Large Amplitude Free Vibration of a Rotating Nonhomogeneous Beam With Nonlinear Spring and Mass System

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
A. Chakrabarti ◽  
P. C. Ray ◽  
Rasajit Kumar Bera
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Mass Effect ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Governing Equations ◽  
Mass System ◽  
Nonlinear Spring ◽  
Reduced Frequency ◽  
Out Of Plane

This paper investigates the free out of plane vibration of a rotating nonhomogeneous beam with nonlinear spring and mass system. The effect of nonhomogeneity of the beam appears both in the governing equations and in the boundary conditions, but the nonlinear spring-mass effect appears in the boundary conditions only. The solution is obtained by applying the method of multiple time scales directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. The effect of the nonhomogeneity of the stiffer beam (β=0.01) reduces the frequencies of vibration of the beam. A possible physical explanation of this reduced frequency of the nonhomogeneous beam is discussed. A subsequent nonlinear study of the nonhomogeneous beam indicates that the mass of the spring and its location also have a pronounced effect on the vibration of the beam. The effect of the nonhomogeneity of the beam on the relative stability of the nonlinear vibration of the beam with spring-mass system is also studied.

Large Amplitude Free Vibration Analysis of a Rotating Beam with Non-linear Spring and Mass System

2005 ◽  
Vol 11 (12) ◽  
pp. 1511-1533 ◽  
Author(s):  
S. K. Das ◽  
P. C. Ray ◽  
G. Pohit
Keyword(s):  
Free Vibration Analysis ◽  
Linear Constraint ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Rotating Beam ◽  
Mass System ◽  
Linear Partial Differential Equations ◽  
Non Linear ◽  
Out Of Plane ◽  
Linear Spring

The free, out-of-plane vibration of a rotating beam with a non-linear spring-mass system has been investigated. The non-linear constraint appears in the boundary condition. The solution is obtained by applying the method of multiple time-scales directly to the non-linear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in the literature. Subsequent non-linear study indicates that there is a pronounced effect of the spring and its mass. The influence of the spring-mass location on frequencies is also investigated for the non-linear frequencies of the rotating beam.

Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

Modeling Geometric Nonlinearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Hamid Moeenfard ◽  
Shorya Awtar
Keyword(s):  
Differential Equations ◽  
Perturbation Technique ◽  
Dynamic Performance ◽  
Nonlinear Partial Differential Equations ◽  
Computational Time ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Analytical Technique ◽  
Numerical Solution Methods ◽  
Tip Mass

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

On the reattachment of a shock layer produced by an instantaneous energy release

1971 ◽  
Vol 48 (2) ◽  
pp. 241-263 ◽  
Author(s):  
H. K. Cheng ◽  
J. W. Kirsch ◽  
R. S. Lee
Keyword(s):  
Shock Layer ◽  
Strong Shock ◽  
Wave Interaction ◽  
Cylindrical Symmetry ◽  
Step Function ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Point Explosion ◽  
Piston Velocity ◽  
Reduced Frequency

The behaviour of a strong shock wave, which is initiated by a point explosion and driven continuously outward by an inner contact surface (or a piston), is studied as a problem of multiple time scales for an infinite shock strength,$\dot{y}_{sh}/a_{\infty}\rightarrow \infty $, and a high shock-compression ratio, ρs/ρ∞∼ 2γ/(γ − 1) ≡ ε−1[Gt ] 1. The asymptotic analyses are carried out for cases with planar and cylindrical symmetry in which the piston velocity is a step function of time. The solution shows that the transition from an explosion-controlled régime to that of a reattached shock layer is characterized by an oscillation with slowly-varying frequency and amplitude. In the interval of a scaled time 1 [Lt ]t[Lt ] ε−2/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)−1t−½(1−ν)and the amplitude varies ast−¼(3+ν)matching the earlier results of Chenget al.(1961). The approach to the large-time limit, ε1/(1+ν)t→ ∞ is found to involve an oscillation with a much reduced frequency, ¼π(1+ν)ε−½t−1, and with an amplitude decaying more rapidly like ε−⅘t−½(4+3ν); this terminal behaviour agrees with the fundamental mode of a shock/acoustic-wave interaction.

Thermoelastic Damping of the Axisymmetric Vibration of Laminated Trilayered Circular Plate Resonators

2013 ◽  
Vol 313-314 ◽  
pp. 600-603 ◽  
Author(s):  
Yu Xin Sun ◽  
Yan Jiang ◽  
Jia Ling Yang
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Plate Theory ◽  
Axisymmetric Vibration ◽  
Laminated Plate ◽  
Thermoelastic Damping ◽  
Governing Equations ◽  
Out Of Plane ◽  
Laminated Circular Plate ◽  
Thermoelastic Problems

In this paper, thermoelastic damping of the axisymmetric vibration of laminated circular plate resonators will be discussed. Based on the classical laminated plate theory assumptions, the governing equations of coupled thermoelastic problems are established for axisymmetric out-of-plane vibration of trilayered circular plate with fully clamped boundary conditions. The analytical expression for thermoelastic damping is obtained and the accuracy is verified through comparison with FEM results.

