Comparison of Static and Dynamic Stiffness for Beams Undergoing Flexural and Torsional Loading With an Intermediate Mass

2000 ◽  
Author(s):  
Arnoldo Garcia ◽  
Arnold Lumsdaine ◽  
Ying X. Yao
Keyword(s):  
Closed Form ◽  
Natural Frequency ◽  
Cross Sections ◽  
Dynamic Stiffness ◽  
Closed Form Solution ◽  
Frequency Equation ◽  
Form Solution ◽  
Torsional Loading ◽  
Intermediate Mass ◽  
Form Equation

Abstract Many studies have been performed to analyze the natural frequency of beams undergoing both flexural and torsional loading. For example, Adam (1999) analyzed a beam with open cross-sections under forced vibration. Although the exact natural frequency equation is available in literature (Lumsdaine et al), to the authors’ knowledge, a beam with an intermediate mass and support has not been considered. The models are then compared with an approximate closed form solution for the natural frequency. The closed form equation is developed using energy methods. Results show that the closed form equation is within 2% percent when compared to the transcendental natural frequency equation.

The Axially Loaded Circular Cylinder

1962 ◽  
Vol 29 (2) ◽  
pp. 318-320
Author(s):  
H. D. Conway
Keyword(s):  
Exact Solution ◽  
Fourier Transform ◽  
Circular Cylinder ◽  
Closed Form ◽  
Cross Sections ◽  
Closed Form Solution ◽  
Form Solution ◽  
Concentrated Force ◽  
Transform Method ◽  
Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

Experimental Analysis of the Transition Between Veering and Crossing on a Two-Beam System

2014 ◽  
Author(s):  
Oliviero Giannini ◽  
Aldo Sestieri
Keyword(s):  
Closed Form ◽  
Natural Frequency ◽  
Experimental Analysis ◽  
Scientific Literature ◽  
Closed Form Solution ◽  
Form Solution ◽  
Experimental Setup ◽  
Modal Interaction ◽  
Coupled Modes ◽  
Beam System

When there is a parameter varying in a system, so that one natural frequency approaches another one, the phenomenon of veering is generally found and highly coupled modes are the emerging characteristic of this dynamic behavior. It is far more difficult to find systems that instead of a veering present crossing between modes, and the crossing phenomenon is almost unreported in the scientific literature, unless the case of uncoupled modes is considered. In this paper, the two-modes interaction is presented. In particular, mode veering and mode crossing are introduced and investigated through a simple analytic 2-dof model that allows for closed-form solution. Then, an experimental setup, appropriately designed to study the two-modes veering and crossing is presented and experimental evidences of both phenomena are measured showing the main characteristics of such modal interaction..

scholarly journals Closed form solution in buckling optimization problem of twisted rod

2021 ◽  
Author(s):  
Vladimir Kobelev
Keyword(s):  
Closed Form ◽  
Cross Section ◽  
Cross Sections ◽  
Optimization Problem ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimal Shape ◽  
Actual Problem ◽  
The Cross ◽  
Simply Connected

Abstract The applications of this method for stability problems are illustrated in this manuscript. In the context of twisted rods, the counterpart for Euler’s buckling problem is Greenhill's problem, which studies the forming of a loop in an elastic bar under torsion (Greenhill, 1883). We search the optimal shape of the rod along its axis. A priori form of the cross-section remains unknown. For the solution of the actual problem the stability equations take into account all possible convex, simply connected shapes of the cross-section. Thus, we drop the assumption about the equality of principle moments of inertia for the cross-section. The cross-sections are similar geometric figures related by a homothetic transformation with respect to a homothetic center on the axis of the rod and vary along its axis. The distribution of material along the length of a twisted rod is optimized so that the rod is of the constant volume T and will support the maximal moment without spatial buckling. The cross section that delivers the maximum or the minimum for the critical eigenvalue must be determined among all convex, simply connected domains. We demonstrate at the beginning the validity of static Euler’s approach for simply supported rod (hinged), twisted by the conservative moment. The applied method for integration of the optimization criteria delivers different length and volumes of the optimal twisted rods. Instead of the seeking for the twisted rods of the fixed length and volume, we directly compare the twisted rods with the different lengths and cross-sections using the invariant factors. The solution of optimization problem for twisted rod is stated in closed form in terms of the higher transcendental functions. In the torsion stability problem, the optimal shape of cross-section is the equilateral triangle.

scholarly journals Closed-Form Formula of the Transverse Dynamic Stiffness of a Shallowly Inclined Taut Cable

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Dan-hui Dan ◽  
Zu-he Chen ◽  
Xing-fei Yan
Keyword(s):  
Closed Form ◽  
Reduction Method ◽  
Dynamic Equilibrium ◽  
Dynamic Stiffness ◽  
Closed Form Solution ◽  
Analytic Form ◽  
Form Solution ◽  
Transverse Dynamic ◽  
Frequency Domains ◽  
Closed Form Formula

The segmented vibration-governed equations and their general solutions for cables acted upon by intermediate transverse forces are derived by applying Hamilton’s principle. Including the effects of sagging, flexible stiffness, clamped boundary conditions, and inclination angle of the cable, the element-wise dynamic stiffness for each cable segment, split into segments having unique transverse forces, is derived. By using methods from the global stiffness assembly process of FEM, the global level of the cables’ dynamic equilibrium equation is obtained, and, as a result, the final closed-form formula of transverse dynamic stiffness is derived. Additionally, the corresponding analytic form, without considering sagging effects, is also obtained. Case studies are conducted on the aspects of accuracy, rationality of the distribution on the spatial field, and frequency domains of dynamic stiffness calculations. By comparison with the Guyan-based static FEM reduction method, it is shown that the result obtained from the proposed closed-form solution, which includes sagging effects, is exact and rational, thus creating a powerful tool in transverse vibration analysis.

scholarly journals Location and Shape Reconstruction of 2D Dielectric Objects by Means of a Closed-Form Method: Preliminary Experimental Results

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Gian Luigi Gragnani ◽  
Maurizio Diaz Mendez
Keyword(s):  
Closed Form ◽  
Cross Sections ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Computation Time ◽  
Shape Reconstruction ◽  
Form Solution ◽  
Scattering Data ◽  
Equivalent Sources ◽  
Value Decomposition

An analytical approach to location and shape reconstruction of dielectric scatterers, that was recently proposed, is tested against experimental data. Since the cross-sections of the scatterers do not depend on the z coordinate, a 2D problem can be formulated. A closed-form singular value decomposition of the scattering integral operator is derived and is used to determine the radiating components of the equivalent source density. This is a preliminary step toward a more complete solution, which will take into account the incident field inside the investigation domain in order to provide the dielectric features of the scatterer and also the nonradiating sources. Reconstructions of the equivalent sources, performed on some scattering data belonging to the Fresnel database, show the capabilities of the method and, thanks to the closed-form solution, results are obtained in a very short computation time.

scholarly journals A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

2017 ◽  
Vol 39 (4) ◽  
pp. 315-328
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Crack Depth ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Frequency Equation ◽  
Form Solution ◽  
Mode Shapes ◽  
Timoshenko Beams

A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations
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