Dunkerley's Formula Research Articles

Lower estimate of the fundamental frequency of natural oscillations of a truss with an arbitrary number of panels

Introduction: the paper deals with oscillations of a statically definable plane, truss with a double lattice of racks and descending braces with massive loads in the nodes of the lower chord. The weight of the truss rods is not taken into account. It is assumed that the freights are moved only vertically. The fundamental frequency of natural oscillations is estimated from the Dunkerley formula by the values of partial frequencies. Materials and methods: an analytical estimate is obtained by generalizing formulas obtained from a series of estimates for trusses with a consistently increasing number of panels. The stiffness of the truss was determined using the Mohr’s integral. The double lattice of the truss does not allow using the cross-section method; therefore, the forces in the rods were calculated (or estimated) in an analytical form using the method of cutting nodes with the compilation of a system of equilibrium equations simultaneously for all rods and three support reactions. The matrix of equilibrium equations was compiled in a software program written in the language of the Maple computer mathematics system based on the coordinates of the nodes and the values of the direction cosines of the forces. For a sequence of coefficients of the desired formula, linear homogeneous recurrent equations were found and solved by means of special operators of the Maple system. Results: the resulting formula estimating the relationship between the fundamental frequency and the panels number has the form of a sixth degree polynomial with coefficients depending on the parity of the number of panels. The analytical result is compared with the smallest frequency obtained numerically from the solution of the problem of oscillation of the cargo system. It is shown that the main frequency, depending on the truss height, has an extremum. Conclusions: the method of generalizing particular solutions using the Maple system operators allowed authors to obtain and analyze a formula for a lower estimate of the fundamental frequency of oscillation of a truss model with an arbitrary number of panels. The resulting estimate can be used as a test for numerically obtained solutions. The formula is especially efficient for systems with a large number of panels; as numerical methods for their calculation are time-consuming require considerable time and have a tendency for accumulating rounding errors.

Fundamental Vibration Period of SW Buildings

This paper focuses on the fundamental period vibration evaluation of civil engineering constructions achieved in reinforced concrete which use the shear walls bracing system. As known, in the vibration modes assessment phase of a given structure, much of the seismic energy is absorbed by the fundamental mode. For this reason, many authors have provided empirical formulas using many approaches. Based on an elastic behavior and the Dunkerley formula, a simplified expression is proposed, which takes into account the geometrical and the mechanical characteristics of the structure. Following this development, the period value is a function of the number of floors of the considered structure. It is in good agreement with that obtained by the FEM that uses numerical methods requiring a sophisticated and more complex process. It may constitute a practical tool for practicing engineers in the preliminary design phase of any SW building.

Bounds for natural frequencies, Dunkerley's formula and application to soil–structure interaction

A new expression for a lower bound for the fundamental natural frequency of a structure is presented and its relationship with Dunkerley's formula is examined. A related improved estimate for the fundamental frequency is also obtained and applied to obtain an approximate expression for the fundamental frequency of a soil–structure system including a 1-DOF structure, a flat rigid foundation, and an elastic soil. The resulting approximation is then compared with the now standard result first obtained by Jennings and Bielak [6]

High-Speed Balancing of Rotors with Overhangs: When Is Overhang Likely to Cause Problems?

Abstract In high-speed balancing of flexible rotors with overhangs, it is important to determine ahead of time if the overhang needs to be restrained while the rotor is being tested in the bunker. This is done by adding a stub shaft to the rotor and a third pedestal to the overall system. Since the bearing properties are rarely known precisely and have a considerable influence on the rotor's dynamics, the model complexity increases tremendously if a third pedestal is needed during the balancing operation. From a balance engineer's viewpoint, it is therefore important to know ahead of time, without modeling the entire rotor, if the overhangs are likely to cause a problem during the balancing operation. This paper presents a criterion for identifying rotors with shaft overhangs that are likely to exhibit dangerous behavior while balancing. The proposed approach provides quick estimates of the L-mode frequency of the overhang using the influence coefficients and Dunkerley's formula. Two additional criteria based on further simplifications of the influence coefficient approach are also presented. Numerical results from four industrial rotors recently balanced indicate that the proposed methods are effective in determining the need for a third pedestal without having to resort to extensive rotor modeling.

A simple model for assessing periods of vibration and modal response quantities in symmetrical buildings

A simple mathematical model is proposed for assessing periods of vibration and mode shapes of common cantilever bents used in concrete structures, such as shear walls, coupled walls, rigid frames and wall-frame assemblies. The bent is treated as a continuum and the proposed model is based on the technique of decomposing a cantilever bent into two complementary subsystems (a flexural and a wall-frame bent) and on the finding that the use of Dunkerley's formula for calculating natural frequencies yields reasonable results for the first three modes of vibration. The objective in proposing this model is to consider the effect of column axial shortenings in the analysis of structural bents. With this model any cantilever bent may be approximated by a simple incompressible shear–flexure system of equal flexural rigidity, but of equivalent modal shear rigidity. This approach has the advantage that the response of different structural bents may be combined in buildings composed of these bents in any arrangement. All bents are approximated by equivalent shear–flexure models and therefore the complete structure may be analysed by a simple methodology, which has been extensively used in the past. Particularly in symmetrical buildings, frequencies may be determined by a simple formula and modal response quantities by available design charts. A quick estimate of these quantities is of particular importance at the early stages of structural design, prior to a full dynamic analysis. In order to illustrate the application of the proposed model a symmetrical building of varying height, composed of different structural bents, is analysed and comparisons are made with more accurate results obtained by 3D computer dynamic analyses. Copyright © 2006 John Wiley & Sons, Ltd.

