Axisymmetric Vibrations of a Piezoelectric Spherical Shell Submerged in a Compressible Viscous Fluid Medium

2010 ◽  
Vol 132 (6) ◽  
Author(s):  
Juxi Hu ◽  
Zhiping Qiu ◽  
Tsung-Chow Su
Keyword(s):  
Viscous Fluid ◽  
Natural Frequencies ◽  
Essential Feature ◽  
Elastic Shell ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Compressible Viscous Fluid ◽  
Previous Theory ◽  
Fluid Medium ◽  
Shell Motion

Axisymmetric vibrations of a hollow piezoelectric sphere submerged in a compressible viscous fluid medium are investigated. The piezoelectric sphere is radially polarized. The differential equations governing the shell motion are obtained by the use of Hamilton’s principle. Based on the classical bending theory of shells, it is shown that all the piezoelectric contributions can be included in the in vacuo natural frequencies and their corresponding mode shapes. As such, the previous theory on elastic shell vibration becomes readily extendable. The flow field, determined by the boundary layer theory, is coupled to the shell motion through no-slip and no-penetrating conditions. It is found that the contribution of the piezoelectric parameters in the thin shell’s free vibration is of small order and is negligible. Natural frequencies and their associated vibration characteristics are numerically obtained and presented for a Polyvinglindene fluoride (PVDF) shell submerged in water. Dynamic responses of a submerged piezoelectric sherical shell, and the associated radiation of sound are investigated. The oscillations are harmonically driven by an axisymmetrically applied electric potential difference across the surface of the shell. The vibrational, fluid loading, and energy flow characteristics are derived and evaluated for a PVDF shell submerged in water. The essential feature of the modal response is determined by various critical frequencies, such as resonant frequencies and vibration-absorbing frequencies. Viscous effect is found noticeable in several cases.

Wave Motion of a Compressible Viscous Fluid Contained in a Cylindrical Shell

1993 ◽  
Vol 115 (3) ◽  
pp. 302-312 ◽  
Author(s):  
J. H. Terhune ◽  
K. Karim-Panahi
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Imaginary Part ◽  
Fluid Velocity ◽  
Wave Number ◽  
Mode Shapes ◽  
Infinite Length ◽  
Complex Frequency ◽  
Compressible Viscous Fluid ◽  
Fluid Velocity Field

The free vibration of cylindrical shells filled with a compressible viscous fluid has been studied by numerous workers using the linearized Navier-Stokes equations, the fluid continuity equation, and Flu¨gge ’s equations of motion for thin shells. It happens that solutions can be obtained for which the interface conditions at the shell surface are satisfied. Formally, a characteristic equation for the system eigenvalues can be written down, and solutions are usually obtained numerically providing some insight into the physical mechanisms. In this paper, we modify the usual approach to this problem, use a more rigorous mathematical solution and limit the discussion to a single thin shell of infinite length and finite radius, totally filled with a viscous, compressible fluid. It is shown that separable solutions are obtained only in a particular gage, defined by the divergence of the fluid velocity vector potential, and the solutions are unique to that gage. The complex frequency dependence for the transverse component of the fluid velocity field is shown to be a result of surface interaction between the compressional and vortex motions in the fluid and that this motion is confined to the boundary layer near the surface. Numerical results are obtained for the first few wave modes of a large shell, which illustrate the general approach to the solution. The axial wave number is complex for wave propagation, the imaginary part being the spatial attenuation coefficient. The frequency is also complex, the imaginary part of which is the temporal damping coefficient. The wave phase velocity is related to the real part of the axial wave number and turns out to be independent of frequency, with numerical value lying between the sonic velocities in the fluid and the shell. The frequency dependencies of these parameters and fluid velocity field mode shapes are computed for a typical case and displayed in non-dimensional graphs.

