Non-dimensional Frequency Parameters Research Articles

Free vibrations of circular cylindrical shell on Winkler and Pasternak foundations

Free vibrations of a thin circular-cylindrical shell in contact with an elastic medium are investigated, employing membrane theory. Response of the elastic medium is represented by Winkler/Pasternak models. Dynamic characteristics involving nondimensional frequency parameters and axial wave parameters are drawn. It is found that the value of the foundation modulus (K), considerably affects radial vibrational mode frequency, while the other two vibrational mode frequencies, i.e. torsional and longitudinal, remain almost unaffected. It is found also that the influence of the nondimensional shear modulus parameter (Ḡ), is more pronounced in radial vibrational mode frequencies, to a lesser extent on torsional mode frequencies and least on longitudinal mode frequencies.

A finite element free vibration analysis of a thermally stressed spinning plate

A Lagrangian quadratic finite element formulation is used to analyse the free vibration problem in spinning isotropic plates exposed to a thermal environment. The plate is oriented arbitrarily with respect to the spinning hub. The inplane component of the distributed centrifugal force produces inplane tensile stresses that add to the bending strain energy. The normal component is added to the inertia force. A finite element steady-state heat conduction analysis of the plate yields the two-dimensional temperature field (convection terms due to spinning inertia are accounted for). In the thermo-structural interface, the inplane thermal forces, and subsequently the inplane thermal stresses, are evaluated. These stresses further affect the bending strain energy. The formulation includes the effects of transverse shear deformation and rotary inertia of the plate. In the numerical examples considered, the effects of the speed of rotation, the hub radius and the high temperature gradients on the non-dimensional frequency parameter, are clearly brought out.

Free Vibrations of Curved and Twisted Cantilevered Cylindrical Panels. Analytical Results.

This paper presents analytical results of nondimensional frequency parameters and mode shapes of vibrations for curved and twisted cantilevered cylindrical panels. The method is based on the three-dimensional elasticity theory with consideration of rotating inertia and shear deformation. Numerical results for curved and twisted cylindrical cantilever panels having typical geometries are compared with those available in the literature. The effects of twisting, aspect ratio of panels and central angles of the cylinder on the results are also discussed.

Analysis of rotating laminated cylindrical shells by different thin shell theories

The natural frequencies of the forward and backward modes of thin rotating laminated cylindrical shells are determined by using four common thin shell theories, namely Donnell’s, Flügge’s, Love’s and Sanders’. The objective is to present a unified analysis, formulated with the use of tracers so that it can be reduced to any of the four shell theories by giving appropriate values to the tracers. For simplicity, results are presented only for the case of simply supported-simply supported boundary conditions, which can be satisfied by expressing the displacement fields in terms of the products of sine and cosine functions. Numerical results presented are the non-dimensional frequency parameters of the forward and backward travelling modes for rotating cylindrical shells and the non-dimensional frequency parameters for non-rotating cylindrical shells. The present analysis is verified by comparing the numerical results with those in the literature and very good agreement is obtained.

Comprehensive exact solutions for free vibrations of thick annular sectorial plates with simply supported radial edges

The first known exact solutions are derived for the free vibrations of thick (Mindlin) annular sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain non-integer order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. Frequency determinant equations are derived for thick annular sectorial plates with circular edges having all nine possible combinations of clamped, simply supported, or free boundary conditions. Extensive amounts of nondimensional frequency parameters are presented for thickness ratio ( h a ) values of ≅ 0,0.1, and 0.2; radii ( b a ) values within the range of 0.1–0.7; and sector angle values of 180° ⩽ α ⩽ 360° for which, in the range of α > 180°, no previously published results are known to exist. Frequency results obtained for thick annular sectorial plates are compared to those determined for classically thin ( h a ≅ 0 ) ones. The frequencies for 360° annular sectorial plates (i.e. annular plates having a hinged crack) are compared with those for complete circular annular plates. The exact solutions presented herein are useful to researchers for determining the correctness of approximate numerical procedures and software packages for thick plate vibration analyses.