A modified Dunkerley formula for eigenfrequencies of beams carrying concentrated masses

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions associated to the masses. The resulting transcendental equations are numerically solved for the eigenvalue. On the other hand, the eigenvalue can be predicted merely from the individual beam system carrying a single mass, by virtue of the Dunkerley's formula. A parametric study on the effects of the two masses and their locations is presented for the beam with different boundary conditions. It is found that the Dunkerley's expression can generally yield good approximation if compared with the result associated with the original characteristic equation. The computation time saved owing to the modified Dunkerley method is also illustrated in a comparison. The Dunkerley's method is recommended for the beam carrying more than two masses at different positions, owing to its good approximation and the saving in computational time.

6角形平面の剛接単層ラチスドームの弾塑性座屈解析 : 形状初期不整が存在しない完全形状ドームについて

The present paper investigates both the elastic and elastic plastic dimple buckling behaviors of a rigidly jointed latticed dome of single layer on a hexagonal plan. The calculation is performed changing several parameters; the number n of grid along line through the apex, the angle θ_0 of the central member at the apex, and the type of boundary conditions. Those buckling loads are expressed in non-dimensionalized form by classical linear buckling load of continuous isotropic spherical shell. And the present study also has revealed that modified Dunkerley's formula is effective to predict the elastic plastic buckling load of the present latticed dome.

Stability of Shear‐Wall Structures

A simple approximate lower bound formula is proposed for the gravity buckling loads of coupled shear‐wall structures using continuous medium assumptions. The deflected shape of the system is decomposed into those of flexural and shear‐flexure cantilevers. For the latter, an accurate approximate buckling load formula is also presented. Inserting the buckling loads of the two systems in Dunkerley's formula yields the required buckling load. The formula is compared with exact results computed by finite differences and good agreement is demonstrated.

Natural Frequencies of Stator Core of Induction Motor : 1st Report, Characteristics of Radial Vibration

The natural frequencies of raidal vibrations in models of the stator core of three induction motors were measured. It is found that two natural frequencies have the same deformation wave numbers and the relation between the deformation wave numbers and natural frequencies is represented by two curves. These curves have two asymptotes, one of which shows the characteristics of the natural frequencies of the ring, and the other shows the characteristics of those of teeth only. The lower of the two natural frequencies can be calculated precisely by Dunkerley's formula. The experimental results are in good agreement with the theoretical results calculated on the assumption that the vibration system is of two degrees of freedom.

A lower bound for Dunkerley's formula in continuous elastic systems

Dunkerley's formula is shown to yield a lower bound for the fundamental frequency of vibration of a continuous elastic system whose flexibility is separable into a number of distinct components. The use of the formula is illustrated by considering a uniform cantilever and a three layer sandwich beam.

Approximate natural frequencies for coupled shear walls

AbstractAn approximate yet accurate formula is proposed for the natural frequencies of coupled shear walls under continuous medium assumptions. First the deflected shape of the structure is represented as the sum of two components: one due to flexural cantilever action and one due to shear‐flexure cantilever action. The natural frequencies of the latter two systems are then combined in Dunkerley's formula to yield the approximate frequency of the structure.

Generalization of Dunkerley's formula for finding the lowest critical velocity of rotating shafts

Glauert, H., The Elements of Aerofoil and Airscrew Theory, Cambridge University Press, New York, 1926. Filotas, L. T., Solution of the Lifting Line Equation for Twisted Elliptic Wings, Journal of Aircraft, Vol. 8, No. 10, Oct. 1971, pp. 835-836. Graham, E. W., A Drag Reduction Method for Wings of Fixed Planform, Journal of the Aeronautical Sciences, Vol. 19, No. 12, 1952, pp. 823-825. Anderson, R. C. and Millsaps, K., Application of the Galerkin Method to the Prandtl Lifting Line Equation, Journal of Aircraft, Vol. 1, No. 3, May-June 1964, pp. 126-128.

The iterative solution of flexural vibration problems based on the Myklestad method

A method of successive approximation for the determination of the natural frequencies and modes of vibration of a flexural system may bo obtained by the use of the Myklestad method, in conjunction with auxiliary formulae for the determination of successive trial solutions. Such auxiliary formulae may be obtained by energy methods, or by consideration of the effect of small changes in the system, or, in the case of the fundamental frequency, by the application of Dunkerley's formula. These alternative methods are discussed and their convergence is examined.