Vibrations of Suspended Cables

1983 ◽  
Vol 50 (3) ◽  
pp. 687-689
Author(s):  
J. G. Gale ◽  
C. E. Smith
Keyword(s):  
Natural Frequencies ◽  
Analytical Investigation ◽  
Plane Motion ◽  
Mode Shapes ◽  
Arc Length ◽  
Power Series Solutions ◽  
Out Of Plane ◽  
Independent Variable ◽  
Suspended Cables ◽  
Frequency Ratios

An analytical investigation of the small, normal-mode motions of a homogeneous, inextensible, perfectly flexible cable suspended in a gravitational field was made. With cable arc length as the independent variable, the differential equations that govern the mode shapes have irrational coefficients. A transformation of the independent position variable yields equations that have polynominal coefficients, which then lend themselves to power series solutions. Natural frequencies of oscillation and corresponding mode shapes are determined from these solutions. Figures showing the natural frequency ratios for a variety of cable support geometries are presented for both in-plane and out-of-plane motion.

In-Plane Vibration Modes of Arbitrarily Thick Disks

1997 ◽  
Author(s):  
Kevin I. Tzou ◽  
Jonathan A. Wickert ◽  
Adnan Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

Abstract The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

Dynamic Characteristics of Tsing Ma Suspension Bridge

1996 ◽  
Vol 1541 (1) ◽  
pp. 43-50 ◽  
Author(s):  
Siu Kui Au ◽  
Neil Mickleborough ◽  
Paul N. Roschke
Keyword(s):  
Analytical Solutions ◽  
Bridge Deck ◽  
Natural Frequencies ◽  
Dynamic Properties ◽  
Suspension Bridge ◽  
Mode Shapes ◽  
Out Of Plane ◽  
Quantitative Results ◽  
Suspension Cable ◽  
Bending Modes

Numerical simulation was carried out to determine the dynamic properties of the Tsing Ma Suspension Bridge. Both the structure as a whole and individual subcomponents were modeled. Classical analytical solutions for simplified models from the available literature were compared with the results obtained from a finite-element code. Quantitative results for static deflection, natural frequencies, and mode shapes were compared with analytical solutions from linear theory. Out-of-plane modes were shown to be dominant. For in-plane antisymmetric and symmetric bending modes, in which the suspension cable and bridge deck vibrate in the same direction, the natural frequency of the main span of the bridge is determined to be approximately equal to the square root of the sum of the squares of the frequencies of the cable and bridge deck.

Managing Vortex Induced Vibration in Well Jumper Systems

2008 ◽  
Author(s):  
Kenneth Bhalla ◽  
Lixin Gong
Keyword(s):  
Natural Frequency ◽  
Natural Frequencies ◽  
Cross Flow ◽  
Mitigation Measures ◽  
Mode Shapes ◽  
Bottom Current ◽  
Vortex Induced Vibration ◽  
Flow Response ◽  
Out Of Plane ◽  
Lock In

The purpose of this paper is to present a method that has been developed to identify if vortex induced vibration (VIV) occurs in well jumper systems. Moreover, a method has been developed to determine when VIV mitigation measures such as strakes are required. The method involves determining the in-plane and out-of-plane natural frequencies and mode shapes. The natural frequencies are then used, in conjunction with the maximum bottom current expected at a given location to determine if suppression is required. The natural frequency of a jumper system is a function of many variables, e.g. span length, leg height, pipe diameter and thickness, buoyancy placement, buoyancy uplift, buoyancy OD, insulation thickness, and contents of the jumper. The suppression requirement is based upon calculating a lower bound lock-in current speed based upon an assumed velocity bandwidth centered about the lock-in current. The out-of-plane VIV cross-flow response is produced by a current in the plane of the jumper; whereas the in-plane VIV cross-flow response is produced by the out-of-plane current. Typically, the out-of-plane natural frequency is smaller than the in-plane natural frequency. Jumpers with small spans have higher natural frequencies; thus small span jumpers may require no suppression or suppression on the vertical legs. Whereas, larger span jumpers may require no suppression, suppression on the vertical legs or suppression on all the legs. The span of jumper systems (i.e. production, water injection, gas lift/injection ...) may vary in one given field; it has become apparent that not all jumper systems require suppression. This technique has allowed us to recognize when certain legs of a given jumper system may require suppression, thus leading to a jumper design whose safety is not compromised while in the production mode, as well as minimizing downtime and identifying potential savings from probable fatigue failures.

