Academic literature on the topic 'Rotating Beam Problem'
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Journal articles on the topic "Rotating Beam Problem"
1
Stephen, N. G., and P. J. Wang. "Stretching and Bending of Rotating Beam." Journal of Applied Mechanics 53, no. 4 (December 1, 1986): 869–72. http://dx.doi.org/10.1115/1.3171873.
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The effect of uniform high-speed rotation on the simplest representation of a rotating blade is analyzed according to the linear theory of elasticity. The blade is modeled as a uniform prismatic beam of general cross section rotating about a principal section axis perpendicular to the centroidal axis. This quasi-elastostatic three-dimensional problem is reduced to a two-dimensional boundary value problem to which solutions for the amenable circular and elliptic cross sections are given. For sections not possessing two axes of cross-sectional symmetry, the theory predicts curvature of the blade center line.
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Bauchau, O. A., and C. H. Hong. "Nonlinear Composite Beam Theory." Journal of Applied Mechanics 55, no. 1 (March 1, 1988): 156–63. http://dx.doi.org/10.1115/1.3173622.
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The modeling of naturally curved and twisted beams undergoing arbitrarily large displacements and rotations, but small strains, is a common problem in numerous engineering applications. This paper has three goals: (1) present a new formulation of this problem which includes transverse shearing deformations, torsional warping effects, and elastic couplings resulting from the use of composite materials, (2) show that the small strain assumption must be applied in a consistent fashion for composite beams, and (3) present some numerical results based on this new formulation to assess its accuracy, and to point out some distinguishing feature of anisotropic beam behavior. First, the predictions of the formulation will be compared with experimental results for the large deflections and rotations of an aluminum beam. Then, the distinguishing features of composite beams that are likely to impact the design of rotating blades (such as helicopter blades) will be discussed. A first type of extension-twisting coupling introduced by the warping behavior of a pretwisted beam is discussed, then, a shearing strain squared term, usually neglected in small strain analyses, is shown to introduce a coupling between axial extension and twisting behavior, that can be significant when the ratio E/G is large (E and G are Young’s and shearing moduli of the beam, respectively). Finally, the impact of inplane shearing modulus changes and torsional warping constraints on the behavior of beams exhibiting elastic couplings is investigated.
3
Chen, W. R., and L. M. Keer. "Transverse Vibrations of a Rotating Twisted Timoshenko Beam Under Axial Loading." Journal of Vibration and Acoustics 115, no. 3 (July 1, 1993): 285–94. http://dx.doi.org/10.1115/1.2930347.
Full textAbstract:
Transverse bending vibrations of a rotating twisted beam subjected to an axial load and spinning about its axial axis are established by using the Timoshenko beam theory and applying Hamilton’s Principle. The equations of motion of the twisted beam are derived in the twist nonorthogonal coordinate system. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations that have gyroscopic terms. A symmetric general eigenvalue problem is formulated and used to study the influence of the twist angle, rotational speed, and axial force on the natural frequencies of Timoshenko beams. The present model is useful for the parametric studies to understand better the various dynamic aspects of the beam structure affecting its vibration behavior.
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Pozhalostin, A. A., and A. V. Panshina. "Self-oscillations in one-dimensional elastic systems with friction." Izvestiya MGTU MAMI 8, no. 4-4 (August 20, 2014): 71–75. http://dx.doi.org/10.17816/2074-0530-67374.
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The possibility of self-oscillations in the case of transverse vibrations of elastic homogeneous beams is investigated taking into account presence of dry friction in system. The problem is solved using examples of a beam suspended by one end on rotating shaft and a beam articulated at ends. The method of replacing original system by equivalent mechanical oscillation system is used.
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Lee, Sen Yung, and Shueei Muh Lin. "Bending Vibrations of Rotating Nonuniform Timoshenko Beams With an Elastically Restrained Root." Journal of Applied Mechanics 61, no. 4 (December 1, 1994): 949–55. http://dx.doi.org/10.1115/1.2901584.