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.

scholarly journals Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7279
Author(s):  
Jin Wei ◽  
Tao Yu ◽  
Dongping Jin ◽  
Mei Liu ◽  
Dengqing Cao ◽  
...  
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Natural Frequencies ◽  
Great Influence ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Orthogonality Relations ◽  
Tip Mass

A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

DYNAMIC ANALYSIS OF FULLY ANCHORED CIRCULAR CYLINDRICAL LIQUIDS’ STORAGE STEEL TANKS USING FINITE ELEMENT METHOD

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Hanadi Abdulridha Lateef ◽  
Abdulamir Atalla
Keyword(s):  
Finite Element ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Elastic Shell ◽  
Shell Element ◽  
Liquid Level ◽  
Mode Shapes ◽  
Rigid Base ◽  
Steel Tanks ◽  
Shell Finite Element

A vibration analysis of circular cylindrical steel liquid storage tanks anchored to rigid base is conducted. Empty, partially and completely liquid filled tanks are considered as well as tanks composed of two courses using ANSYS 11.0 finite element package. The tank wall is modeled using linear elastic shell finite element and a new method, based on the added mass approach, is developed to model the effect of the contained liquid. In this method the properties of the shell element is modified to include the effect of the contained liquid. The analysis includes four tank case studies which are empty, fully filled with water, and filled with changeable liquid level in addition to study the effect of the variable thickness of tank on the natural frequencies and mode shapes. The results show that the natural frequency of completely filled tall tank may be less by 70.7% than the natural frequency of empty tank. It is also found that a maximum value of natural frequency can be obtained when the lower thick course consists 0.75 of tank height and its thickness is four times that of the upper one. The natural frequencies decrease with the increasing in liquid level for tall tank. The natural frequency of completely filled tank is less by 70.7% than the natural frequency of empty tank.

scholarly journals Influence of Excitation on Dynamic System Identification for a Multi-Span Reinforced Concrete Bridge

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
M. Alwash ◽  
B. F. Sparling ◽  
L. D. Wegner
Keyword(s):  
Reinforced Concrete ◽  
Natural Frequencies ◽  
Harmonic Excitation ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Concrete Bridge ◽  
Random Loading ◽  
Reinforced Concrete Bridge ◽  
Random Forcing ◽  
Vehicle Loading

In vibration-based damage detection, changes to structural modal properties are tracked over time in order to infer the current state of damage or deterioration. As such, the ability to obtain reliable estimates of modal parameters, particularly natural frequencies and mode shapes, is of critical importance. In the present study, the influence of the dynamic excitation source on the accuracy and statistical uncertainty of modal property estimates for a three span reinforced concrete bridge was investigated experimentally and numerically. Comparisons were made between the dynamic responses due to vehicle loading, harmonic and random forcing, impact, and environmental excitation. It was demonstrated that natural frequencies and mode shapes extracted from the free vibration response following vehicle and random loading events were of higher quality than corresponding values determined during the forcing phase of those events. Harmonic excitation at resonant frequencies and impact were also found to produce statistically reliable results.

scholarly journals Axisymmetric Free Vibration of Thick Orthotropic Hemispherical Shells Under Various Edge Conditions

1991 ◽  
Author(s):  
C. C. Chao ◽  
T. P. Tung ◽  
Y. C. Chern
Keyword(s):  
Free Vibration ◽  
Shear Strain ◽  
Transverse Shear ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Elastic Shell ◽  
Mode Shapes ◽  
Transverse Shear Strain ◽  
Equilibrium Equations ◽  
Hemispherical Shells

Abstract Axisymmetric free vibration of moderately thick polar orthotropic hemispherical shells are studied under the various boundary conditions of sliding, guided pin, clamped and hinged edges. Based on the improved linear elastic shell theory with the transverse shear strain and rotatory inertia taken into account, the dynamic equilibrium equations are formulated and transformed into the displacement form in terms of mid-surface meridian and radial displacements and parallel circle cross-section rotation. These partial differential equations are solved by the Galerkin method using proper Legendre polynomials as admissible displacement functions with the aid of the orthogonality and a number of special integral relations. Numerical results of the present theory compare well with existing data, which is available only in the isotropic theories. Good convergence is obtained for natural frequencies and mode shapes. Study of the effects of thickness and modulus ratio reveals higher frequencies for the thicker and/or stiffer shells with E\ oriented parallel to the meridians. Ranking of the natural frequencies descends in the order of guided pins, sliding, clamped and hinged edges in general. Also seen are the effects of transverse shear strain from the mode shapes with clamped and sliding edges on the slant. For the guided pin and sliding edges, frequencies increase fast as thickness increases so that new fundamental modes are generated in filling up the “frequency gap”. These are the new discoveries in the field of anisotropic shells, as a result of polar orthotropy of shell material and construction.