Free Vibrations of Twisted Cantilevered Thin Plates Having Thickness Varying in Two Directions

Free vibrations of twisted cantilevered thin plates having thickness varying in two directions are investigated analytically and experimentally in this study. A numerical procedure based on the Rayleigh-Ritz method is presented using algebraic polynomials as displacement functions. To demonstrate the procedure, nondimensional frequency parameters and mode shapes are analyzed for typical twisted plates. The convergence of frequency parameters is studied and effects of plate thickness variation on vibrating characteristics are discussed. Experimental tests are carried out for three kinds of aluminum specimens. On the basis of comparison between experimentally obtained resonance frequencies and mode shapes and those obtained from the analysis, it appears that the numerical procedure proposed in the present study is useful for analyzing the free vibrations of twisted thin plates with variable thickness.

Vibratory behaviour of shallow conical shells by a global Ritz formulation

The development of a new approach based on the energy principle is presented to study the free vibration of shallow conical shells. In the numerical procedure, a set of orthogonally generated and kinematically oriented shape functions, which satisfies the geometric boundary conditions at the outset, is proposed to overcome the mathematical complexity in expressing the geometry and variable surface curvature of these shells. To establish the accuracy of this method, convergence and comparison studies have been carried out. In this study, the effects of various shell parameters on the fundamental and higher mode frequencies are investigated. It is found that monotonic increases in the fundamental nondimensional frequency parameter occurs when the cone vertex angle or base subtended angle is increased independently for the cantilever conical shell. The fundamental frequency parameter also becomes higher for the fully clamped conical shell with higher panel-length to cone-length ( a/ s) ratio. A set of first published vibration mode shapes is also presented.

Investigations into the Precessing Vortex Core Phenomenon in Cyclone Dust Separators

Experimental investigations have been carried out to examine the effect of downstream pipework configurations on the precessing vortex core (PVC) generated within the exhaust region of a cyclone dust separator. Characterization of the PVC using a non-dimensionalized frequency parameter (NDFP) was used to determine the relationship between Reynolds number and geometrical swirl number of the cyclone. The results show that the NDFP tends towards an asymptotic value for Reynolds numbers of about 50 000 and high swirl numbers (> 3.043). This value is reached earlier with lower swirl numbers. It was concluded that any exhaust pipework configuration produced a significant drop in the PVC frequency, and certain configurations either delayed or promoted the development of the PVC.

Exact analytical solutions for free vibrations of thick sectorial plates with simply supported radial edges

The first known exact analytical solutions are derived for the free vibrations of thick (Mindlin) sectorial plates having simply supported radial edges and arbitrary conditions along the circular edge. The general solutions to the Mindlin differential equations of motion contain noninteger order ordinary and modified Bessel functions of the first and second kinds, and six arbitrary constants of integration. By exercising a careful limiting process, three regularity conditions at the vertex of the radial edges are invoked to yield three equations of constraint among the six constants for sector angles exceeding 180° (re-entrant corners). Three additional linearly independent equations among the six constants are obtained by satisfying the three boundary conditions along the circular edge. Frequency determinant equations are derived for Mindlin sectorial plates with circular boundaries which are clamped, simply supported, or free. Nondimensional frequency parameters are presented for over a wide range of salient and re-entrant sector angles (30° ⩽ α ⩽ 360°), and thickness-to-radius ratios of 0.1, 0.2 and 0.4. Frequency results obtained for Mindlin sectorial plates are compared to those determined for classically thin sectorial plates, and the results are found to be considerably different than those derived from thin plate theory, particularly for the fundamental frequencies of plates having sector angles slightly greater than 180° when the circular boundary is free. The frequencies for 360° sectorial plates (i.e. circular plates having a hinged crack) are compared with those for complete circular ones.

Influence of Varying Cam Profile and Follower Motion Event Types on Parametric Vibration and Stability of Flexible Cam-Follower Systems

A method to study parametric stability of flexible cam-follower systems, based on Floquet theory, as well as a closed-form numerical algorithm to compute periodic response of the system, have been developed in a companion paper. These are applied to an automotive valve train, modeled as a single-degree-of-freedom vibration system. The inclusion of the transverse and rotational flexibilities of the camshaft results in a system that is governed by a linear, second-order, ordinary differential equation with time-dependent coefficients. In this paper, the parametric stability of the system is studied, and the results are presented in the form of parametric stability charts. The regions of instability are plotted on the nondimensionalized frequency and excitation (amplitude) parameter plane. The maximum positional error of the follower motion, analyzed by the closed-form numerical algorithm, enables a novel presentation of three-dimensional stability and response charts. Stability of the system is investigated for three types of follower motion events and four different cam profiles. The effect of damping on parametric instability is also studied. A comparative study of these event and cam profile types reveals some very interesting and hitherto unknown results.