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Without considering the Coriolis force, the governing differential equations for the pure bending vibrations of a rotating nonuniform Timoshenko beam are derived. The two coupled differential equations are reduced into two complete fourth-order differential equations with variable coefficients in the flexural displacement and in the angle of rotation due to bending, respectively. The explicit relation between the flexural displacement and the angle of rotation due to bending is established. The frequency equations of the beam with a general elastically restrained root are derived and expressed in terms of the four normalized fundamental solutions of the associated governing differential equations. Consequently, if the geometric and material properties of the beam are in polynomial forms, then the exact solution for the problem can be obtained. Finally, the limiting cases are examined. The influence of the coupling effect of the rotating speed and the mass moment of inertia, the setting angle, the rotating speed and taper ratio on the natural frequencies, and the phenomenon of divergence instability (tension buckling) are investigated.
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Bazoune, A. "Effect of Tapering on Natural Frequencies of Rotating Beams." Shock and Vibration 14, no. 3 (2007): 169–79. http://dx.doi.org/10.1155/2007/865109.
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The problem of free vibration of a rotating tapered beam is investigated by developing explicit expressions for the mass, elastic and centrifugal stiffness matrices in terms of the taper ratios. This investigation takes into account the effect of tapering in two planes, the effect of hub radius as well as the stiffening effect of rotation. The equations of motion are derived; the associated generalized eigenvalue problem is defined in conjunction with a suitable Lagrangian form and solved for a wide range of parameter changes. The effect of tapering on the natural frequencies of the beam is examined with all parameter changes present. Results are compared with those available in literature and are found to be in excellent agreement.
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Yang, Dahao, Zhong-Rong Lu, and Li Wang. "Detection of Structural Damage in Rotating Beams Using Modal Sensitivity Analysis and Sparse Regularization." International Journal of Structural Stability and Dynamics 20, no. 08 (July 2020): 2050086. http://dx.doi.org/10.1142/s0219455420500868.
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Rotating beams are often encountered in the wind turbines and the rotors, and detection of the damages in rotating beams as earlier as possible is central to ensuring the safety and serviceability of practical structures. To this end, a modal sensitivity approach in conjunction with the sparse regularization is proposed in this paper. First, the eigen equations for the flap-wise and chord-wise vibrations of a rotating beam are established upon Hamilton’s principle. Then, damage detection is formulated as a nonlinear least-squares problem that finds the damage coefficients to minimize the error between the measured and calculated data. To solve the nonlinear least-squares problem, the sensitivity method that requires the modal sensitivity analysis is developed. In real applications, damage detection is usually an ill-posed problem and to circumvent the ill-posedness, the sparse regularization is introduced due to the fact that the numbers of actual damage locations are often scarce. Numerical examples are studied and results show that the proposed approach is more accurate than the enhanced sensitivity approach and the flap-wise modal data outperforms the chord-wise modal data in damage detection of rotating beams.
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Nourifar, Mostafa, Ali Keyhani, and Ahmad Aftabi Sani. "Free Vibration Analysis of Rotating Euler–Bernoulli Beam with Exponentially Varying Cross-Section by Differential Transform Method." International Journal of Structural Stability and Dynamics 18, no. 02 (February 2018): 1850024. http://dx.doi.org/10.1142/s0219455418500244.
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In this paper, the free vibration analysis of non-uniform rotating Euler–Bernoulli beam is carried out. It is assumed that the beam has exponentially decaying circular cross-section. In order to solve the problem, the differential transform method (DTM) is utilized. Based on our knowledge, we claim that the recurrence relation presented herein is an elaborate recurrence relation which has been obtained for ordinary differential equations. Non-dimensional natural frequencies of the beam are obtained and tabulated for different values of the beam parameters such as taper ratio and rotating speed. Furthermore, the finite element method (FEM) is employed to solve the problem. Comparison of the results obtained by DTM and FEM indicates the accuracy of proposed solutions.
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Zolkiewski, Sławomir. "Diagnostics and Transversal Vibrations Control of Rotating Beam by Means of Campbell Diagrams." Key Engineering Materials 588 (October 2013): 91–100. http://dx.doi.org/10.4028/www.scientific.net/kem.588.91.
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The paper concerns problem of diagnostics and transversal vibrations control of rotating beamlike system. The considered system is consisted of a simple prismatic beam. The beam is homogeneous and is being rotated round its end. The beam is fixed on the rotational disk. The most common method of analyzing rotational systems is the Campbell diagram. It gives the short and precise information about resonance points and critical angular velocities. In literature it is a very popular method, but used for shaft systems or rotors rather than for beams rotating round the axis of revolution perpendicular to its own axis of symmetry. In this work the exemplary Campbell diagrams for considered systems derived from the dynamic flexibility of beams are presented. In the used mathematical model the Coriolis forces and centrifugal forces were taken into consideration. Also the different types of boundary conditions were applied in this work. The results after proper adaptation can be used in practical applications such as pumps, turbines or wind power plants.
