A Numerical Solution for Vibration Analysis of the Stiffened Variable-Thickness Conical Shells

2015 ◽  
Vol 757 ◽  
pp. 121-125
Author(s):  
Wei Ning ◽  
Feng Sheng Peng ◽  
Nan Wang ◽  
Dong Sheng Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variable Thickness ◽  
Conical Shell ◽  
Thickness Distribution ◽  
Ritz Method ◽  
Free Vibrations ◽  
Conical Shells ◽  
Linear Eigenvalue Problem ◽  
Element Method

The free vibrations of the stiffened hollow conical shells with different variable thickness distribution modes are investigated in detail in the context of Donnel-Mushtari conical shell theory. Two sets of boundary conditions have been considered. The algebraic energy equations of the conical shell and the stiffeners are established separately. The Rayleigh-Ritz method is used to equate maximum strain energy to maximum kinetic energy which leads to a standard linear eigenvalue problem. Numerical results are presented graphically for different geometric parameters. The parametric study reveals the characteristic behavior which is useful in selecting the shell thickness distribution modes and the stiffener type. The comparison between the present results and those of finite element method shows that the present results agree well with those of finite element method.

Free Vibration Analysis of Stiffened Conical Shell with Variable Thickness Distribution

2014 ◽  
Vol 614 ◽  
pp. 7-11 ◽  
Author(s):  
Wei Ning ◽  
Dong Sheng Zhang ◽  
Ji Ling Jia
Keyword(s):  
Variable Thickness ◽  
Conical Shell ◽  
Thickness Distribution ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Conical Shells ◽  
Linear Eigenvalue Problem ◽  
Standard Linear ◽  
Energy Equations

The free vibrations of the stiffened hollow conical shells with different variable thickness distribution modes are investigated in detail in the context of Donnel-Mushtari conical shell theory. Two sets of boundary conditions have been considered. The algebraic energy equations of the conical shell and the stiffeners are established separately. The Rayleigh-Ritz method is used to equate maximum strain energy to maximum kinetic energy which leads to a standard linear eigenvalue problem. Numerical results are presented graphically for different geometric parameters. The parametric study reveals the characteristic behavior which is usefulis inuseful in selecting the shell thickness distribution modes and the stiffener type.

scholarly journals Three-Dimensional Free Vibration Analysis of Gears with Variable Thickness Using the Chebyshev–Ritz Method

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yi Li ◽  
Ming Lv ◽  
Shi-ying Wang ◽  
Hui-bin Qin ◽  
Jun-fan Fu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Variable Thickness ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Admissible Function ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Element Method

To reflect vibration more comprehensively and to satisfy the machining demand for high-order frequencies, we presented a three-dimensional free vibration analysis of gears with variable thickness using the Chebyshev–Ritz method based on three-dimensional elasticity theory. We derived the eigenvalue equations. We divided the gear model into three annular parts along the locations of the step variations, and the admissible function was a Ritz series that consisted of a Chebyshev polynomial multiplying boundary function. The convergence study demonstrated the high accuracy of the present method. We used a hammering method for a modal experiment to test two annular plates and one gear’s eigenfrequencies in a completely free condition. We also applied the finite element method to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, we found that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. The relative errors of four sets of data were greater than 4%, and the errors of the other 48 sets were less than 4%. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of gears with variable thickness.

A numerical approach based on finite element method for the wrinkling analysis of dielectric elastomer membranes

2021 ◽  
pp. 1-37
Author(s):  
Guoyong Mao ◽  
Wei Hong ◽  
Martin Kaltenbrunner ◽  
Shaoxing Qu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Thickness Distribution ◽  
Numerical Approach ◽  
Dielectric Elastomer ◽  
Equivalent Model ◽  
Maxwell Stress ◽  
Buckling Mode ◽  
Post Buckling ◽  
Element Method

Abstract Dielectric elastomer (DE) actuators are deformable capacitors capable of a muscle-like actuation when charged. When subjected to voltage, DE membranes coated with compliant electrodes may form wrinkles due to the Maxwell stress. Here, we develop a numerical approach based on the finite element method (FEM) to predict the morphology of wrinkled DE membranes mounted on a rigid frame. The approach includes two steps, I) pre-buckling and II) post-buckling. In step I, the first buckling mode of the DE membrane is investigated by substituting the Maxwell stress with thermal stress in the built-in function of the FEM platform SIMULIA Abaqus. In step II, we use this first buckling mode as an artificial geometric imperfection to conduct the post-buckling analysis. For this purpose, we develop an equivalent model to simulate the mechanical behavior of DEs. Based on our approach, the thickness distribution and the thinnest site of the wrinkled DE membranes subjected to voltage are investigated. The simulations reveal that the crests/troughs of the wrinkles are the thinnest sites around the center of the membrane and corroborate these findings experimentally. Finally, we successfully predict the wrinkles of DE membranes mounted on an isosceles right triangle frame with various sizes of wrinkles generated simultaneously. These results shed light on the fundamental understanding of wrinkled dielectric elastomers but may also trigger new applications such as programmable wrinkles for optical devices or their prevention in DE actuators.

