Natural Frequencies and Critical Buckling Loads of Marine Risers

1988 ◽  
Vol 110 (1) ◽  
pp. 2-8 ◽  
Author(s):  
Y. C. Kim
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Simple Procedure ◽  
Variable Cross Section ◽  
Mode Shapes ◽  
Variable Cross ◽  
Bending Rigidity ◽  
Buckling Loads ◽  
Closed Form Solutions ◽  
Marine Risers

Natural frequencies, mode shapes and critical buckling loads of marine risers simply supported at both ends are given in closed form by using the WKB method. These solutions allow variable cross section, bending rigidity, tension and mass distribution along the riser length. Furthermore, a simple procedure to predict natural frequencies for other boundary conditions is described. Some special forms of these closed-form solutions are compared with existing solutions in the literature.

An Extended Separation-of-Variable Method for Free Vibration of Rectangular Mindlin Plates

2021 ◽  
pp. 2150154
Author(s):  
Gen Li ◽  
Yufeng Xing ◽  
Zekun Wang
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Closed Form ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Arbitrary Boundary ◽  
Unknown Variable ◽  
Closed Form Solutions ◽  
Solution Methods ◽  
Separation Of Variable

For rectangular thick plates with non-Levy boundary conditions, it is important to explore analytical free vibration solutions because the classical inverse and semi-inverse exact solution methods are not applicable to this category of problems. This work is to develop an extended separation-of-variable (SOV) method to find closed-form analytical solutions for the free vibration of rectangular Mindlin plates with arbitrary homogeneous boundary conditions. In the extended SOV method, characteristic differential equations and boundary conditions in two directions are obtained by employing the Rayleigh principle and the assumption that the mode functions are in the SOV form, and two transcendental eigenvalue equations are achieved through boundary conditions. But these two eigenvalue equations cannot be solved simultaneously since there are two equations and only the natural frequency is the unknown variable. Considering this, the second assumption in this method is that the natural frequencies corresponding to two-direction mode functions are independent of each other in the mathematical sense, thus there are two unknowns in two transcendental eigenvalue equations, and the closed-form solutions for plates with arbitrary boundary conditions can be obtained non-iteratively. From the physical sense, the natural frequencies pertaining to different direction mode functions should be the same, and this conclusion is validated analytically and numerically. The present natural frequencies and mode shapes agree well with those obtained by other analytical and numerical methods. Especially, for the plates with at least two opposite sides simply supported, the present solutions are exact.

Vibration analysis of sandwich beams with variable cross section on variable Winkler elastic foundation

2013 ◽  
Vol 20 (4) ◽  
pp. 359-370 ◽  
Author(s):  
Ersin Demir ◽  
Hasan Çallioğlu ◽  
Metin Sayer
Keyword(s):  
Elastic Foundation ◽  
Cross Section ◽  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Sandwich Beam ◽  
Variable Cross Section ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Variable Cross ◽  
Winkler Elastic Foundation

AbstractIn this study, free vibration behavior of a multilayered symmetric sandwich beam made of functionally graded materials (FGMs) with variable cross section resting on variable Winkler elastic foundation are investigated. The elasticity and density of the functionally graded (FG) sandwich beam vary through the thickness according to the power law. This law is related to mixture rules and laminate theory. In order to provide this, a 50-layered beam is considered. Each layer is isotropic and homogeneous, although the volume fractions of the constituents of each layer are different. Furthermore, the width of the beam varies exponentially along the length of the beam, and also the beam is resting on an elastic foundation whose coefficient is variable along the length of the beam. The natural frequencies are computed for conventional boundary conditions of the FG sandwich beam using a theoretical procedure. The effects of material, geometric, elastic foundation indexes and slenderness ratio on natural frequencies and mode shapes of the beam are also computed and discussed. Finally, the results obtained are compared with a finite-element-based commercial program, ANSYS®, and found to be consistent with each other.

A Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes of the Rods with Variable Cross Section

2011 ◽  
Vol 110-116 ◽  
pp. 2537-2547
Author(s):  
M. Khoshbayani Arani ◽  
N. Rasekh Saleh ◽  
M. Nikkhah Bahrami
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Exact Analytical Solution ◽  
Finite Difference Methods ◽  
Variable Cross Section ◽  
Mode Shapes ◽  
Variable Cross ◽  
Approximate Methods ◽  
Modified Method

Analytical solutions for vibration analysis of the rods with variable cross section are in general complex and in many cases impossible. On the other hand, approximate methods such as the weighted residual, Rayleigh-Ritz and finite difference methods also have their own shortcomings such as a limited number of natural frequencies and accuracy. Using the wave propagation method, the structure is partitioned into several continuous segments with constant cross-section, for which there exists an exact analytical solution. Waves in positive and negative directions at the entrance of each segment are obtained in terms of waves at the initial segment. Then, by satisfying the boundary conditions, the characteristic equation is obtained and all natural frequencies are calculated. By adding waves in positive and negative directions at each point, the shape modes are obtained. To verify this modified method, the frequencies and mode shapes of a rod with polynomial cross section, which has an exact analytical solution, are compared and have proven to be of highly accuracy. Therefore, this method can also be used to calculate the natural frequencies and its mode shapes of the rods with variable cross section for which no analytical solution is available.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

A Discretization Approach to Modeling Capacitive MEMS Filters

2007 ◽  
Author(s):  
Bashar K. Hammad ◽  
Ali H. Nayfeh ◽  
Eihab Abdel-Rahman
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Wide Range

We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.

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