Vibration frequency of a curved beam with tip mass

In flexible blade auto cooling fans, the first vibration frequency is of fundamental importance. These fan blades are usually curved and have a tip mass in the form of a strip along one edge. For the first frequence, the blade can be modelled as a curved beam with a tip mass. This paper reports on an investigation of the vibration frequency of a curved beam with a tip mass, in which both theoretical finite element and experimental methods were used. In the finite element methods, both the normal and tangential displacements are approximated by cubic polynomials to ensure that rigid body displacements are closely represented. The effect of the tip mass is incorporated into the mass matrix. The results show that the curvature has a slight effect on the first mode natural frequencies but has great influence on the higher frequencies, and that the coupling effect between the tip mass and the curvature is insignificant.     

Analysis of mode shapes of a rigidly backed cylindrical foam using three-dimensional finite element acoustical analysis

In this study, the three-dimensional finite element frequency domain acoustical analysis is used to determine the modal shapes of cylindrical foam with a rigid backing and subjected to a unit normal incidence impulsive sound pressure loading while placed in the impedance tube. The acoustic results predicted for the foam are validated by data from the two-microphone acoustic measurements, and good agreement between the measured and predicted acoustic results is observed. The mode shapes of the incident face of the foam at a low frequency, resonant and anti-resonant frequencies as well as the frequency that occurring the peak loss modulus are illustrated. It is found that the modal behaviors of the cylindrical foam are dominated by the fluid, although the acoustic properties of the cylindrical foam are also influenced by the circumferential edge constraints and the modal movements of the solid skeleton.     

Improved eigenvalues for combined dynamical systems using a modified finite element discretization scheme

New approaches are presented to discretize an arbitrarily supported linear structure carrying various lumped attachments. Specifically, the exact eigendata, i.e., the exact natural frequencies and mode shapes, of the linear structure without the lumped attachments are first used to modify its finite element mass and stiffness matrix so that the eigensolutions of the discretized system coincide with the exact modes of vibration. This is achieved by identifying a set of minimum changes in the finite element system matrices and enforcing certain constraint conditions. Once the updated matrices for the linear structure are found, the finite element assembling technique is then used to include the lumped attachments by adding their parameters to the appropriate elements in the modified mass and stiffness matrices. Numerical experiments show that for the same number of elements, the proposed scheme returns higher natural frequencies that are substantially more accurate than those given by the finite element model. Alternatively, the proposed discretization scheme allows one to efficiently and accurately determine the higher natural frequencies of a combined system without increasing the number of elements in the finite element model.     

Smearing technique for vibration analysis of simply supported cross-stiffened and doubly curved thin rectangular shells

Plates stiffened with ribs can be modeled as equivalent homogeneous isotropic or orthotropic plates. Modeling such an equivalent smeared plate numerically, say, with the finite element method requires far less computer resources than modeling the complete stiffened plate. This may be important when a number of stiffened plates are combined in a complicated assembly composed of many plate panels. However, whereas the equivalent smeared plate technique is well established and recently improved for flat panels, there is no similar established technique for doubly curved stiffened shells. In this paper the improved smeared plate technique is combined with the equation of motion for a doubly curved thin rectangular shell, and a solution is offered for using the smearing technique for stiffened shell structures. The developed prediction technique is validated by comparing natural frequencies and mode shapes as well as forced responses from simulations based on the smeared theory with results from experiments with a doubly curved cross-stiffened shell. Moreover, natural frequencies of cross-stiffened panels determined by finite element simulations that include the exact cross-sectional geometries of panels with cross-stiffeners are compared with predictions based on the smeared theory for a range of different panel curvatures. Good agreement is found.     

Vibration analysis of circular segment shaped plates

Circular segment shaped plates are analyzed to determine their natural frequencies and mode shapes of vibration. The analysis is based on the finite element approach. The curved sided triangular plate bending element is used for solving the problem. The effect of variation of the size of the plate on the vibrational characteristics is studied and several important conclusions are made.     

DYNAMIC ANALYSIS OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT FLEXURAL RIGIDITY

For the purpose of developing a vibration-based tension force evaluation procedure for bridge cables using measured multimode frequencies, an investigation on accurate finite element modelling of large-diameter sagged cables taking into account flexural rigidity and sag extensibility is carried out in this paper. A three-node curved isoparametric finite element is formulated for dynamic analysis of bridge stay cables by regarding the cable as a combination of an “ideal cable element” and a fictitious curved beam element in the variational sense. With the developed finite element formulation, parametric studies are conducted to evaluate the relationship between the modal properties and cable parameters lying in a wide range covering most of the cables in existing cable-supported bridges, and the effect of cable bending stiffness and sag on the natural frequencies. A case study is eventually provided to compare the measured natural frequencies of main cables of the Tsing Ma Bridge and the computed frequencies with and without considering cable bending stiffness. The results show that ignoring bending stiffness gives rise to unacceptable errors in predicting higher order natural frequencies of the cables, and the proposed finite element formulation provides an accurate baseline model for cable tension identification from measured multimode frequencies.     

