Forced Vibration of Flexible Body Systems: A Dynamic Stiffness Method

1993 ◽  
Vol 115 (4) ◽  
pp. 468-476 ◽  
Author(s):  
T. S. Liu ◽  
J. C. Lin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elastic Deformation ◽  
Forced Vibration ◽  
Dynamic Stiffness ◽  
Equations Of Motion ◽  
Flexible Body ◽  
Shape Functions ◽  
Stiffness Matrices ◽  
Dynamic Shape

Due to the development of high speed machinery, robots, and aerospace structures, the research of flexible body systems undergoing both gross motion and elastic deformation has seen increasing importance. The finite element method and modal analysis are often used in formulating equations of motion for dynamic analysis of the systems which entail time domain, forced vibration analysis. This study develops a new method based on dynamic stiffness to investigate forced vibration of flexible body systems. In contrast to the conventional finite element method, shape functions and stiffness matrices used in this study are derived from equations of motion for continuum beams. Hence, the resulting shape functions are named as dynamic shape functions. By applying the dynamic shape functions, the mass and stiffness matrices of a beam element are derived. The virtual work principle is employed to formulate equations of motion. Not only the coupling of gross motion and elastic deformation, but also the stiffening effect of axial forces is taken into account. Simulation results of a cantilever beam, a rotating beam, and a slider crank mechanism are compared with the literature to verify the proposed method.

Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

2003 ◽  
Vol 03 (02) ◽  
pp. 299-305 ◽  
Author(s):  
F. W. Williams ◽  
D. Kennedy
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stiffness Matrix ◽  
Infinite Order ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Structural Stiffness ◽  
Trial And Error ◽  
Stiffness Matrices ◽  
Vibration Problems

Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.

scholarly journals Modal analysis of damaged structures by the modified finite element method

1998 ◽  
Vol 20 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Nguyen Cao Menh ◽  
Nguyen Tien Khiem ◽  
Dao Nhu Mai ◽  
Nguyen Viet Khoa
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Standard Procedure ◽  
Frame Structures ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Stiffness Matrices ◽  
Flexible Supports ◽  
Cubic Polynomial ◽  
Element Method

The classical 3D beam element has been modified and developed as a new finite element for vibration analysis of frame structures with flexible connections and cracked members. The mass and stiffness matrices of the modified elements are established basing on a new form of shape functions, which are obtained in investigating a beam with flexible supports and crack modeled through equivalent springs. These shape functions remain the cubic polynomial form and contain flexible connection (or crack) parameters. They do not change standard procedure of the finite element method (FEM). Therefore, the presented method is easy for engineers in application and allows to analyze Eigen-parameters of structures as functions of the connection (or crack) parameters. The proposed approach has been applied to calculate natural frequencies and mode shape of typical frame structures in presented examples.

Torsional Vibration of a Damped Shaft System Using the Analytical-and-Numerical-Combined Method

2001 ◽  
Vol 38 (04) ◽  
pp. 250-260
Author(s):  
Jong-Shyong Wu ◽  
Mang Hsieh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Torsional Vibration ◽  
Forced Vibration ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Combined Method ◽  
Rotating Shaft ◽  
Shaft System ◽  
Element Method

Torsional vibration analysis of the propulsive shaft system of a marine engine—one of the most important tasks in preliminary ship design—is carried out today by either the Holzer method, the transfer matrix method (TMM), or the finite-element method (FEM). Of the three methods, Holzer is the most popular and is adopted by shipyards worldwide. The purpose of this paper is to present an analytical-and-numerical-combined method (ANCM) to improve the drawbacks of existing methods. In comparison with the Holzer method (or TMM), the presented ANCM has the following merits: the mass of the rotating shaft is inherently considered, the damping effect is easily tackled, and the forced vibration responses due to various external excitations are obtained with no difficulty. Since the order of the overall property matrices for the equations of motion derived from the ANCM is usually lower than that derived from the conventional finite-element method (FEM), the CPU time required by the former is usually less than that required by the latter, particularly in the forced vibration analysis. Besides, the sizes (and the total number) of the elements for the FEM have a close relationship with the locations of the disks and the dampers and so does the accuracy of the FEM, but various distributions (or locations) of the disks and the dampers will not create any problems for the ANCM.

