Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-460
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-461
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

scholarly journals Lateral-torsional buckling of compressed and highly variable cross section beams

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Ida Mascolo ◽  
Mario Pasquino
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Equilibrium Shape ◽  
Variable Cross Section ◽  
Thin Wall ◽  
Variable Cross ◽  
Total Potential ◽  
Energetic Formulation ◽  
Rayleigh Method ◽  
Minimum Total Potential Energy

AbstractIn the critical state of a beam under central compression a flexural-torsional equilibrium shape becomes possible in addition to the fundamental straight equilibrium shape and the Euler bending. Particularly, torsional configuration takes place in all cases where the line of shear centres does not correspond with the line of centres of mass. This condition is obtained here about a z-axis highly variable section beam; with the assumptions that shear centres are aligned and line of centres is bound to not deform. For the purpose, let us evaluate an open thin wall C-cross section with flanges width and web height linearly variables along z-axis in order to have shear centres axis approximately aligned with gravity centres axis. Thus, differential equations that govern the problem are obtained. Because of the section variability, the numerical integration of differential equations that gives the true critical load is complex and lengthy. For this reason, it is given an energetic formulation of the problem by the theorem of minimum total potential energy (Ritz-Rayleigh method). It is expected an experimental validation that proposes the model studied.

scholarly journals SERIES SOLUTION OF LARGE DEFORMATION OF A BEAM WITH ARBITRARY VARIABLE CROSS SECTION UNDER AN AXIAL LOAD

2009 ◽  
Vol 51 (1) ◽  
pp. 10-33 ◽  
Author(s):  
SHIJUN LIAO
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Axial Load ◽  
Nonlinear Differential Equations ◽  
Variable Cross Section ◽  
Variable Coefficients ◽  
Buckling Load ◽  
Moment Of Inertia ◽  
Variable Cross ◽  
Critical Buckling Load

AbstractA general analytic approach is proposed for nonlinear eigenvalue problems governed by nonlinear differential equations with variable coefficients. This approach is based on the homotopy analysis method for strongly nonlinear problems. As an example, a beam with arbitrary variable cross section acted on by a compressive axial load is used to show its validity and effectiveness. This approach provides us with great freedom to transfer the original nonlinear buckling equation with variable coefficients into an infinite number of linear differential equations with constant coefficients that are much easier to solve. More importantly, it provides us with a convenient way to guarantee the convergence of solution series. As an example, the beam displacement and the critical buckling load can be obtained for arbitrary variable cross sections. The influence of nonuniformity of moment of inertia is investigated in detail and the optimal distributions of moment of inertia are studied. It is found that the critical buckling load of a beam with the optimal distribution of moment of inertia can be approximately 21–22% larger than that of a uniform beam with the same average moment of inertia. Mathematically, this approach is rather general and thus can be used to solve many other linear/nonlinear differential equations with variable coefficients.

scholarly journals Closed-form solutions for vibrations of a magneto-electro-elastic beam with variable cross section by means of Green’s functions

2018 ◽  
Vol 30 (1) ◽  
pp. 82-99 ◽  
Author(s):  
Xuan Ling Zhang ◽  
Xiao Chao Chen ◽  
Echuan Yang ◽  
Hai Feng Li ◽  
Jian Bo Liu ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Cross Section ◽  
Electric Potential ◽  
Green's Functions ◽  
Green’S Functions ◽  
Elastic Beam ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Cross Sectional ◽  
Closed Form Solutions

In this article, closed-form solutions are obtained for vibrations of a magneto-electro-elastic beam with variable cross section. Based on Timoshenko beam assumptions, governing equation for the non-uniform beam with exponentially varying width is obtained. Laplace transform approach applied to the governing equation results in the corresponding Green’s functions for the beams with various boundary conditions. The equations, which are solved to obtain Green’s functions, are degenerated for the analyses of the characters of free vibration. For free vibrations of the beams under different mechanical boundary conditions, the effects of the non-uniformly cross-sectional parameters and magneto-electric boundary conditions on the dynamic characters are studied. In addition, the magneto-electric potential modal variables’ distributions through the thickness are presented. In the discussions of forced vibration, two points in the beam are selected to investigate frequency responses in terms of displacement and magneto-electric potential. Moreover, the influences of excitation frequency and cross-sectional parameter on through-the-thickness distributions of electric potentials are investigated.

scholarly journals On the Derivation of Exact Solutions of a Tapered Cantilever Timoshenko Beam

2019 ◽  
Vol 21 (2) ◽  
pp. 89-96 ◽  
Author(s):  
Foek Tjong Wong ◽  
Junius Gunawan ◽  
Kevin Agusta ◽  
Herryanto Herryanto ◽  
Levin Sergio Tanaya
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Bending Moment ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Minimum Potential Energy ◽  
Concentrated Force ◽  
Practical Applications ◽  
Minimum Potential

A tapered beam is a beam that has a linearly varying cross section. This paper presents an analytical derivation of the solutions to bending of a symmetric tapered cantilever Timoshenko beam subjected to a bending moment and a concentrated force at the free end and a uniformly-distributed load along the beam. The governing differential equations of the Timoshenko beam of a variable cross section are firstly derived from the principle of minimum potential energy. The differential equations are then solved to obtain the exact deflections and rotations along the beam. Formulas for computing the beam deflections and rotations at the free end are presented. Examples of application are given for the cases of a relatively slender beam and a deep beam. The present solutions can be useful for practical applications as well as for evaluating the accuracy of a numerical method.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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