Approximate Frequency Equation Research Articles

Vibration and stability of a spinning functionally graded cylinder in a liquid-filled concentric drum

The vibration and stability of an axially functionally graded (AFG) cylinder with whirl motion in the annular liquid environment are investigated. The model of the performed system is given by the spinning Rayleigh beam assumptions with the rotary inertia and the gyroscopic effects. The fluid forces exerted on the cylinder, as a result of the external fluid, are calculated analytically. The coupled governing equation of motion for the system is developed via Hamilton's principle. The exact and approximate whirl frequency equations are presented for vibration and stability analysis of the AFG cylinder. The validity of the proposed model is confirmed by comparing it with the numerical solutions available in the literature. Detailed parameter discussions are conducted to evaluate the effects of the density ratio, outer-to-inner radius ratio, hollowness ratio, and slenderness ratio on the whirl characteristics and stability of the system. The results show that the whirl characteristics and instability of the AFG cylinder are strongly dependent on the external fluid.

Dynamic Behavior of a Spinning Exponentially Functionally Graded Shaft With Unbalanced Load

Abstract The dynamic behaviors of a pinned–pinned spinning exponentially functionally graded shaft with unbalanced loads are investigated. The shaft is simulated in the Rayleigh beam model considering rotary inertia and gyroscopic effects. The governing equation for the flexural vibration of the shaft is derived via the Hamilton principle. Based on the boundary conditions, both the exact and approximate whirl frequency equations of the system are obtained analytically. Also, the validity of the proposed model is confirmed by comparing it with the results reported in the literature. Finally, a numerical study on the basis of the analytical solutions is performed to evaluate the main parameters, including slenderness ratio (α), gradient index (β), mass ratio (μ), and eccentric distance (γ) on the whirl frequency, critical spinning speed, mode shapes, and stability of the system. The results reveal that the vibration and instability of the spinning shaft are strongly dependent on the unbalanced load and material gradient.

Forced Vibration of Axially Accelerating three Parameter Beam Constituted by Fraction Alderivative Model

The forced vibration characteristics of axially variable speed viscoelastic beams are studied. The material is described by the three parameter model of Poynting-Thompson beam. According to Newton's second law, the fractional derivative is introduced to deduce the governing equation of the beam. The approximate analytical solution and amplitude frequency equation are obtained by multi-scale method. The amplitude frequency equation is divided into real part and imaginary part by separating variables. According to the Routh-Hurwitz criterion, the Jacobi matrix and characteristic equation are established to determine the stable region and unstable region obtained under different parameters. Numerical examples show that the instability region shows a downward trend with the increase of the order of fractional derivative.

Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects

A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia, in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro- and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short, explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.

Simultaneous measurement of elastic constants of full-size engineered wood-based panels by modal testing

Abstract Engineered wood-based panels are widely used in structural applications. Accurate measurement of their elastic properties is of great importance for predicting their mechanical behavior during structural design. In this study, an efficient non-destructive test method for measurement of effective elastic constants of orthotropic wood-based panels is proposed based on a modal testing technique. An algorithm was developed based on an improved approximate frequency equation of transverse vibration of orthotropic plates under the boundary condition, in which two opposite sides are simply supported and the other two are free (SFSF). The method is able to predict the frequency ranges and mode indices as well as corresponding normalized sensitivity to elastic constants based on initial estimates of orthotropic ratios with uncertainties and measured fundamental natural frequency. Full-size engineered wood-based panels including cross laminated timber (CLT), oriented strand board (OSB), and medium density fiberboard (MDF) were tested with the proposed method. In general, the measured elastic constants of the three types of panel based on modal test agreed well with those corresponding values measured by static tests. More tests are needed with a range of panel sizes and types for further validation of the proposed test method.

Vibration of Visco-elastic Orthotropic Parallelogram Plate with Linear Thickness Variation in Both Directions

A simple model presented here for vibration of a visco-elastic, orthotropic parallelogram plate with linear thickness variation in both directions. Using the separation of variables method, the governing differential equation has been solved for the vibration of a visco-elastic, orthotropic parallelogram plate having clamped boundary conditions on all four of the edges. An approximate but convenient frequency equation is derived by using the Rayleigh-Ritz technique with a two-term deflection function. The time period and deflection function at different points for the first two modes of vibration are calculated for various values of taper constants, aspect ratio and skew angle.