Dynamics of a beam-column element on an elastic foundation

2016 ◽  
Vol 43 (8) ◽  
pp. 685-701
Author(s):  
Çiğdem Dinçkal ◽  
Bülent N. Alemdar ◽  
Polat H. Gülkan
Keyword(s):  
Exact Analytical Solution ◽  
Dynamic Stiffness ◽  
Winkler Foundation ◽  
Shape Functions ◽  
Numerical Examples ◽  
Closed Form Solutions ◽  
Stiffness Matrices ◽  
Column Element ◽  
Foundation Parameter ◽  
Dynamic Shape

An exact analytical solution of a harmonically vibrating beam-column element resting on an elastic Winkler foundation is derived. The solution covers four cases comprised of constant compressive or tensile axial force with the restrictions ks – mω2 < 0 and ks – mω2 > 0. The proposed solution is not restricted to a particular range of magnitudes of the foundation parameter. Closed form solutions of dynamic shape functions are explicitly derived for each sub-case to obtain frequency-dependent dynamic stiffness terms that constitute the exact dynamic stiffness matrices. Four numerical examples are provided to demonstrate the merits of the present study.

The study of the elastic deformation of a flexible wheel by a cam wave generator

2020 ◽  
Vol 65 (1) ◽  
pp. 51-58
Author(s):  
Sava Ianici
Keyword(s):  
Numerical Simulation ◽  
Finite Element Method ◽  
Finite Element ◽  
Elastic Deformation ◽  
Theoretical Study ◽  
Elastic Field ◽  
The Finite Element Method ◽  
Elastic Deformations ◽  
Wave Generator ◽  
Two Stages

The paper presents the results of research on the study of the elastic deformation of a flexible wheel from a double harmonic transmission, under the action of a cam wave generator. Knowing exactly how the flexible wheel is deformed is important in correctly establishing the geometric parameters of the wheels teeth, allowing a better understanding and appreciation of the specific conditions of harmonic gearings in the two stages of the transmission. The veracity of the results of this theoretical study on the calculation of elastic deformations and displacements of points located on the average fiber of the flexible wheel was subsequently verified and confirmed by numerical simulation of the flexible wheel, in the elastic field, using the finite element method from SolidWorks Simulation.

scholarly journals Higher Order Multiscale Finite Element Method for Heat Transfer Modeling

Materials ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 3827
Author(s):  
Marek Klimczak ◽  
Witold Cecot
Keyword(s):  
Heat Transfer ◽  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Shape Functions ◽  
Multiscale Finite Element Method ◽  
Multiscale Finite Element ◽  
Steady State Heat ◽  
Steady State Heat Transfer

In this paper, we present a new approach to model the steady-state heat transfer in heterogeneous materials. The multiscale finite element method (MsFEM) is improved and used to solve this problem. MsFEM is a fast and flexible method for upscaling. Its numerical efficiency is based on the natural parallelization of the main computations and their further simplifications due to the numerical nature of the problem. The approach does not require the distinct separation of scales, which makes its applicability to the numerical modeling of the composites very broad. Our novelty relies on modifications to the standard higher-order shape functions, which are then applied to the steady-state heat transfer problem. To the best of our knowledge, MsFEM (based on the special shape function assessment) has not been previously used for an approximation order higher than p = 2, with the hierarchical shape functions applied and non-periodic domains, in this problem. Some numerical results are presented and compared with the standard direct finite-element solutions. The first test shows the performance of higher-order MsFEM for the asphalt concrete sample which is subject to heating. The second test is the challenging problem of metal foam analysis. The thermal conductivity of air and aluminum differ by several orders of magnitude, which is typically very difficult for the upscaling methods. A very good agreement between our upscaled and reference results was observed, together with a significant reduction in the number of degrees of freedom. The error analysis and the p-convergence of the method are also presented. The latter is studied in terms of both the number of degrees of freedom and the computational time.

scholarly journals Forced Vibration Analysis of Laminated Composite Shells Reinforced with Graphene Nanoplatelets Using Finite Element Method

2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Trung Thanh Tran ◽  
Van Ke Tran ◽  
Pham Binh Le ◽  
Van Minh Phung ◽  
Van Thom Do ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Forced Vibration ◽  
Vibration Analysis ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Thermal Environments ◽  
Forced Vibration Analysis ◽  
Laminated Composite Shells ◽  
Element Method

This paper carries out forced vibration analysis of graphene nanoplatelet-reinforced composite laminated shells in thermal environments by employing the finite element method (FEM). Material properties including elastic modulus, specific gravity, and Poisson’s ratio are determined according to the Halpin–Tsai model. The first-order shear deformation theory (FSDT), which is based on the 8-node isoparametric element to establish the oscillation equation of shell structure, is employed in this work. We then code the computing program in the MATLAB application and examine the verification of convergence rate and reliability of the program by comparing the data of present work with those of other exact solutions. The effects of both geometric parameters and mechanical properties of materials on the forced vibration of the structure are investigated.

