Transverse Vibration of a Rectangularly Orthotropic Spinning Disk, Part 1: Formulation and Free Vibration

1999 ◽  
Vol 121 (3) ◽  
pp. 273-279 ◽  
Author(s):  
A. Phylactopoulos ◽  
G. G. Adams
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Circular Disk ◽  
Polar Coordinates ◽  
Fourier Series Expansion ◽  
Vibration Modes ◽  
Classical Formulation ◽  
Modes Of Vibration ◽  
Orthotropic Disk

The transverse vibration of a spinning circular disk with rectangular orthotropy is investigated. Two dimensionless parameters are established in order to characterize the degree of disk anisotropy and solutions are sought for a range of these parameters. The orthotropic bending stiffness is transferred into polar coordinates and is found to differ from a classical formulation for a stationary disk. A Fourier series expansion is used in the circumferential direction. Unlike the isotropic disk, the Fourier components determining the transverse vibration modes of the orthotropic disk do not separate. This condition results in an eigenvalue problem involving a coupled set of ordinary differential equations which are solved by a combination of numerical integration and iteration. Thus the natural frequencies and normal modes of vibration are determined. Because each eigenfunction contains contributions from more than one Fourier component, the normal modes do not possess distinct nodal diameters or nodal circles. Furthermore, disk orthotropy causes the natural frequencies corresponding to the sine and cosine modes to split; the degree of splitting decreases as the rotational speed increases.

Free Vibration of Continuous Skin-Stringer Panels

1960 ◽  
Vol 27 (4) ◽  
pp. 669-676 ◽  
Author(s):  
Y. K. Lin
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Time Dependent ◽  
Experimental Measurements ◽  
Structural Configuration ◽  
Modes Of Vibration ◽  
Fuselage Skin

The determination of the natural frequencies and normal modes of vibration for continuous panels, representing more or less typical fuselage skin-panel construction for modern airplanes, is discussed in this paper. The time-dependent boundary conditions at the supporting stringers are considered. A numerical example is presented, and analytical results for a particular structural configuration agree favorably with available experimental measurements.

The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

1952 ◽  
Vol 211 (1105) ◽  
pp. 204-224 ◽  
Keyword(s):  
Lower Bounds ◽  
Natural Frequencies ◽  
Hermitian Operator ◽  
Normal Modes ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Subsequent Paper ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

The Normal Modes of Vibrations of Beams Having Noncollinear Elastic and Mass Axes

1940 ◽  
Vol 7 (3) ◽  
pp. A97-A105
Author(s):  
Clyne F. Garland
Keyword(s):  
Physical Properties ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Longitudinal Axis ◽  
Cantilever Beams ◽  
Amplitude Ratios ◽  
Numerical Example ◽  
Modes Of Vibration ◽  
Axis Passing

Abstract This analysis deals with vibration characteristics of cantilever beams in which the longitudinal axis, passing through the mass centers of the elementary sections, is not collinear with the longitudinal axis about which the beam tends to twist under the influence of an applied torsional couple. Expressions are derived from which the natural frequencies and normal modes of vibration of such a beam can be determined. The Rayleigh-Ritz method is employed to determine the frequencies and amplitude ratios. Following the development of the general expressions, more specific equations are derived which express the natural frequencies and relative amplitudes of motion in each of two normal modes of vibration. The theoretical relationships of the several physical properties of the beam to the natural frequencies of vibration are shown graphically. Finally a numerical example is presented for a particular beam, and the computed natural frequencies and normal modes are compared with those determined experimentally.

Modal Analysis of Circular Plates

2014 ◽  
Vol 611 ◽  
pp. 245-251
Author(s):  
Jozef Bocko ◽  
Peter Sivák ◽  
Ingrid Delyová ◽  
Štefánia Šelestáková
Keyword(s):  
Circular Plate ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Engineering Practice ◽  
Structural Elements ◽  
Circular Plates ◽  
Structural Element ◽  
Thin Circular Plate ◽  
Outer Periphery ◽  
Modes Of Vibration

In engineering practice, some of the structural elements take the form of a thin planar plate. For such elements, it is sometimes important to consider dangerous condition of resonance. A structural element cannot operate in the range of resonant frequencies. It is therefore necessary to determine natural frequencies and normal modes of vibration of such structural elements. Parts of the paper are the results of the analysis of natural frequencies and normal modes of vibration using FEM program Cosmos. The subject of the analysis was a thin flat circular plate considered in three modifications, i.e. free thin circular plate without hole, a thin circular plate without hole, clamped on the outer periphery, a thin circular plate with a hole, clamped on the outer and inner circumference. At the same time, Chladni patterns were obtained. They were created using the Matlab system and extraction of the outputs of the Cosmos program.

scholarly journals Revised dynamic characteristics of uniform beams with free - hinged and free - free end conditions