Dynamic analysis of practical blades with shear center effect

Shear center effects on the natural frequencies and mode shapes of rotating and non-rotating practical blades are considered. An 18 degrees of freedom thick beam finite element is developed. Bending and shear force displacements and slopes, and torsional displacements are taken as degrees of freedom at both ends of the element. Total blade deflection slopes are considered as composed of bending and shear force deflection slopes in calculations of blade strain and kinetic energy. This element is compared with the existing thin and thick beam finite elements, and theoretical models. Results obtained for the vibration characteristics of rotating and non-rotating non-uniform aerofoil cross-sectioned blades are compared with the available calculated and experimental values. In all cases considered the element exhibits good convergence characteristics and produces accurate results.     

Dynamic behaviour of a cylindrical shell with a cutout

This paper presents the results of analytical and experimental investigations connected with the dynamic behaviour of a cylindrical shell with a rectangular cutout. The finite element method is used to predict the vibration frequencies and mode shapes. The resulting eigenvalue problems are solved by using a simultaneous iteration technique. The analytical study shows the influence of the cutout on the natural frequencies and mode shapes of the shell. The subtended angle of the cutout ranges from 40° to 120°. Experimental verification was performed on a machined mild steel shell having welded end rings bolted on to sturdy supports. A reasonably good agreement is obtained, with the discrepancies of the order of less than 10 %. The cutout is found to have very little influence on the natural frequencies.     

Acoustic characteristics of annular cavities with locally non-uniform media

A theoretical approach is developed to obtain the natural frequencies and the mode shapes of annular cavities that have locally non-uniform media. The equation of motion is derived based on a special form of the wave equation that is capable of representing the variation of material properties with position, and the unit step function is used in the equation to express the local non-uniformity of the media. The Laplace transform is adopted in eigenvalue analysis to calculate the natural frequencies and the normal mode shapes of the annular cavities. The validity of the presented method is verified through finite element analysis and experiments. Parametric studies are performed to find out the relation between the acoustic characteristics of the cavity and the local deviation of the media, and the acoustic characteristics are explained in terms of the mass and stiffness effect of the local deviation in an annular cavity upon the natural vibration characteristics.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

Studies on vibration of some rib-stiffened cantilever plates

An experimental study was carried out to determine the resonant mode shapes and frequencies of some rib-stiffened skew cantilever plates by holographic interferometry. The influences of varying the sweep back angle, the rib stiffness and the aspect ratio, and the effect of varying the boundary conditions at the root chord, on the frequencies and mode shapes were also investigated. Results of the above investigation and also those of a comparative study with the finite element solution obtained for some of the cases studied are presented and discussed.     

A study of the modal characteristics of curved pipes

Finite element procedures are employed to investigate the dynamic behavior of curved pipes, specifically resonance frequencies and modeshapes. Pipe geometries are chosen to reflect industry norms and the study is limited to frequencies below 1600 Hz. Experimental and theoretical procedures are employed to verify finite element results where appropriate. A parametric study is undertaken to observe the effect of changing various geometric parameters on the resonances and modeshapes of a curved pipe. The focus of the parametric study is upon the effects of changing the pipe's toroidal bend angle. As the bend angle is increased, each mode responds differently, some becoming monotonically more stiff, some becoming monotonically less so and others remaining at an approximately constant stiffness. Finite element procedures also readily provide a graphical representation of the change in various modeshapes as the toroidal bend angle is varied. Increasing the wall thickness, reducing the sectional radius or reducing the bend radius of the pipe each yield an overall increase in the stiffness of all modes studied. Further to these results, the finite element procedures suggest that, for some modes, magnitudes of in-plane components are of a similar order to those of the out-of-plane (flexural) components, an observation confirmed by subsequent experimental procedures.     

Vibration of a rotating beam with tip mass

The vibration frequency of a rotating beam with tip mass is investigated. The finite element method is used, a third order polynomial being assumed for the variation of the lateral displacement. The effects of the root radius, the setting angle and the tip mass are incorporated into the finite element model. The results are compared with results from previous authors utilizing Myklestad and extended Galerkin methods. The results show that the setting angle has a significant effect on the first mode frequencies but not on the high frequencies. The tip mass tends to depress the frequencies at low speeds of rotation but it tends to increase the frequencies at high speeds of rotation. The results of this work have applications in wind turbine rotors, helicopter rotors, etc., and the method used here can be extended to investigate the vibration frequency of flexible blade auto cooling fans.     

