Response of Fluid-Elastically Coupled Coaxial Cylindrical Shells to External Flow

1977 ◽  
Vol 99 (2) ◽  
pp. 319-324 ◽  
Author(s):  
M. K. Au-Yang
Keyword(s):  
Vibration Analysis ◽  
Natural Frequencies ◽  
Numerical Solutions ◽  
Dynamic Pressure ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Coupled System ◽  
Coupling Coefficients ◽  
Coupled Equations

The dynamics of a system of two fluid-elastically coupled coaxial cylindrical shells is studied theoretically. The general equations of motion for free and forced-damped vibration are derived in terms of virtual mass, coupling coefficients, and uncoupled natural frequencies of the individual cylindrical shells. For free vibration, numerical solutions to the coupled equations of motion are given as a function of these parameters. For forced-damped vibration, solution is given to the special case when the external force is a normal one acting on the surface of the outer shell, such as the dynamic pressure forces arising from an external turbulent axial flow. It is shown that the coupled system can then be reduced to an equivalent single cylindrical shell. However, the effective force acting on the equivalent single cylinder, as well as its natural frequencies and effective damping ratios, are all modified from the corresponding uncoupled values. The response of the system can then be predicted by established methods in flow-induced random vibration analysis. Curves are included. The study aims mainly at applications to the vibration analysis of hydraulically coupled internal components of a pressurized nuclear reactor but is general enough to find application in other engineering disciplines.

Flow-Induced Vibration of Anisotropic Cylindrical Shells

2002 ◽  
Author(s):  
M. H. Toorani ◽  
A. A. Lakis
Keyword(s):  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Coupled System ◽  
Flow Induced Vibration ◽  
Critical Flow Velocity ◽  
Flowing Fluid ◽  
Coupled Equations ◽  
Stiffness Matrices ◽  
Analytical Integration

This paper deals with the vibration analysis of anisotropic laminated cylindrical shells conveying fluid. We focus on the axi-symmetric (n=0) and lateral (beam-like, n=1) vibration modes of the anisotropic cylindrical shells. Particularly important in this study is to obtain the natural frequencies of the fluid-structure coupled system and also to estimate the critical flow velocity at which the structure loses its stability. The coupled equations between the shell and the fluid are derived from a refined shell theory by taking into account the shear deformation effects. The displacement functions are obtained from the exact solution of refined shell equations and therefore the mass and stiffness matrices of the shell are determined by precise analytical integration. The added mass, stiffness and damping matrices of the fluid are obtained by an analytical integration of the fluid pressure over the liquid element. Thereafter, these matrices are coupled with the dynamic equation of the empty shell. The natural frequencies obtained with the shell partially or completely filled with liquid are in good agreement with those obtained experimentally and from other theories. The stability of the shell subjected to a flowing fluid is also studied. The shell’s anisotropy is discussed.

Small-amplitude free vibration analysis of active multistable shells under static multifield loading

2020 ◽  
pp. 107754632092393
Author(s):  
Dimitris Varelis
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Small Amplitude ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Shell Element ◽  
Curvilinear Coordinates ◽  
Coupled Equations ◽  
Modal Response

This study considers the small-amplitude free vibrational response performed on top of the quasi-static snap through buckling, which is accompanied by large displacements and rotations of shallow doubly curved laminated piezoelectric shells under multifield loading. The mechanics incorporate coupling between mechanical, electric, and thermal fields and encompass geometric nonlinearity effects due to large quasi-static displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. Based on the above mechanics and adopting the finite element methodology, an eight-node nonlinear shell element is developed to yield the linearized discrete coupled small-amplitude dynamic equations of motion. Initially, the nonlinear coupled equations are linearized and solved quasi-statically using an extended cylindrical arc-length method in combination with the Newton–Raphson iterative technique, and subsequently the free vibration analysis is performed at each solution point. Validation and evaluation cases on laminated cylindrical shells demonstrate the accuracy of the present method and its robust capability to predict the modal response on top of the nonlinear quasi-static response of active multistable shells subject to combined thermo–piezo–electromechanical loads. Numerical cases show the feasibility to develop smart shell structures to detect, via the monitoring of natural frequencies, the onset of snap-through instability. The capability of smart shells to actively modify its natural frequencies such as to promote or mitigate snap-through instabilities is quantified. Additional results quantify the effect of thermomechanical loads on actuation capability. The influence of geometric parameters (curvature and thickness) on the modal response is finally investigated.

