Vibration Analysis Of Beam Structures Research Articles

A Screw Approach to Vibration Analysis of Beam Structures With Nonstandard Conditions

Abstract A major advantage of the screw theory is that translations and rotations are treated simultaneously, which can provide greater insight into the vibration phenomena, such as vibration centers and axes. The present study describes how these concepts are extended into beam theory. The stiffness matrix of a beam was derived by incorporating different types of vibrations, such as extension, compression, torsion, and bending, at the same time, which was then used to obtain the equations of motion, including nonstandard forms of boundary conditions. We then presented an analytical method to solve these equations by focusing on two distinct examples, namely the cantilever and robot link. In the first numerical example, the mode shapes of the beam could be regarded as rotations about the vibration centers or axes of the rigid bodies in a discrete system. In the second example, the analytical solutions of mode shapes and natural frequencies of a robot link, for which the revolute joints at both sides are not parallel, were presented to demonstrate the utility of the screw theory. We demonstrated that the screw approach could accurately describe the vibrations of both discrete and continuous systems and that the geometric meaning of the vibration modes of discrete systems can be extended into continuous systems.

Modified transfer matrix method for vibration analysis of beam structures including branches and rigid bodies

Beam frames can be found in many application scenarios, ranging from big-scale structure engineering of buildings, bridges to small systems of micromachines, sensors, robotics and metamaterials. However, several challenging issues, e.g. serial-parallel configurations, beam and rigid body coupling along with inconsistent dimension of state variables, often lead to a complicated dynamic modeling. In this paper, we show that the free vibration analysis of a class of beam structures with serial-parallel branches and rigid bodies can be significantly streamlined by modifying the transfer matrix method in a graphic manner. Treating general beam structures as a topological graph with standard building blocks enables a modular modeling process and hence offers a new way to divide the complex modeling issue into much easier steps. The transfer matrix of typical building blocks, such as a beam connected to rigid bodies, tree branches and parallel clamped sub-chains, are derived and standardized providing new perspectives and distinct forms in contrast to previous studies. The transfer matrix method exhibits several advantages of easy programming, accurate solution and small degrees-of-freedom for serial structures. The presented approach extends these characteristics to a range of multi-body systems consisting of rigid bodies and beams. Comparative validation with four application case studies confirms these advantages.

Vibration analysis of beam structures with localized nonlinearities by a wave approach

This paper presents a general wave propagation approach for the vibration analysis of structures consisting of beam elements with localized nonlinearities. The scattering of a wave at a nonlinear discontinuity creates multi-harmonic waves. This forms the basis of the model formulation in which the displacement in each element composing the structure is defined as a sum of waves associated with the fundamental frequency and its harmonics. The coupling between separate elements is expressed using transmission coefficients function of the harmonic number and includes propagating and decaying near field waves whose amplitudes are the unknowns of the problem. With the use of harmonic balance, a characteristic algebraic equation of motion is derived. The general systematic methodology is demonstrated and detailed on beam-like systems with nonlinear boundaries for validation against a conventional finite element analysis. To illustrate the effectiveness of the approach on a more practical case, the method is also applied to a simplified model of a turbomachinery blade with underplatform friction dampers. The predictions of stick-slip effect and global nonlinear damping behavior made using the proposed approach are shown to be in excellent agreement with the results of numerical time integration with the advantage of reduced computational costs.

Timoshenko beam theory: A perspective based on the wave-mechanics approach

This paper reviews Timoshenko beam theory from the point of view of wave mechanics. Vibration of beam structures can be studied in terms of either normal modes or propagating waves. The latter wave approach has two distinct features: first, it gives rise to clear physical understanding of beam vibration; second, it leads to exact methods for vibration analysis of beam structures, especially in the mid-frequency range. In this paper, the work on wave solutions of an infinite Timoshenko beam is first discussed. The work on the splitting effect of spinning on wave solutions is also reviewed. The wave is treated as constitutive components of standing waves (i.e. normal modes), and a discussion on how the wave components formulate various standing waves is presented. Finally, several numerical examples are presented to illustrate the pros and cons of using different wave approaches to tackle vibration analysis of finite-length Timoshenko beams.

Discrete singular convolution element method for static, buckling and free vibration analysis of beam structures

A new version of discrete singular convolution (DSC) algorithm, called the DSC element method, is proposed. The new method combines the high computational efficiency and accuracy of the original DSC algorithm with the convenience and generality of the finite element method in application to structural analysis. The shortage existing in the original DSC algorithm has been overcome. Besides, the proposed method provides a unique way to apply various boundary conditions in using DSC algorithm. Two types of DSC beam element are presented and a variety of problems, such as static, buckling and free vibrations with discontinuities in geometry or load, have been successfully solved by the proposed method. Numerical results show that the proposed DSC element method is an accurate and efficient method for analyzing beam structures.

Refined free vibration analysis of one-dimensional structures with compact and bridge-like cross-sections

The free vibration analysis of beam structures with compact and bridge-like sections is presented in this paper. Refined beam models are used. Their derivation is embedded in the Carrera Unified Formulation (CUF) framework, which allows us to obtain any-order theories with no need for ‘ad hoc’ formulations. Up to fifth-order models are used; classical (Euler–Bernoulli and Timoshenko) beam theories are considered as particular cases of the linear expansion. The finite element (FE) formulation is used in order to analyze arbitrary geometries and boundary conditions. Comparisons with analytical and solid 3D models from commercial code analyses are given. Natural frequencies and modal shapes are investigated. The results have revealed excellent accuracy in the bending frequency computation. Higher-order theories have shown enhanced capabilities to predict more complex phenomena such as torsional frequencies, bending/torsional coupling due to non-symmetrical cross-section and ‘shell-like’ modal shapes characterized by significant cross-section distortions. Moreover, the proposed formulation has shown it is capable in reducing the computational cost significantly.