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Lu, Tianle, Zhongmin Wang, and Dongdong Liu. "Analysis of Complex Modal Characteristics of Fractional Derivative Viscoelastic Rotating Beams." Shock and Vibration 2019 (October 16, 2019): 1–10. http://dx.doi.org/10.1155/2019/5715694.
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For the transverse vibration problem of a fractional derivative viscoelastic rotating beam, the differential equation of the system is obtained based on the Euler–Bernoulli beam theory and Hamilton principle. Then, introducing dimensionless quantities to differential equations and boundary conditions, the generalized complex eigenvalue equations of the system are obtained by the differential quadrature method. The effects of the slenderness ratio, the viscoelastic ratio, the hub radius-beam length ratio, and dimensionless hub speed and fractional order on the vibration characteristics of fractional derivative viscoelastic rotating beams are discussed by numerical examples. Numerical calculations show that when the dimensionless hub speed is constant, the real part of complex frequency increases with the increase of the fractional order, and the higher-order growth trend is more obvious. Through the study of displacement response at different points on the beam, it can be seen that the closer to the free end, the larger the response amplitude. And, the amplitude of response has been attenuated, which is also consistent with the vibration law of free vibration considering damping.
Dissertations / Theses on the topic "Rotating Beam Problem"
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Sarkar, Korak. "Closed-form Solutions For Rotating And Non-rotating Beams : An Inverse Problem Approach." Thesis, 2012. https://etd.iisc.ac.in/handle/2005/1832.
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Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions.
Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.
Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency.
Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.
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Sarkar, Korak. "Closed-form Solutions For Rotating And Non-rotating Beams : An Inverse Problem Approach." Thesis, 2012. http://etd.iisc.ernet.in/handle/2005/1832.
Full textAbstract:
Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions.
Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.
Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency.
Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.
3
Panchore, Vijay. "Analysis of Rotating Beam Problems using Meshless Methods and Finite Element Methods." Thesis, 2016. http://etd.iisc.ac.in/handle/2005/3209.
Full textAbstract:
A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.
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Panchore, Vijay. "Analysis of Rotating Beam Problems using Meshless Methods and Finite Element Methods." Thesis, 2016. http://hdl.handle.net/2005/3209.
Full textAbstract:
A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.
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Sarkar, Korak. "Inverse Problems in Free Vibration Analysis of Rotating and Non-Rotating Beams and its Application to Random Eigenvalue Characterization." Thesis, 2016. http://etd.iisc.ac.in/handle/2005/3139.
Full textAbstract:
Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using different beam theories. In most cases, the governing differential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions.
The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six different boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three different axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded.
We assume simple polynomial mode shapes which satisfy the different boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiffness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The effect of different parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases.
The derived analytical solutions can also serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the derived stiffness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.
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Sarkar, Korak. "Inverse Problems in Free Vibration Analysis of Rotating and Non-Rotating Beams and its Application to Random Eigenvalue Characterization." Thesis, 2016. http://hdl.handle.net/2005/3139.
Full textAbstract:
Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using different beam theories. In most cases, the governing differential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions.
The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six different boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three different axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded.
We assume simple polynomial mode shapes which satisfy the different boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiffness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The effect of different parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases.
The derived analytical solutions can also serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the derived stiffness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.
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Gagan, Versha. "Two Problems on the Parametric Resonances and Bifurcations in Rotating Beams." Thesis, 2018. http://etd.iisc.ac.in/handle/2005/4199.