A New Approach to the Rigid Finite Element Method in Modeling Spatial Slender Systems

2018 ◽  
Vol 18 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Iwona Adamiec-Wójcik ◽  
Łukasz Drąg ◽  
Stanisław Wojciech
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Dynamic Analysis ◽  
Equations Of Motion ◽  
Nonlinear Dynamic Analysis ◽  
New Approach ◽  
Static And Dynamic Analysis ◽  
Rigid Finite Element Method ◽  
Longitudinal Flexibility ◽  
Element Method

The static and dynamic analysis of slender systems, which in this paper comprise lines and flexible links of manipulators, requires large deformations to be taken into consideration. This paper presents a modification of the rigid finite element method which enables modeling of such systems to include bending, torsional and longitudinal flexibility. In the formulation used, the elements into which the link is divided have seven DOFs. These describe the position of a chosen point, the extension of the element, and its orientation by means of the Euler angles Z[Formula: see text]Y[Formula: see text]X[Formula: see text]. Elements are connected by means of geometrical constraint equations. A compact algorithm for formulating and integrating the equations of motion is given. Models and programs are verified by comparing the results to those obtained by analytical solution and those from the finite element method. Finally, they are used to solve a benchmark problem encountered in nonlinear dynamic analysis of multibody systems.

Implementation of p and h-p Versions of the Finite Element Method

1992 ◽  
Author(s):  
Ye-Chen Lai ◽  
Timothy C. S. Liang ◽  
Zhenxue Jia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elastic Analysis ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Shape Functions ◽  
Linear Elastic ◽  
Finite Element Software ◽  
Analysis Process ◽  
Adaptive Analysis

Abstract Based on hierarchic shape functions and an effective convergence procedure, the p-version and h-p adaptive analysis capabilities were incorporated into a finite element software system, called COSMOS/M. The range of the polynomial orders can be varied from 1 to 10 for two dimensional linear elastic analysis. In the h-p adaptive analysis process, a refined mesh are first achieved via adaptive h-refinement. The p-refinement is then added on to the h-version designed mesh by uniformly increasing the degree of the polynomials. Some numerical results computed by COSMOS/M are presented to illustrate the performance of these p and h-p analysis capabilities.

Study of Snubbing Mechanism Using Finite Element Method

2014 ◽  
Author(s):  
Jingming Chen ◽  
Paolo Pennacchi ◽  
Dongxiang Jiang ◽  
Steven Chatterton
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Forced Vibration ◽  
Excitation Frequency ◽  
Relative Displacement ◽  
Rotor Blades ◽  
Motion Pattern ◽  
Motion Characteristics ◽  
The Time Domain ◽  
Element Method

In the rotating machineries, large vibrations of a blade would result in fatigue crack, which is a great threaten to the safety. Therefore, it is of great importance to reduce the blade vibrations. Snubbing technique is a possible solution to this problem. A tiny gap is left between the shrouds of adjacent blades. While the forced vibration makes the relative displacement between two neighboring blades exceed the gap, the contact happens at the contact face of the shrouds, accompanied with friction and energy dissipation, which restricts the vibration. In this paper, a simplified model for a set of rotor blades is established, by using finite element method. The contact between the adjacent shrouds is considered. In this way, snubbing phenomenon can occur under forced vibration. Based on the model, modal analysis has been conducted. The 8x rev. frequency has been chosen as the excitation frequency. Under a certain amplitude of sine excitation, the circumferential vibration of the blades has been simulated. The vibration has been analyzed in the time domain. As expected, the blade motion is divided into four different states in one period. They are: non-contact, rebounding, sticky and escaping state. The four states had different mechanical and motion characteristics. The motion pattern for the set of blades has been also analyzed and the wave spreading along the bladerow has been described. Because of the snubbing mechanism, the waveform was distorted into serrated shape.

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