac{\left|{k}_n\right|}{2\pi T}\right) $$ ω n 2 πT k n 2 πT at small ω and k in the Lifshitz background. We verify that the position of the pole-skipping points does not depend on the standard quantization or alternative quantization of the boundary theory in AdS2× ℝd−1 geometry. In the Rindler geometry, we cannot find the corresponding Green’s function to calculate pole-skipping points because it is difficult to impose the boundary condition. However, we can still obtain “special points” near the horizon where bulk equations of motion have two incoming solutions. These “special points” correspond to the nonuniqueness of the Green’s function in physical meaning from the perspective of holography.

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.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

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.

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