Vibration of a Beam With Concentrated Mass, Spring, and Dashpot

1948 ◽  
Vol 15 (1) ◽  
pp. 65-72
Author(s):  
Dana Young
Keyword(s):  
Analytical Method ◽  
Composite System ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Concentrated Mass ◽  
Orthogonal Functions ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Concentrated Masses ◽  
Mass Spring

Abstract An analytical method is developed for determining the natural frequencies of a composite system which consists of a uniform beam with a concentrated mass, spring, and dashpot attached at any point along the length of the beam. The method makes use of a series expansion in terms of the set of orthogonal functions which represent the normal modes of vibration of the beam alone. Numerical examples are given for a beam with a combined concentrated mass and spring, for a beam with two concentrated masses, and for a beam with an attached dashpot.

Vibrations of a Rectangular Plate With Concentrated Mass, Spring, and Dashpot

1963 ◽  
Vol 30 (1) ◽  
pp. 31-36 ◽  
Author(s):  
Y. C. Das ◽  
D. R. Navaratna
Keyword(s):  
Rectangular Plate ◽  
Composite System ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Double Series ◽  
Concentrated Mass ◽  
Orthogonal Functions ◽  
Isotropic Rectangular Plate ◽  
Mass Spring

An analytical method, developed by Young [1], is here extended to the determination of the natural frequencies of a composite system which consists of an isotropic rectangular plate with a concentrated mass, spring, and dashpot attached at any point of the plate. This method makes use of a double series expansion in terms of two sets of orthogonal functions which represent the normal modes of the vibration of the plate alone. Numerical examples for a square plate with (a) a combined concentrated mass and spring, (b) two concentrated masses, and (c) an attached dashpot have been presented.

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 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 Vibrations of Constrained Beams

1952 ◽  
Vol 19 (4) ◽  
pp. 471-477
Author(s):  
Winston F. Z. Lee ◽  
Edward Saibel
Keyword(s):  
Natural Frequencies ◽  
Degree Of Approximation ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Continuous Beam ◽  
Concentrated Mass ◽  
Simple Manner ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Rigid Supports

Abstract A general expression is developed from which the frequency equation for the vibration of a constrained beam with any combination of intermediate elastic or rigid supports, concentrated masses, and sprung masses can be found readily. The method also is extended to the case where the constraint is a continuous elastic foundation or uniformly distributed load of any length. This method requires only the knowledge of the natural frequencies and natural modes of the beam supported at the ends in the same manner as the constrained beam but not subjected to any of the constraints between the ends. The frequency equation is obtained easily and can be solved to any desired degree of approximation for any number of modes of vibration in a quick and simple manner. Numerical examples are given for a beam with one concentrated mass, for a beam with one sprung mass, and a continuous beam with one sprung mass.

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 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.

Vibration Frequencies for a Uniform Beam With One End Elastically Supported and Carrying a Mass at the Other End

1975 ◽  
Vol 42 (4) ◽  
pp. 878-880 ◽  
Author(s):  
D. A. Grant
Keyword(s):  
Normal Modes ◽  
Frequency Equation ◽  
The Other ◽  
Concentrated Mass ◽  
Mass Moment ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Mass Moment Of Inertia ◽  
Elastically Supported ◽  
Range Of Values

In this paper the author obtains the frequency equation for the normal modes of vibration of uniform beams with linear translational and rotational springs at one end and having a concentrated mass at the other free end. The eigenfrequencies for the fundamental mode are given for a wide range of values of mass ratio, mass moment of inertia ratios, and stiffness ratios.

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