An improved dynamic stiffness method and modal analysis for beam-like structures

Chen, Shilin; Géradin, Michel; Lamine, E.

doi:10.1016/0045-7949(95)00448-3

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…[ ] is flexural vibration displacement trial function for dynamic beam element. With (13), it can be expressed as…”

Section: Analytical Trial Function Ofmentioning

confidence: 99%

“…To gain more accurate results, stiffness matrix of tensile torsion bar and Euler beam about frequency, that is, dynamic stiffness matrix, was firstly derived from analytical solution for studying vibration characteristics of plane truss. A lot of work on the research and development of dynamic stiffness matrix method was also done by Long and Bao [11], Hashemi and Richard [12], Chen et al [13], Banerjee et al [14], and Banerjee et al [15]. Shavezipur and Hashemi [16] put forward an accurate finite element method.…”

Section: Introductionmentioning

confidence: 99%

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Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

Cao

Deng

Jiang

et al. 2015

Mathematical Problems in Engineering

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For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about1×10-9and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.

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“…[ ] is flexural vibration displacement trial function for dynamic beam element. With (13), it can be expressed as…”

Section: Analytical Trial Function Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

Cao

Deng

Jiang

et al. 2015

Mathematical Problems in Engineering

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“…However, during application of this spherical capacitive probing sensor on CMMs and other measuring machines, stylus vibration caused by low stiffness and low structure damping comes to be a new problem, which is a certain result of ultra-large length to diameter ratio of the stylus [3][4][5][6]. On one hand, usually the natural frequency of driving structure of a CMM or other measuring machine is around 20 Hz, so stiffness of the stylus should be as high as possible to make its natural frequency far larger than 20 Hz to avoid resonance.…”

Section: Introductionmentioning

confidence: 99%

Optimal design of spherical capacitive probe with ultra-large aspect ratio based on modal analysis

Cui¹,

Lu²,

He³

et al. 2014

Tm – Technisches Messen

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In order to solve the problem of stylus vibration of the spherical capacitive probe with ultra-large aspect ratio, modal analysis of the stylus is done and optimal design of stylus structure is carried out. The Euler-Bernoulli Beam model is introduced to build the dynamic equation of lateral vibration of the stylus. Based on results of modal analysis and simulation, the first order natural frequency of the stylus is optimized to near 100 Hz, and the resolution of the probing sensor is improved from 20 nm to 4 nm with adoption of damping material. Zusammenfassung: Das Problem der Vibrationen bei Tastköpfen bei kapazitiven, kugelförmigen Sensoren mit übergroẞem Formfaktor beruht auf geringer Steifigkeit und geringem Dämpfungsaufbau. Um dieses Problem zu lösen, werden Modalanalysen und Simulationen des Tastkopfes durchgeführt und ein optimales Design der Taster-Struktur aufgezeigt. Mit Hilfe der Euler-BernoulliBalkentheorie wird die dynamische Gleichung für die Quervibration des Tasters entwickelt. Basierend auf den Ergebnissen zur Eigenwertanalyse und den Simulationen wird die Grundfrequenz erster Ordnung auf nahezu 100 Hertz optimiert. Die Ausgangsstabilität des Tastkopfes wird durch Einsetzen von Dämpfermaterialien von 20 nm auf 4 nm (Spitze-zu-Spitze-Wert) verbessert. Schlüsselwörter: Kugelförmig-kapazitiver Tastkopf, über-groẞer Formfaktor, Euler-Bernoulli-Balkentheorie, Eigenwertanalyse.

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Calculation of natural and forced vibrations of a piping system. Part 1. Analysis of vibrations of a 3D beam system

2007

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Dynamic behavior of piping as a beam system has been analyzed with the use of the dynamic stiffness method. According to this method, the equations describing the relation between unknown parameters are written by the method of initial parameters, therefore, the solution procedure is similar to that for a static problem. It is shown that for curvilinear beams it is simpler and more efficient to apply a model that consists of straight segments and inertia-free rotation elements. To determine natural frequencies of 3D beam systems, it is proposed to use a method of disconnection of displacements, which makes it possible to discern the frequencies corresponding to different vibration modes (transverse, longitudinal, etc.). The approach allows a correct simulation of the system behavior under forced vibrations induced by a harmonic exciting force.

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An improved dynamic stiffness method and modal analysis for beam-like structures

Cited by 5 publications

References 11 publications

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

Optimal design of spherical capacitive probe with ultra-large aspect ratio based on modal analysis

Calculation of natural and forced vibrations of a piping system. Part 1. Analysis of vibrations of a 3D beam system

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