A beam–ring circular truss antenna restrained by means of the negative speed feedback procedure

2021 ◽  
pp. 107754632110036
Author(s):  
Ashraf T EL-Sayed Taha ◽  
Hany S Bauomy
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Nonlinear Partial Differential Equations ◽  
Time Frame ◽  
Ring Structure ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Averaged Equations ◽  
Fractional Differential

The present article contemplates the nonlinear powerful exhibitions of affecting dynamic vibration controller over a beam–ring structure for demonstrating the circular truss antenna exposed to mixed excitations. The dynamic controller comprises the included negative speed input added to the framework’s idea. By using the statue, Hamilton, the nonlinear fractional differential administering conditions of movement and the limit conditions have inferred for the shaft ring structure. Through Galerkin’s method, the nonlinear partial differential equations referred to overseeing the movement of the shaft ring structure have diminished to a coupled normal differential equations extending the nonlinearities square terms. Multiple time scales have helped in acquiring (getting) the four-dimensional averaged equations for measuring the primary and 1:2 internal resonances. This article’s controlled assessment is useful for controlling the nonlinear vibrations of the considered framework. Likewise, the controller dispenses with the framework’s oscillations in a brief time frame. The demonstrations of the numerous coefficients and the framework directed at the examined resonance case have been determined. Using MATLAB 7.0 programs has aided in completing the simulation results. At last, the numerical outcomes displayed an admirable concurrence with the methodical ones. A comparison with recently available articles has also indicated good results through using the presented controller.

Stress analysis in transverse loading of soft core sandwich plates with various boundary conditions

2018 ◽  
Vol 22 (8) ◽  
pp. 2692-2734 ◽  
Author(s):  
Isa Ahmadi
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Stress Singularity ◽  
Sandwich Plates ◽  
Local Concentration ◽  
Governing Equations ◽  
Transverse Loading ◽  
Weak Formulation ◽  
Various Boundary Conditions ◽  
Out Of Plane

In this paper, the transverse loading of sandwich plate is formulated to study the three-dimensional stress field in the sandwich plates for various edge conditions. The formulation is based on the weak formulation approach. A complete three-dimensional displacement field is considered and the weak formulation approach is employed to obtain the governing equations of the plate using the three dimensional equilibrium equations of elasticity. An analytical solution is presented for governing equations when two opposite edges of plate are simply supported. A one-step stress recovery scheme is used to compute the out-of-plane stresses in the sandwich plates. A comparison is made with the predictions of exact elasticity solutions in the open literature and very good agreements are achieved. The distribution of stresses is investigated for various boundary conditions and the log-linear procedure is employed to study the order of stress singularity at free and clamped edge of the plate. It is seen that the present approach accurately predicts the distribution of out-of-plane stresses and local concentration of stresses in the vicinity of free and clamped edges of sandwich structures.

scholarly journals CLASSIFICATION OF INTERNAL RESONANCES IN NONLINEAR FRACTIONALY DAMPED UFLYAND-MINDLIN PLATES

2020 ◽  
Vol 16 (3) ◽  
pp. 60-77
Author(s):  
Marina Shitikova ◽  
Elena Osipova
Keyword(s):  
Power Series ◽  
Small Parameter ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Governing Equations ◽  
Internal Resonances ◽  
Shear Deformations

In the present paper, the nonlinear free vibrations of fractionally damped plates are studied, equations of motion of which take the rotary inertia and shear deformations into account and involve fivecoupled nonlinear differentialequations in terms of three mutually orthogonal displacements and two angles of rotation. The procedure resulting in decoupling linear parts of equations has been proposed with further utilization of the generalized method of multiple time scales for solving nonlinear governing equations of motion, in so doing the amplitude functions have been expanded into power series in terms of the small parameter and depend on differenttime scales. The occurrence of the internal or combinational resonances in Uflyand-Mindlinplates has been revealed and classified

scholarly journals A New Exact Solution for the Flow of a Fluid through Porous Media for a Variety of Boundary Conditions

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 125 ◽  
Author(s):  
U. Mahabaleshwar ◽  
P. Vinay Kumar ◽  
K. Nagaraju ◽  
Gabriella Bognár ◽  
S. Nayakar
Keyword(s):  
Boundary Conditions ◽  
Similarity Transformation ◽  
Exact Analytical Solution ◽  
Nonlinear Partial Differential Equations ◽  
Porous Solid ◽  
Brinkman Equation ◽  
Governing Equations ◽  
Viscous Fluid Flow ◽  
Navier’S Slip ◽  
Flow Through

The viscous fluid flow past a semi-infinite porous solid, which is proportionally sheared at one boundary with the possibility of the fluid slipping according to Navier’s slip or second order slip, is considered here. Such an assumption takes into consideration several of the boundary conditions used in the literature, and is a generalization of them. Upon introducing a similarity transformation, the governing equations for the problem under consideration reduces to a system of nonlinear partial differential equations. Interestingly, we were able to obtain an exact analytical solution for the velocity, though the equation is nonlinear. The flow through the porous solid is assumed to obey the Brinkman equation, and is considered relevant to several applications.

Plane-Stress Deformation in Strain Gradient Plasticity

1999 ◽  
Vol 67 (1) ◽  
pp. 105-111 ◽  
Author(s):  
J. Y. Chen ◽  
Y. Huang ◽  
K. C. Hwang ◽  
Z. C. Xia
Keyword(s):  
Boundary Conditions ◽  
Plane Stress ◽  
Strain Gradient ◽  
Three Dimensional ◽  
Strain Gradient Plasticity ◽  
Plane Boundary ◽  
Gradient Plasticity ◽  
Governing Equations ◽  
Stress Deformation ◽  
Out Of Plane

A systematic approach is proposed to derive the governing equations and boundary conditions for strain gradient plasticity in plane-stress deformation. The displacements, strains, stresses, strain gradients and higher-order stresses in three-dimensional strain gradient plasticity are expanded into a power series of the thickness h in the out-of-plane direction. The governing equations and boundary conditions for plane stress are obtained by taking the limit h→0. It is shown that, unlike in classical plasticity theories, the in-plane boundary conditions and even the order of governing equations for plane stress are quite different from those for plane strain. The kinematic relations, constitutive laws, equilibrium equation, and boundary conditions for plane-stress strain gradient plasticity are summarized in the paper. [S0021-8936(00)02301-1]

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