scholarly journals Modelling Plates and Shells by Means of the Rigid Finite Element Method

2015 ◽  
Vol 62 (1) ◽  
pp. 101-114 ◽  
Author(s):  
Iwona Adamiec-Wójcik ◽  
Andrzej Nowak ◽  
Stanisław Wojciech
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibrations ◽  
The Finite Element Method ◽  
Hybrid Finite Element Method ◽  
Hybrid Finite Element ◽  
Rigid Finite Element Method ◽  
Plates And Shells ◽  
Single Set ◽  
Element Method

Abstract The rigid finite element method (RFEM) has been used mainly for modelling systems with beam-like links. This paper deals with modelling of a single set of electrodes consisting of an upper beam with electrodes, which are shells with complicated shapes, and an anvil beam. Discretisation of the whole system, both the beams and the electrodes, is carried out by means of the rigid finite element method. The results of calculations concerned with free vibrations of the plates are compared with those obtained from a commercial package of the finite element method (FEM), while forced vibrations of the set of electrodes are compared with those obtained by means of the hybrid finite element method (HFEM) and experimental measurements obtained on a special test stand.

The Finite-Element Method—Part I

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Plane Elasticity ◽  
Heat Transfer Analysis ◽  
Deformation Analysis ◽  
The Finite Element Method ◽  
Element Method

The concept of the finite-element procedure may be dated back to 1943 when Courant approximated the warping function linearly in each of an assemblage of triangular elements to the St. Venant torsion problem and proceeded to formulate the problem using the principle of minimum potential energy. Similar ideas were used later by several investigators to obtain the approximate solutions to certain boundary-value problems. It was Clough who first introduced the term “finite elements” in the study of plane elasticity problems. The equivalence of this method with the well-known Ritz method was established at a later date, which made it possible to extend the applications to a broad spectrum of problems for which a variational formulation is possible. Since then numerous studies have been reported on the theory and applications of the finite-element method. In this and next chapters the finite-element formulations necessary for the deformation analysis of metal-forming processes are presented. For hot forming processes, heat transfer analysis should also be carried out as well as deformation analysis. Discretization for temperature calculations and coupling of heat transfer and deformation are discussed in Chap. 12. More detailed descriptions of the method in general and the solution techniques can be found in References [3-5], in addition to the books on the finite-element method listed in Chap. 1. The path to the solution of a problem formulated in finite-element form is described in Chap. 1 (Section 1.2). Discretization of a problem consists of the following steps: (1) describing the element, (2) setting up the element equation, and (3) assembling the element equations. Numerical analysis techniques are then applied for obtaining the solution of the global equations. The basis of the element equations and the assembling into global equations is derived in Chap. 5. The solution satisfying eq. (5.20) is obtained from the admissible velocity fields that are constructed by introducing the shape function in such a way that a continuous velocity field over each element can be denned uniquely in terms of velocities of associated nodal points.

Free Vibration Characteristics of Sandwich Conical Shells With FGM Face Sheets: A Finite Element Approach

2019 ◽  
Author(s):  
Tripuresh Deb Singha ◽  
Apurba Das ◽  
Gopal Agarwal ◽  
Tanmoy Bandyopadhyay ◽  
Amit Karmakar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Conical Shell ◽  
Vibration Characteristics ◽  
Mode Shapes ◽  
Iteration Algorithm ◽  
Operational Life ◽  
Sandwich Type ◽  
Element Method

Abstract This paper presents an analytical investigation on the free vibration characteristics of symmetric sandwich conical shell with functionally graded material (FGM) face sheets using finite element method. Sandwich-type structures offer higher stiffness to weight ratio with excellent thermal barrier in high temperature application extending the operational life of the component. The sandwich-type conical structure used in the advanced supersonic and hypersonic space vehicles. The material properties of FGM face sheets are considered to be varied in thickness direction as per simple power law distribution in terms of the volume fractions of the FGM constituents. The core layer is considered as homogeneous and made of an isotropic material (Titanium alloy-Ti–6Al–4V). A finite element method is used to reduce the governing equations of vibration problem. The QR iteration algorithm used to solve the standard eigen value problem for determine the natural frequencies. Convergence studies are performed in respect of mesh sizes to substantiate the accuracy of the proposed method. Computer codes developed to obtain the numerical results for the combined effects of twist angle and rotational speed on the free vibration characteristics of symmetric sandwich conical shell with FGM face sheets. A detailed numerical study is carried out to examine the influence of the sandwich plate type, volume fraction index on the free vibration characteristics. The typical mode shapes are also illustrated for different cases.

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