2017 ◽  
Vol 6 (2) ◽  
pp. 41
Author(s):  
Alexander Shulemovich
Keyword(s):  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Structural Strength ◽  
Normal Modes ◽  
Rigorous Analysis ◽  
Physical Evidence ◽  
End Conditions ◽  
Modes Of Vibration

The uniform beams with free-hinged ends and with free-free ends have very slack bonds and, therefore, in accordance with Rayleigh’s theorem, their lowermost eigenvalue must be lesser compared to the lower most eigenvalue of beams with clamped-free and clamped-hinged ends. In spite of the physical evidence, the magnitudes of lowermost eigenvalue of beams with slack bonds, available in all publications, are larger. This contradiction signifies that there are the missing modes of vibration with the lowermost eigenvalue for beams with free-free and free-hinged ends. The rigorous analysis of uniform beams vibration with free-free and free-hinged ends conditions defines these missing lowermost natural frequencies and normal modes and ascertains the frequencies and modes for all uniform beams with various end conditions into the ordered system. The lowermost mode of vibration of a beam with free-free ends, caused by ocean choppiness and determined in this investigation, is paramount for estimation of the ships structural strength, particularly important for the tankers.

Design and Optimization of Multi-Modal Vibration Energy Harvesters Using Slitted Beams

2014 ◽  
Author(s):  
Mina Dawoud ◽  
Hesham Hegazi ◽  
Mustafa Arafa
Keyword(s):  
Permanent Magnets ◽  
Natural Frequencies ◽  
Geometrical Parameters ◽  
Vibration Modes ◽  
Cantilever Beams ◽  
Outer Contour ◽  
Design And Optimization ◽  
Energy Harvesters ◽  
Modes Of Vibration ◽  
Modal Vibration

The objective of this work is to design cantilever beams possessing close vibration modes to enable harvesting energy from variable frequency sources of base motion. In this context, the geometry of two-dimensional cantilever beams is designed to obtain closely spaced harvestable modes of vibration. A number of internal slits are made inside the beam, whose outer contour and mass distribution are altered in such a way to obtain the desired frequency spacing. The beam carries two permanent magnets that oscillate past stationary pickup coils in order to convert the mechanical motion into electric power. Optimum design results of the shape and geometrical parameters of the system are presented towards controlling the natural frequencies, their spacing and the output power. Simulations of the system dynamics are supported by experimental validation.

Thermal Stresses in Rectangular Plates

1959 ◽  
Vol 10 (1) ◽  
pp. 65-78 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Thermal Stresses ◽  
Plate Thickness ◽  
Normal Modes ◽  
Thermoelastic Stress ◽  
Characteristic Functions ◽  
Vibration Modes ◽  
Rectangular Plates ◽  
Practical Application ◽  
Flat Plates ◽  
Modes Of Vibration

SummaryThe characteristic functions for beam vibration modes are used to derive an approximate solution for the calculation of thermal stresses in rectangular isotropic flat plates subjected to arbitrary temperature distributions in the plane of the plate and constant temperatures through the plate thickness. The thermal stresses are obtained in the form of generalised Fourier expansions in terms of the characteristic functions, and their derivatives, representing normal modes of vibration of a clamped-clamped beam. Since these functions have recently been tabulated, the practical application of this new method to the thermoelastic stress analysis of plates presents no difficulty.

scholarly journals Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
D. V. Bambill ◽  
C. A. Rossit
Keyword(s):  
Transverse Vibration ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Mechanical Performance ◽  
Ritz Method ◽  
Normal Modes ◽  
Thin Plates ◽  
Transverse Vibrations ◽  
Edge Conditions ◽  
Distributed Forces

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

The Determination of Vibration Modes

1949 ◽  
Vol 53 (468) ◽  
pp. 1095-1099
Author(s):  
N. F. Harpur
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Dynamic Loads ◽  
Vibration Modes ◽  
Resonance Modes ◽  
Modes Of Vibration

At some stage in the design of every aeroplane it is necessary to estimate or to measure the resonance modes of vibration. This has not always been the case, but the problems of flutter, control reversal and dynamic loads have increased in importance as speeds have risen. Nowadays, it is an airworthiness requirement that these effects be considered and the aircraft made safe for all conditions of flight. A knowledge of the normal modes of vibration is essential for all accurate estimates of these aeroelastic effects.Taking flutter as an example, the technique of flutter investigations consists of first determining which combinations of the various possible degrees of freedom are liable to excite dangerous oscillations. Typical degrees of freedom for a wing are bending and twist in each normal mode, aileron deflection and tab deflection; for a tailplane and elevator we might consider tailplane bending or twist, elevator deflection, tab deflection, fuselage bending and twist, and pitching of the whole aeroplane.

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