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1.
A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics. 相似文献
2.
Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems 总被引:2,自引:0,他引:2
The literature regarding the free vibration analysis of single-span beams carrying a number of spring-mass systems is plenty, but that of multi-span beams carrying multiple spring-mass systems is fewer. Thus, this paper aims at determining the “exact” solutions for the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple spring-mass systems. Firstly, the coefficient matrices for an intermediate pinned support, an intermediate spring-mass system, left-end support and right-end support of a uniform beam are derived. Next, the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. In this paper, the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made, thus, the last solutions are the exact ones. The effects of attached spring-mass systems on the free vibration characteristics of the 1-4-span beams are studied. 相似文献
3.
Y.A. DUGUSHM. EISENBERGER 《Journal of sound and vibration》2002,254(5):911-926
The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli-Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results. 相似文献
4.
An analytical method for free vibration analysis of functionally graded beams with edge cracks 总被引:1,自引:0,他引:1
In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler–Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures. 相似文献
5.
C.G. Balachandran 《Journal of sound and vibration》1976,48(4):559-560
The natural frequencies and mode shapes of a number of box beams are calculated by using the finite element displacement method. The structures are considered as assemblages of plates, and in general it is necessary to consider both the in-plane and transverse motion of the plates. A method of representing these two types of motion in the analysis of the vibrations of box beams is presented. A number of box beams of varying sectional parameters are analysed as systems of plates and the results compared with the predictions of Euler and Timoshenko beam theories. The comparisons show that for short beams constructed of thin plates, the new method can successfully represent the localized plate deformations, which cannot be described by beam theory. 相似文献
6.
In general, the exact solutions for natural frequencies and mode shapes of non-uniform beams are obtainable only for a few types such as wedge beams. However, the exact solution for the natural frequencies and mode shapes of an immersed wedge beam is not obtained yet. This is because, due to the “added mass” of water, the mass density of the immersed part of the beam is different from its emerged part. The objective of this paper is to present some information for this problem. First, the displacement functions for the immersed part and emerged part of the wedge beam are derived. Next, the force (and moment) equilibrium conditions and the deflection compatibility conditions for the two parts are imposed to establish a set of simultaneous equations with eight integration constants as the unknowns. Equating to zero the coefficient determinant one obtains the frequency equation, and solving the last equation one obtains the natural frequencies of the immersed wedge beam. From the last natural frequencies and the above-mentioned simultaneous equations, one may determine all the eight integration constants and, in turn, the corresponding mode shapes. All the analytical solutions are compared with the numerical ones obtained from the finite element method and good agreement is achieved. The formulation of this paper is available for the fully or partially immersed doubly tapered beams with square, rectangular or circular cross-sections. The taper ratio for width and that for depth may also be equal or unequal. 相似文献
7.
The natural frequencies and mode shapes of beams with constant width and linearly tapered depth (or thickness) carrying any number of point masses at arbitrary positions along the length of the beams were investigated using the Euler-Bernoulli equation. Use of the closed-form (exact) solutions for the natural frequencies and mode shapes of the unconstrained single-tapered beam (without carrying any point masses) and incorporation of the expansion theorem, the equation of motion for the associated constrained beam (carrying any point masses) were derived. Solution of the last equation will yield the desired natural frequencies and mode shapes of the constrained single-tapered beam. The bending vibrations of a single-tapered beam with six kinds of boundary conditions were investigated. Comparison with the existing literature or the traditional finite element method results reveals that the adopted approach has excellent accuracy and simple algorithm. 相似文献
8.
This paper provides an approximate method to determine the stiffness and the fundamental frequency of a cracked beam. The cracked beam is first represented as an un-cracked beam with equivalent reduced sections around the cracks. The effect of the cracks is explained, visualised and quantified using the equivalence concept developed for stepped beams with periodically variable cross-sections. Then an alternative expression of the improved Rayleigh method is provided to calculate the natural frequencies of a beam with a variable stiffness distribution along its length. As the method is insensitive to the assumed mode shapes, it avoids the difficulty in choosing appropriate mode shapes and yields accurate results. This is shown using several examples to compare the results determined using the proposed method and the Finite Element method (FEM). The method greatly simplifies the calculation of cracked beams with complicated configurations, such as a beam with several cracks, a cracked beam with concentrated masses, a beam with cracks close to each other, and a beam with periodically distributed cracks. 相似文献
9.
Qibo Mao 《Journal of sound and vibration》2012,331(11):2532-2542
The Adomian modified decomposition method (AMDM) is employed in this paper to investigate the free vibrations of N elastically connected parallel Euler–Bernoulli beams, which are continuously joined by a Winkler-type elastic layer. The proposed AMDM method can be used to analyze the vibration of beam system consisting of an arbitrary number of beams. By using boundary conditions the natural frequencies and corresponding mode shapes can be easily obtained simultaneously. The numerical results for different boundary conditions, beam numbers and the stiffness of the Winkler-type elastic layer are presented. It is shown that the AMDM offers an accurate and effective method of free vibration analysis of multiple-connected beams with arbitrary boundary conditions. 相似文献
10.
The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated. 相似文献
11.
In this paper, free vibration response of stepped beams made from functionally graded materials (FGMs) is investigated. The beams are supported by various types of elastically end constraints. The differential transformation method (DTM) is employed to solve the governing differential equations of such beams in order to obtain natural frequencies and mode shapes. The power law distribution is used and modified to describe material compositions across the thickness of the beams made of FGMs. Two main types of the stepped FGM beams in which their material compositions can be described by using the modified power law distribution are selected to investigate their vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions, spring constants and material volume fraction are taken into investigation. 相似文献
12.
