Static–Dynamic Relationship for Flexural Free Vibration of Extensible Beams

2018 ◽  
Vol 18 (09) ◽  
pp. 1871010 ◽  
Author(s):  
Zhenyu Chen ◽  
C. W. Lim
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Technical Note ◽  
Structural Systems ◽  
Vibration System ◽  
Dynamic Relationship ◽  
Static Approach ◽  
Vibration Problems

This technical note presents a static–dynamic relationship for the flexural free vibration analysis of beams in tension with some specific boundary conditions. It is shown to be possible that a free vibration system can be solved via a static analysis approach to determine the natural frequencies of the beam with tension forces. The key idea of this study is to substitute the real natural frequency parameters with zero or negative elastic foundation stiffness, thereby allowing one to obtain the natural frequencies by analyzing the case with negative foundation elastic constant. This static approach for vibration problems can be extended for more complicated engineering structural systems.

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.

Uncertain Structural Free Vibration Analysis With Non-Probabilistic Spatially Varying Parameters

2019 ◽  
Vol 5 (2) ◽  
Author(s):  
Jinwen Feng ◽  
Qingya Li ◽  
Alba Sofi ◽  
Guoyin Li ◽  
Di Wu ◽  
...  
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Spatial Dependency ◽  
The Finite Element Method ◽  
Engineering Structures ◽  
Spatially Varying ◽  
Computational Strategy ◽  
Bounded System

The uncertain free vibration analysis of engineering structures with the consideration of nonstochastic spatially dependent uncertain parameters is investigated. A recently proposed concept of interval field is implemented to model the intrinsic spatial dependency of the uncertain-but-bounded system parameters. By employing the appropriate discretization scheme, evaluations of natural frequencies for engineering structures involving interval fields can be executed within the framework of the finite element method. Furthermore, a robust, yet efficient, computational strategy is proposed such that the extreme bounds of natural frequencies of the structure involving interval fields can be rigorously captured by performing two independent eigen-analyses. Within the proposed computational analysis framework, the traditional interval arithmetic is not employed so that the undesirable effect of the interval overestimation can be completely eliminated. Consequently, both sharpness and physical feasibility of the results can be guaranteed to a certain extent for any discretized interval field. The plausibility of the adopted interval field model, as well as the feasibility of the proposed computational scheme, is clearly demonstrated by investigating both academic-sized and practically motivated engineering structures.

scholarly journals Free Vibration Analysis of Rings via Wave Approach

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wang Zhipeng ◽  
Liu Wei ◽  
Yuan Yunbo ◽  
Shuai Zhijun ◽  
Guo Yibin ◽  
...  
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Classical Method ◽  
Free Vibration Analysis ◽  
Material Parameters ◽  
Wave Approach ◽  
Out Of Plane ◽  
Vibration Transmissibility ◽  
Ring Structures

Free vibration of rings is presented via wave approach theoretically. Firstly, based on the solutions of out-of-plane vibration, propagation, reflection, and coordination matrices are derived for the case of a fixed boundary at inner surface and a free boundary at outer surface. Then, assembling these matrices, characteristic equation of natural frequency is obtained. Wave approach is employed to study the free vibration of these ring structures. Natural frequencies calculated by wave approach are compared with those obtained by classical method and Finite Element Method (FEM). Afterwards natural frequencies of four type boundaries are calculated. Transverse vibration transmissibility of rings propagating from outer to inner and from inner to outer is investigated. Finally, the effects of structural and material parameters on free vibration are discussed in detail.

FREE VIBRATION ANALYSIS OF NON-UNIFORM TIMOSHENKO BEAMS USING DIFFERENTIAL TRANSFORM

1998 ◽  
Vol 22 (3) ◽  
pp. 231-250 ◽  
Author(s):  
Cha’o Kuang Chen ◽  
Shing Huei Ho
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Present Method ◽  
Series Solution ◽  
Free Vibration Analysis ◽  
Timoshenko Beams ◽  
Differential Transform ◽  
Vibration Problems ◽  
Algebraic Operations

This study introduces using differential transform to solve the free vibration problems of a general elastically end restrained non-uniform Timoshenko beam. First, differential transform is briefly introduced. Second, taking differential transform of a non-uniform Timoshenko beam vibration problem, a set of difference equations is derived. Doing some simple algebraic operations on these equations, we can determine any i-th natural frequency, the closed form series solution of any i-th normalized mode shape. Finally, three examples are given to illustrate the accuracy and efficiency of the present method.