A computational approach to calculating frequency-dependent structural operators using the spectral finite element method and a wavenumber-based gridding technique

Spectral finite element methods are used to compute exact vibration solutions of structural models at specific frequencies. The applicability of these methods to certain areas of structural dynamics is limited by two major factors: the lack of separate structural operators (mass, damping, and stiffness matrices), and the subsequent difficulty in computing mode shapes via eigenvalue decomposition. In the work presented in this article, a method is investigated to accurately calculate spectral finite elements while overcoming these limitations. The approach incorporates a two-dimensional, discrete solution utilizing a wavenumber-based gridding technique to compute frequency-dependent local mass, damping, and stiffness matrices which can be assembled into the global structural operators. Computed models are able to be used for precise vibration analysis as well as modal analysis via eigenvalue decomposition of the structural operators.     

Free vibrations of spatial Timoshenko arches

This paper addresses the evaluation of the exact natural frequencies and vibration modes of structures obtained by assemblage of plane circular arched Timoshenko beams. The exact dynamic stiffness matrix of the single circular arch, in which both the in-plane and out-of-plane motions are taken into account, is derived in an useful dimensionless form by revisiting the mathematical approach already adopted by Howson and Jemah (1999 [18]), for the in plane and the out-of-plan natural frequencies of curved Timoshenko beams. The knowledge of the exact dynamic stiffness matrix of the single arch makes the direct evaluation of the exact global dynamic stiffness matrix of spatial arch structures possible. Furthermore, it allows the exact evaluation of the frequencies and the corresponding vibration modes, for the distributed parameter model, through the application of the Wittrick and Williams algorithm. Consistently with the dimensionless form proposed in the derivation of the equations of motion and the dynamic stiffness matrix, an original and extensive parametric analysis on the in-plane and out-of-plane dynamic behaviour of the single arch, for a wide range of structural and geometrical dimensionless parameters, has been performed. Moreover, some numerical applications, relative to the evaluation of exact frequencies and the corresponding mode shapes in spatial arched structures, are reported. The exact solution has been numerically validated by comparing the results with those obtained by a refined finite element simulation.     

Natural frequencies of shallow cylindrically curved panels

The natural frequencies of cylindrically curved panels are available in closed form for only two boundary condition sets. This paper demonstrates how Sewall's shallow shell formation can be recast in a relatively simple form to allow direct computation of the natural frequencies and mode shapes of cylindrical panels with a wide range of boundary conditions.     

Vibration analysis of circular cylindrical double-shell structures under general coupling and end boundary conditions

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed.     

Vibrations and buckling of a beam on a variable winkler elastic foundation

Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.     

Multiple modes in the vibration of cantilevered shells

The term multiple modes describes pairs of modes which are similar in shape but occur at different frequencies. This phenomenon has been observed in holographic vibration test results for a turbine blade. Pairs of modes were found, such as two modes which both resembled first torsional modes. In this investigation holographic interferometry was used to verify the earlier results for the turbine blade and to investigate three shell segments simulating blades. The shells ranged in size from moderately to very thick with length to thickness ratios of 16, 8 and 5·6. The blade geometry is characterized by a circumferential angle of 142° and a ratio of length to inner radii arc length near 1·0. In addition, a NASTRAN finite element analysis was performed on these simulated blades. Both mode shapes and frequencies were found to be in good agreement with the results from the experiment. The multiple mode phenomenon was found to be an artifact of the holographic experiment. Pairs of modes were found in the NASTRAN results for the simulated blades in which the out-of-plane displacements (those seen in the hologram) were very similar, but for which the displacements in the plane of the hologram differed significantly. Thus, the two modes which appeared in the experimental results as first torsional modes were seen to include quite different in-plane displacements. The two modes are therefore quite different and do not contradict the normal result, which may be justified from such elementary considerations as a Rayleigh quotient, that similar modes must produce similar frequencies.     

Analytical formulation and evaluation for free vibration of naturally curved and twisted beams

An analytical study for free vibration of naturally curved and twisted beams with uniform cross-sectional shapes is carried out using spatial curved beam theory based on the Washizu's static model. In the governing equations of motion of the beams, all displacement functions and the generalized warping coordinate are defined at the centroid axis and also the effects of rotary inertia, transverse shear deformations and torsion-related warping are included in the proposed model. Explicit analytical expressions are derived for the vibrating mode shapes of a curved, bending-torsional-shearing coupled beam under clamped-clamped boundary condition with the help of symbolic computing package Mathematica, and a process of searching is used to determine the natural frequencies. Comparisons of the present results with the FEM results using beam elements in ANSYS code show good accuracy in computation and validity of the model. Further, the present model is used for cylindrical helical springs with circular cross-section fixed at both ends, and the results indicate that the natural frequencies agree well with the theoretical and experimental results available.