Thermal Effect on Vibration of Parallelogram Plate of Bi-Direction Linearly Varying Thickness

In this paper, the effect of thermal gradient on the vibration of parallelogram plate with linearly varying thickness in both direction having clamped boundary conditions on all the four edges is analyzed. Thermal effect on vibration of such plate has been taken as one-dimensional distribution in linear form only. An approximate but quiet convenient frequency equation is derived using Rayleigh-Ritz technique with a two-term deflection function. The frequencies corresponding to the first two modes of vibration of a clamped parallelogram plate have been computed for different values of aspect ratio, thermal gradient, taper constants and skew angle. The results have been presented in tabular forms. The results obtained in this study are reduced to that of unheated parallelogram plates of uniform thickness and have generally been compared with the published one.

Thermal Effect on Free Vibration of Non-Homogeneous Orthotropic Visco-Elastic Rectangular Plate of Parabolically Varying Thickness

A simple model presented here is to study the thermal effect on vibration of non-homogeneous orthotropic visco-elastic rectangular plate of parabolically varying thickness having clamped boundary conditions on all the four edges. For non-homogeneity of the plate material, density is assumed to vary linearly in one direction. Using the separation of variables method, the governing differential equation has been solved for vibration of non-homogeneous orthotropic viscoelastic rectangular plate. An approximate frequency equation is derived by using Rayleigh-Ritz technique with a two-term deflection function. Results are calculated for time period and deflection at different points, for the first two modes of vibration, for various values of temperature gradients, non-homogeneity constant, taper constant and aspect ratio and shown by graphs.

INFLUENCES OF SMALL CURVATURES ON THE MODAL CHARACTERISTICS OF THE JOINED HERMETIC SHELL STRUCTURES

From a practical engineering point of view, there is interest in understanding the influences of small curvatures on the modal characteristics of a plate–shell combination. However, research reports in this area are rarely found in literature. This paper is the first one to study the influences of small curvatures on free vibration of a plate–shell combined structure, consisting of a circular barrel-like shell and two spherically curved end plates, using the line receptance concepts and approximate frequency equations of barrel-shaped shell and spherically curved plate. It illustrates in detail information about natural frequencies and mode shapes of the combined structure under different geometric configurations. Consequently, it provides useful information for setting up criteria for designing hermetic plate-shell joined structures, such as the hermetic shell of a refrigeration compressor. The receptance method, which is basically a component mode synthesis technique, is adopted as the solution procedure in this study. It predicts the responses and frequencies of a combined system in terms of the responses and frequencies of its subsystems. This technique, for cases to which it can be applied successfully, is a powerful tool in creating an understanding of the physics of combined structures.

On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass in-span

The present study is concerned with the derivation of the eigenfrequencies and their sensitivity of a cantilevered Bernoulli—Euler beam carrying a tip mass (primary system) to which a spring-mass (secondary system) is attached in-span. After establishing the exact frequency equation of the combined system, a Dunkerley-based approximate formula is given for the fundamental frequency. Using the normal mode method, a second approximate frequency equation is established which is then used for the derivation of a sensitivity formula for the eigenfrequencies. The frequency equations of some simpler systems are obtained from the general equation as special cases. These frequency equations are then numerically solved for various combinations of physical parameters. The comparison of the numerical results with those from exact frequency equations indicate clearly that the eigenfrequencies of the combined system described above can be accurately determined by the present method.

On the eigenfrequencies of a cantilevered beam with a tip mass and in-span support

The present study addresses essentially to the investigation of the bending eigenfrequencies of a cantilevered Bernoulli-Euler beam with a tip mass which has an in-span support. First, an exact frequency equation is established via a boundary value formulation and then an approximate frequency equation is derived using the Lagrange's multipliers method. For a design engineer who is interested in a quick but accurate determination of the fundamental eigenfrequency of the system, an approximate formula is also established by using the Dunkerley's procedure. Finally, a formula for the sensitivity of the eigenfrequencies with respect to small changes in the location of the in-span support is derived. Numerical results on a digital computer justify the usefulness of the obtained analytical results.

Three-dimensional analysis of bending vibrations of isotropic homogeneous circular plates.

