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.

scholarly journals Effect of intermediate supports location on natural frequencies of multiple cracked continuous beams

2018 ◽  
Author(s):  
Do Nam ◽  
Nguyen Tien Khiem ◽  
Le Khanh Toan ◽  
Nguyen Thi Thao ◽  
Pham Thi Ba Lien
Keyword(s):  
General Solution ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Continuous Beam ◽  
Cracked Beam ◽  
Continuous Beams ◽  
Simplified Method ◽  
Rigid Supports ◽  
Natural Frequency Analysis ◽  
Eigenfrequency Spectrum

The present paper deals with free vibration of multiple cracked continuous beams with intermediate rigid supports. A simplified method is proposed to obtain general solution of free vibration in cracked beam with intermediate supports that is then used for natural frequency analysis of the beam in dependence upon cracks and support locations. Numerical results show that the support location or ratio of span lengths in combination with cracks makes a significant effect on eigenfrequency spectrum of beam. The discovered effects of support locations on eigenfrequency spectrum of cracked continuous beam are useful for detecting not only cracks but also positions of vanishing deflection on the beam.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

Node Control of Uniform Beams Subject to Various Boundary Conditions

1992 ◽  
Vol 59 (4) ◽  
pp. 983-990 ◽  
Author(s):  
L. Weaver ◽  
L. Silverberg
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Controlled System ◽  
Free Boundary Conditions ◽  
Various Boundary Conditions ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Control Forces ◽  
Direct Feedback

This paper introduces node control, whereby discrete direct feedback control forces are placed at the nodes of the N+1th mode (the lowest N modes participate in the response). Node control is motivated by the node control theorem which states, under certain conditions, that node control preserves the natural frequencies and natural modes of vibration of the controlled system while achieving uniform damping. The node control theorem is verified for uniform beams with pinned-pinned, cantilevered, and free-free boundary conditions, and two cases of beams with springs on the boundaries. A general proof of the node control theorem remains elusive.

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.

Numerical Analysis of Forced Vibrations of Beams

1953 ◽  
Vol 20 (1) ◽  
pp. 53-56
Author(s):  
N. O. Myklestad
Keyword(s):  
Natural Frequencies ◽  
Driving Forces ◽  
Bending Moments ◽  
Internal Damping ◽  
Shear Forces ◽  
Natural Modes ◽  
Tabular Method ◽  
Modes Of Vibration ◽  
The One ◽  
Phase Angles

Abstract In this paper a simple tabular method is developed by which the vibration amplitudes, bending moments, and shear forces of a beam of variable but symmetrical cross section, carrying any number of concentrated masses and acted on by any number of harmonically varying forces, can be found. The driving forces must all have the same frequency but the phase angles may be different. The method is an extension of the one employed by the author to find natural modes of vibration of beams, but in the case of forced vibration only one application of the tabular calculations is necessary, making it essentially a far simpler problem than that of finding the natural modes. Internal damping of the beam material is easily considered and should always be taken into account if there is any danger that the forced frequency is near any one of the natural frequencies.

A Theoretical and Experimental Study of Vibrations of Thick Circular Cylindrical Shells and Rings

1988 ◽  
Vol 110 (4) ◽  
pp. 533-537 ◽  
Author(s):  
R. K. Singal ◽  
K. Williams
Keyword(s):  
Cylindrical Shells ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Theory Of Elasticity ◽  
Frequency Equation ◽  
Experimental Investigations ◽  
Resonant Frequencies ◽  
Modes Of Vibration ◽  
Circular Cylindrical Shells ◽  
Experimental Values

The free vibrations of thick circular cylindrical shells and rings are discussed in this paper. The well-known energy method, which is based on the three-dimensional theory of elasticity, is used in the derivation of the frequency equation of the shell. The frequency equation yields resonant frequencies for all the circumferential modes of vibration, including the breathing and beam-type modes. Experimental investigations were carried out on several models in order to assess the validity of the analysis. This paper first describes briefly the method of analysis. In the end, the calculated frequencies are compared with the experimental values. A very close agreement between the theoretical and experimental values of the resonant frequencies for all the models was obtained and this validates the method of analysis.

scholarly journals Dynamic testing of a modern concrete bridge

1979 ◽  
Vol 6 (3) ◽  
pp. 447-455 ◽  
Author(s):  
J. H. Rainer ◽  
G. Pernica
Keyword(s):  
Natural Frequencies ◽  
Dynamic Testing ◽  
Dynamic Properties ◽  
Mode Shapes ◽  
Total Width ◽  
Concrete Bridge ◽  
Reinforced Concrete Bridge ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Good Agreement