Free Vibration Analysis of Cable Structures Using Isogeometric Approach

2017 ◽  
Vol 14 (03) ◽  
pp. 1750033 ◽  
Author(s):  
Son Thai ◽  
Nam-Il Kim ◽  
Jaehong Lee
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Penalty Method ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Basis Functions ◽  
Displacement Fields ◽  
Numerical Examples ◽  
B Splines ◽  
Cable Structures

This paper presents a free vibration analysis of cable structures based on the isogeometric approach. The nonuniform rational B-splines (NURBS) basis functions are employed to represent both the exact geometry of cable and displacement fields. In order to enrich the basis functions, the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-refinement strategies are implemented. Therefore, these refinement schemes increase the accuracy of solution fields. For determining the static configuration of slack cables as a reference configuration, the well-known penalty method is used. Three numerical examples for slack and taut cable structures are presented in which different refinement schemes are utilized to obtain the converged results. The accuracy and reliability of the present numerical method are verified by comparing the natural frequencies with the results given by other researchers.

Free Vibration Analysis of an Inflated Toroidal Shell

2002 ◽  
Vol 124 (3) ◽  
pp. 387-396 ◽  
Author(s):  
Akhilesh K. Jha ◽  
Daniel J. Inman ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Circular Cross Section ◽  
Free Vibration Analysis ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Toroidal Shell

Free vibration analysis of a free inflated torus of circular cross-section is presented. The shell theory of Sanders, including the effect of pressure, is used in formulating the governing equations. These partial differential equations are reduced to ordinary differential equations with variable coefficients using complete waves in the form of trigonometric functions in the longitudinal direction. The assumed mode shapes are divided into symmetric and antisymmetric groups, each given by a Fourier series in the meridional coordinate. The solutions (natural frequencies and mode shapes) are obtained using Galerkin’s method and verified with published results. The natural frequencies are also obtained for a circular cylinder with shear diaphragm boundary condition as a special case of the toroidal shell. Finally, the effects of aspect ratio, pressure, and thickness on the natural frequencies of the inflated torus are studied.

Free vibration analysis of a rectangular plate carrying multiple various concentrated elements

2003 ◽  
Vol 217 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J-S Wu ◽  
H-M Chou ◽  
D-W Chen
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Normal Mode ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Mode Shapes

The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.

Matched Interface and Boundary Method for Free Vibration Analysis of Irregular Membranes

2020 ◽  
pp. 2041006
Author(s):  
Zhiwei Song ◽  
Xiaoqiao He ◽  
Wei Li ◽  
De Xie
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Numerical Results ◽  
Irregular Domains ◽  
Topological Relations ◽  
Geometric Shapes ◽  
Cartesian Mesh ◽  
Boundary Method ◽  
Vibration Problems

Matched interface and boundary (MIB) method is introduced for free vibration analysis of irregular membranes. Two distinct schemes-on-interface and off-interface schemes are used to deal with the topological relations between edges of irregular domains and the Cartesian mesh lines. Different geometric shapes such as triangle and quadrilateral are dealt with by using MIB procedures. A number of examples are chosen to demonstrate the accuracy and convergence of MIB method. Numerical results show that MIB method is an efficient and highly accurate approach to solve free vibration problems of irregular membranes. This study further extends the application of MIB.

scholarly journals Free Vibrations of a Reddy-Bickford Multi-Span Beam Carrying Multiple Spring-Mass Systems

2011 ◽  
Vol 18 (5) ◽  
pp. 709-726 ◽  
Author(s):  
Yusuf Yesilce
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Structural Elements ◽  
Mass System ◽  
Assembly Technique

The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.

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