Free Vibration of Damaged Frame Structures Considering the Effects of Axial Extension, Shear Deformation and Rotatory Inertia: Exact Solution

2017 ◽  
Vol 17 (10) ◽  
pp. 1750111
Author(s):  
Ugurcan Eroglu ◽  
Ekrem Tufekci
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Shear Deformation ◽  
Natural Frequencies ◽  
Plane Motion ◽  
Mode Shapes ◽  
Frame Structures ◽  
Crack Identification ◽  
Rotatory Inertia ◽  
Axial Extension

In this paper, a procedure based on the transfer matrix method for obtaining the exact solution to the equations of free vibration of damaged frame structures, considering the effects of axial extension, shear deformation, rotatory inertia, and all compliance components arising due to the presence of a crack, is presented. The crack is modeled by a rotational and/or translational spring based on the concept of linear elastic fracture mechanics. Only the in-plane motion of planar structures is considered. The formulation is validated through some examples existing in the literature. Additionally, the mode shapes and natural frequencies of a frame with pitched roof are provided. The variation of natural frequencies with respect to the crack location is presented. It is concluded that considering the axial compliance, and axial-bending coupling due to the presence of a crack results in different dynamic characteristics, which should be considered for problems where high precision is required, such as for the crack identification problems.

Dynamic Analysis of an Elevator Traveling Cable Using a Singularity-Free Beam Formulation

2017 ◽  
Vol 84 (4) ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Dynamic Analysis ◽  
Multibody Dynamics ◽  
Natural Frequencies ◽  
Vertical Motion ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Out Of Plane ◽  
Forced Responses ◽  
Free Beam ◽  
Good Agreement

A round elevator traveling cable is modeled using a singularity-free beam formulation. Equilibria of the traveling cable with different elevator car positions are studied. Natural frequencies and the corresponding mode shapes of the traveling cable are calculated and they are in excellent agreement with those calculated by abaqus. In-plane natural frequencies of the traveling cable do not change much with the car position compared with its out-of-plane ones. Dynamic responses of the traveling cable are calculated and they are in good agreement with those from commercial multibody dynamics software recurdyn. Effects of vertical motion of the car on free responses of the traveling cable and those of in-plane and out-of-plane building sways on forced responses are investigated.

Buckling Loads and Natural Frequencies of Drill Bits and Fluted Cutters

1984 ◽  
Vol 106 (3) ◽  
pp. 196-204 ◽  
Author(s):  
E. B. Magrab ◽  
D. E. Gilsinn
Keyword(s):  
Natural Frequencies ◽  
Twist Angle ◽  
Axial Loading ◽  
Basis Functions ◽  
Mode Shapes ◽  
Euler Beam ◽  
Buckling Loads ◽  
Drill Bits ◽  
Out Of Plane ◽  
Beam Type

The buckling loads, natural frequencies and mode shapes of twist-drill bits and certain fluted cutters under a variety of combinations of twist angle, cross-section geometry, and axial loading have been obtained. The drill bit is modelled as a twisted Euler beam under axial loading that is clamped at both ends. The governing system of differential equations is solved by the Galerkin procedure. Explicit forms for the basis functions used to generate the Galerkin coefficients are presented in general form in an appendix. They may be used for obtaining numerical results for that class of problems which use the Rayleigh-Ritz-Galerkin methods with beam-type functions as the basis functions. The representative set of modes obtained exhibit a complex out-of-plane twisting-type motion that suggests a possible explanation for the out-of-roundness of certain drilled holes.

Role of Dynamic Testing in Assessment of Bridges

1997 ◽  
Vol 1594 (1) ◽  
pp. 115-124 ◽  
Author(s):  
P. C. Das ◽  
J. S. Owen ◽  
B. J. Eccles ◽  
M. A. Woodings ◽  
B. S. Choo
Keyword(s):  
Reinforced Concrete ◽  
Natural Frequencies ◽  
Concrete Beams ◽  
Dynamic Testing ◽  
Reinforced Concrete Beams ◽  
Mode Shapes ◽  
Ambient Conditions ◽  
Frequency Changes ◽  
Frequency Ratios

Six reinforced concrete beams were loaded incrementally up to failure. After each increment the load was removed and measurements of the modal properties of the beams were made by impulse testing. The variation of the natural frequencies, frequency ratios, mode shapes, and the level of damage were investigated. It was found that on completion of the tests the natural frequencies of the beams had been reduced by an average of 25 percent in each mode. However, changes in mode shape were very small, and appreciable differences were only observed when the damage was highly localized. Modeling of the beam by using finite elements predicted trends that compared well with experimental observations. It is concluded that if dynamic testing were used in monitoring reinforced concrete structures, then the changes in frequency due to initial concrete cracking or yield of the reinforcement could be detected. More useful information associated with the spread and type of cracking through a structure may be detectable, although the level of the frequency changes is of the same order as those due to changes in ambient conditions.