A posttensioned reinforced concrete bridge, slated for demolition, was tested to obtain its dynamic properties. The 10 year old bridge consisted of a continuous flat slab deck of variable thickness having a total width of 103 ft (31.39 m) and spans of 28 ft 6 in. (8.69 m), 71 ft 0 in. (21.64 m), and 42 ft 6 in. (12.95 m). The entire bridge was skewed 10°50′ and the deck was slightly curved in plan.The mode shapes, natural frequencies, and damping ratios for the lowest five natural modes of vibration were determined using sinusoidal forcing functions from an electrohydraulic shaker. These modes, located at 5.7, 6.4, 8.7, 12.0, and 17.4 Hz, were found to be highly dependent on the lateral properties of the bridge deck. Damping ratios were determined from the widths of resonance peaks. The modal properties from the steady state excitation were compared with those obtained from measurements of traffic-induced vibrations and good agreement was found between the two methods.

On the Nature of Natural Control

1989 ◽  
Vol 111 (4) ◽  
pp. 412-422 ◽  
Author(s):  
L. Silverberg ◽  
M. Morton
Keyword(s):  
Control Systems ◽  
Structural Control ◽  
Natural Frequencies ◽  
Dynamic Performance ◽  
Natural Control ◽  
Space Truss ◽  
Natural Modes ◽  
Modal Damping ◽  
Modes Of Vibration ◽  
Control Forces

This paper examines families of structural control systems and reveals inherent properties that provide the essential motivation behind the theory of Natural Control. It is determined that the associated fuel consumed by the controls is near minimal when the natural frequencies are identical to the controlled modal frequencies, and when the natural modes of vibration are identical to the controlled modes of vibration. Also, by casting the objective to suppress vibration in the form of an exponential stability condition, it is found that vibration is most efficiently suppressed when the modal damping rates are identical to a designer chosen decay rate. The use of a limited number of control forces over distributed control is characterized by a change in fuel consumed by the controls and by a deterioration in the dynamic performance reflected by changes in the modal damping rates. The Natural Control of a space truss demonstrates the results.

scholarly journals Flexural vibrations of the walls of thin cylindrical shells having freely supported ends

1949 ◽  
Vol 197 (1049) ◽  
pp. 238-256 ◽  
Keyword(s):  
Experimental Investigation ◽  
Natural Frequency ◽  
Energy Method ◽  
Natural Frequencies ◽  
Frequency Equation ◽  
Diameter Ratio ◽  
Frequency Range ◽  
Thin Cylinder ◽  
Nodal Pattern ◽  
Modes Of Vibration

The paper deals with the general equations for the vibration of thin cylinders and a theoretical and experimental investigation is made of the type of vibration usually associated with bells. The cylinders are supported in such a manner that the ends remain circular without directional restraint being imposed. It is found that the complexity of the mode of vibration bears little relation to the natural frequency; for example, cylinders of very small thicknessdiameter ratio, with length about equal to or less than the diameter, may have many of their higher frequencies associated with the simpler modes of vibration. The frequency equation which is derived by the energy method is based on strain relations given by Timoshenko. In this approach, displacement equations are evolved which are comparable to those of Love and Flugge, though differences are evident due to the strain expressions used by each author. Results are given for cylinders of various lengths, each with the same thickness-diameter ratio, and also for a very thin cylinder in which the simpler modes of vibration occur in the higher frequency range. It is shown that there are three possible natural frequencies for a particular nodal pattern, two of these normally occurring beyond the aural range.

Natural Modes and Natural Frequencies of Uniform, Circular, Free-Edge Plates

1979 ◽  
Vol 46 (2) ◽  
pp. 448-453 ◽  
Author(s):  
K. Itao ◽  
S. H. Crandall
Keyword(s):  
Circular Plate ◽  
Isotropic Material ◽  
Natural Frequencies ◽  
Free Edge ◽  
Poisson's Ratio ◽  
Thin Circular Plate ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Homogeneous Isotropic Material ◽  
Free Edges

The natural modes and natural frequencies for the first 701 modes of vibration of a uniform thin circular plate with free edges are tabulated for a homogeneous isotropic material with Poisson’s ratio ν = 0.330.

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