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The current research work addresses two interesting systems related to rotating beams. The first dynamical system under consideration is an articulated uniform beam rotating in vacuum under vertical hub excitation along the axis of rotation. The study considers the ensuing parametric instabilities in the considered system. Parametric excitation manifests in the form of time-varying parameters of a dynamical system. Interestingly, more often than not, these time-varying parameters are time periodic in nature and possess inherent frequency/frequencies of oscillation. In an externally forced system, response grows increasingly large when the frequency of the external forcing is close to the system natural frequency/frequencies. In contrast, a parametrically excited system may show large responses under the effect of parametric resonance even when the excitation frequency does not coincide with the system natural frequency/frequencies. In the current study, the effect of parametric excitation on a simple articulated uniform rotating beam in vacuum is considered. The autonomous form (in the absence of base excitation) of the system possesses a non-trivial equilibrium point which is a function of rotation speed and the acceleration due to gravity. In the absence of damping, the considered equilibrium point is neutrally stable and happens to be stabilize in the presence of positive structural damping. However, on application of the hub excitation, the equilibrium point is annihilated, but the system still oscillates in the near neighborhood of the equilibrium point (corresponding to the autonomous system). The ensuing oscillations can be rendered unstable owing to parametric resonance through tuning of the excitation. The linearization of the system close to the equilibrium point leads to the celebrated Mathieu equation with external forcing. The stability surface dividing the stability-instability region of the resulting Mathieu equation is derived by invoking the Floquet theory.
The nonlinear stability boundaries are constructed using Poincaré maps, and are compared with the linear stability boundaries. In order to validate the theoretical study, an experimental model is designed. Although, the primary resonance is captured sufficiently well, the higher and lower order parametric resonances are nuanced due to the presence of damping. However, the experimental results do in fact indicate the existence of these parametric resonances on the parameter plane.
The second part of the study is concerned with the existence and bifurcations of nonlinear normal modes (NNMs) of a flexible beam rotating in vacuum with both transverse and longitudinal deflection. The nonlinear strain-displacement relation is considered while deriving the equations of motion which are coupled through quadratic and cubic nonlinearities. The discretized form of nonlinear coupled equations of motion is derived considering coupling between the first mode of transverse and longitudinal oscillations resulting in nonlinearly coupled oscillators. The effect of slenderness ratio, rotation speed and the system energy is considered on the evolution of the ensuing NNMs and sub-harmonics and the cascade of bifurcations experienced by these modes are explored.
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DENG, XI-YUAN, and 鄧錫遠. "Static and dynamic condensation of matrices applied to the problem of rotating elastic beams." Thesis, 1987. http://ndltd.ncl.edu.tw/handle/74574286932361110509.
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Reddy, Basireddy Sandeep. "A Study of Two Problems in Nonlinear Dynamics Using the Method of Multiple Scales." Thesis, 2015. http://etd.iisc.ac.in/handle/2005/3959.
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This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts.
In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable.
The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.
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Reddy, Basireddy Sandeep. "A Study of Two Problems in Nonlinear Dynamics Using the Method of Multiple Scales." Thesis, 2015. http://etd.iisc.ernet.in/2005/3959.
Full textAbstract:
This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts.
In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable.
The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.
Books on the topic "Rotating Beam Problem"
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Ganguli, Ranjan, and Vijay Panchore. The Rotating Beam Problem in Helicopter Dynamics. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-10-6098-4.
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Ganguli, Ranjan, and Vijay Panchore. Rotating Beam Problem in Helicopter Dynamics. Springer Singapore Pte. Limited, 2018.
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Ganguli, Ranjan, and Vijay Panchore. The Rotating Beam Problem in Helicopter Dynamics. Springer, 2017.
Find full textBook chapters on the topic "Rotating Beam Problem"
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Metral, E., G. Rumolo, and W. Herr. "Impedance and Collective Effects." In Particle Physics Reference Library, 105–81. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-34245-6_4.
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AbstractAs the beam intensity increases, the beam can no longer be considered as a collection of non-interacting single particles: in addition to the “single-particle phenomena”, “collective effects” become significant. At low intensity a beam of charged particles moves around an accelerator under the Lorentz force produced by the “external” electromagnetic fields (from the guiding and focusing magnets, RF cavities, etc.). However, the charged particles also interact with themselves (leading to space charge effects) and with their environment, inducing charges and currents in the surrounding structures, which create electromagnetic fields called wake fields. In the ultra-relativistic limit, causality dictates that there can be no electromagnetic field in front of the beam, which explains the term “wake”. It is often useful to examine the frequency content of the wake field (a time domain quantity) by performing a Fourier transformation on it. This leads to the concept of impedance (a frequency domain quantity), which is a complex function of frequency. The charged particles can also interact with other charged particles present in the accelerator (leading to two-stream effects, and in particular to electron cloud effects in positron/hadron machines) and with the counter-rotating beam in a collider (leading to beam–beam effects). As the beam intensity increases, all these “perturbations” should be properly quantified and the motion of the charged particles will eventually still be governed by the Lorentz force but using the total electromagnetic fields, which are the sum of the external and perturbation fields. Note that in some cases a perturbative treatment is not sufficient and the problem has to be solved self consistently. These perturbations can lead to both incoherent (i.e. of a single particle) and coherent (i.e. of the centre of mass) effects, in the longitudinal and in one or both transverse directions, leading to beam quality degradation or even partial or total beam losses. Fortunately, stabilising mechanisms exist, such as Landau damping, electronic feedback systems and linear coupling between the transverse planes (as in the case of a transverse coherent instability, one plane is usually more critical than the other).