Chebyshev Collocation Solutions for Vibration Analysis of Circular Cylindrical Shells with Arbitrary Boundary Conditions

2017 ◽  
Vol 17 (02) ◽  
pp. 1750020 ◽  
Author(s):  
Nuttawit Wattanasakulpong ◽  
Sacharuck Pornpeerakeat ◽  
Arisara Chaikittiratana
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Chebyshev Collocation Method ◽  
Arbitrary Boundary ◽  
Chebyshev Collocation ◽  
Various Boundary Conditions ◽  
Circular Cylindrical Shells

This paper applies the Chebyshev collocation method to finding accurate solutions of natural frequencies for circular cylindrical shells. The shells with different boundary conditions are considered in the parametric study. By using the method to solve the coupled differential equations of motion governing the vibration of the shell, numerical results are obtained from the algebraic eigenvalue equation using the Chebyshev differentiation matrices. And the results satisfy both the geometric and force boundary conditions. Based on the numerical examples, the proposed method shows its capacity and reliability in predicting accurate frequency results for circular cylindrical shells with various boundary conditions as compared to some exact solutions available in the literature.

scholarly journals The Normal Modes of Nonlinear n-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
R. M. Rosenberg
Keyword(s):  
Linear System ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Minimum Problem ◽  
Infinite Class ◽  
Coupled Equations ◽  
Geometrical Problem ◽  
Maximum Minimum

A system of n masses, equal or not, interconnected by nonlinear “symmetric” springs, and having n degrees of freedom is examined. The concept of normal modes is rigorously defined and the problem of finding them is reduced to a geometrical maximum-minimum problem in an n-space of known metric. The solution of the geometrical problem reduces the coupled equations of motion to n uncoupled equations whose natural frequencies can always be found by a single quadrature. An infinite class of systems, of which the linear system is a member, has been isolated for which the frequency amplitude can be found in closed form.

scholarly journals Natural Frequncies of Coupled Blade-Bending and Shaft-Torsional Vibrations

2007 ◽  
Vol 14 (1) ◽  
pp. 65-80 ◽  
Author(s):  
B.O. Al-Bedoor
Keyword(s):  
Finite Element ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Coupled Model ◽  
Small Deformation ◽  
Torsional Vibrations ◽  
Reduced Order ◽  
Rotating Speed ◽  
Coupled Equations ◽  
Stiffening Effect

In this study, the coupled shaft-torsional and blade-bending natural frequencies are investigated using a reduced order mathematical model. The system-coupled model is developed using the Lagrangian approach in conjunction with the assumed modes method to discretize the blade bending deflection. The model accounts for the blade stagger (setting) angle, the system rotating speed and its induced stiffening effect. The coupled equations of motion are linearized based on the small deformation theory for the blade bending and shaft torsional deformation to enable calculation of the system natural frequencies for various combinations of system parameters. The obtained coupled eignvalue system is ready for use as a reference for comparison for larger size finite element simulations and for the use as a fast check on natural frequencies for the coupled blade bending and shaft torsional vibrations in the design and diagnostics processes. Some results on the predicted natural frequencies are graphically presented and discussed pertinent to the coupling controlling factors and their effects. In addition, the predicted coupled natural frequencies are validated using the Finite Element Commercial Package (Pro-Mechanica) where good agreements are found.

Theoretical Analysis of Free Vibrations Based on a New High Order Shell Theory for Cylindrical Shells Conveying Fluid

2017 ◽  
Author(s):  
Ming Ji ◽  
Kazuaki Inaba
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Free Vibrations ◽  
High Order ◽  
Out Of Plane ◽  
Conveying Fluid

The natural frequencies of free vibrations for thick cylindrical shells with clamped-clamped ends conveying fluid are investigated. Equations of motion and boundary conditions are derived by Hamilton’s principle based on the new high order shell theory. The hydrodynamic force is derived from the linearized potential flow theory. Besides, fluid pressure acting on the shell wall is gotten by the assumption of non-penetration condition. The out-of-plane and in-plane vibrations are coupled together due to the existence of fluid-solid-interaction (FSI). Under the assumption of harmonic motion, the dispersion relationships are presented. Using the method of frequency sweeping, the natural frequencies of symmetric modes and asymmetric modes corresponding to each flow velocity are found by satisfying the dispersion relationship equations and boundary conditions. Several numerical examples with different flow velocities and thickness are presented compared with previous thin shell theory and FEM results and show reasonable agreement. The effects of thickness are discussed.

Dynamic Behavior of Damaged Bridge with Multi-Cracks Under Moving Vehicular Loads

2017 ◽  
Vol 17 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Xinfeng Yin ◽  
Yang Liu ◽  
Lu Deng ◽  
Xuan Kong
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Interaction Force ◽  
Vehicle Model ◽  
Coupled Equations ◽  
Contact Points ◽  
Local Flexibility ◽  
Bridge Structures ◽  
Road Surface Roughness

When studying the vibration of a bridge–vehicle coupled system, most researchers mainly focus on the intact or original bridge structures. Nonetheless, a large number of bridges were built long ago, and most of them have suffered serious deterioration or damage due to the increasing traffic loads, environmental effect, material aging, and inadequate maintenance. Therefore, the effect of damage of bridges, such as cracks, on the vibration of vehicle–bridge coupled system should be studied. The objective of this study is to develop a new method for considering the effect of cracks and road surface roughness on the bridge response. Two vehicle models were introduced: a single-degree-of-freedom (SDOF) vehicle model and a full-scale vehicle model with seven degrees of freedom (DOFs). Three typical bridges were investigated herein, namely, a single-span uniform beam, a three-span stepped beam, and a non-uniform three-span continuous bridge. The massless rotational spring was adopted to describe the local flexibility induced by a crack on the bridge. The coupled equations for the bridge and vehicle were established by combining the equations of motion for both the bridge and vehicles using the displacement relationship and interaction force relationship at the contact points. The numerical results show that the proposed method can rationally simulate the vibrations of the bridge with cracks under moving vehicular loads.

