Dynamic Response of a Timoshenko Beam to a Moving Force

2008 ◽  
Vol 75 (2) ◽  
Author(s):  
Paweł Śniady
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Constant Velocity ◽  
Free Vibrations ◽  
Dynamical Response ◽  
Transverse Displacement ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Moving Force

We consider the dynamical response of a finite, simply supported Timoshenko beam loaded by a force moving with a constant velocity. The classical solution for the transverse displacement and the rotation of the cross section of a Timoshenko beam has a form of a sum of two infinite series, one of which represents the force vibrations (aperiodic vibrations) and the other one free vibrations of the beam. We show that one of the series, which represents aperiodic (force) vibrations of the beam, can be presented in a closed form. The closed form solutions take different forms depending if the velocity of the moving force is smaller or larger than the velocities of certain shear and bar velocities.

Elastic–plastic deformation and residual stresses in helical springs

2019 ◽  
Vol 16 (3) ◽  
pp. 448-475
Author(s):  
Vladimir Kobelev
Keyword(s):  
Residual Stresses ◽  
Closed Form ◽  
Cross Section ◽  
Circular Cross Section ◽  
Content Type ◽  
Closed Form Solutions ◽  
Helical Springs ◽  
First Case ◽  
Setting Process ◽  
The Wire

Purpose The purpose of this paper is to develop the method for the calculation of residual stress and enduring deformation of helical springs. Design/methodology/approach For helical compression or tension springs, a spring wire is twisted. In the first case, the torsion of the straight bar with the circular cross-section is investigated, and, for derivations, the StVenant’s hypothesis is presumed. Analogously, for the torsion helical springs, the wire is in the state of flexure. In the second case, the bending of the straight bar with the rectangular cross-section is studied and the method is based on Bernoulli’s hypothesis. Findings For both cases (compression/tension of torsion helical spring), the closed-form solutions are based on the hyperbolic and on the Ramberg–Osgood material laws. Research limitations/implications The method is based on the deformational formulation of plasticity theory and common kinematic hypotheses. Practical implications The advantage of the discovered closed-form solutions is their applicability for the calculation of spring length or spring twist angle loss and residual stresses on the wire after the pre-setting process without the necessity of complicated finite-element solutions. Social implications The formulas are intended for practical evaluation of necessary parameters for optimal pre-setting processes of compression and torsion helical springs. Originality/value Because of the discovery of closed-form solutions and analytical formulas for the pre-setting process, the numerical analysis is not necessary. The analytical solution facilitates the proper evaluation of the plastic flow in torsion, compression and bending springs and improves the manufacturing of industrial components.

A Conical Beam Finite Element for Rotor Dynamics Analysis

1985 ◽  
Vol 107 (4) ◽  
pp. 421-430 ◽  
Author(s):  
L. M. Greenhill ◽  
W. B. Bickford ◽  
H. D. Nelson
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Cross Section ◽  
Rotor Dynamics ◽  
General Purpose ◽  
Closed Form Expression ◽  
Dynamics Analysis ◽  
Closed Form Solutions ◽  
Analysis Computer ◽  
Rotor Dynamic

The development of finite element formulations for use in rotor dynamics analysis has been the subject of many recent publications. These works have included the effects of rotatory inertia, gyroscopic moments, axial load, internal damping, and shear deformation. However, for most closed-form solutions, the element geometry has been limited to a cylindrical cross-section. This paper extends these previous works by developing a closed-form expression including all of the above effects in a linearly tapered conical cross-section element. Results are also given comparing the formulation to previously published examples, to stepped cylinder representations of conical geometry, and to a general purpose finite element elasticity solution. The elimination of numerical integration in the generation of the element matrices, and the ability of the element to represent both conical and cylindrical geometries, make this formulation particularly suited for use in rotor dynamic analysis computer programs.