Free Vibrations of Curved and Twisted Cantilevered Thin Plates. Analytical and Experimetal Results.

Free vibrations of curved and twisted cantilevered thin plates are investigated analytically and experimentally. A numerical method based on the Rayleigh-Ritz procedure is presented using algebraic polynomials as displacement functions and an approximate integration, dividing the plate into elements. Convergency studies of nondimensional frequency parameters are made with increase of the number of elements dividing the plate and of terms in the assumed functions. Then, effects of initial twist and curvature on vibrating characteristics are discussed for typical curved and twisted cantilevered plates. Finally, experimental tests are carried out and resonance frequencies and mode shapes obtained are compared with analytical results.

Exact Analytical Solutions for the Vibrations of Sectorial Plates With Simply-Supported Radial Edges

The first known exact analytical solutions are derived for the free vibrations of sectorial thin plates having their radial edges simply supported, with arbitrary conditions along their circular edges. This requires satisfying: (1) the differential equation of motion, (2) boundary conditions along the radial and circular edges, and (3) proper regularity conditions at the vertex of the radial edges. The solution to the differential equation involves ordinary and modified Bessel functions of the first and second kinds, of non-integer order, and four constants of integration. Utilizing a careful limiting process, the regularity conditions are invoked to develop two equations of constraint among the four constants for sector angles exceeding 180 deg (re-entrant corners). Moment singularities for re-entrant corners are shown to be the same as the ones determined by Williams (1952) for statically loaded sectorial plates. Frequency determinants and equations are generated for circular boundaries which are clamped, simply-supported, or free. Nondimensional frequency parameters are presented for all three types of configurations for sector angles of 195, 210, 270, 330, and 360 deg (i.e., re-entrant corners).

Effect of model cooling on periodic transonic flow

Experimental investigations were conducted in a small transonic intermittent tunnel on 14 and 18% thick biconvex airfoils to study the effect of model temperature on periodic transonic flows. The Reynolds number based on the model chord was in the range 0.7-0.9 x 106. Tests were conducted in the Mach number range of 0.7-0.85, and the model-to-tunnel stagnation temperature ratio range of 0.6-1.0. The models were cooled in liquid nitrogen to the required temperature before a run. The measurements included model surface mean pressure and temperatures and tunnel side wall dynamic pressures. The results showed a large effect of model temperature on mean pressure distribution and periodic buffet excitation levels. The effects observed are thought to correspond with what would happen to adiabatic flows at greatly increased Reynolds numbers. Nomenclature Cp = pressure coefficient Cpte = pressure coefficient measured at x/c =0.95 c = model chord length, mm / = frequency, Hz M = freestream Mach number MI = Mach number just upstream of the shock n = nondimensional frequency parameter, 2irfc/u p = rms pressure fluctuations, N/M2 q = kinetic pressure, N/M2 Re = Reynolds number based on model chord TW, Tad = wall and adiabatic recovery temperature, K / = times, s u - freestream velocity, m/s x,y = coordinates from the leading edge of model 5 = boundary-layer thickness

Torsional vibrations of layered composite paraboloidal shells

An analysis is presented for the torsional vibration characteristics of layered composite paraboloidal shells. The conditions for the uncoupling of the torsional modes from the bending and extensional modes are first determined for a layered shell of revolution. A finite difference scheme is developed for the solution of the resulting governing equation for the uncoupled torsional frequencies. The results of the finite difference solution for the paraboloids are compared with the analytical solution of uncoupled torsional frequencies of circular plates, in the limit as the curvature of the paraboloid goes to zero. As a result of the analysis of different paraboloids, the non-dimensional uncoupled torsional frequency parameters of the paraboloids are found to be inversely proportional to the ratio of the meridional arclength to the thickness of the paraboloid for a given edge angle.