Three-dimensional analyses of bending vibrations of circular plates have been made by Pickett and Hutcinson, but their investigations were insufficient concerning both the theoretical treatment and the numerical results. The analytical method presented in this paper is based on the superposition of the guided wave modes in an infinite plate. Therefore the solutions used in the analysis satisfy exactly the traction free condition on the parallel boundaries. On the other hand, the boundary conditions on the curved surface can not be satisfied exactly. Then the equations of the boundary conditions on the exterior edge are integrated piecewisely in the thickness direction to derive an approximate frequency equation. Some numerical results of natural, frequencies are shown and discussed.

Analytical and experimental investigation on vibrating rectangular plates with two opposite free edges and internal supports while the remaining are elastically restrained against rotation

This paper deals with the determination of an approximate fundamental frequency equation for a rectangular plate with two edges ( y = 0, b) elastically restrained against rotation while the edges x=0, a are free but internal supports parallel to the free edges are present in the structural system. The algorithmic procedure can be easily implemented on a micro computer or even a hand, programmable calculator. Good engineering agreement with experimental results is shown to exist in the case of a plate with edges y=0,b rigidly clamped.

A note on the determination of the fundamental frequency of vibration of a rectangular plate with a free, semicircular cut-out along the edge

The title problem is tackled by using polynomial co-ordinate functions and the Ritz method in order to generate an approximate frequency equation. It is shown that in the case of a clamped square plate the results are in good engineering agreement with frequency values obtained by means of the finite element method when the cut-out size is small relative to the plate dimensions. Experimental results are also presented and it is shown that they are in excellent agreement with finite element values.

Fundamental frequency of vibration of thin elastic plates elastically supported along internal segments

The title problem is approximately solved in a unified manner in the case of rectangular and circular plates elastically restrained against rotation along the boundary. Polynomial coordinate functions and a variational approach are used in order to obtain an approximate frequency equation. Judging from the excellent accuracy attained using the present methodology in the case where no internal supports are present, it is reasonable to expect that the results presented in this study will be sufficiently accurate for engineering purposes in the situations considered.

Anomaly observed in the approximate analysis of vibrating rectangular plates with edges elastically restrained against rotation and translation

This paper deals with the solution of the title problem in the case of an orthotropic plate with three edges elastically restrained against rotation whilst transverse displacement is prevented. It is assumed that the fourth edge is elastically restrained against rotation and translation. The natural boundary conditions are satisfied approximately at the fourth edge. The Rayleigh-Ritz method is used to derive an approximate fundamental frequency equation and a rather curious anomaly is shown to exist in the frequency values for a certain range of values of the rotational and translational spring parameters which govern the motion of the fourth edge.

Vibrations of orthotropic rectangular plates with edges possessing different rotational flexibility and subjected to in-plane forces

The title problem is solved using polynomial approximations and a weighted residuals approach. It is shown that a very simple, yet accurate, approximate fundamental frequency equation can be generated. A basic, forced vibrations problem is also analyzed. The procedure is quite convenient for design purposes since all the resulting expressions for frequency coefficients, displacements and stress resultants can be easily implemented for numerical evaluation in a programmable, pocket calculator.

Dispersion Relation of Elastic Waves in Bars of Rectangular Cross Section

A formal solution for travelling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented. The solution is obtained by crosswise superposition of two single series, each term of which gives zero shearing stresses on the lateral surfaces. Frequency equations, which define the dispersion relations of elastic waves, are given by infinite determinants. Based on the approximate frequency equations, dispersion relations for a longitudinal mode, a torsional mode, and bending modes were calculated with the aid of a digital computer. Numerical results were compared with those given by other authers.

Antisymmetric modes of vibration of a circular plate elastically restrained against rotation and subjected to a hydrostatic state of in‐plane stress

The analysis of flexural vibrations of plates with edges elastically restrained against rotation is of interest to the design engineer since ideal supports or clamps are difficult to obtain in practice. A survey of the literature reveals that several investigations have been performed in the past on (a) vibrating circular edges (without in‐plane loading) and (b) vibrating simply supported and clamped circular plates subjected to hydrostatic in‐plane loading. The present paper deals with the determination of very simple, approximate frequency equations which allow prediction of natural frequencies in the case of a vibrating circular plate which executes antisymmetric modes. It is shown that use of simple polynomial expression and a variational approach leads, in some cases, to numerical values which are more accurate than those available in the technical literature. The approach followed in the present investigation can be extended in a straightforward fashion to the case where the edge has translational flexibility.