Non-Linear Fluid-Structure Interaction in Cylindrical Shells

2002 ◽  
Author(s):  
M. H. Toorani ◽  
A. A. Lakis
Keyword(s):  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Axial Flow ◽  
Present Theory ◽  
Nuclear Plant ◽  
Nuclear Industry ◽  
Linear Potential ◽  
Critical Flow Velocity ◽  
Fluid Structure ◽  
Non Linear

Nuclear plant reliability depends directly on its component performance. The higher heat transfer performance of nuclear plant components often requires higher flow velocities through the shell and tube heat exchangers. So, these cylindrical structures are subjected to either axial or cross flow, while the excessive flow-induced vibrations, (which are a major cause of machinery downtime; fatigue failure and high noise), limit the performance of these structures. On the other hand, these shell components often experience large amplitude vibrations that are greater than the shell thickness. Therefore, the evaluation of complex vibrational behavior of these structures is highly desirable in the nuclear industry. A semi-analytical approach has been developed in the present theory to predict the geometrical non-linearity influence on the natural frequencies of anisotropic cylindrical shells conveying axial flow. Particular important in this study is to obtain the natural frequencies of the coupled system of the fluid-structure, taking into account the geometrical non-linearity of the structure, and also estimating the critical flow velocity at which the structure loses its stability. The displacement functions, mass and stiffness matrices, linear and non-linear ones, of the structure are obtained by exact analytical integration over a hybrid element developed in this work. Linear potential flow theory is applied to describe the fluid effect that leads to the inertial, centrifugal and Coriolis forces. Numerical results are given and compared with those of experiment and other theories to demonstrate the practical application of the present method.

Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Masoud Ansari ◽  
Ebrahim Esmailzadeh ◽  
Nader Jalili
Keyword(s):  
Frequency Analysis ◽  
Time Domain ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Bending Vibrations ◽  
Resonant Condition ◽  
Mass System ◽  
Coupled Equations ◽  
Tip Mass ◽  
Gyroscopic Systems

An exact frequency analysis of a rotating beam with an attached tip mass is addressed in this paper while the beam undergoes coupled torsional-bending vibrations. The governing coupled equations of motion and the corresponding boundary condition are derived in detail using the extended Hamilton principle. It has been shown that the source of coupling in the equations of motion is the rotation and that the equations are linked through the angular velocity of the base. Since the beam-tip-mass system at hand serves as the building block of many vibrating gyroscopic systems, which require high precision, a closed-form frequency equation of the system should be derived to determine its natural frequencies. The frequency analysis is the basis of the time domain analysis, and hence, the exact frequency derivation would lead to accurate time domain results, too. Control strategies of the aforementioned gyroscopic systems are mostly based on their resonant condition, and hence, acquiring knowledge about their exact natural frequencies could lead to a better control of the system. The parameter sensitivity analysis has been carried out to determine the effects of various system parameters on the natural frequencies. It has been shown that even the undamped systems undergoing base rotation will have complex eigenvalues, which demonstrate a damping-type behavior.

A New Static Analysis Approach for Free Vibration of Beams

2018 ◽  
Vol 10 (01) ◽  
pp. 1850004 ◽  
Author(s):  
C. W. Lim ◽  
Zhenyu Chen
Keyword(s):  
Free Vibration ◽  
Static Analysis ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Numerical Solutions ◽  
Free Vibration Analysis ◽  
Concentrated Force ◽  
Static Approach ◽  
Beam Bending ◽  
The Relationship

This study deals with a new method for the free vibration analysis of beams under different boundary conditions. We show that it is possible to apply a static approach for solving free vibration systems, i.e., we obtain natural frequencies for free vibration of beams by analyzing static beam bending problems. Specifically, the basic governing equation for beams with harmonic loadings and resting on an elastic foundation is solved and the solutions are used directly to yield the beam free vibration solutions. In the free vibration analysis, the natural frequency can be a real number or an imaginary number while in the static analysis, the foundation stiffness can be either positive or negative. We show that one can solve the deflection of a beam subjected to a given concentrated force and subsequently deduce the possible infinite deflection when the stiffness becomes zero or negative. In such cases, there exists an equivalent relationship between the free vibration frequencies and the negative stiffness. Consequently, determining the natural frequencies becomes a problem of determining an appropriate negative foundation elastic constant. In general, the numerical vibration solutions can be obtained by analyzing the relationship between loadings and frequencies. For comparison, a comparison with the classical free vibration solutions is presented and excellent agreement is illustrated. We further show that this static approach for free vibration solutions has a clear edge over the classical free vibration approach in computational beam vibration solutions. Very accurate and convergent numerical solutions can be obtained using a very simple numerical solution method. This static approach for free vibration problems can be extended for plate, shell and other structural systems.

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