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Ranjan Ganguli - One of the best experts on this subject based on the ideXlab platform.
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modal tailoring and closed form solutions for Rotating non uniform euler bernoulli Beams
International Journal of Mechanical Sciences, 2014Co-Authors: Korak Sarkar, Ranjan GanguliAbstract:In this paper, the free vibration of a Rotating Euler-Bernoulli Beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a Rotating Beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the Beam which are typical of helicopter blades. Thus, it is found that an infinite number of such Beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a Rotating cantilever Beam and the first and second elastic modes of a Rotating pinned-free Beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of Rotating Beam numerical methods and may also guide nodal tailoring. (C) 2014 Elsevier Ltd. All rights reserved.
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modal tailoring and closed form solutions for Rotating Beams
AHS International Forum 69, 2013Co-Authors: Korak Sarkar, Ranjan GanguliAbstract:In this paper, the free vibration of a Rotating Euler-Bernoulli Beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a Rotating Beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and quadratic stiffness variations of the Beam which are typical of helicopter blades. Thus, it is found that an infinite number of such Beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the second and third mode of a Rotating cantilever Beam and the first and second elastic mode of a Rotating pinned-free Beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these four cases. The derived results can be used as benchmark solutions for the validation of Rotating Beam numerical methods and may also guide nodal tailoring.
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new rational interpolation functions for finite element analysis of Rotating Beams
International Journal of Mechanical Sciences, 2008Co-Authors: Jagadish Babu Gunda, Ranjan GanguliAbstract:A Rotating Beam finite element in which the interpolating shape functions are obtained by satisfying the governing static homogenous differential equation of Euler–Bernoulli Rotating Beams is developed in this work. The shape functions turn out to be rational functions which also depend on rotation speed and element position along the Beam and account for the centrifugal stiffening effect. These rational functions yield the Hermite cubic when rotation speed becomes zero. The new element is applied for static and dynamic analysis of Rotating Beams. In the static case, a cantilever Beam having a tip load is considered, with a radially varying axial force. It is found that this new element gives a very good approximation of the tip deflection to the analytical series solution value, as compared to the classical finite element given by the Hermite cubic shape functions. In the dynamic analysis, the new element is applied for uniform, and tapered Rotating Beams with cantilever and hinged boundary conditions to determine the natural frequencies, and the results compare very well with the published results given in the literature.
Jiazhen Hong - One of the best experts on this subject based on the ideXlab platform.
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modal characteristics of a Rotating flexible Beam with a concentrated mass based on the absolute nodal coordinate formulation
Nonlinear Dynamics, 2017Co-Authors: Xiaoshun Zhang, Dingguo Zhang, Sijia Chen, Jiazhen HongAbstract:A new dynamic model of a Rotating flexible Beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the Beam, a Dirac’s delta function is used to express the mass per unit length of the Beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a Beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the Rotating Beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a Rotating flexible Beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible Beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the Rotating Beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.
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model study and active control of a Rotating flexible cantilever Beam
International Journal of Mechanical Sciences, 2004Co-Authors: Guoping Cai, Jiazhen Hong, Simon X YangAbstract:Abstract For a dynamic system of a Rotating flexible cantilever Beam, the traditional model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the Beam are uncoupled. This traditional hybrid coordinate model is referred as the zero-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of a flexible cantilever Beam with a high rotational speed. In this paper, a first-order approximation coupling model is presented to analyze the dynamics of Rotating flexible Beam system, which is based on the Hamilton theory and the finite element discretization method. The proposed model for the system considers the second-order coupling quantity of the axial displacement caused by the transverse displacement of the Beam. The dynamic characteristics of the Rotating Beam system when using the zero-order approximation coupling model are compared with those when using the first-order approximation coupling models through numerical simulations. In addition, the applicability of the two dynamic models for control design are studied by using the classical optimal control method. Simulation and comparison studies show that, for the case without control for the system, there exists big difference between the result using the zero-order approximation coupling model and that using the first-order approximation coupling model even for the case of small angular velocity of the system. The larger is the angular velocity, the bigger is the difference. Vibration frequency of the Beam by using the first-order approximation coupling model is higher than that by using the zero-order approximation coupling model. When the angular velocity of the system is close to or is larger than the fundamental frequency of the Beam without rotation motion, the zero-order approximation coupling results in a wrong result, while the first-order approximation coupling model is valid. For the case with control for the system, the applicability of the zero-order approximation coupling model can be much broadened. The critical angular velocity of the system for validity of the zero-order approximation coupling model is much larger than that without control for the system. The first-order approximation coupling model is available not only for the case of small angular velocity but also for the case of large angular velocity of the system, and is applicable to the cases with or without control for the system.
F J Belzunce - One of the best experts on this subject based on the ideXlab platform.
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effect of coverage and double peening treatments on the fatigue life of a quenched and tempered structural steel
Surface & Coatings Technology, 2014Co-Authors: A T Vielma, V Llaneza, F J BelzunceAbstract:Abstract The effects of shot peening treatments comprising different degrees of coverage and also the influence of double treatments applied to a quenched and tempered medium-carbon alloyed steel were analyzed. The latter consisted, in one case, of high intensity peening followed by a lower intensity peening treatment and, in the other, of the removal of the damaged surface layer. Surface roughness, subsurface hardening and the residual stress profiles were determined and compared. Furthermore, the fatigue life corresponding to the different shot peening treatments was assessed on a Rotating Beam machine under alternative stresses of 50% of the tensile strength of the steel. The full coverage peening treatment gave rise to the best fatigue behavior, as under-coverage produces a heterogeneous surface stress field, while overlong shot peening treatments probably lead to surface damage. On the other hand, double surface treatments are at least partially able to mitigate the surface damage produced by a first high intensity peening treatment. Nevertheless, none of the applied double treatments was able to exceed the fatigue life of the optimal single shot peening.