Power absorption by thin wave energy devices

Power absorption by thin wave energy devices is analyzed in this paper. Themathematical preliminaries are taken from other sources to obtain the analytic expressions for wave-making coefficient and maximum constrained capture width in the cases of two- and three-dimensional cylindrical devices. Their variations with the nondimensional frequency parameter kL‵ 1 for different values of the parameter α 1 dependent on the constraint are computed and shown grapically. On comparison it is observed that the cylindrical device with circular cross section presented here is a better wave maker and hence a more efficient energy absorber than the two devices, spheroid and lenticular, studied earlier for certain range of values of kL‵ 1 . It is further noticed that the amplitude of the displacement that the cylindrical device undergoes when tuned to respond optimally to the incident wave is much less than that of spheroid and lenticular devices. Also, the effect of constraint on absorbing abilities of cylinder is low in comparison to that for the lenticular device, reducing maximum capture width by 85.7% (for the cylinder) and 92.3% (for the lenticular device) of the unconstrained values when kL‵ 1 = 1.0. In the case of elongated two-dimensional devices with the waves incident normal to the surface, the cylinder with rectangular cross section is a better wave maker (and hence a better energy absorber) than that with elliptic cross section. From the present analysis it can be concluded that, in particular, the two- and three-dimensional cylindrical devices analyzed here are better absorbers than those studied earlier in the particular range of values of kL‵ 1 .

Free vibrations of twisted and tapered cantilever plates.

A numerical method, based on the Rayleigh-Ritz approach, for the analysis of the free vibration behavior of twisted and tapered cantilever plates is presented in this paper, together with some numerical results. Displacement functions in the analysis are assumed by algebraic polynomials. Numerical results obtained are compared with those available in the literature and a good agreement is found between them. Nondimensional frequency parameters and modes of vibration are computed for typical twisted and tapered cantilever plates and the effects of pretwist angles and rates of variable thickness on them are discussed.

Vibration and buckling of bimodulus laminated plates according to a higher-order plate theory

This paper presents the governing equations of rectangular simply supported single-layer and two-layer cross-ply plates of bimodulus materials related to the higher order shear deformation terms. Natural frequency parameters and buckling coefficients obtained by the iteration method are compared with published results for ordinary thick plates. The Mindlin plate theory result can be obtained by simplifying the governing equation of the higher-order plate theory formulation. The effects of higher-order shear deformation terms on neutral surface locations, buckling coefficients and natural frequencies are evaluated by observing the difference between the present higher-order theory results and the Mindlin plate theory results. Also, the bending stress is shown to affect significantly the buckling coefficient K cr and the non-dimensional frequency parameter Ω.

Investigation of Fixed-Free Annular Plate Resonant Frequency Predictions

Nondimensional frequency parameters for predicting the resonant frequencies of annular plates with fixed-free boundary conditions as the plate inner to outer radius ratio approaches unity have been investigated experimentally. Frequency parameters have been determined by using modal analysis to measure resonant frequencies for annular plates of varied materials, thicknesses, and with radius ratios of 0.5 to 0.9. The data are compared to two different analytical frequency predictions which have been presented as solutions for resonance of fixed-free annular plates based on classical elastic plate theory.

Free lateral vibrations of a beam under tension with a concentrated mass at the mid-point

An analytical procedure to determine the fundamental vibrational natural frequency of a uniform beam (flexible support) under axial tension is presented. The beam is built in at both ends, and has a concentrated mass (rigid body) attached at the mid-point. Based on this procedure, a graph is developed which gives a non-dimensionalized fundamental vibrational natural frequency parameter in terms of non-dimensionalized mass and tension parameters. This graph can be employed to determine changes in natural frequency which result from varying only the tensile force. The analytical solution of a particular geometry is verified through comparison with experimental results.

Determination of the natural frequencies of transverse vibration for conveyor belts with orthotropic properties

A non-dimensional frequency parameter γ has been determined for each mode of vibration of plate-like structures with orthotropic properties. This parameter depends on the type of plate boundary, the plate rigidity and the aspect ratio. Expressions for γ are derived for plates with simple supports in the direction of the applied load and with clamped and free boundaries on the other edges. Values of γ are computed as a function of plate aspect ratio. From computer plots of γ(r), approximate expressions for γ are developed and applied to the calculation of modal frequencies of steel cord conveyor belts.