A Linear Model of Stationary Elevator Traveling and Compensation Cables

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Linear Model ◽  
Nonlinear Model ◽  
Natural Frequencies ◽  
Full Scale ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Circular Loop ◽  
Full Scale Experiment ◽  
Scale Experiment ◽  
Lower Loop

Based on a recent asymptotic analysis of a nonlinear model of elevator traveling and compensation cables, a computationally efficient, linear model is developed for calculating the natural frequencies, mode shapes, and dynamic responses of stationary elevator traveling and compensation cables. The linear cable model consists of two vertical cable segments connected by a half circular loop at the bottom. The two vertical cable segments are modeled as a string with a variable tension due to the weight of the cable. The horizontal displacements of the cable segments consist of boundary induced displacements and internal displacements, where the boundary induced displacements are interpolated from the displacements of the two ends of the cable segments, and the internal displacements satisfy the corresponding homogeneous boundary conditions of the cable segments. The horizontal displacement of the lower loop is interpolated from those of the two lower ends of the two cable segments, and the bending stiffness of the lower loop is represented by a spring with a constant stiffness, which can be calculated from the nonlinear model. Given a car position, the natural frequencies and mode shapes of an elevator traveling or compensation cable are calculated using the linear model and compared with those from the nonlinear model. The calculated natural frequencies are also compared with those from a full-scale experiment. In addition, the dynamic responses of a cable under a boundary excitation are calculated and compared with those from the nonlinear model. There is a good agreement between the predictions from the linear and nonlinear models and between the measured natural frequencies from a full-scale experiment and the corresponding calculated ones.

Modal Analysis and Dynamic Responses of a Hysteretically Damped Axle Shaft With a Transverse Crack

2019 ◽  
Author(s):  
C. Shravankumar ◽  
Yash K. Sarda ◽  
V. Thamarai Selvan
Keyword(s):  
Modal Analysis ◽  
Condition Monitoring ◽  
Crack Detection ◽  
Natural Frequencies ◽  
Fatigue Cracks ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Fast Fourier Transformation ◽  
Transverse Loads ◽  
Eigen Value

Abstract An axle shaft supports rotating elements, and is fitted to the housing by means of bearings. It mostly does not transmit torque, with exceptions such as in train axles. Non-rotating axles are subjected to bending moments due to dynamic transverse loads. Axles such as in automobiles are marked with occasional failures due to fatigue cracks, which can prove serious, if the cracks are not detected early. Vibration based condition monitoring is the field concerned with crack detection based on the dynamic responses of the system. In this light, the present paper discusses the vibration analysis of a cracked axle. The cracked shaft is modelled using finite element method, for transverse vibration conditions. The shaft is modelled based on Euler-Bernoulli theory for bending, while the crack is modelled based on fracture mechanics approach. After modelling, modal analysis of the system is carried out, with the consideration of proportional hysteretic damping. The Eigen value problem provides the natural frequencies and mode shapes. The Frequency Response Functions (FRF’s) magnitude and phase plots are obtained, from which the natural frequencies and structural damping loss factors can be calculated. Further, the free vibration and forced vibration system time responses are obtained, using numerical integration methods. The corresponding responses in frequency domain are obtained using Fast Fourier Transformation (FFT). The FRF’s and dynamic responses of the shaft without and with crack are comparatively studied. The study provides the platform for condition monitoring of shaft cracks.

scholarly journals Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method

2017 ◽  
Vol 2017 ◽  
pp. 1-26 ◽  
Author(s):  
Taehyun Kim ◽  
Ilwook Park ◽  
Usik Lee
Keyword(s):  
Modal Analysis ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Moving Loads ◽  
Analysis Method ◽  
Simply Supported

The modal analysis method (MAM) is very useful for obtaining the dynamic responses of a structure in analytical closed forms. In order to use the MAM, accurate information is needed on the natural frequencies, mode shapes, and orthogonality of the mode shapes a priori. A thorough literature survey reveals that the necessary information reported in the existing literature is sometimes very limited or incomplete, even for simple beam models such as Timoshenko beams. Thus, we present complete information on the natural frequencies, three types of mode shapes, and the orthogonality of the mode shapes for simply supported Timoshenko beams. Based on this information, we use the MAM to derive the forced vibration responses of a simply supported Timoshenko beam subjected to arbitrary initial conditions and to stationary or moving loads (a point transverse force and a point bending moment) in analytical closed form. We then conduct numerical studies to investigate the effects of each type of mode shape on the long-term dynamic responses (vibrations), the short-term dynamic responses (waves), and the deformed shapes of an example Timoshenko beam subjected to stationary or moving point loads.

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