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Neustupa, Jiří. "Stability of a Vibrating and Rotating Beam." In Numerical Treatment of Eigenvalue Problems Vol. 5 / Numerische Behandlung von Eigenwertaufgaben Band 5, 171–75. Basel: Birkhäuser Basel, 1991. http://dx.doi.org/10.1007/978-3-0348-6332-2_13.
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Korobov, V. I., W. Krabs, and G. M. Sklyar. "On the Solvability of Trigonometric Moment Problems Arising in the Problem of Controllability of Rotating Beams." In Optimal Control of Complex Structures, 145–56. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8148-7_12.
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Xiaoqing, Sun. "The Physics of a Bead Rolling on a Rotating Hoop—A Problem in Nonlinear Dynamics." In IRC-SET 2021, 201–16. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-16-9869-9_16.
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Gaute-Alonso, Alvaro, and David Garcia-Sanchez. "Simplified Matrix Calculation for Analysis of Girder-Deck Bridge Systems." In Applied Methods in Bridge Design Optimization - Theory and Practice [Working Title]. IntechOpen, 2022. http://dx.doi.org/10.5772/intechopen.102362.
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In the design of girder-deck bridge systems, it is necessary to determine the cross-sectional distribution of live loads between the different beams that make up the cross section of the deck. This article introduces a novel method that allows calculating the cross-sectional distribution of live loads on beam decks by applying a matrix formulation that reduces the structural problem to 2 degrees of freedom for each beam: the deflection and the rotation of the deck slab at the center of the beam’s span. To demonstrate the proposed method, the procedures are given through three different examples by applying loads to a bridge model. Deflection, bending moment, and shear force of the bridge girders are calculated and discussed through the given examples. The use of the proposed novel method of analysis will result in significant savings in material resources and computing time and contributes in the minimization of total costs, and it contributes in the smart modeling process for girder bridge behavior analysis allowing to feed a bridge digital twin (DT) model based on Inverse Modeling holding the latest updated information provided by distributed sensors. The presented methodology contributes also to speed up real-time decision support system (DSS) demands.
Conference papers on the topic "Rotating Beam Problem"
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Garci´a Vallejo, Daniel, and Juan S. Valverde Garci´a. "Stability and Bifurcation Analysis of a Rotating Beam Substructured Model." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86210.
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One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. The substructuring technique was proven to be a valid strategy to account for this problem. The earliest developments of the absolute nodal coordinate formulation were focused on the use of a substructured schem, in which each element uses a local element frame for the elastic force implementation. Later on, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated in the literature. In a recent study of the rotating beam, it was found the existence of a critical angular velocity, beyond which the system becomes unstable, that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.
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Jaroslaw Wozniak. "On smoothness of end states in the problem of controllability of a rotating beam." In 2006 14th Mediterranean Conference on Control and Automation. IEEE, 2006. http://dx.doi.org/10.1109/med.2006.236129.
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Wozniak, Jaroslaw. "On smoothness of end states in the problem of controllability of a rotating beam." In 2006 14th Mediterranean Conference on Control and Automation. IEEE, 2006. http://dx.doi.org/10.1109/med.2006.328803.
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Xie, Kai, Laith K. Abbas, Dongyang Chen, and Xiaoting Rui. "Free Vibration Characteristic of a Rotating Cantilever Beam With Tip Mass." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-85120.