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shot peening intensity optimization to increase the fatigue life of a quenched and tempered structural steel
Procedia Engineering, 2014Co-Authors: A T Vielma, V Llaneza, F J BelzunceAbstract:Abstract A quenched and tempered medium-carbon alloyed steel was subjected to different shot peening treatments of varying intensities, from a low intensity 8A to a high intensity 21A, with 100% coverage. The surface roughness, subsurface hardening and residual stress profiles thus obtained were determined and compared. In addition, the fatigue lifes corresponding to the different shot peening treatments were evaluated on a Rotating Beam machine under alternative stresses of 45 and 50% of the tensile strength of the steel. Although all the shot peening treatments improved the cyclic behavior of the untreated specimens, the best fatigue behavior corresponded to the 10A treatment. High intensity shot peening treatments gives rise to worse fatigue behavior, in spite of an increase in surface hardening and deeper compressive residual stress fields, due to surface damage. This damage was not appreciated under the scanning electron microscope, but was indirectly detected by means of the relaxation of the surface residual stress.
G Pohit - One of the best experts on this subject based on the ideXlab platform.
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free vibration analysis of a Rotating Beam with nonlinear spring and mass system
Journal of Sound and Vibration, 2007Co-Authors: S K Das, P C Ray, G PohitAbstract:Abstract Free, out of plane vibration of a Rotating Beam with nonlinear spring–mass system has been investigated. The nonlinear constraint is connected to the Beam between two points on the Beam through a rigid rod. Formulation of the equation of motion is obtained starting from transverse/axial coupling through axial strain. Solution is obtained by applying method of multiple time scale directly to the nonlinear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in open literature. Subsequent nonlinear study indicates that there is a pronounced effect of spring and its mass. The influence of rigid rod location on frequencies is also investigated on nonlinear frequencies of Rotating Beam.
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large amplitude free vibration analysis of a Rotating Beam with non linear spring and mass system
Journal of Vibration and Control, 2005Co-Authors: S K Das, P C Ray, G PohitAbstract:The free, out-of-plane vibration of a Rotating Beam with a non-linear spring-mass system has been investigated. The non-linear constraint appears in the boundary condition. The solution is obtained by applying the method of multiple time-scales directly to the non-linear partial differential equations and the boundary conditions. The results of the linear frequencies match well with those obtained in the literature. Subsequent non-linear study indicates that there is a pronounced effect of the spring and its mass. The influence of the spring-mass location on frequencies is also investigated for the non-linear frequencies of the Rotating Beam.
Sunil K Sinha - One of the best experts on this subject based on the ideXlab platform.
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combined torsional bending axial dynamics of a twisted Rotating cantilever timoshenko Beam with contact impact loads at the free end
Journal of Applied Mechanics, 2007Co-Authors: Sunil K SinhaAbstract:In this paper, consideration is given to the dynamic response of a Rotating cantilever twisted and inclined airfoil blade subjected to contact loads at the free end. Starting with the basic geometrical relations and energy formulation for a Rotating Timoshenko Beam constrained at the hub in a centrifugal force field, a system of coupled partial differential equations are derived for the combined axial, lateral and twisting motions which includes the transverse shear, rotary inertia, and Coriolis effects, as well. In the mathematical formulation, the torsion of the thin airfoil also considers a very general case of shear center not being coincident with the CG (center of gravity) of the cross section, which allows the equations to be used also for analyzing eccentric tip-rub loading of the blade. Equations are presented in terms of axial load along the longitudinal direction of the Beam which enables us to solve the dynamic pulse buckling due to the tip being loaded in the longitudinal as well as transverse directions of the Beam column. The Rayleigh-Ritz method is used to convert the set of four coupled-partial differential equations into equivalent classical mass, stiffness, damping, and gyroscopic matrices. Natural frequencies are computed for Beams with varying "slenderness ratio" and "aspect ratio" as well as "twist angles." " Dynamical equations account for the full coupling effect of the transverse flexural motion of the Beam with the torsional and axial motions due to pretwist in the airfoil. Some transient dynamic responses of a Rotating Beam repeatedly rubbing against the outer casing is shown for a typical airfoil with and without a pretwist.
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non linear dynamic response of a Rotating radial timoshenko Beam with periodic pulse loading at the free end
International Journal of Non-linear Mechanics, 2005Co-Authors: Sunil K SinhaAbstract:Abstract Consideration is given to the dynamic response of a Timoshenko Beam under repeated pulse loading. Starting with the basic dynamical equations for a Rotating radial cantilever Timoshenko Beam clamped at the hub in a centrifugal force field, a system of equations are derived for coupled axial and lateral motions which includes the transverse shear and rotary inertia effects, as well. The hyperbolic wave equation governing the axial motion is coupled with the flexural wave equation governing the lateral motion of the Beam through the velocity-dependent skew-symmetric Coriolis force terms. In the analytical formulation, Rayleigh–Ritz method with a set of sinusoidal displacement shape functions is used to determine stiffness, mass and gyroscopic matrices of the system. The tip of the Rotating Beam is subjected to a periodic pulse load due to local rubbing against the outer case introducing Coulomb friction in the system. Transient response of the Beam with the tip deforming due to rub is discussed in terms of the frequency shift and non-linear dynamic response of the Rotating Beam. Numerical results are presented for this vibro-impact problem of hard rub with varying coefficients of friction and the contact-load time. The effects of Beam tip rub forces transmitted through the system are considered to analyze the conditions for dynamic stability of a Rotating blade with intermittent rub.
