Free Vibration of a Simply Supported Bar With a Linearly Variable Height of Cross Section

1962 ◽  
Vol 29 (3) ◽  
pp. 497-501 ◽  
Author(s):  
E. Krynicki ◽  
Z. Mazurkiewicz
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Exact Solutions ◽  
Cross Section ◽  
Approximate Solutions ◽  
Variable Cross ◽  
Simply Supported ◽  
Special Cases ◽  
Formal Integration ◽  
Simple Integration

The problem of vibration of nonhomogeneous bars or bars of variable cross section1 leads to differential equations, which are generally unsolvable by formal integration. It is known that functional coefficients occur in these equations, which make it difficult, if not impossible, to obtain exact solutions by simple integration. Several exact solutions obtained for a few special cases and also some interesting approximate solutions are mentioned in the paper.

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.

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.

Free Vibration of Thin-Walled Open Section Beams With Unconstrained Damping Treatment

1981 ◽  
Vol 48 (1) ◽  
pp. 169-173 ◽  
Author(s):  
S. Narayanan ◽  
J. P. Verma ◽  
A. K. Mallik
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Vibration Characteristics ◽  
Numerical Results ◽  
Simply Supported ◽  
Clamped Beams ◽  
Thin Walled ◽  
Open Cross Section

Free-vibration characteristics of a thin-walled, open cross-section beam, with unconstrained damping layers at the flanges, are investigated. Both uncoupled transverse vibration and the coupled bending-torsion oscillations, of a beam of a top-hat section, are considered. Numerical results are presented for natural frequencies and modal loss factors of simply supported and clamped-clamped beams.

Imprecise Solutions of Ordinary Differential Equations for Boundary Value Problems Using Metaheuristic Algorithms

2016 ◽  
pp. 401-421
Author(s):  
Ali Sadollah ◽  
Joong Hoon Kim
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Water Cycle ◽  
Error Function ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Metaheuristic Algorithms ◽  
General Strategy ◽  
Optimization Task

In this chapter, a general strategy is recommended to solve variety of linear and nonlinear ordinary differential equations (ODEs) with boundary value conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic algorithms, ODEs can be represented as an optimization problem. The purpose is to reduce the weighted residual error (error function) of the ODEs. Boundary values of ODEs are considered as constraints for the optimization model. Inverted generational distance metric is utilized for evaluation and assessment of approximate solutions versus exact solutions. Four ODEs having different orders and features are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization and the water cycle algorithm. The optimization results obtained show that the proposed method equipped with metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs.

Analytical Treatment of In-Plane Parametrically Excited Undamped Vibrations of Simply Supported Parabolic Arches

2003 ◽  
Vol 125 (1) ◽  
pp. 73-79 ◽  
Author(s):  
Dimitris S. Sophianopoulos ◽  
George T. Michaltsos
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Dynamic Stability ◽  
Parametric Excitation ◽  
Stability Problem ◽  
Axial Loading ◽  
Time Dependent ◽  
Analytical Treatment ◽  
Simply Supported ◽  
Forced Motion

The present work offers a simple and efficient analytical treatment of the in-plane undamped vibrations of simply supported parabolic arches under parametric excitation. After thoroughly dealing with the free vibration characteristics of the structure dealt with, the differential equations of the forced motion caused by a time dependent axial loading of the form P=P0+Pt cos θt are reduced to a set of Mathieu-Hill type equations. These may be thereafter tackled and the dynamic stability problem comprehensively discussed. An illustrative example based on Bolotin’s approach produces results validating the proposed method.

Solution of Initial and Boundary Value Problems by an Effective Accurate Method

2017 ◽  
Vol 14 (06) ◽  
pp. 1750069 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Taylor Series Expansion ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Linear Differential ◽  
Appl Math

The newly proposed analytic approximate solution method in the recent publications [Turkyilmazoglu, M. [2013] “Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane-Emden-Fowler type,” Appl. Math. Mod. 37, 7539–7548; Turkyilmazoglu, M. [2014] “An effective approach for numerical solutions of high-order Fredholm integro-differential equations,” Appl. Math. Comput. 227, 384–398; Turkyilmazoglu, M. [2015] “Parabolic partial differential equations with nonlocal initial and boundary values,” Int. J. Comput. Methods, doi: 10.1142/S0219876215500243] is extended in this paper to solve initial and boundary value problems governed by any order linear differential equations whose exact solutions are hard to obtain. Exact solutions are found from the method when the solutions are themselves polynomials. Better accuracies are achieved within the method by increasing the number of polynomials. Comparisons with some available methods show the ability of the proposed technique, even performing much better than the traditional Taylor series expansion.

Two-dimensional quasi-stationary flows in gas dynamics

1968 ◽  
Vol 64 (4) ◽  
pp. 1099-1108 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Differential Equations ◽  
Unsteady Flow ◽  
Exact Solutions ◽  
Gas Dynamics ◽  
Two Dimensional ◽  
Independent Variables ◽  
Stationary Flows ◽  
First Case ◽  
Special Cases ◽  
Flow Round

In this paper we are concerned with the two-dimensional, unsteady flow of an inviscid, polytropic gas whose adiabatic index γ lies between 1 and 3. We recall that comparatively early in the study of gas dynamics we encounter two exact solutions of gas dynamic problems. One, in one-dimensional unsteady flow, is the expansion of a semi-infinite column of gas which is initially at rest behind a piston which, at time t = 0, begins to move with constant speed away from the gas. The second, in two-dimensional, steady, supersonic flow, is the Prandtl–Meyer flow round a sharp convex corner. Both of those flows may be regarded as special cases of more general exact solutions which are obtained by the method of characteristics (see, for example, Courant and Friedrichs(1)). On the other hand, each may be obtained directly from the appropriate equations by making use of the fact that, in so far as neither problem contains any characteristic length parameter in its formulation, the principle of dynamic similarity can be used to reduce the system of partial differential equations to one of ordinary differential equations. In the first case the independent variables x and t occur only in the combination x/t and in the second the independent variables x and y occur only in the combination x/y. Interesting and instructive as the derivation of these solutions from such principles may be, it is an unfortunate fact that they are the only non-trivial solutions of the respective equations. This is not altogether surprising as the equations are ordinary with (in this case) a limited number of non-trivially distinct solutions.

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