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In many cases, vibration is a serious problem, undesirable, wasting energy and creating unwanted sound. Transfer Matrix Method for Multibody Systems (MSTMM) is one of the sophisticated methods that can be used efficiently to (1) Model large systems with a large number of subsystems and rigid-flexible structures, and (2) Calculate the vibration characteristics and dynamic responses of the multibody systems. The size of matrices in MSTMM remains small regardless of the number of elements in the model. Having smaller matrix sizes helps to have less computational expense leading to a faster answer. Based on the MSTMM advantages, vibration characteristic of a rotating cantilever beam with an attached tip mass is modeled and simulated in the present paper while the beam undergoes flapwise vibration. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The overall transfer equation in the MSTMM context only involves boundary state vectors, whereas the state vectors at all other connection points do not appear. The state vectors at the boundary include the displacements, rotation angles, bending moments and shear forces. These are partly known and partly unknown. The eigenvalue problem is solved by using Frobenius method of solution in power series. Recursive eigenvalue search algorithm is used to determine the system frequencies. Numerical examples are performed to validate with those published in the literature and produced by Workbench ANSYS.
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Karch, Gerald, and Jörg Wauer. "Rotating, Axially Loaded Timoshenko Shaft: Modeling and Stability." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0206.
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Abstract For studying the stability of a rotating shaft subject to axial load, the derivation of correct stability equations is the essential preliminary problem. Here, the model of a uniform non-circular Timoshenko shaft under a compressive end load of constant magnitude is dealt with. Starting point is the nonlinear boundary value problem for coupled extensional-bending-torsional oscillations where a finite strain beam theory in a floating reference frame following the rigid body rotation is applied. First, the equation set describing the stationary shaft configuration is deduced. Next, the variational equations for small superimposed perturbations are derived. The only interesting stability problem for usual properties of the shaft cross section is constituted by a linear boundary value problem describing the bending vibrations. The corresponding characteristic equation is evaluated finally to find the critical buckling load also for the case of an oval shaft not considered before.
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Lee, Y. C. E., E. H. K. Fung, J. Q. Zou, and H. W. J. Lee. "A Computational Optimal Control Approach to the Design of a Flexible Rotating Beam With Active Constrained Layer Damping." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-62205.
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In this paper, a computational approach is adopted to solve the optimal control and optimal parameter selection problems of a rotating flexible beam fully covered with active constrained layer damping (ACLD) treatment. The beam rotates in a vertical plane under the gravitational effect with variable angular velocity and carries an end mass. Tangent coordinate system and the moving coordinate system are used in the system modeling. Due to the highly nonlinear and coupled characteristics of the system, a relative description method is used to represent the motion of the beam and the motion equations are set up by using relative motion variables. Finite element shape functions of a cantilever beam [1] are used as the displacement shape functions in this study. Lagrangian formulation and Raleigh-Ritz approach [2] are employed to derive the governing equations of motion of the nonlinear time-varying system. The problem is posed as a continuous-time optimal control problem. The control function parameters are the control gains. The two system parameters are the thickness of the constraining layer and the viscoelastic material layer. The software package MISER3.2, which is based on the Control Parametrization and the Control Parametrization Enhancing Transform (CPET) techniques is used to solve the combined problems. The optimal solution takes the end deflection, control voltage and the total weight into account. Results show that substantial improvements are obtained with ACLD as compared to the passive constrained layer damping (PCLD) treatment.
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Rahaeifard, M., S. A. Moeini, M. H. Kahrobaiyan, and M. T. Ahmadian. "Vibration Analysis of a Rotating FGM Cantilever Arm." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-11062.
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Functionally graded materials (FGMs) are inhomogeneous composites which are usually made of a mixture of metals and ceramics. Properties of these kinds of materials vary continuously and smoothly from a ceramic surface to a metallic surface in a specified direction of the structure. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses that are induced when two dissimilar materials with large differences in material properties are bonded. FGMs have attracted much attention as advanced structural materials in recent years. In this paper, free vibration of a rotating FGM cantilever arm is studied. The arm is modeled by an Euler-Bernoulli beam theory in which rotary inertia and shear deformation are neglected. The cross section area of the beam is rectangular with properties varying through the thickness following a simple power law exponent (n). This variation is a function of the volume fraction of the beam material constituents. The beam is composed of a mixture of aluminum and alumina. The deformation of the beam is considered to be in the plane of rotation. The equations of motion are derived using Hamilton’s principle and assumed mode method. Ten lowest polynomial functions are considered as mode shapes of the rotating beam. Natural frequencies of the arm are obtained and compared with the literature and verification is presented. Finally effects of various parameters on the natural frequencies and mode shapes are investigated.