In-Plane Vibration Modes of Arbitrarily Thick Disks

1998 ◽  
Vol 120 (2) ◽  
pp. 384-391 ◽  
Author(s):  
K. I. Tzou ◽  
J. A. Wickert ◽  
A. Akay
Keyword(s):  
Natural Frequencies ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Three Dimensional Model ◽  
Bending Vibration ◽  
Annular Disk ◽  
Out Of Plane ◽  
Plane Bending

The three-dimensional vibration of an arbitrarily thick annular disk is investigated for two classes of boundary conditions: all surfaces traction-free, and all free except for the clamped inner radius. These two models represent limiting cases of such common engineering components as automotive and aircraft disk brakes, for which existing models focus on out-of-plane bending vibration. For a disk of significant thickness, vibration modes in which motion occurs within the disk’s equilibrium plane can play a substantial role in-setting its dynamic response. Laboratory experiments demonstrate that in-plane modes exist at frequencies comparable to those of out-of-plane bending even for thickness-to-diameter ratios as small as 10−1. The equations for three-dimensional motion are discretized through the Ritz technique, yielding natural frequencies and mode shapes for coupled axial, radial, and circumferential deformations. This treatment is applicable to “disks” of arbitrary dimension, and encompasses classical models for plates, bars, cylinders, rings, and shells. The solutions so obtained converge in the limiting cases to the values expected from the classical theories, and to ones that account for shear deformation and rotary inertia. The three-dimensional model demonstrates that for geometries within the technologically-important range, the natural frequencies of certain in- and out-of-plane modes can be close to one another, or even identically repeated.

Effects of Temperature Variation on Vibration of a Cable-Stayed Beam

2017 ◽  
Vol 17 (10) ◽  
pp. 1750123 ◽  
Author(s):  
Yaobing Zhao ◽  
Zhiqian Wang ◽  
Xiaoyu Zhang ◽  
Lincong Chen
Keyword(s):  
Thermal Effect ◽  
Temperature Variation ◽  
Natural Frequencies ◽  
Dominant Role ◽  
Plane Motion ◽  
Tension Force ◽  
Effect Of Temperature ◽  
Stiffness Ratio ◽  
Initial Tension ◽  
Out Of Plane

This paper is concerned with the temperature effect on the vibration of a cable-stayed beam. The thermal effect is considered by using two non-dimensional factors for the cable tension force and sag. The nonlinear in-plane and out-of-plane vibration equations of motion of the cable-stayed beam with thermal effect are derived by using the extended Hamilton’s principle. Eigenvalue analysis is performed to obtain closed-form eigenvalue solutions. It is shown that the effect of temperature variation plays a dominant role on the vibration behavior of the cable-stayed beam, and the effect is closely related with the initial tension force and the stiffness ratio. As to the in-plane motion, both positive and negative correlations between the temperature variations and natural frequencies are found, which depend on the mode order and the stiffness ratio of the cable-stayed beam. However, as to the out-of-plane motion, there only exist negative correlations between the temperature variation and natural frequencies, and the effect of temperature dropping condition seems more obvious. Moreover, both for the in-plane and out-of-plane motions, the locations of veering points between two natural frequencies are shifted under the thermal effect, which can significantly affect the internal resonance between different modes of the cable-stayed beam.

Free Out-of-Plane Vibration of a Ring Elastically Supported at Several Points

1982 ◽  
Vol 49 (4) ◽  
pp. 854-860 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
H. Okada
Keyword(s):  
Differential Equation ◽  
Infinite Series ◽  
Natural Frequencies ◽  
Circular Ring ◽  
Mode Shapes ◽  
Matrix Differential Equation ◽  
Plane Vibration ◽  
Out Of Plane ◽  
Matrix Differential ◽  
Elastically Supported

An analysis is presented for the free out-of-plane vibration of a circular ring elastically supported against deflection, rotation, and torsion at several points located at equal angular intervals. The equations of out-of-plane vibration of the ring is expressed as a matrix differential equation by using the transfer matrix, the solution to which is conveniently given by infinite series. The vibrations arising in the ring are classified into several types, for each of which the natural frequencies and the mode shapes are calculated numerically up to higher modes.

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