Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package

Horr, A.M.; Schmidt, Lewis C.

doi:10.1016/0045-7949(95)98867-p

“…The theory developed above is applied to a light aircraft, which is representative of a high aspect ratio sailplane. The data used for the wing are, E/=l-5xl0 5 Nm 2 , G/=4-5xl0 4 Nm 2 , m=15 kg/m, 4=0-35 kgm, x a =0-08 m and L(semi-span)=7-5 m. The mass and inertia properties about the X and Y axes of the fuselage, tail-plane, fin and rudder assembly taken about the fuselage centre line and wing elastic axis intersection are respectively 300 kg, 20 kgm 2 and 500 kgm 2 . These values were halved because only one symmetric half of the aircraft was considered when dealing with the symmetric and antisymmetric cases separately.…”

Section: Resultsmentioning

confidence: 99%

Exact modal analysis of an idealised whole aircraft using symbolic computation

Banerjee

¹

2000

Aeronaut. j.

3

0

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Exact expressions for the frequency equations and mode shapes of an idealised whole aircraft are derived using the symbolic computation package REDUCE. The aircraft is idealised as a bending-torsion coupled beam for the wing with the associated mass and inertia of the fuselage, tail-plane, fin and rudder assembly being concentrated at the root. The governing differential equation of motion of the aircraft is then solved in its most general form. The analysis is split into symmetric and anti-symmetric cases to formulate the equations from which the natural frequencies and the corresponding mode shapes of the unrestrained aircraft are derived. The analytical approach used here is of great significance, particularly from an aeroelastic optimization point of view, for which repetitive calculations of natural frequencies and mode shapes are often required. The application of the theory is demonstrated by numerical results. These results have been compared with those obtained from the dynamic stiffness and finite element methods.

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“…The complete spectral solution of any differential equation is obtained by summing all modes for each frequency values [6,19,20,21]: Therefore, the general solution for a differential equation with non constant coefficient can be expressed as [22]:…”

Section: Offshore Steel Jacket Platformmentioning

confidence: 99%

“…This addition of shear deformation leads to the invalidity of the Euler Bernoulli theory that plane sections remain plane. This means that the slope of any section along the length of the beam simply cannot be obtained by differentiation of the transverse displacement v. This then creates two independent motions , v. These corrections are important when studying the modes of vibrations of higher frequencies [22] when a vibrating beam is subdivided into comparatively short length portions. Timoshenko [23] gave the equation of motion for a beam with tubular section as:…”

Section: Offshore Steel Jacket Platformmentioning

confidence: 99%

Design of a Test Rig for Vibration Control of Oil Platforms Using Magneto-Rheological Dampers

Karkoub

¹

,

Lamont

²

,

Chaar

³

2011

Journal of Offshore Mechanics and Arctic Engineering

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Offshore steel structures are widely used around the world, e.g., in Gulf of Mexico, the Middle East, and the North Sea. These structures are costly to build, and any damage to them could be catastrophic financially and environmentally. There are many factors that could possibly affect the health of these steel structures, particularly hydrodynamic forces. These forces cause the structure to vibrate and in the long run could lead to fatigue failure. Therefore, measures have to be taken in order to prevent these structures from failing. Magneto-rheological (MR) dampers are proven as a feasible alternative to reducing structural vibrations. An experimental setup is built in our laboratory to improve the dynamic modeling of steel jacketlike structures and study the effectiveness of MR dampers in decreasing the hydrodynamically induced vibrations. The structure is analyzed theoretically and experimentally, and the results are presented here.

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“…According to the commonly accepted definition, shape factor k is the ratio of the average shear strain on a section to the shear strain at the centroid depend only the shape of the beam cross‐section (Bathe, 1982). However, there is strong evidence that the accepted definition of k leads to unsatisfactory results for the high‐frequency spectrum (Horr and Schmidt, 1995). The problem arises when the distribution of shear strain over a cross‐section depends on the mode of vibration of the beam.…”

Section: Shear Coefficient Factormentioning

confidence: 99%

A simple approximate method for free vibration analysis of framed tube structures

Kamgar

¹

,

Rahgozar

²

2010

Structural Design Tall Build

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A simple approximate method is developed for determining natural frequency of tall buildings. Timoshenko's beam model, which considers the infl uence of shear and fl exural deformation, was used in modelling framed tube structures. In this paper, natural frequency and mode shape of framed tube structures were calculated based on the fl exural and shear rigidities along with the effects of rotational inertia. Dynamic model of Timoshenko's beam can be obtained by writing equilibrium equations of forces acting on an infi nite element. The solution of the dynamic model was obtained by fi rst applying the separation of variables to time and space, followed by the assumption of harmonic motion in time, the steady state eigen system was obtained. Natural frequencies of framed tube structures were calculated by solving the eigenproblem. A numerical example has been presented to demonstrate the ease of application and accuracy of the proposed method. The structure's fundamental frequency was computed using ETABS V9.0.0 (Computer and Structures, Berkeley, California, USA) and compared with the result obtained from proposed method, which shows that the percentage of error was low and acceptable. The proposed method can be adopted as an alternative procedure to evaluate the natural frequency of framed tube in the preliminary stages of structural design. 22. R. KAMGAR AND R. RAHGOZAR that the transverse displacement is a harmonic vibration, a power series solution that represents the mode shape function of tube-in-tube tall buildings was derived. Dym and Williams (2007) used two beam models for estimating fundamental frequencies of tall buildings. The models included the coupled two beam model and Timoshenko's beam model. In their work, it was established that the fundamental frequency of tall buildings vary inversely with the building's height. In comparison of the two beam models, it was concluded that Timoshenko's model is appropriate for describing the behaviour of shear wall buildings, while the coupled two-beam model is better suited for shear wall frame buildings. Kuang and Ng (2004) used matrix method to determine frequencies of coupled vibration in tall building structures. In their work, based on the continuum model and D'Alembert's principle, the governing equation of motion and the corresponding eigenvalue problem were derived, and by applying the Galerkin technique, a general solution method was proposed for coupled vibration analysis of general tall building structures. Free vibration analysis is presented for asymmetric plan frame structures by Kuang and Ng (2009). The analysis includes the frequency and mode shape determinations for the coupled lateral defl ection due to the lateral shear deformation and torsional rotation due to St Venant torsion deformation of the structures. Swaddiwudhipong et al. (2002) studied on effects of axial deformation and axial force on vibration characteristic of tall buildings. In their work, a tall building composed of frames and shear walls coupled together was idealized as a s...

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Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package

Cited by 32 publications

References 1 publication

Exact modal analysis of an idealised whole aircraft using symbolic computation

Exact modal analysis of an idealised whole aircraft using symbolic computation

Design of a Test Rig for Vibration Control of Oil Platforms Using Magneto-Rheological Dampers

A simple approximate method for free vibration analysis of framed tube structures

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