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Atmeh, Ghassan M., and Zeaid Hasan. "Parameter Estimation of a Rotating Beam Under a Kalman Filtering Framework." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-88720.
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The current study presents the problem of state and parameter estimation of flexible structures under a Kalman filtering framework. Inspired from a previous publication by the authors, the work presented here tackles the issue of practically acquiring the natural frequency of a flexible structure for structural health monitoring purposes. The Kalman filter theory is introduced where the linear Kalman filter and the unscnted Kalman filter algorithms are explained. An example of estimating the dynamics of a simply supported beam, modeled using the finite element method, is first discussed for the purpose of establishing the effectiveness of the Kalman filtering approach in dynamic structural systems. A more complicated system consisting of a flexible appendage attached at one end to a rotating hub is then introduced. The system dynamics are modeled using the finite element method, which is incorporated in a computer simulation where the Kalman filter is applied to estimate not only the appendage dynamics, but its parameters as well; specifically its natural frequency. The purpose of the work is to establish a practical method of acquiring the natural frequency for a flexible structure to accommodate a structural health monitoring system. Results show that the Kalman filter is a viable option for estimating the natural frequency of flexible structure.
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Hoskoti, Lokanna, Ajay Misra, and Mahesh Manchakattil Sucheendran. "Vortex Induced Vibration of a Rotating Blade." In ASME 2017 Gas Turbine India Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/gtindia2017-4709.
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The vortex-induced vibration (VIV) of a rotating blade is studied in this paper. Euler-Bernoulli beam equation and the nonlinear oscillator satisfying Van der Pol equation are used to model the rotating blade and vortex shedding, respectively. While the fluctuating lift due to vortex shedding acts on the blade and the blade is coupled with fluid through a linear inertial coupling, resulting in a fluid-structure interaction problem. The coupled equations are discretized by using modes which satisfy the Eigenvalue problem. The work attempts to understand the instabilities associated with the frequency lock-in phenomenon. The method of multiscale is used to obtain the frequency response equation and frequency bifurcation diagrams of the coupled system. They are obtained for the primary (1:1) resonance for different values of the coupling parameter. The stability of the solution is presented by examining the nature of the Eigenvalues of the Jacobian matrix.
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Gaul, Lothar, and Nils Wagner. "A Boundary Tracing Method and Its Application to Inward-Oriented Rotating Cantilever Beam." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84411.
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Every physical system is described by parameters, and one goal of the present contribution is to study the movements of eigenvalues in the complex plane depending on the system inherent parameters. A main focus lies in tracing the boundary curve which separates unstable from marginally stable domains in the parameter space. Hence, there is no need to study the whole parameter space but a certain subset which can be characterized by a zero eigenvalue. The method is illustrated by means of a static stability problem arising in the study of rotating blades.
Reports on the topic "Rotating Beam Problem"
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MOMENT-ROTATION MODEL OF EXTERNAL COVER PLATE JOINTS BETWEEN STEEL BEAMS AND CONCRETE-FILLED SQUARE STEEL TUBULAR COLUMNS WITH INNER I-SHAPED CFRP PROFILE. The Hong Kong Institute of Steel Construction, June 2023. http://dx.doi.org/10.18057/ijasc.2023.19.2.5.
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As a new type of beam-column joint, external cover plate joints can be used in concrete-filled square steel tubular (CFSST) structures. To accurately analyze the mechanical characteristics of this novel joint during structural design, it is necessary to investigate the moment-rotation relationships. Based on the analysis of the force-transferring mechanism, the formulas to decide the initial rotation stiffness and ultimate bending moment are founded by using the component analysis method, while the finite element analysis results are also utilized to verify these formulas. Considering the advantages and disadvantages of the existing typical moment-rotation models, a new representation for calculating the moment-rotation curve of the external cover plate joints is proposed using the ultimate bending moment and initial rotation stiffness as two basic parameters. The research reveals that the moment-rotation model proposed in this paper is able to take all loading stages of this joint into account, which facilitates the analysis of yield and ultimate loads. In addition, this model is smooth and continuous at the piecewise points to avoid numerical problems that may be caused in the calculation. Comparing the moment-rotation curves obtained by the calculation model and finite element simulation, the results show good consistency, demonstrating that the moment-rotation model presented in this paper is applicable to the analysis and design of the external cover plate joints.