M. Haterbouch 《Journal of sound and vibration》2003,265(1):123-154
The effects of large vibration amplitudes on the first two axisymmetric mode shapes of clamped thin isotropic circular plates are examined. The theoretical model based on Hamilton's principle and spectral analysis developed previously by Benamar et al. for clamped-clamped beams and fully clamped rectangular plates is adapted to the case of circular plates using a basis of Bessel's functions. The model effectively reduces the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. Numerical results are given for the first and second axisymmetric non-linear mode shapes for a wide range of vibration amplitudes. For each value of the vibration amplitude considered, the corresponding contributions of the basic functions defining the non-linear transverse displacement function and the associated non-linear frequency are given. The non-linear frequencies associated to the fundamental non-linear mode shape predicted by the present model were compared with numerical results from the available published literature and a good agreement was found. The non-linear mode shapes exhibit higher bending stresses near to the clamped edge at large deflections, compared with those predicted by linear theory. In order to obtain explicit analytical solutions for the first two non-linear axisymmetric mode shapes of clamped circular plates, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. for beams and rectangular plates, has been adapted to the case of clamped circular plates, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values of 0.5 and 0.44 for the first and second axisymmetric non-linear mode shapes, respectively. 相似文献
13.
Q S Li 《The Journal of the Acoustical Society of America》2001,110(4):1958-1966
In this paper, shear-type structures such as frame buildings, etc., are treated as nonuniform shear beams (one-dimensional systems) in free-vibration analysis. The expression for describing the distribution of shear stiffness of a shear beam is arbitrary, and the distribution of mass is expressed as a functional relation with the distribution of shear stiffness, and vice versa. Using appropriate functional transformation, the governing differential equations for free vibration of nonuniform shear beams are reduced to Bessel's equations or ordinary differential equations with constant coefficients for several functional relations. Thus, classes of exact solutions for free vibrations of the shear beam with arbitrary distribution of stiffness or mass are obtained. The effect of taper on natural frequencies of nonuniform beams is investigated. Numerical examples show that the calculated natural frequencies and mode shapes of shear-type structures are in good agreement with the field measured data and those determined by the finite-element method and Ritz method. 相似文献
14.
The dynamic transfer matrix method for determining natural frequencies and mode shapes of the bending-torsion coupled vibration of axially loaded thin-walled beams with monosymmetrical cross sections is developed by using a general solution of the governing differential equations of motion based on Bernoulli-Euler beam theory. This method takes into account the effect of warping stiffness and gives allowance to the presence of axial force. The dynamic transfer matrix is derived in detail. Two illustrative examples on the application of the present theory are given for bending-torsion coupled beams with thin-walled open cross sections. The effects of axial load and warping stiffness on coupled bending-torsional frequencies are discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the proposed method. 相似文献
15.
Floor vibrations annoying to humans often occur in lightweight constructions. A number of methods to solve the problem of resonant vibrations are reported in the literature. Tuned mass damper, semi-active tuned vibration absorber and active control system are all examples of existing methods. A new method has been tested in laboratory environment on a prefabricated floor containing a resilient ceiling with a size up to 6.8×4.8 m2. The method takes advantage of small pieces of visco-elastic material connected between the ceiling joists and the primary beams. A finite element model is used to calculate the correct amount of visco-elastic material. The new damper is especially effective in damping mode shapes where the ceiling oscillates out of phase relative to the floor but shows improvements for other mode shapes as well. 相似文献
16.
This paper aims at investigating the interaction of two flexible permanent magnet beams facing each other. The governing equations of motion are obtained based on the Euler–Bernoulli beam model along with Hamilton's principle. Assuming that the beams' tips are far enough, each magnet beam is considered as a series of dipole segments and the external force and moment distributions over each beam due to the magnetic field of the other one is calculated in the deformed configuration. The transverse deflections of the beams are written as series expansions of the mode shapes of an unloaded cantilever beam and the Galerkin method is applied to determine the stability and resonance frequencies. Using the obtained model, the stability regions of the beams for both cases of opposite poles and same poles facing each other are obtained. Also the effect of magnet's strength and flexibility of the beams on the stability boundaries are illustrated. 相似文献
17.
The torsion-flexure coupling effect of generally orthotropic beams is dependent on reinforcing fibre orientation and mode order. At higher ranks of vibration, this coupling effect is principally contributed by the twisting moment induced by bending. The influence of fibre orientation on normal mode shapes is more significant for small values of fibre orientation. The mode shapes can change suddenly with a small increase in fibre orientation. 相似文献
18.
A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included. 相似文献
19.
Y. AYVAZK. ÖZGAN 《Journal of sound and vibration》2002,255(1):111-127
The purpose of this paper is to apply the modified Vlasov model to the free vibration analysis of beams resting on elastic foundations and to analyze the effects of the subsoil depth, the beam length, their ratio and the value of the vertical deformation parameter within the subsoil on the frequency parameters of beams on elastic foundations. This analysis has been carried out by the aid of a computer program based on the finite element method. The first ten frequency parameters are presented in tabular and graphical forms to evaluate the effects of the parameters considered in this study. Then mode shapes corresponding to the first six of the frequency parameters are given in figures. It is concluded that the effect of the subsoil depth on the frequency parameters of beams on an elastic foundation is generally larger than those of the other parameters considered in this study. 相似文献
20.
The theory of differential and integral inequalities is applied to obtain upper and lower bounds to the transfer matrix for beams with varying geometry. Various techniques of generating and refining these bounds are investigated. Numerical results indicate that these bounds can be refined to produce numerical agreement of the upper and the lower bound to a given number of significant digits.Proceeding from bounds on the transfer matrix elements a theory is developed for determining upper and lower bounds on the natural frequencies and mode shapes and on the solution state vector for static loading of such beams. This procedure is then extended to the analysis of multispan beams with varying geometry. Numerical results are presented for various configurations. 相似文献