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
S. H. Mirtalaie ◽  
M. A. Hajabasi
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Timoshenko Beam ◽  
Variable Coefficient ◽  
Slenderness Ratio ◽  
Beam Theory ◽  
Free Vibrations ◽  
Timoshenko Beam Theory ◽  
Euler Angles ◽  
Shear Deformations

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

Torsion of Composite Elastic Bars of Arbitrary Cross Section

1968 ◽  
Vol 90 (3) ◽  
pp. 435-440 ◽  
Author(s):  
E. M. Sparrow ◽  
H. S. Yu
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Elastic Properties ◽  
Cross Section ◽  
Excellent Agreement ◽  
Solution Method ◽  
Circular Cross Section ◽  
High Accuracy ◽  
Arbitrary Cross Section ◽  
Closed Form Solutions

A method of analysis is presented for determining closed-form solutions for torsion of inhomogeneous prismatic bars of arbitrary cross section, the inhomogeneity stemming from the layering of materials of different elastic properties. It is demonstrated that the solution method is very easy to apply and provides results of high accuracy. As an application, solutions are obtained for the torsion of a bar of circular cross section consisting of two materials separated by a plane interface. The results are compared with those of various limiting cases and excellent agreement is found to exist. Among the limiting cases, an exact solution was derived by Green’s functions for the problem in which the interface between the materials coincides with a diameter of the circular cross section.

Closed form solutions for the dynamic response of Euler–Bernoulli beams with step changes in cross section

2006 ◽  
Vol 295 (1-2) ◽  
pp. 214-225 ◽  
Author(s):  
Michael A. Koplow ◽  
Abhijit Bhattacharyya ◽  
Brian P. Mann
Keyword(s):  
Dynamic Response ◽  
Closed Form ◽  
Cross Section ◽  
Closed Form Solutions

Closed Form Solutions for the Motion of Electrically Excited Micro-Cantilever Beams

2006 ◽  
Author(s):  
Seyed Babak Ghaemi Oskouei ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mode Shapes ◽  
Dynamical Response ◽  
Accurate Analysis ◽  
Nonlinear Part ◽  
Closed Form Solutions ◽  
Micro Cantilever ◽  
The Galerkin Method

The differential equation governing the motion of an electrically excited capacitive microcantilever beam is a nonlinear PDE [1]. Accurate analysis about its motion is of great importance in MEMS' dynamical response. In this paper first the nonlinear 4th order 2 point boundary value problem (ODE) governing the static deflection of the system is solved using three methods. 1. The nonlinear part is linearized and its exact solution is obtained. 2. For low applied DC voltages (not near pull-in) the solutin is found using the direct straight forward perturbation analysis. 3. Numerical computer solutions which are used for the previous solution's verifications. The next parts are devoted to the dynamic solution. The nonlinear time variant 4th order PDE governing the dynamic deflection of an electrically excited microbeam is scrutinized. First using the Galerkin Method the mode shapes and the first three mode temporal equations of the linearized equation are found. Considering no damping, using the perturbations method the temporal equations are solved in three states: far from resonance, near 1:1 resonance and near 1:2 resonance. Finally the damped equation is solved using the aforementioned method. In the literature no closed form solution for this problem is presented.

Exact Solutions for the Free Vibrations and Buckling of Rectangular Plates With Linearly Varying In-Plane Loading

2001 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jae-Hoon Kang
Keyword(s):  
Solution Procedure ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Simply Supported ◽  
Edge Conditions ◽  
Elastically Supported

Abstract An exact solution procedure is formulated for the free vibration and buckling analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies and buckling loads, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and buckling loads, and their mode shapes, are presented for a variety of edge conditions and in-plane loadings, especially pure in-plane moments.

Transverse Vibration of a Two-Span Beam Under Action of a Moving Constant Force

1950 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
R. S. Ayre ◽  
George Ford ◽  
L. S. Jacobsen
Keyword(s):  
Mechanical Model ◽  
Constant Velocity ◽  
Natural Mode ◽  
Theoretical Studies ◽  
Constant Force ◽  
Bending Stress ◽  
Transient Vibration ◽  
Simply Supported ◽  
Moving Force ◽  
Theory And Experiment

Abstract The problem relates to the transient vibration of a symmetrical, continuous, simply supported two-span beam which is traversed by a constant force moving with constant velocity. The beam is of slender proportions, flexure alone being considered. Damping is zero, and there is no mass associated with the moving force. Exact theoretical solutions for bending stress have been derived in general form. They consist of three infinite series, each related to one of three time eras as follows: (a) Where force is crossing first span; (b) is crossing second span; (c) has left the beam. Each term of a series is related to a natural mode of vibration. Quantitative theoretical studies show the variation in individual terms of the series, and also in summations of the first five terms, as the traversing velocity is varied. A mechanical model with electrical recording of stress was employed to obtain a more complete quantitative solution than was feasible analytically. The agreement between theory and experiment was reasonably good. Large magnifications of stress (of the order of 2.5) were found in the neighborhood of resonance with the fundamental mode.

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