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.

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

1968 ◽  
Vol 72 (688) ◽  
pp. 341-344 ◽  
Author(s):  
B. Dawson
Keyword(s):  
Shear Deformation ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Rotary Inertia ◽  
Characteristic Functions ◽  
Cantilever Beams ◽  
Dynamic Displacement ◽  
Modes Of Vibration ◽  
Good Agreement

Summary The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Vibration of Triangular Cantilever Plates by the Ritz Method

1954 ◽  
Vol 21 (4) ◽  
pp. 365-370
Author(s):  
B. W. Andersen
Keyword(s):  
Cantilever Beam ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Plan View ◽  
Modes Of Vibration ◽  
Accuracy Of Results ◽  
Isosceles Triangles

Abstract Using the method published by Ritz in 1909, natural frequencies and corresponding node lines have been determined for two symmetric and two antisymmetric modes of vibration of isosceles triangular plates clamped at the base and having length-to-base ratios of 1, 2, 4, and 7 and for the two lowest modes of right triangular plates clamped along one leg and having ratios of the length of the free leg to that of the clamped one of 2, 4, and 7. A nonorthogonal co-ordinate system was used which gave constant limits of integration over the area of the triangle. The co-ordinate transformation made it necessary to modify the functions used by Ritz in approximating deflections and to consider cross products in the integration. The integration was done numerically, using tables compiled by Young and Felgar in 1949. To check the accuracy of results, a solution was obtained to the problem of a vibrating cantilever beam of uniform depth and triangular plan view. The results obtained were found to be consistent with those obtained for the plates by using an eight-term series to approximate the deflections of the symmetric plates (isosceles triangles) and a six-term series to approximate the deflections of the unsymmetric plates (right triangles).

The Natural Frequencies of Free and Constrained Non-Uniform Beams

1960 ◽  
Vol 64 (599) ◽  
pp. 697-699 ◽  
Author(s):  
R. P. N. Jones ◽  
S. Mahalingam
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Iteration Process ◽  
Conservative System ◽  
Basic System ◽  
Orthogonal Functions ◽  
Added Masses ◽  
The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

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.

Normal Vibrations of a Uniform Plate Carrying Any Number of Finite Masses

1959 ◽  
Vol 26 (2) ◽  
pp. 210-216
Author(s):  
W. F. Stokey ◽  
C. F. Zorowski
Keyword(s):  
Natural Frequencies ◽  
Physical Characteristics ◽  
Normal Vibrations ◽  
Simply Supported ◽  
Numerical Example ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
General Method

Abstract A general method is presented for determining approximately the natural frequencies of the normal vibrations of a uniform plate carrying any number of finite masses. Its application depends on knowing the frequencies and natural modes of vibration of the unloaded plate and the physical characteristics of the mass loadings. A numerical example is presented in detail in which this method is applied to a simply supported plate carrying two masses. Results also are included of experimentally measured frequencies for this configuration and several additional cases along with the frequencies computed using this method for comparison.

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.

The Bending Vibration Simulation of Fixed Arm Crane Luffing

2013 ◽  
Vol 694-697 ◽  
pp. 425-429
Author(s):  
Hao Wang ◽  
Yan Wei ◽  
Hui Li
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Natural Frequencies ◽  
Displacement Curve ◽  
Cantilever Beams ◽  
Bending Vibration ◽  
Beam System ◽  
Modes Of Vibration ◽  
The Finite Difference Method ◽  
Vibration Simulation

The cranes simplified as the cantilever beam system. The bending vibration differential equations were built using Hamilton principle.According to discrete differential equations by the finite difference method, the modes of vibration and the natural frequencies can be obtained. Application of discrete time and MATLAB vibration toolbox, the displacement curve and phase on the crane arm can be gotten. And the effect of amplitude angles and lifting loads on the arm vibration had been discussed.The work of the vibration on the crane arm has an universal meaning in the cantilever beams.

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.

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